Quantum state generation device, quantum state measurement device, and quantum key distribution device

Abstract

The present invention discloses a state generation device and a state measurement device that are for high-dimensional quantum states and use mutually unbiased bases using a finite field. The mutually unbiased bases in the high-dimensional quantum states are implemented by a calculation method using the finite field. The present invention discloses a device that uses mutually unbiased bases in high-dimensional quantum states using a finite field such as an optical time-bin quantum state and a frequency-bin quantum state and in other optical modes. Each of the generation device and the measurement device is implemented by units that are equivalent to a phase modulator and a matrix transform operation. The measurement device includes a phase modulation unit that is a first unit on the front stage side and corresponds to diagonal unitary transform on computational bases and a high-dimensional Hadamard transform measurement unit or a Fourier transform measurement unit that is a second unit.

Description

Technical Field

[0001] The present invention relates to the generation, measurement, and application communication of high-dimensional quantum states.

Background Art

[0002] Quantum communication can use photons as information carriers to achieve communication with high confidentiality and high transmission rates that are impossible with conventional communication technologies. Quantum key distribution (QKD) is also realized, which shares an encryption key that cannot be leaked to a third party in principle by taking advantage of the fact that measurement of a quantum state inevitably causes a change in that state.

[0003] In recent years, research on the high-dimensionalization of quantum states used has been actively conducted to make quantum communication and quantum information processing more advanced. Since high-dimensional quantum states can take a plurality of mutually orthogonal states, it is possible to increase the amount of information that can be transmitted by one particle. The time-bin quantum state of photons is a stable quantum state that is less affected by disturbances on fiber transmission, and is widely used in quantum communication because the state degradation during transmission is small. The time-bin quantum state also has the advantage that it can be easily high-dimensionalized by simply increasing the time slots that make up the quantum state.

Prior Art Documents

Non-Patent Documents

[0004]

Non-Patent Document 1

Non-Patent Document 2

Non-Patent Document 8

Non-Patent Document 9

Summary of the Invention

Problems to be Solved by the Invention

[0005] In quantum information processing technologies such as quantum key distribution using time-bin quantum states and quantum state tomography for measuring quantum states in detail, in addition to the computational basis, it is necessary to use mutually unbiased bases (MUBs). The computational basis is composed of physically orthogonal states of light as a reference, and the MUB is a basis non-orthogonal to the computational basis. As a method for implementing this MUB, one using the Fourier transform basis is known. However, the Fourier transform basis requires very high-resolution and high-precision phase modulation as the dimension d increases. The requirement for high phase resolution in the state of the MUB using the Fourier transform basis has a problem of degrading the accuracy and efficiency of state measurement. In addition, since a quantum state measurement device requires a large number of interferometers and photon detectors, there is also a problem that the scale of the measurement device increases as the dimension d increases.

Means for Solving the Problems

[0006] One embodiment of the present invention is a measuring device that performs a projective measurement on a high-dimensional quantum state defined by one computational basis {|m>|m ∈ {0, 1, ···, d - 1}} of a d-dimensional quantum state consisting of orthogonal light states and a mutually unbiased basis of label r (an integer greater than or equal to 0) in which a quantum state of label n (0, 1, ···, d - 1) is defined and is non-orthogonal to the computational basis, where d = 2 N (N is a natural number greater than or equal to 2), and the quantum state of the label n is represented by the following formula TIFF0007705068000001.tif11150 Probability amplitude B mn (r) When decomposed into a diagonal unitary matrix and a Hadamard transformation matrix, it corresponds to the diagonal unitary matrix and includes a phase modulation unit that gives a phase modulation to each state of the computational basis of the received d-dimensional quantum state, and a measurement unit that determines the label n of the d-dimensional quantum state corresponding to the Hadamard transformation matrix. It is a measuring device for a high-dimensional quantum state.

[0007] Another embodiment of the present invention is a measuring device that performs a projective measurement on a high-dimensional quantum state defined by one computational basis {|m>|m ∈ {0, 1, ···, d - 1}} of a d-dimensional quantum state consisting of orthogonal light states and a mutually unbiased basis of label r (an integer greater than or equal to 0) in which a quantum state of label n (0, 1, ···, d - 1) is defined and is non-orthogonal to the computational basis, where p is an odd prime number and d = p N (N is a natural number), and the quantum state of the label n is represented by the following formula TIFF0007705068000002.tif11150 Probability amplitude B mn (r) When decomposed into a diagonal unitary matrix and a tensor product of Fourier transformation matrices, it corresponds to the diagonal unitary matrix and includes a phase modulation unit that gives a phase modulation to each state of the computational basis of the received d-dimensional quantum state, and a measurement unit that determines the label n of the d-dimensional quantum state corresponding to the Fourier transformation matrix. It is a measuring device for a high-dimensional quantum state.

[0008] A further embodiment of the present invention is a generator of a high-dimensional quantum state defined by one computational basis {|m>|m∈{0,1,··,d - 1}} of a d-dimensional quantum state consisting of states of orthogonal light, and a mutually unbiased basis of label r (an integer greater than or equal to 0) that is non-orthogonal to the computational basis and in which a quantum state of label n (0,1,··,d - 1) is defined. d = 2 N (where N is a natural number greater than or equal to 2), and the quantum state of the label n is represented by the following formula Let f be a primitive element that is the basis of a finite field of order d of TIFF0007705068000003.tif11150 i and when the symmetric matrix A (j) satisfies the following formula The probability amplitude of TIFF0007705068000004.tif14150 is represented by TIFF0007705068000005.tif18150, and it is a state generator that takes only four phase states.

[0009] A further alternative embodiment of the present invention is a generator of a high-dimensional quantum state defined by one computational basis {|m>|m∈{0,1,··,d - 1}} of a d-dimensional quantum state consisting of states of orthogonal light, and a mutually unbiased basis of label r (an integer greater than or equal to 0) that is non-orthogonal to the computational basis and in which a quantum state of label n (0,1,··,d - 1) is defined, where d = p N (where N is a natural number and p is an odd prime number), and the quantum state of the label n is represented by the following formula Let f be a primitive element that is the basis of a finite field of order d of TIFF0007705068000006.tif11150 i and when the symmetric matrix A (j) satisfies the following formula The probability amplitude of TIFF0007705068000007.tif14150 is represented by TIFF0007705068000008.tif18150, and it is a state generator that takes only p phase states.

Advantages of the Invention

[0010] Provided is a high-dimensional state generation / state measurement apparatus that relaxes and simplifies the requirements for phase resolution.

Brief Description of the Drawings

[0011]

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Best Mode for Carrying Out the Invention

[0012] In the following disclosure, a quantum state generation device and a quantum state measurement device based on mutually unbiased bases (MUBs) using a finite field are presented. Also presented is high-dimensional quantum key distribution applying the quantum state generation device and the quantum state measurement device. First, the problems in conventional quantum state generation devices and quantum state measurement devices using Fourier transform bases are described. Subsequently, the features and various implementation forms of the quantum state generation device and the quantum state measurement device using MUBs with a finite field of the present disclosure are described. Furthermore, a quantum key distribution system using the quantum state generation device and the quantum state measurement device is also described. For the sake of simplicity, in the following description, when simply referring to the "generation device" and the "measurement device", they shall respectively mean a quantum state generation device and a quantum state measurement device.

[0013] High-dimensional MUBs are realized by a computational method using finite fields. In the following disclosure, as an example of a high-dimensional quantum state using a finite field, the mutually unbiased bases in the time-bin quantum state of light using orthogonal modes of time in the computational basis will be described. As will be described later, the generation apparatus and the measurement apparatus of the present disclosure can be realized by a phase modulator unit and a unit equivalent to a matrix transformation operation, respectively. In this regard, the following disclosure regarding the time-bin quantum state can be similarly applied to each apparatus of high-dimensional quantum states using other optical modes. Examples of measurement apparatuses and generation apparatuses using other optical modes other than the time-bin quantum state using orthogonal modes of time will be described at the end.

[0014] [Time-bin quantum state, mutually unbiased basis (MUB)] As described above, the time-bin quantum state is a stable quantum state that is hardly affected by disturbances on fiber transmission. In explaining the MUB for the time-bin quantum state, the computational basis will first be described.

[0015] FIG. 1 is a conceptual diagram for explaining the computational basis in the time-bin quantum state. As shown in FIG. 1(a), consider several time positions (d: t0, t2, ··· t d-1 ) of the optical pulse 1 that is continuous on the time axis. Consider a state in which a single photon exists at one of these time positions. In FIG. 1(b), it represents a state in which a single photon exists only in the optical pulse 2 at the time position t0. As shown by the dotted triangle 3, at time positions other than t0, there is no optical pulse. Similarly, in FIG. 1(c), it represents a state in which a single photon exists only in the optical pulse 4 at the time position t d-1 . As shown in FIGS. 1(b) and 1(c), |i> (|0>, |1>, ···, |d - 1>) is a state in which a photon exists only at time t i , that is, a state of the computational basis of dimension d.

[0016] In quantum key distribution, it is necessary to generate (using a generating device) and perform projective measurement (using a measuring device) on the states of mutually unbiased bases (MUBs) that are not only in the computational basis state where photons definitely exist at a specific time as shown in FIG. 1, but also non-orthogonal to the computational basis state and in a superposition state.

[0017] FIG. 2 is a conceptual diagram showing a simplified system for information transmission by quantum communication. The quantum communication system 10 includes a generating device 11, an optical transmission line 12 for transmitting a quantum state, and a measuring device 13. In the generating device 11 continuous light or pulsed light 14 is input, and intensity and phase modulation are applied to the optical pulse train 17 based on the computational basis and MUB 16 to generate the necessary quantum state. In the measuring device 13, for example, projective measurement is performed on the received optical pulse train 18 to measure the quantum state.

[0018] MUB is a basic component that appears in various scenarios of quantum information processing such as the above-mentioned quantum communication, quantum key distribution, quantum state tomography which is a technique for measuring a state density operator, and others. In recent years, in order to improve the secret key generation rate in quantum key distribution, methods using MUBs in high-dimensional quantum states have been reported.

[0019] [MUB by Fourier transform basis] The MUB with respect to the computational basis is a basis in which all basis states are non-orthogonal to the basis states of the computational basis and are defined as superposition states of the basis states of the computational basis, and the square of the absolute value of the inner product between any two states, one being an arbitrary basis state included in the MUB and the other being an arbitrary basis state of the computational basis, is 1 / d with respect to the dimension d. According to another definition, MUB is a non-orthogonal quantum state in which, in two orthonormal bases, the basis states included in one basis are superposition states of all the basis states included in the other basis, and for any combination of taking out one basis state from each of the two bases, the square of the absolute value of the inner product between the two quantum states is 1 / d with respect to the dimension d.

[0020] As an implementation method of the MUB required for high-dimensional quantum key distribution, the Fourier transform basis |f n > has been mainly used. TIFF0007705068000009.tif9150In the above equation, m is a label (integer) that defines the state of the computational basis, n is a label (integer) that indicates the state of the Fourier transform basis, and d is the dimension of the computational basis and the Fourier transform basis.

[0021] This Fourier transform basis |f n > has a phase proportional to 1 / d with respect to the dimension d, as can be seen from the term of e in Equation (1). Therefore, for the state generation of the MUB using the Fourier transform basis in Equation (1), phase modulation with a very high resolution is required as the dimension d increases. In the time-bin quantum state using the Fourier transform basis, which is also disclosed in Non-Patent Document 1, a high phase resolution is required, which reduces the accuracy and efficiency of state measurement. In the implementation of the measurement device, d - 1 interferometers and d photon detectors are also required, and the scale of the measurement device increases as the dimension d increases.

[0022] In order to solve the above problems that a high phase resolution is required as the dimension d increases, the inventors introduced another MUB using a finite field into a device that handles high-dimensional quantum states. By using the MUB using a finite field, the required conditions for the phase resolution are significantly relaxed, the device configuration is simplified, and it becomes possible to implement a scalable generation device and measurement device.

[0023] [MUB by Finite Field] A finite field with order d = 2 N is considered, and when the elements of the finite field are represented in bit form, the addition becomes the exclusive OR of each component (Condition 1). Furthermore, in the above bit representation, the element of the finite field in which only the i-th bit becomes 1 is f i and a symmetric matrix A (j) is defined to satisfy the following equation (Condition 2). Note that the operator on the left side of the following equation represents the binary operation of the product on the finite field. TIFF0007705068000010.tif13150Satisfy the above conditions 1 and 2, and set the dimension to d = 2 N When doing so, consider a basis defined by the following equation. TIFF0007705068000011.tif13150 TIFF0007705068000012.tif23150

[0024] The above equation (3) is a basis using a finite field and represents a mutually unbiased basis (MUB) as will be described later. Equation (4) represents the probability amplitude of this basis. Also, the computational basis |m> is represented by the following equation. TIFF0007705068000013.tif6150Furthermore, m, n, and r are also defined as follows. m: A label that defines the state of the computational basis n: A label that defines the state of the mutually unbiased basis (MUB) r: A label of the MUB In the parentheses including Σ in Equation (4), r, m, n (in bold) represent vectors when r, m, n are represented in bit form respectively. Also r j (in bold) is a scalar quantity indicating the j-th bit when r is represented in bit form. m T (in bold) is a horizontal vector, A (j) is a square matrix, and m (in bold) becomes a vertical vector.

[0025] The basis defined by the above equations (2) to (4) is mutually unbiased with the computational basis |m>. Also, the bases for different r are mutually unbiased with each other. That is, the basis defined by equations (2) to (4) forms a mutually unbiased basis (Non-Patent Document 2). Also, for the new (d + 1) bases obtained by taking the complex conjugate simultaneously for all the probability amplitudes of the states of the MUB generated in this way, they also form (d + 1) MUBs. Therefore, in the following description, the same discussion holds for the method based on the equations obtained by taking the complex conjugate of equations (3) to (4).

[0026] In the above definition, an element of the finite field in which only the \(i\)-th bit is 1 in the bit string representation is denoted as \(f\). i There are multiple other methods for constructing the finite field itself, and it can be shown that finite fields of the same order are equivalent to only one finite field by permuting the labels representing their elements. Therefore, it can be immediately seen that there are multiple states having exactly the same probability amplitudes as in Equation (4) by permuting the labels. In addition, it is obvious that MUBs obtained simply by independently permuting \(m\), \(n\), and \(r\) respectively are equivalent if they are associated by the inverse transformation of permutation again. For such equivalent MUBs obtained by permuting the labels, the examples of MUBs described below are valid as they are, and similar discussions hold.

[0027] Figure 3 is a diagram conceptually showing the relationship between MUBs and computational bases. \(d\) MUBs \(21 - 0\) to \(21-(d - 1)\) are generated from the computational basis \(20\). All the bases in Figure 3 are non-orthogonal to each other. For example, MUB \(21 - 0\) (label \(r = 0\)) is non-orthogonal to the other \((d - 1)\) MUBs and is also non-orthogonal to the computational basis \(20\). In Figure 3, for the sake of clarity, the non-orthogonal display is shown only for MUB \(21 - 0\), but the relationship between MUB \(21 - 0\) and the other MUBs holds for all the other MUBs respectively.

[0028] In the MUB defined by Equations (2) to (4), for any dimension \(d = 2\) N at most four phases of each probability amplitude appear. This can be easily understood from the fact that the expression inside the parentheses containing \(\sum\) of the \(e\) term in Equation (4) is always an integer, and the possible phases of the probability amplitude are integer multiples of \(\pi / 2\). Therefore, the MUB defined by Equations (2) to (4) can avoid the problem that higher phase resolution is required with the increase of dimension \(d\) in the Fourier transform basis by Equation (1).

[0029] In the MUB defined by equations (2) to (4), up to (d + 1) mutually unbiased bases can be generated. That is, in equation (4), the labels of the MUB are r = 0, 1, ··· d - 1, and d MUBs are generated by equation (3). By adding one more computational basis { |m>}, all (d + 1) bases become mutually unbiased bases.

[0030] In the following description, the configurations of a quantum state generation device and a measurement device using the MUB defined by equations (2) to (4) above are disclosed.

[0031] [MUB State Generation Device Using a Finite Field] FIG. 4 is a diagram showing the configuration of the MUB state generation device using the finite field of the present disclosure. The generation device 30 is configured to generate the states of the MUB based on equations (3) and (4) and also generate the states of the computational basis. An intensity modulator 31 and a phase modulator 32 are connected in cascade. A modulation signal generator 33 that supplies a modulation signal 36 to the phase modulator 32 is also provided. Continuous light or continuous pulse light 34 is input to the intensity modulator 31, and when generating the state of the MUB, d continuous pulses 35 are output. The continuous pulses 35 have not been phase-modulated yet, and the relative phase between the pulses is 0 at any time.

[0032] The modulation signal generator 33 generates a modulation signal (control signal) 36 for applying phase modulation according to the probability amplitude of equation (4) to each pulse of the continuous pulses 35 based on the information of the basis r and the information of the label n of the state in the basis. By the modulation signal 36, phase modulation is applied to each pulse of the continuous pulses 35 input to the phase modulator 32, and an output pulse train 37 is obtained.

[0033] The intensity modulator 31 is also used to generate the computational basis |m> represented by Equation (5). That is, the intensity modulator 31 operates to transmit only the m-th pulse among the d pulses to the phase modulator 32 and suppress the other pulses, instead of the d consecutive pulses for generating the states of the MUB. The input light 34 to the intensity modulator 31 may be continuous light or pulsed light repeatedly generated at a predetermined interval. The generating device with the configuration shown in FIG. 4 can generate a high-dimensional quantum state of the MUB using a finite field, where the relative phase with respect to the leading pulse in the d consecutive pulses has only four values.

[0034] FIG. 5 is a diagram showing another configuration of a state generating device using the MUB using a finite field of the present disclosure. Similar to the generating device 30 of FIG. 4, the generating device 40 is also configured to generate the states of the MUB based on Equations (3) and (4) and to be able to generate the states of the computational basis. The generating device 40 includes an optical IQ modulator 41 and a modulation signal generator 42. The modulation signal generator 42 generates two modulation signals (I signal, Q signal) 45 for applying phase modulation according to Equation (4) to each pulse of the continuous pulse 44 based on the information r of the basis and the information n of the label of the state in the basis. The optical IQ modulator 41 applies phase modulation to each pulse of the continuous pulse 44 by the I signal and the Q signal 45, and an output pulse train 46 is obtained.

[0035] If the optical IQ modulator 41 is used as in the configuration of FIG. 5, the same operation as the generating device shown in FIG. 4 can be achieved with only a single modulator. Since the optical IQ modulator can be compactly implemented as a single modulator, the configuration of the generating device can be further simplified. In FIG. 5, an example of d consecutive pulses is shown as the input light 43, but even if continuous light is input, any state of the computational basis and the MUB can be generated.

[0036] The above-mentioned FIGS. 4 and 5 directly generate the MUB defined by formulas (2) to (4) and the states of the computational basis according to formula (5). In addition to such a generating device, a configuration in which the configuration of the measuring device is inverted is also conceivable. These other configurations will be touched upon again after the configuration and operation of the following measuring device are described in detail.

[0037] [State Measuring Device for MUB Using Finite Field] Here, the configuration and operation of a state measuring device using a finite-field-based MUB will be described. In the following disclosure, the state measurement of the MUB represented by formulas (2) to (5) will be described using the time-bin quantum state as an example. However, in terms of realizing the measuring device by dividing it into two units, the following disclosure can be applied not only to the time-bin quantum state, which is a mode orthogonal in time, but also to the quantum states of other optical modes. As will be described later, the measuring device for the finite-field-based MUB of the present disclosure includes a first unit corresponding to a diagonal unitary transformation with respect to the computational basis and a second unit corresponding to an Hadamard-transform state measurement that decomposes the received high-dimensional quantum state into a lower-dimensional quantum state equivalent thereto and performs a projective measurement. Such a configuration that decomposes the probability amplitude of such a MUB into partial systems of two matrices can also be applied to state measurement when other optical modes such as frequency modes orthogonal in frequency, spatial modes using the orbital angular momentum of light, optical paths, etc. are used as the computational basis.

[0038] Hereinafter, taking the time-bin quantum state as an example, the basic principle and specific configuration example of a measuring device using a finite-field-based MUB will be described.

[0039] [Basic Configuration of Measuring Device with Two Units] FIG. 6 is a diagram for explaining the basic configuration of the MUB state measuring device using the finite field of the present disclosure. (a) of FIG. 6 shows the most conceptual measuring device. The measuring device for the quantum state, for example, for the input pulse train 55 to be targeted by the time-bin quantum state, the probability amplitude B represented by formula (4) mn (r) having the quantum state |ψ n (r)It can be understood as a measuring device 50-1 that identifies the label n of >.

[0040] Probability amplitude B of MUB using a finite field mn (r) The expression (4) representing can be decomposed into two matrices as shown in the following expression. TIFF0007705068000014.tif6150 TIFF0007705068000015.tif18150

[0041] In expression (6), D on the left side of the right side mm (r) is a diagonal unitary matrix. And referring to expression (7), D mm (r) consists only of the exp term. In the case of the time-bin quantum state, it can be seen that D mm (r) corresponds to the phase modulation for each pulse. Therefore, the unit corresponding to the diagonal unitary matrix on the left side of the right side of expression (6) can be realized as a phase modulator for the computational basis.

[0042] Referring to both sides of expression (6), the relationship defined by expression (6) represents the conversion between different bases, that is, the conversion between bases from the MUB with a specific label r = 0 to the MUB with an arbitrary label r. Therefore, if a measurement unit corresponding to B mn (0) on the right side of the right side of expression (6) can be realized, state measurement in the MUB with an arbitrary label r can be realized by combining it with the unit corresponding to the diagonal unitary matrix. The operation by B mn (0) on the right side of the right side of expression (6) corresponds to a projective measurement onto the Hadamard transform basis, as will be described later. Therefore, the state measurement device can be configured to include a unit that performs phase modulation according to expression (7) on the front stage side of the measurement unit that performs the operation of the Hadamard transform matrix. Probability amplitude B of MUB mn (r) By the units corresponding to the two matrices decomposed from, the probability amplitude B of expression (4) mn (r)Measurements on the basis (MUB of label r) can be realized.

[0043] Figure 6(b) shows a measuring device 50-2 with a configuration combining the above two units. The measuring device 50-2 includes a phase modulation unit 51 corresponding to the first unit on the front stage, which is D mm (r) and a measuring unit 52 corresponding to the second unit, which is B mn (0) The modulation signal generator 53 supplies a modulation signal to the phase modulator 54. However, in the phase modulation unit 51, instead of the conversion from the 0th basis (label 0) to the rth basis (label r), the conversion from the rth basis to the 0th basis is required. Equation (6) decomposes the probability amplitude B mn (r) in the generation of MUB. Therefore, in the measuring device, the inverse conversion of the conversion represented by Equation (6), that is, the phase modulation that is the complex conjugate of D mm (r) on the left side of the right side of Equation (6) is performed. In principle, the phase modulator 54 in Figure 6(b) can also be replaced with an optical IQ modulator as in Figure 5.

[0044] Following the phase modulation unit 51, which is the first unit of the above measuring device, the specific implementation method of the measuring unit 52 corresponding to the second unit, which is B mn (0) is the key to the realization of the measuring device. Next, the principle and specific implementation example of the Hadamard transform for B mn (0) will be described in detail together with the drawings.

[0045] [High-Dimensional Hadamard Transform] The matrix B, which is the second unit shown in Figure 6(b) mn (0)The measurement corresponding to [the above] can be implemented as a projective measurement on the state obtained by decomposing the state to be measured into equivalent lower-dimensional two-dimensional quantum states (qubits) and performing a two-dimensional Hadamard transformation on each of these states. Each two-dimensional quantum state obtained by the decomposition does not necessarily correspond to a physical particle, but since it can be treated as a virtual equivalent particle in terms of mathematical formulas, it will be simply referred to as a "particle" below, and when it can be decomposed into N states, it will be referred to as an "N-particle". Here, first, the high-dimensional Hadamard transformation will be explained.

[0046] Figure 7 conceptually explains the high-order original Hadamard transformation operation for quantum states using a finite field. (a) of Figure 7 explains the replacement from a d-dimensional quantum state to the state of a plurality of equivalent particles. Consider the case where the dimension d = 4, that is, a 4-dimensional quantum state 71. Each position of the pulse train arranged at time intervals τ corresponds to the four quantum states of the computational basis, corresponding to the states |0>, |1>, |2>, |3> of the computational basis. Here, this 4-dimensional quantum state is considered as an N-particle two-dimensional quantum state with an equivalent dimension. That is, the four quantum states 71 are replaced and considered as the four quantum states 72 of |00>, |01>, |10>, |11> in the bit representation of 2 bits (N = 2).

[0047] Figure 7(b) explains, as the next step, the operation of decomposing the N-particle two-dimensional quantum state bit by bit. The four quantum states 72, which are two-particle two-dimensional states, can be divided into two blocks with different delay times when decomposed into two states for each digit by focusing on each digit of the bit. For the of 2 first digit, it can be divided into the block 73-1 of the state |0> and the block 73-2 of the state |1>, and the time difference between the two blocks is τ. Also, for the of 2 second digit, it can be divided into the block 74-1 of the state |0> and the block 74-2 of the state |1>, and the time difference between the two blocks is 2τ. B of the second unit in Figure 6 mn (0)The transformation is an operation of decomposing an N - particle two - dimensional quantum state equivalent to the quantum state of the measurement target shown in Fig. 7(b) into bits, that is, performing a two - dimensional Hadamard transformation on all of the equivalent qubits.

[0048] The Hadamard transformation H in the two - dimensional case is defined by the following matrix. TIFF0007705068000016.tif10150

[0049] Here, if Equation (8) is regarded as the probability amplitude B of Equation (3) showing MUB mn (r) then the necessary projective measurements in the two - dimensional case are the following two states |+> and |->. TIFF0007705068000017.tif8150 TIFF0007705068000018.tif8150

[0050] In the case of a high - dimensional Hadamard transformation, the basis states of Equation (3) showing MUB are represented as the tensor product of the following two states |+> and |->. TIFF0007705068000019.tif9150 TIFF0007705068000020.tif9150 TIFF0007705068000021.tif9150 TIFF0007705068000022.tif9150

[0051] Based on the relationships from Equation (9) to Equation (11 - 4) above, some implementation configuration examples of a projective measurement device to a high - dimensional Hadamard transformation basis for realizing the transformation of B in Fig. 6(b) mn (0) will be described. In the following description, a projective measurement device to a high - dimensional Hadamard transformation basis for realizing the transformation of B mn (0) which is the second unit shown in Fig. 6(b) is called a Hadamard transformation state measurement unit or, for simplicity, a Hadamard transformation unit.

[0052] [Implementation of High-Dimensional Hadamard Transform: Configuration Example 1] FIG. 8 is a diagram for explaining the configuration of a Hadamard transform state measurement unit having a tree structure. The Hadamard transform unit 80 is composed of a two-layer interferometer and four photodetectors 83-1 to 83-4. Specifically, a Mach-Zehnder interferometer (MZI) is used as the interferometer for each layer, and different delay times are set for the MZIs for each layer. The MZI is composed of an input coupler 84, two arm waveguides 86-1 and 86-2 having different lengths, and an output coupler 85. In the first-layer MZI 81, the delay time difference between the long arm waveguide 86-1 and the short arm waveguide 86-2 is set to 2τ, and the relative phase difference is set to 0. In the second-layer MZIs 82-1 and 82-2, the delay time difference between the long arm waveguide and the short arm waveguide is set to τ, and the relative phase difference is set to 0, respectively.

[0053] In all the following explanations, in the part equivalent to an interferometer including waveguides having different lengths, although it is shown as 2τ, 0, it is assumed that the left side of the comma indicates the delay time difference between the arm waveguides, and the right side of the comma indicates the relative phase between the arm waveguides.

[0054] Four consecutive pulses 87 of a target time interval τ in a time-bin quantum state are input to the first-layer MZI 81 of the Hadamard transform unit 80. In the MZI arranged in the two-layer tree structure of FIG. 8, taking the output ports of the second-layer MZI as port a, port b, port c, and port d, photons from each port are detected by corresponding photodetectors 83-1 to 83-4. As will be described later, at a specific time 89, projective measurements onto the superposition state of all input pulses are realized from the above four ports. The detection of photons at a certain output port means that projective measurements onto either of the two states |+> and |-> on the equivalent qubit corresponding to that port are being performed. For example, the information on whether it is output to port a' or port b' of the interferometer 81 with a first-layer delay time of 2τ corresponds to projective measurements onto either the |+> or |-> state of the equivalent qubit with a delay time of 2τ in the second digit (q1 of |q1, q0>) when decomposed into the two-particle two-dimensional state of FIG. 7(b). Furthermore, for the superposition state of all input states, it will be described together with the delay operation up to the output points by each MZI arranged in the tree structure.

[0055] FIG. 9 is a diagram for explaining the relationship between the delay of each MZI and the measurement pulse at the output port. The Hadamard transform unit 80 shown in FIG. 9 is the same as that shown in FIG. 8. In FIG. 9, attention is paid to how the 4D continuous pulse train 87 to be measured input to the Hadamard transform unit 80 is output from each of the four output ports after passing through the delay of the two-layer MZI. The pulse train 87 is sequentially input to the first-layer MZI from the pulse 87-1 that arrives first in time to the pulse 87-4 that arrives last. Here, consider the pulse 88 observed at each time at port a. Among the plurality of pulses 88 shown in FIG. 9, the four pulses arranged along the time axis at the bottom show the pulses observed at each time through the path with a delay time of 0. Similarly, the four pulses arranged along the time axis second from the bottom show the pulses observed at each time through the path with a delay time of τ. Similarly, the four pulses arranged third and fourth from the bottom along the time axis show the pulses observed at each time through the paths with delay times of 2τ and 3τ, respectively.

[0056] When the first pulse 87-1 is input to the multi-stage (multi-layer) MZI, it propagates through the shortest paths of the short arms 86-2 and 86-4 of each MZI in the two layers, and at time t0 after a certain initial delay time, it is observed as the output pulse 88-1. At the same port a, at time t1 when a delay time of τ has elapsed from time t0, the second pulse 87-2 propagates through the above-mentioned shortest path and appears as the output pulse 88-2. At the same time, the first pulse 87-1 appears as the pulse 88-1(+τ) that has propagated through the long-arm waveguide 86-3 of the second-layer MZI and is delayed by time τ. Thus, at time t1, two pulses appear at port a, and the two input pulses are in a superimposed state.

[0057] At time t2 when a delay time of 2τ has elapsed from time t0, the third pulse 87-3 propagates along the above-described shortest path and appears as the output pulse 88-3. At the same time, the second pulse 87-2 propagates along the long-arm waveguide 86-3 of the second-layer MZI and appears as the pulse 88-2(+τ) delayed by time τ, and further, the first pulse 87-1 propagates along the long-arm waveguide 86-1 of the first-layer MZI and appears as the pulse 88-1(+2τ) delayed by time 2τ. Thus, at time t2, three pulses appear at port a, resulting in a state where the three input pulses are superimposed.

[0058] Furthermore, at time t3 when a delay time of 3τ has elapsed from time t0, the last pulse 87-4 propagates along the above-described shortest path and appears as the output pulse 88-4. At the same time, the third pulse 87- 3 propagates along the long-arm waveguide 86-3 of the second-layer MZI and appears as the pulse 88-3(+τ) delayed by time τ, the second pulse 87-2 propagates along the long-arm waveguide 86-1 of the first-layer MZI and appears as the pulse 88-2(+2τ) delayed by time 2τ, and further, the first pulse 87-1 propagates along the long-arm waveguides 86-1 and 86-3 of the two-layer MZI and appears as the pulse 88-1(+3τ) delayed by time 3τ. At this time t3, the four pulses shown in the dotted-line region 89 appear at port a simultaneously, resulting in a state where the four input pulses are superimposed.

[0059] As described above, at each time, a plurality of pulses in a superimposed state appear at port a of the second-layer MZI, and at a specific time t3, a superimposed state of all four input pulses 87-1 to 87-4 is obtained. Also, in each MZI, the relative phase of the output light from the two output ports is set to 0. No phase change is given to the pulses output from the MZI, regardless of the presence or absence of delay. Therefore, no phase fluctuation is given to the four output pulses that appear at output port a at time t3, and a state where all the input continuous pulses 87 are superimposed in their original phase relationship is observed.

[0060] At time t3, the superposition state of the above four input pulses is similarly observed at ports b, c, and d. Due to the delay caused by the MZI arranged in these two-layer structures in the photon detector, in the dotted area 89, projective measurement onto the superposition state of all input states is realized. The interference light passing through the MZI, which is an interferometer, is superimposed with a phase relationship determined by the MZI optical path, and the probability amplitudes of photons at different times on the input side are superimposed according to the interference pattern. Considering the phase during interference in the MZI, it can be known whether projective measurement onto one of the two states |+> and |-> on the equivalent qubit is performed according to which output port of each MZI the photon is observed at.

[0061] At each output coupler in the MZI shown in FIG. 8, an interference state with a relative phase of 0 (in-phase) is obtained at ports a, a´, and c. On the other hand, an interference state with a relative phase of π (anti-phase) is obtained at ports b, b´, and d. Therefore, it can be understood that the interference state and the interference pattern are different depending on which output port of the output coupler of the MZI the output is from. It should also be noted that at the input of the interferometer, if the input ports are swapped, the above in-phase and anti-phase relationships are reversed.

[0062] Referring to FIG. 8 again, the information on which of ports a´ and b´ of the MZI81 with a delay time of 2τ the photon is output to corresponds to the measurement of the second digit of the equivalent qubit in FIG. 7, which is either the |+> or |-> state. Therefore, at time t3, it is possible to determine which projective measurement state was made based on which port the photon was detected by the photon detector at. In the case of the two-layer interferometer tree structure for the four-dimensional quantum state in FIG. 8, the relationship between photon detection and the measured state is as follows.

[0063] * Photon detection by the photodetector 83-1 -> the |+,+> state in Equation (11-1) * Photon detection by the photodetector 83-2 -> the |+,- > state in Equation (11-2) * Photon detection by the photodetector 83-3 -> the |-,+> state of Equation (11-3) * Photon detection by the photodetector 83-4 -> the |-,-> state of Equation (11-4) The configuration of the Hadamard transformation unit by the interferometers arranged in a tree shape in Fig. 8 decomposes the high-dimensional time-bin quantum state of the measurement target into equivalent qubits, and realizes it by setting a delay time equal to the delay time for identifying the equivalent qubits in each layer of the interferometer in the tree structure. Therefore, for an 8-dimensional (d = 2 3 = 8) time-bin quantum state with 8 consecutive pulses, for the configuration of Fig. 8, an interferometer with a delay time of 4τ is added as the 0th layer, and the number of MZIs in the 1st layer and the 2nd layer is doubled. As a result, the Hadamard transformation unit for the 8-dimensional time-bin quantum state is composed of 7 MZIs in 3 layers. Similarly, by adding MZIs (interferometers), it is possible to implement a measurement device for an arbitrary 2 N -dimensional time-bin quantum state.

[0064] In the configuration by the interferometers arranged in a tree shape in Fig. 8, the interferometers within one layer may have the same delay time. Therefore, instead of decreasing the delay time in order from the upper layer, equivalent state measurement is possible even if the delay times are interchanged between layers (for example, in the order of 2τ, 4τ, τ). However, generally, interferometers with longer delay times have problems in terms of manufacturing difficulty and stability. For this reason, arranging the delay times in descending order (for 8 dimensions, in the order of 4τ, 2τ, τ) reduces the number of unstable interferometers, resulting in a stable configuration for the entire measurement device.

[0065] Furthermore, the configuration of the tree-shaped Hadamard transform unit in FIG. 8 can set the relative phases of all the interferometers to 0. In order to measure the MUB using the Fourier transform basis of the prior art in the configuration by MZIs, it is necessary to incorporate, for example, a configuration designed such that the phase difference between the arm waveguides is π / 2 or the like into a part of the tree-shaped configuration. Also, the types of set values of the phase values of this interferometer increase with the dimension d. If this phase difference can be set to 0, it is possible to adjust all the interferometers only by adjusting some measurement quantity for the MZIs, for example, so that the extinction ratio becomes maximum or minimum. Since the adjustment of each layer of the interferometer can be a uniform one with a phase difference of 0 and can be simplified, there is an advantage that the adjustment of the interferometer is easy even when the dimension d becomes large.

[0066] [Implementation of High-Dimensional Hadamard Transform: Configuration Example 2] FIG. 10 is a diagram for explaining the configuration of a Hadamard transform state measurement unit using a delay line. In the Hadamard transform state measurement unit with the tree-shaped interferometer described in FIG. 8, as the dimension increases and the number of layers increases, a large number of interferometers are required. However, the interferometers in the same layer only need to have the same delay time and the same relative phase. Therefore, if one interferometer can be shared as an interferometer in the same layer by some method, the number of interferometers can be reduced.

[0067] (a) of FIG. 10 shows an Hadamard transform state measurement unit of Configuration Example 2 using a delay line. The Hadamard transform unit 90 of Configuration Example 2 receives a continuous pulse train 96 in a 4-dimensional time-bin quantum state, similar to the 4-dimensional Hadamard transform unit shown in FIG. 8, but the overall configuration is much smaller. The first-layer MZI 91-1 and the second-layer MZI 91-2 are connected in cascade, and the two output ports of MZI 91-2 are equipped with corresponding photon detectors 95-1 and 95-2. The first-layer MZI 91-1 has a delay time difference of 2τ between the arm waveguides and a relative phase set to 0. The second-layer MZI 91-2 has a delay time difference of τ between the arm waveguides and a relative phase set to 0. Therefore, the configuration of the MZI in each layer is the same as that of the MZI in the case of the tree-like arrangement in FIG. 8. The difference from the configuration in FIG. 8 is that a delay line 92 is connected between the ports not used in the cascade connection of the two-layer MZI, that is, from another port of the output-side coupler of the first-layer MZI to another port of the input-side coupler of the second-layer MZI.

[0068] The delay line 92 is set to a delay time τ' shorter than the delay τ of the second-layer MZI. Therefore, compared with the pulse passing through the path connected in cascade from one output of the first-layer MZI 91-1, the pulse passing through the delay line 92 from the other output port of the output coupler of MZI 91-1 is delayed by a time τ' and input to the second-layer MZI 91-2.

[0069] Here, if the delay time τ' of the delay line 92 is set appropriately, by using one MZI 91-2 at different times, the functions of the two MZIs arranged in parallel in the second layer of FIG. 8 can be achieved. In the configuration of (a) of FIG. 10, the delay time τ' of the delay line 92 is set to a time shorter than the delay τ, but it is only necessary that the inputs at different times to one interferometer 91-2 do not interfere with each other. If they are set to be shifted so that the inputs at different times do not interfere with each other, the delay time of the delay line can be set to any value.

[0070] (b) of FIG. 10 shows the state of output pulse 99 observed by the Hadamard transform unit 90 in (a). Output pulse 99 corresponds to output pulse 88 in the tree-like configuration of FIG. 9. By comparing the two, the operation of delay line 92 can be understood. In output pulse 99 of FIG. 10(b), triangular pulse 96-1 is a pulse output by the two-layer MZI through a path directly connected to the cascade. Similar to the case of the tree-like configuration example 1 in FIG. 8, it is observed in the superposed state of four input consecutive pulses 96 at time t3 within the dotted line region 98.

[0071] The four consecutive pulses 96 that are delayed and input to the second-layer MZI 91-2 via delay line 92 correspond to the dotted triangular pulse 96-2 in FIG. 10(b). Except for being delayed by a delay time τ' compared to pulse 96-1, it is the same as the output pulse of the path directly connected to the cascade. Therefore, for example, if τ' is selected to be half of the time interval τ of input pulse 96, output pulses arranged at intervals of τ / 2 can be obtained by any photon detector. Four types of interference patterns can be obtained by the combination of information on which of the two output ports of the second-layer MZI and the two measurement times t3 and t3 + τ'. These combinations can be associated with the four states of |+,+>, |-,+>, |+,->, and |-,-> in the same way as configuration example 1 in FIG. 8.

[0072] As described above, in the Hadamard transform unit 90 of configuration example 2 in FIG. 10, the single second-layer MZI is repeatedly used with a time shift, and it can be considered that a single MZI is reused multiple times in a time-division manner. Such repeated use of a single interferometer can be performed independently in each layer except for the first layer even when the number of layers increases. By repeatedly using (reusing) this interferometer multiple times, the number of required interferometers can be significantly simplified to logd, and the number of photon detectors can be simplified to two.

[0073] (c) of FIG. 10 is for 16 dimensions (d = 2 4) shows a configuration example of the Hadamard transform unit in the case of . Four layers of MZIs 91-1 to 91-4 are cascade-connected. The delay time differences between the arm waveguides of the MZIs in each layer are set to 8τ, 4τ, 2τ, and τ in order, and the phase difference between the arm waveguides is set to 0. Between the first-layer MZI and the second-layer MZI, a delay line 92-1 with a delay time of τ´ is connected in parallel with the cascade connection. Between the second-layer MZI and the third-layer MZI, a delay line 92-2 with a delay time of τ´´ is similarly connected in parallel. Between the third-layer MZI and the fourth-layer MZI, a delay line 92-3 with a delay time of τ´´´ is also similarly connected in parallel.

[0074] The delay time of each delay line may be set at a timing when a plurality of pulses (solid-line triangles and dotted-line triangles) in adjacent superimposed states in Fig. 10(b) do not collide at the output port.

[0075] When setting τ´ of the delay line that connects the interferometers of adjacent layers in parallel as shown in Fig. 10(b) so that photons delayed to the middle time among the original detection times (t0, t1, t2··) at the photodetector are detected, there may be a limit to the number of times one interferometer can be repeatedly used in a time-division manner. This is because there are certain limitations when narrowing the measurement time interval due to the influence of timing jitter of the photodetector and subsequent analysis devices. In such a case, the repeated use of the interferometer can be partially performed on some layers in the entire Hadamard transform unit. That is, a hybrid configuration can also be adopted by combining the tree-like configuration example in Fig. 8 and the configuration example in Fig. 10, and performing repeated use of the interferometer using a delay line on some layers.

[0076] For example, 8 dimensions (d = 2 3) In the case of , if it is realized with the tree-like configuration of Configuration Example 1, seven MZIs and eight photon detectors are required in three layers. When considering the same eight-dimensional hybrid configuration of Configuration Example 1 and Configuration Example 2, as a configuration that reuses the first and second layers, a configuration can be considered where only the third layer has a tree-like shape including two MZIs and uses four photon detectors. In the case of this hybrid configuration as well, projective measurement onto eight states can be realized by the combination of two measurement timings and four photon detectors (2×4). In each configuration shown in FIG. 10, it should be noted that the delay line functions not to cause interference to the input continuous pulse, but only to cause a delay. Therefore, unlike the relative phase of the long arm waveguide in the interferometer of each layer, the change in the relative phase occurring in the delay line can be ignored.

[0077] [Implementation of High-Dimensional Hadamard Transformation: Configuration Example 3] FIG. 11 is a diagram for explaining the configuration of a Hadamard transformation state measurement unit using an optical switch (optical SW). Compared with the Hadamard transformation state measurement unit 90 in FIG. 10, it is similar in that interference structures including two arm waveguides are cascade-connected, and the spaces (between layers) between the interference structures are connected in parallel by delay lines. The Hadamard transformation unit 100 in FIG. 11 has a configuration in which one layer is added to make the dimension d = 8 with respect to Configuration Example 2 in FIG. 10 where adjacent MZIs in different layers are connected in parallel by delay lines, and the input coupler of the MZI which is an interferometer is replaced with an optical SW.

[0078] Specifically, the Hadamard transformation unit 100 of Configuration Example 3 in FIG. 11 replaces the input couplers of the MZI parts 101-1 to 101-3 in each layer in Configuration Example 2 with dimension d = 8 with optical SWs 102-1 to 102-3, and further adds an optical SW 104 at the final stage. If the input coupler is replaced with an optical SW, strictly speaking, the MZI parts 101-1 to 101-3 in FIG. 11 are not Mach-Zehnder interferometers (MZIs). However, since it includes two arm waveguides whose delay time and phase are set to predetermined values and can cause interference, it is called an "MZI part".

[0079] Another difference between the Hadamard transform unit 100 in FIG. 11 and Configuration Example 2 is that the delays of the delay lines 103-1 to 103-3 connected in parallel to the path connecting the MZI parts between the layers in a cascade connection are set to the same values as the delay times of the respective preceding-stage MZI parts. Specifically, the delay time of the delay line 103-1 connected in parallel between the first layer and the second layer is set to the same 4τ as the delay time of the MZI part 101-1 on the preceding stage side. The delay time of the delay line 103-2 connected in parallel between the second layer and the third layer is set to the same 2τ as the delay time of the MZI part 101-2 on the preceding stage side. Similarly, the delay time of the delay line 103-3 connected in parallel between the third layer and the optical switch 104 is set to the same τ as the delay time of the MZI part 101-3 on the preceding stage side. The photon detector 106 detects photons with the output of the final-stage optical switch 104.

[0080] In each layer, the optical switch replaced from the input coupler of a normal MZI operates to switch one or more inputs to one or more outputs. For example, the optical switch 102-1 outputs the input continuous pulse 105 to either of the two arm waveguides of the MZI part 101-1 in synchronization with the time interval τ. Also, the optical switch 102-2 represented by a solid line outputs the pulse from the and a pulse from a path connecting to the MZI section 101-2, or the delay line 103-1 represented by a dotted line MZI part 101-1 of the two arm waveguides to either of the MZI parts 101-2 of the two arm waveguides in synchronization with the time interval τ. Similarly, the optical switches 102-3 and 104 also perform the switching operation of the input and output relationship in synchronization with the time interval τ. At this time, the optical switches are operated so that the output destinations of the optical switches in each layer are determined according to |0> and |1> of each qubit when the high-dimensional quantum state is made to correspond to an equivalent two-dimensional quantum state.

[0081] In Configuration Example 3, the delay time of the delay line is set to the same value as the delay time of each MZI section on the previous stage side, and the delay time is an integer multiple of the time interval τ of the input continuous pulse. Therefore, at the output points of the MZI sections of each layer, pulses that have propagated through different paths appear simultaneously, reaching a timing where they can collide. However, due to the switching operation of the input-output relationship of each optical SW, no collision occurs. As a result, at the output point of the final-stage optical SW104, the superposition state due to different interference patterns of the input continuous pulse 105 interfered by the output coupler of each MZI section is realized at different times.

[0082] The Hadamard transform unit 100 in FIG. 11 switches each of the above-described optical SWs of each layer while synchronizing with the time interval τ of the input continuous pulse 105, so that at the output point of the optical SW104, the superposition state of the input continuous pulse 105 is realized at all measurement times (t0 to t7). Note that the first measurement time t0 is after all eight input continuous pulses 105 have been input to the Hadamard transform unit 100, so at least 8τ of time has elapsed since the first pulse input. At any detection time (t0, t1,... t7), the eight input pulses are superimposed and interfered, and the interference pattern can be made to correspond to a high-dimensional Hadamard transform.

[0083] FIG. 11(b) shows the pulse 107 observed at each time at the output point of the Hadamard transform unit 100 of Configuration Example 3. At any detection time from the earliest t0 to t7, the eight pulses of the input continuous pulse 105 appear simultaneously. Therefore, according to the information on at which detection time a photon was detected, when an 8-dimensional quantum state is converted into a qubit representation of an equivalent 3-particle 2-dimensional quantum state, it is possible to determine which projective measurement onto a state of a combination of the 2-dimensional quantum states |+> and |-> on the equivalent qubit was performed.

[0084] * Photon detection at time t0 -> |+, +, +> state * Photon detection at time t1 -> |+, +, -> state *Photon detection at time t2 -> |+,-,-> state *Photon detection at time t3 -> |+,-,+> state *Photon detection at time t4 -> |-,-,+> state *Photon detection at time t5 -> |-,-,-> state *Photon detection at time t6 -> |-,+,-> state *Photon detection at time t7 -> |-,+,+> state In the Hadamard transformation unit 100 of FIG. 11, since the waveguides and delay lines connecting the MZI parts of adjacent layers can also be regarded as an interference circuit, similar to the MZI shown after FIG. 8, it shows the delay time difference between the arm waveguides and the relative phase between the arm waveguides.

[0085] The structural configuration of the Hadamard transformation unit 100 of FIG. 11 is also disclosed in, for example, Non-Patent Document 3. However, for the quantum state measurement of MUB using a finite field, it is significantly different from the configuration of Non-Patent Document 3 in that the relative phase in each delay line can be set separately and to arbitrary values. In the configuration of FIG. 11, the relative phase generated by the delay of the delay lines 103-1 to 103-3 appears as the relative phase between the output pulses at different detection times, like the pulses within the dotted line 108 at time t1 and the pulses within the dotted line 109 at time t0 in FIG. 11(b). In other words, the relative phase generated by the delay line has no relation to the phase between the pulses (between the eight pulses within the dotted line region 108) that appear as the superposition state of the input continuous pulses at the same detection time.

[0086] The fact that the phase of the above-mentioned delay line can be arbitrary can be explained as follows. In the Hadamard transform unit 100, photons are detected by the photon detector 106 immediately after a series of cascaded MZI sections (interferometers). However, the measurement result of the photon detector 106 is not affected by the relative phase between the pulses in the superposition state corresponding to different detection times (between region 108 and region 109). Specifically, in Fig. 11(b), the relationship between the phase of the pulse in region 109 at time t0 and the phase of the pulse in region 108 at time t1 does not affect the measurement result of the photon detector 106.

[0087] Therefore, the phases generated by the delay lines 103-1 to 103-3 can be set arbitrarily. In the Hadamard transform unit 100 of Fig. 11, the value of this phase difference may vary among the three delay lines, and may be any value not limited to 0.

[0088] In the Hadamard transform unit 100 of Fig. 11, the fact that the phase in the delay line can be set arbitrarily means that it is not necessary to stabilize the phases of these delay lines like the MZI sections 101-1 to 101-3. In the MZI sections 101-1 to 101-3, phase modulation should not be applied to each of the input continuous pulses. Therefore, the relative phase between the arm waveguides in the MZI sections 101-1 to 101-3 must be accurately set to 0. On the other hand, for the delay lines 103-1 to 103-3 of the Hadamard transform unit 100 in Fig. 11, the conditions for the delay line design in the actual device can be greatly relaxed.

[0089] Also, in each of the Hadamard transform units of Configuration Example 1 in Fig. 8 and Configuration Example 2 in Fig. 10 described above, only the measurement results at some detection times where all input pulses interfere can be used in the output of the interferometer. For this reason, the decrease in detection efficiency becomes a problem. On the other hand, in Configuration Example 3 of Fig. 11, the detection results of photons at all detection times correspond to the projective measurement of any of the Hadamard transform basis states. Therefore, in principle, it is possible to realize a state measurement device of MUB using a finite field without a decrease in detection efficiency. Furthermore, in Configuration Example 3, the number of photon detectors can be reduced to one.

[0090] [Implementation of High-Dimensional Hadamard Transform: Configuration Example 4] FIG. 12 is a diagram for explaining the configuration of a Hadamard transform state measurement unit that utilizes a loop structure. The Hadamard transform state measurement unit 110 in FIG. 12 of Configuration Example 4 deforms the configuration of Configuration Example 3 into a loop shape and repeatedly uses the MZI section as in Configuration Example 2, thereby realizing a Hadamard transform with a more compact configuration.

[0091] In the Hadamard transform state measurement unit 110, similar to Configuration Example 3, the MZI section 111 includes an optical switch 112, two arm waveguides 113 with different lengths, and an output coupler 114. One output port of the output coupler 114 is further connected to the input port of the optical switch 116. The other output port of the output coupler 114 is connected to another input port of the optical switch 116 via the delay line 115. One output of the optical switch 116 is input to the optical switch 112 in order to repeatedly use the MZI section 111. The other output of the optical switch 116 is given to the photon detector 117 as the final output after the repeated use of the MZI section 111 for a certain number of times. In this way, the MZI section 111, the delay line 115, and the optical switch 116 are configured in a loop shape so that they are repeatedly used.

[0092] The delay time of the MZI section 111 is set to Δ(t) and takes values such as τ, 2τ, 4τ,... as in Configuration Example 3. The delay time Δ(t) changes temporally in synchronization with the time interval τ of the input continuous pulse 118 so that the MZI section 111 can be repeatedly used with different configurations. Similarly, the delay time of the delay line 115 is set to the same Δ(t) as the MZI section 111 and changes temporally in synchronization with the time interval τ of the input continuous pulse 118. By switching the relationship between the inputs and outputs in the two optical switches 112 and 116 in the same way as in Configuration Example 3, the same MZI section 111 is repeatedly used to form an example 3 andThe same operations become possible. In the case of 8 dimensions, for the final output, at all detection times, as shown in Fig. 11(b), the 8 input pulses are superimposed and an interfering state is obtained. According to the information on the time when the photon is observed, it is possible to determine which state's projective measurement has been performed when the 8-dimensional quantum state is decomposed into 3-particle 2-dimensional quantum states.

[0093] From the above description, it can be understood that the configuration of the Hadamard transform state measurement unit 110 in Fig. 12 repeats the use of the configuration of the MZI part and the delay line for one layer linearly expanded in Fig. 11 as different layers by forming a loop structure with the feedback path. Also in the Hadamard transform state measurement unit using the loop structure in Fig. 12, the same discussion as in Configuration Example 3 of Fig. 11 holds for the phase difference in the delay line 115. Therefore, the phase difference applied to the delay line 115 can be an arbitrary value, and there is no need to stabilize the phase difference with respect to the optical path of the delay line. Furthermore, since the feedback path between the optical switch 116 and the optical switch 112 is formed by the same waveguide and the single MZI part is repeatedly used, there is no need to align the phase values in this feedback path. Therefore, when configuring the MUB state measurement device using the Hadamard transform unit 110 of Configuration Example 4, it is only necessary to stabilize the accuracy of the phase difference between the two arm waveguides of the MZI part 111. By forming a feedback path in a single MZI part to form a loop structure, the design and manufacturing conditions of the state measurement device can be significantly relaxed.

[0094] As described in the above four configuration examples, the Hadamard transform state measurement unit in the time-bin quantum state is a plurality of optical interferometers arranged in a tree shape in N layers, and each of the plurality of optical interferometers is given a delay time corresponding to the layer position of the N layers, a first configuration (configuration example 1) including a plurality of optical interferometers, a plurality of optical interferometers cascade-connected in N layers, and each of the plurality of optical interferometers is given a delay time corresponding to the layer position of the N layers. And a second configuration (configuration example 2, configuration example 3) including one or more delay lines in parallel with the connection between two adjacent layers and having a delay time set corresponding to the delay time of the optical interferometer in the previous layer, or a third configuration (configuration example 4) including an optical interferometer connected in a loop shape and having a variable delay time corresponding to the number of turns. It can be implemented as including any of the above.

[0095] [State measurement device of MUB using finite field: Specific configuration 1] Configuration example 1 in FIG. 8, configuration example 2 in FIG. 10, configuration example 3 in FIG. 11, and configuration example 4 in FIG. 12 are examples of the Hadamard transform state measurement unit that performs projective measurement onto the Hadamard transform basis when the state measurement device of MUB using a finite field is decomposed into two units. Here, the specific overall configuration of the state measurement device including the state measurement in the computational basis is shown.

[0096] FIG. 13 is a diagram showing a configuration example of a state measurement device of MUB including the computational basis. The MUB state measurement device 120 using a finite field has the same configuration as the measurement device 50-2 obtained by combining the two units shown in FIG. 6(b). It includes a phase modulation unit 51 which is the first unit and a Hadamard transform state measurement unit 90 corresponding to B on the right side of the right side of the equation (6) which is the second unit. mn (0) The Hadamard transform state measurement unit 90 in FIG. 13 is the same as the Hadamard transform state measurement unit 90 in configuration example 2 using the delay line shown in FIG. 10(a).

[0097] The measurement device 50-2 shown in Fig. 6(b) is a projective measurement device for the quantum states of the maximum d (r = 0, 1, ···, d-1) MUBs defined by Eqs. (3) and (4). The state measurement device 120 in Fig. 13 can measure the computational basis in addition to the d MUB states, realizing a total of (d + 1) MUB measurements. Since the projective measurement onto the computational basis is a measurement of when the photon is present, it can be directly measured by a photon detector without using an interferometer.

[0098] For the projective measurement onto the computational basis, in the state measurement device of Fig. 13, an optical SW122 is provided on the input side, and if a photon is sent to the photon detector 123-3, a projective measurement onto the computational basis is performed. When a photon (input continuous pulse 121) is sent to the phase modulation unit 51 side by the optical SW122, a projective measurement onto the state of the MUB of label r described by Eqs. (3) and (4) is performed according to the modulation by the phase modulator 54. Therefore, in the state measurement device 120 of Fig. 13, the optical SW122 and the phase modulator 54 serve to switch and select different (d + 1) MUBs.

[0099] Needless to say, in the configuration of the state measurement device of Fig. 13, the Hadamard transform state measurement unit 90 of the second unit can be replaced with other configurations. The Hadamard transform state measurement unit 90 of the units of the state measurement device of Fig. 13 can be replaced by any of the configuration examples 1 to 4 described in Figs. 8 to 12. Also, if the switching between the computational basis and other bases can be random, such as for applications to QKD, the optical SW can be replaced with an optical beam splitter with an appropriate splitting ratio.

[0100] [State Measurement Device for MUB Using Finite Field: Specific Configuration 2] Fig. 14 is a diagram showing another configuration example of a state measurement device for MUB using a finite field. The state measurement device 130 in Fig. 14 also has the same configuration as the measurement device 50-2 shown in Fig. 6(b), including a phase modulation unit 51 as the first unit and a B as the second unit mn (0)It is provided with a Hadamard transform state measurement unit 131 corresponding to

[0101] Among the optical SWs used in the Hadamard transform state measurement units of Configuration Example 3 and Configuration Example 4, there are those that can not only switch the input and output relationships but also play the same role as a beam splitter by adjusting the control conditions. Such an optical SW has three states as output states, including transmission, blocking (reflection), and distribution (half-mirror state).

[0102] The Hadamard transform state measurement unit 131 in the state measurement apparatus 130 of FIG. 14 has three-layer MZI units 132-1 to 132-3 connected in cascade, and delay lines are connected in parallel between the layers, which has the same configuration as Configuration Example 3. However, the output-side couplers of the MZI units in each layer are replaced with optical SWs 133-1 to 133-3 including the output light distribution state. The replaced optical SWs perform substantially the same operation as the output couplers of the MZIs with a phase difference set to 0 in the distribution (half-mirror) state. Therefore, the Hadamard transform state measurement unit 131 in FIG. 14 and the Hadamard transform state measurement unit 100 in FIG. 11 can perform the same operation. Furthermore, if we consider a state where the state measurement device in Fig. 14 sends pulses only along one fixed path without performing the time-switching operation of all optical SWs, it can be seen that the MZI section in this state functions as just a delay line. In such a state, since no interference occurs in the input continuous pulse 135, the measurement by the state measurement device in Fig. 14 corresponds to a projective measurement onto the computational basis. Therefore, with the configuration of the state measurement device 130 in Fig. 14, it is possible to measure (d + 1) MUBs including the computational basis. Also, according to the configuration of the state measurement device 130, it is possible to measure the states of all MUBs including the computational basis with just one photon detector 134. The advantages such as the value of the phase difference being arbitrary for the delay line shown by the dotted line and the stabilization of the phase difference in the delay line being unnecessary are the same as those of the Hadamard transform state measurement unit in Configuration Example 3 of Fig. 11. As shown in Fig. 14, the number of states that each optical SW should take only needs to be two values (transmission / reflection states) on the input side of the MZI section of each layer and three values (transmission / reflection / half-mirror state) on the output side.

[0103] Needless to say, replacing the output coupler in the MZI section with an optical SW that realizes a distribution state as described above can also be applied to the Hadamard transform state measurement unit 110 according to Configuration Example 4 with the loop-like structure shown in Fig. 12.

[0104] [Another configuration of the MUB state generation device using a finite field] In Figs. 6 to 14, various configuration examples of the MUB state measurement device using a finite field have been presented. Here, regarding the MUB state generation device again, an approach starting from the configuration of the state measurement device will be presented. In the case of the most basic two-dimensional time-bin qubit, a configuration realized by an interferometer and a phase modulator is widely used as a method for generating the states of MUBs.

[0105] FIG. 15 is a diagram for explaining the generation of MUB states of two-dimensional time-bin qubits. (a) of FIG. 15 shows the configuration of a generation apparatus currently used for generating MUB states of two-dimensional time-bin qubits. Comparing the configuration of the state generation apparatus 30 shown in FIG. 4 with the configuration of (a) of FIG. 15, in (a) of FIG. 15, the intensity modulator 31 in FIG. 4 is replaced with an interferometer 140-1. This interferometer is an MZI and is composed of an input-side coupler 141, two arm waveguides with a delay time set to τ, and an output-side coupler 142. Using this interferometer 140-1, a superposition state of two consecutive pulses is realized at a time interval τ.

[0106] The state generation apparatus 30 shown in FIG. 4 can be used for generating states with pseudo single photons using weak coherent light allowed in QKD etc. However, when using a true single photon source in the state generation apparatus 30 shown in FIG. 4, a separate apparatus for creating a superposition state regarding the time position is required. The method using the interferometer 140-1 in (a) of FIG. 15 has the advantage that it can be used for a true single photon source because the interferometer creates the superposition state itself regarding the time position.

[0107] In the configuration of (a) of FIG. 15, the photons output to port b' are lost, so even when using an ideal apparatus, the state generation fails with a probability of 1 / 2. Therefore, as shown in (b) of FIG. 15, by replacing the subsequent output-side coupler (beam splitter) 142 with an optical SW143 and not outputting to port b', in principle, the state generation is possible with a probability of 1.

[0108] Furthermore, as shown in (c) of FIG. 15, when the input-side coupler is also replaced with an optical SW144 and all the couplers of the interferometer 140-3 are replaced with optical SWs, it is possible to generate all MUBs including not only the superposition state regarding the time position but also the time position states |0> and |1>.

[0109] What should be noted in the configuration of the apparatus for generating the states of the above-mentioned two-dimensional MUB is that the arrangement order of the components of the state measurement apparatus for the high-dimensional MUB of the present disclosure described in FIGS. 6 to 14 is reversed in the similar configuration in the two-dimensional case for each configuration in FIG. 15.

[0110] FIG. 16 is a diagram for explaining an MUB state generation apparatus for an 8-dimensional time-bin qubit. The MUB state generation apparatus 200 using a finite field is an extension of the MUB state generation apparatus for the two-dimensional time-bin qubit in FIG. 15(c) to an 8-dimensional time-bin quantum state. At the same time, the components of the state measurement apparatus described so far are arranged in the reverse order. That is, the state generation apparatus 200 in FIG. 16 arranges the components of the state measurement apparatus 130 in FIG. 14 in the reverse order. Specifically, it is arranged in the order of a delay processing part 201 including an MZI part 203-1 with a delay time τ, an MZI part 203-2 with a delay time 2τ, and an MZI part 203-3 with a delay time 4τ, and a phase modulation part 202. The state generation apparatus 200 in FIG. 16 can generate the states of all nine MUBs including the computational basis in the 8-dimensional case for the input of a single photon 206.

[0111] In the state generation apparatus 200 in FIG. 16, the delay line of the state measurement apparatus 130 in FIG. 14 is not included. This is because even if there is a delay line, due to the switching operation of the optical SW, it will be in the same state as if there is no delay line. Therefore, the components of the measurement apparatus in FIG. 14 can be arranged in the reverse order and directly used as the generation apparatus, or the unnecessary delay line can be deleted. Similarly, the state measurement apparatuses in FIGS. 6 to 13 can be used as the state generation apparatus in the same way as in FIG. 14 by reversing the order of arrangement of the components as they are or in a configuration with the delay line removed.

[0112] For example, by reversing the arrangement of the Hadamard transform state measurement unit 80 in the tree-like configuration example 1 shown in FIG. 8 and replacing the delay processing section 201 in FIG. 16 with a configuration consisting of a plurality of optical interferometers arranged in an inverted tree shape in N layers from the input side to the output side, light may be input to one port of the interferometer closest to the input side. Similarly, in the configuration obtained by reversing the arrangement of the Hadamard transform state measurement unit 100 in the cascade-connected interferometer configuration example 3 shown in FIG. 11, the delay processing section 201 in FIG. 16 may be replaced. At this time, the delay lines between the layers of the interferometers may be omitted. Regarding the Hadamard transform state measurement unit 110 in the configuration example 4 shown in FIG. 12, the delay processing section 201 in FIG. 16 can be similarly replaced with a configuration in which the input-output relationship is reversed.

[0113] However, in the case of the Hadamard transform state measurement unit 90 in the configuration example 2 using cascade-connected interferometers without using the optical SW in FIG. 10, the delay lines must necessarily be removed. This is because due to the influence of the delay lines, a number of pulses exceeding the assumed dimension are generated. As described above, when using the optical SW, the effect of the delay lines can be eliminated, so the delay lines can be left and the exact same device can be used. In the case of the tree-like configuration example 1, there are no delay lines in the first place, so this is not a problem.

[0114] In addition, in the configuration consisting of a plurality of optical interferometers arranged in an inverted tree shape in the above-mentioned N layers, there are unused optical interferometers in the lower layers of the tree. For example, in combination with an optical circulator or the like, one tree-shaped optical interferometer can be used for both the generating device and the measuring device to achieve overall compactness.

[0115] Therefore, another state generation device of the present disclosure is a time-bin quantum state in which the states of orthogonal lights correspond to the pulses of each of d consecutive pulse trains and utilize orthogonal modes of time, and is a plurality of optical interferometers arranged in an inverse tree shape in N layers from the input side to the output side, and each is provided with a delay time corresponding to the layer position of the N layers. A first configuration (inverse arrangement of Example 1 of the Hadamard transform state measurement unit configuration) including a plurality of optical interferometers, a plurality of optical interferometers cascade-connected in N layers, and each having a delay time corresponding to the layer position of the N layers. A second configuration (inverse arrangement of Example 3 of the same configuration) including a plurality of optical interferometers, or a third configuration including an optical interferometer connected in a loop shape and provided with a variable delay time corresponding to the number of turns (inverse arrangement of Example 4 of the same configuration). It can be implemented as including any one of the configurations 201 and a phase modulator 202 connected to the last stage of any one of the configurations, and adding phase modulation to each of the pulses based on the information r of the basis and the information n of the label of the state in the basis.

[0116] [Quantum key generation system using MUB state generation / measurement device using finite field] FIG. 17 is a diagram showing a quantum key distribution system using a MUB quantum state using a finite field. The state generation device and measurement device of the high-dimensional MUB described so far can all be used as the following high-dimensional quantum key distribution system 300. The high-dimensional quantum key distribution system 300 includes a transmitter-side block 301, a receiver-side block 302, and classical communication channels 303 and quantum communication channels 304 connecting the two. The transmitter-side block 301 includes a high-dimensional MUB state generation device 305 and an error correction / confidentiality enhancement processing unit 306. The receiver-side block 302 includes a high-dimensional MUB state measurement device 307 and an error correction / confidentiality enhancement processing unit 308. In this high-dimensional quantum key distribution system 300, quantum key distribution is performed according to the following procedure.

[0117] Step 1: The sender Alice 301 randomly selects (d + 1) MUBs from the basis r = r a and further randomly selects the state n = n a from among them.

[0118] Step 2: Input the information of the pair (r a , n a ) into the MUB state generation device 305 using a finite field, and send the generated state to the recipient Bob 302 through the quantum communication channel 304.

[0119] Step 3: The recipient Bob 302 randomly selects the basis r = r b in the same way as Alice, and inputs it together with the state sent from Alice into the MUB state measurement device 307 using a finite field, so as to obtain the measurement result n = n b .

[0120] Step 4: Repeat from the state generation to the state measurement in Steps 1 - 3. Among the obtained sequence of states n, Alice and Bob disclose the information of the basis r to each other via the classical communication channel 303, and only keep the n when they match, thereby obtaining a shift key.

[0121] Step 5: Alice and Bob disclose the results of a small number of test bits in the shift key to estimate the distribution of their n a , n b , or the simplified non - matching probability (error rate).

[0122] Step 6: Based on the estimated distribution or non - matching probability, perform bit error correction and privacy amplification used to obtain a secure and matching key on the remaining shift key, and generate a secret key for use in encrypted communication.

[0123] The advantage of this quantum key distribution system 300 over the prior - art high - dimensional quantum key distribution system is that all of the MUBs can be utilized in any two n dimensions. Compared with a quantum key distribution system that uses only two types of MUBs (for example, Non - Patent Document 4), the quantum key distribution system 300 can obtain an improved error rate tolerance effect.

[0124] The quantum key distribution protocol of the above steps 1 to 6 is the most basic one. However, as long as the MUB state generation device and measurement device using the finite field of the present disclosure are used, extensions such as the decoy method widely used in two-dimensional quantum key distribution can be applied, and the above-described effect of improving error rate tolerance can be obtained.

[0125] Also, the number of options for basis selection in the above steps 1 to 6 can be reduced from a maximum of (d + 1) to a minimum of 2. In this case, the effect of improving error rate tolerance due to the number of basis options decreases. Instead, for example, by not using the time basis, the photon detector 123-3 in the state measurement device 120 shown in FIG. 13 can be made unnecessary, and the quantum key generation system can be further simplified. Even in this case, as the dimension d increases, advantages can be obtained compared to a quantum key distribution system that uses only two-dimensional MUBs.

[0126] [Quantum Key Distribution System Using Entangled States and MUB State Measurement Devices Utilizing Finite Fields] FIG. 18 is a diagram showing another example of a quantum key distribution system using an MUB quantum state utilizing a finite field. As a QKD method, there is known a method in which a third party Charlie shares a maximum quantum entangled state with Alice and Bob, and both of them perform measurements to share a secret key. The high-dimensional MUB state measurement devices described so far can also be used as the following high-dimensional quantum key distribution system 400. The quantum key distribution system 400 includes a sender-side block 401, a receiver-side block 402, a third-party block 403, and classical communication channels 407 and quantum communication channels 406a and 406b that connect the three parties. The sender-side block 401 includes a high-dimensional MUB state measurement device 404 and an error correction / privacy amplification processing unit 405. The receiver-side block 402 also includes a high-dimensional MUB state measurement device 408 and an error correction / privacy amplification processing unit 409.

[0127] In this high-dimensional quantum key distribution system 400, the MUB state measurement device using the finite field described so far can be used in the following procedure.

[0128] Step 1: Alice and Bob each prepare high-dimensional MUB state measurement devices 404 and 408. However, for one of them (Alice), D mm (r)* is used as the modulation signal to the phase modulator, and for the other (Bob), D mm (r) is used as the modulation signal to the phase modulator.

[0129] Step 2: Charlie generates a d-dimensional maximally entangled state according to the following formula, and sends one photon to Alice and the other photon to Bob through quantum communication channels 406a and 406b. TIFF0007705068000023.tif13150

[0130] Step 3: Alice and Bob each randomly select (d + 1) MUBs from the bases r = r a , r = r b . By inputting the selected basis information and the photons received from Charlie into the high-dimensional MUB state measurement devices 404 and 408, measurement results n a , n b are obtained respectively.

[0131] Step 4: Repeat from the state generation to the measurement in Steps 2 - 3. Among the obtained columns of states n, Alice and Bob disclose the information of the basis r to each other via the classical communication channel, and keep only the n when they match, thus obtaining a shift key.

[0132] Step 5: Alice and Bob disclose the results of a small number of test bits in the shift key to estimate the distribution of their respective n a , n b , or the simplified non - matching probability (error rate).

[0133] Step 6: Based on the estimated distribution or error rate, perform bit error correction and privacy amplification used to obtain a secure and matching key on the remaining shift key, and generate a secret key for encrypted communication.

[0134] In the conventional high-dimensional quantum key distribution apparatus, in the case of a prime dimension, QKD using entanglement with all MUBs has been implemented using the orbital angular momentum of light (for example, Non-Patent Document 5). The advantage of the high-dimensional quantum key distribution system 400 over this conventional quantum key distribution system is that, similar to the quantum key distribution system 300 in FIG. 17, all MUBs can be used in any two n dimensions, and a high effect of improving the error rate tolerance can be expected.

[0135] Also, by using entanglement, the number of random numbers required to operate the quantum key distribution system can be reduced, or the light source can be prepared by an untrusted third party, etc., and the same advantages as QKD using quantum entanglement in the case of two dimensions can be obtained.

[0136] [High-Dimensional Quantum States by MUB Using a Finite Field of the Power of an Odd Prime Number p] The state generation device and state measurement device using MUB with a finite field up to now have been described for the case of dimension d = 2 N . Here, considering the case where p is an odd prime number and the dimension d = p N , the probability amplitude B mn (r) is given by the following equation (Non-Patent Document 2). Similar to Equation (4) in TIFF0007705068000024.tif18150, m, n, and r are defined as follows. m: Label defining the state of the computational basis n: Label defining the state of the mutually unbiased basis (MUB) r: Label of MUB In the parentheses including Σ in Equation (13), r, m, and n (boldface) represent vectors when the integers r, m, and n are expressed in p-adic notation such that each element is an element of a finite field of order p. Also, r j (boldface) is a scalar quantity indicating the j-th element when r is vectorized in p-adic notation. m T (boldface) is a horizontal vector, A (j) is a square matrix, and m (boldface) becomes a vertical vector.

[0137] Also, for the number of bits p such that only the i-th element becomes 1 N Let the element that is the basis of the finite field be f i and the symmetric matrix A (j) is defined as a matrix that satisfies equation (2) in the same way as in the case of the finite field of order 2 N Here, even if p = 2 is substituted into equation (13), it does not become the same as equation (4) in the case of the finite field of order 2, and a different treatment is required when p is an odd prime number. Since the calculation inside the parentheses in equation (13) is a calculation in the finite field of order p, simply perform the product and sum of integers and calculate mod p. Furthermore, since the phase of this exponential function is an integer multiple of 2π / p, the calculation of mod p can also be omitted, and simply the product and sum calculation of integers is sufficient N According to equation (13), for d = p where p is an odd prime number

[0138] the phase of the probability amplitude B N takes values that are integer multiples of 2π / p. For example, in the case of dimension mn (r) being 3 d it can be seen that it only takes three values: 0, 2π / 3, and 4π / 3. Furthermore, as the dimension 1 increases, even when it is 9 or 27, it still only takes three values, and the phase resolution only needs to be 1 / 3 of 2π. In the MUB based on the Fourier transform basis represented by equation (1), which is a prior art, the problem of extremely high phase resolution being required as the dimension d increases can be significantly solved. Also, the probability amplitude is represented by equation (4) and takes four values as the phase value (resolution 1 / 4) in the case of dimension d = 2 d In the case of dimension d = 3 with odd prime p = 3 2 = 9, 3 3 = 27, the phase resolution is further relaxed compared to the case of dimension d = 2, and the state generation of the MUB with odd prime p = 3 and dimension d = 3 N is excellent. As will be described later, generally, the state generation of the MUB with odd prime p and dimension d = p N is also superior to the state generation of the MUB based on the Fourier transform basis represented by equation (1). N Regarding the state generation of the MUB with odd prime p and dimension d = p

[0139] [State generation device using MUB with a finite field of a power of an odd prime number p] The probability amplitude is represented by Equation (13), and the dimension d = p N (p: odd prime number), the configuration of the MUB state generation device using a finite field may be the same as that of the state generation device 30 in FIG. 4 and the state generation device 40 in FIG. 5. In FIGS. 4 and 5, since the set phases to the phase modulators 32 and 41 are different, only the modulation signals 36 and 45 applied from the modulation signal generators 33 and 42 are different. Therefore, the configuration of the state generation device can directly use the quantum state generation device using the MUB defined by Equations (2) to (4).

[0140] [State Measurement Device Using MUB with a Finite Field of the Power Dimension of an Odd Prime Number p] Next, consider the state measurement device. For the MUB using a finite field of dimension d = p N (p odd prime number), similar to the discussion in Equation (6), the probability amplitude B mn (r) can be decomposed into two matrix components, and each matrix can be divided into corresponding units to realize the state measurement device.

[0141] Specifically, the probability amplitude B according to Equation (13) mn (r) can also be decomposed into two elements as shown in Equation (6), that is, D on the left side of the right side of Equation (6) mm (r) and B on the right side of the right side mn (0) When considering the case of p = 2 and d = 2 N similar to the above, D that becomes the first unit mm (r) will be p types of phase modulation units.

[0142] On the other hand, for the probability amplitude B according to Equation (13) mn (r) in, for B that becomes the second unit mn (0) for any p N the p -adic number representation corresponding to 0 of the integer r in the dimension is a vector with N 0s arranged. Therefore, since r j in the Σ term in the parentheses of Equation (13) becomes 0, the Σ term disappears, and B mn(0) It becomes the following equation. TIFF0007705068000025.tif11150

[0143] Equation (14) is for the case of dimension d = 2 N of B mn (0) and corresponds to it, but this is p N For the quantum state of dimension, considering the N - particle p - dimensional quantum state by the equivalent p - dimensional quantum state (qudit), for each p - dimensional quantum state, it is equivalent to the operation of performing Fourier transform. Therefore, for d = 2 N replace the two - dimensional Hadamard transform in the case of with the p - dimensional Fourier transform, and replace the tensor product of |+> and |-> states in projective measurement with any N tensor products among p states |f0>, |f1>, ··· |f p-1 >, then the same discussion as the state measurement of the MUB in dimension d = 2 N holds.

[0144] As an example, consider the case of dimension 9 (d = 3 2 ), N = 2, p = 3). The quantum states of the nine - dimensional computational basis at this time are nine states of |0>, |1>, |2>, |3>, |4>, |5>, |6>, |7>, |8>. For these quantum states, consider the equivalent two - particle three - dimensional quantum state (N - particle p - dimensional state). That is, using the equivalent two - particle p - adic representation, associate the nine quantum states of the computational basis with the states of |00>, |01>, |02>, |10>, |11>, |12>, |20>, |21>, |22> of the two - particle qudit quantum state. Note that for the qubit which is two - dimensional and binary in the case of the MUB in dimension d = 2 N , when dealing with quantum states which are p - adic numbers of three dimensions or more, it is called qudit.

[0145] In the MUB over the finite field of dimension d = p N (p is an odd prime number) represented by Equation (13), the necessary projective measurement in the case of p = 3 is the tensor product state of the three - dimensional Fourier transform basis. Specifically, the necessary projections of the three - dimensional Fourier transform basis are the following three states |f0>, |f1>, |f2>. TIFF0007705068000026.tif13150 TIFF0007705068000027.tif13150 TIFF0007705068000028.tif13150

[0146] In the case of the Fourier transform basis, the basis state represented by Equation (14) will be represented by the tensor product of the above three states |f0>, |f1>, |f2>. Therefore, the dimension d = p N In the case of the MUB over the finite field of (p prime number), B of the second unit in FIG. 6 mn (0) The transformation of is an operation of decomposing the N-particle p-dimensional quantum state equivalent to the quantum state to be measured into each digit in the p-adic representation, that is, performing a p-dimensional Fourier transform on all of the equivalent qudits. Therefore, the dimension d = p N In the state measurement device of the MUB over the finite field of (p prime number), the second unit 52 in FIG. 6(b) becomes a p-dimensional Fourier transform state measurement unit.

[0147] FIG. 19 is a diagram for explaining the configuration of the p-dimensional Fourier transform state measurement unit. FIG. 19(a) shows the configuration of a 3-dimensional Fourier transform state measurement unit using a multi-arm delay interferometer. It is known that the Fourier transform measurement for a p-dimensional time-bin quantum state can be configured using a multi-arm delay interferometer having p arms (Non-Patent Document 6). The Fourier transform state measurement unit 500 is a multi-arm delay interferometer having three arms and includes three input waveguides 501, an input coupler 502, three arm waveguides 503 having different lengths, an output coupler 504, and three output waveguides 505. The three arm waveguides 503 have delay times of 0, τ, and 2τ.

[0148] In the Fourier transform state measurement unit 500, an input continuous pulse 506 with a time interval τ is input, and from any output port, an output pulse 507 in an overlapping state with a time interval of τ is obtained corresponding to the delay time of the arm waveguide 503. In the dotted line area 508, all the input pulses of the input continuous pulse 506 appear, and a superposition state of three input pulses is formed. That is, the input photons are projection-measured into the state of the following formula, in which the time position states |0>, |1>, |2> of the input pulses are superposed equally with a relative phase of 0. TIFF0007705068000029.tif12150

[0149] The state of the above formula (16) is the same as the base state |f0> of the formula (15-1) among the three states of the above-mentioned three-dimensional Fourier transform basis. At this time, according to the phase relationship between the input port and the output port of the three-input three-output interferometer 500, projection measurements onto the base states of the remaining Fourier transform bases are realized at the other two output ports. Therefore, at the time of the dotted line area 508, it is possible to determine which state among the three states of the formulas (15-1) to (15-3) has been projection-measured based on the information on which output port the photon is detected by the photon detector.

[0150] To extend the p-dimensional Fourier transform measurement by the above multi-arm delay interferometer to p N dimensions, for example, if it is nine dimensions, as described in the configuration of the Hadamard transform unit in FIG. 8, a plurality of multi-arm delay interferometers may be arranged and connected in a two-layer (N-layer) tree shape.

[0151] FIG. 19(b) shows a two-particle three-dimensional Fourier transform state measurement unit by a multi-arm delay interferometer with a tree structure. The two-particle three-dimensional Fourier transform state measurement unit 510 is composed of a first-layer multi-arm delay interferometer 511 and a second-layer multi-arm delay interferometer 512. The first-layer multi-arm delay interferometer 511 includes three arm waveguides with relative delays of 6τ, 3τ, and 0. To each of the three output ports of the multi-arm delay interferometer 511, a multi-arm delay interferometer 512 composed of three arm waveguides with relative delays of 2τ, τ, and 0 is connected.

[0152] When realizing the three-input multi-arm delay MZI in (b) of FIG. 19 with a two-layer structure in the case of a dimension d = 9 (3 2 ) configuration, nine photon detectors 1 to 9 (not shown) are provided at the output ports of the three MZIs 512 in the second layer. The relationship between the photon detection at each photon detector and the projected state is as follows.

[0153] According to the information on which photon detector detected a photon, when expressing the nine-dimensional quantum state as an equivalent two-particle three-dimensional quantum state in qudit, it is possible to determine which state of the tensor product of the three-dimensional quantum states shown in equations (15-1) to (15-3) on the equivalent qudit was subjected to the projective measurement.

[0154] * Photon detected by photodetector 0 -> |f0, f0> state * Photon detected by photodetector 1 -> |f0, f1> state * Photon detected by photodetector 2 -> |f0, f2> state * Photon detected by photodetector 3 -> |f1, f0> state * Photon detected by photodetector 4 -> |f1, f1> state * Photon detected by photodetector 5 -> |f1, f2> state * Photon detected by photodetector 6 -> |f2, f0> state * Photon detected by photodetector 7 -> |f2, f1> state * Photon detected by photodetector 8 -> |f2, f2> state As described above, the two-particle three-dimensional Fourier transform state measurement unit in (b) of FIG. 19 corresponds to Configuration Example 1 of the tree-structured Hadamard transform state measurement unit in the state measurement of MUB by a finite field of dimension d = 2 N . Other configuration examples of the Hadamard transform state measurement unit (FIGS. 8, 10, 11, 12) can also be applied to the Fourier transform state measurement unit for state measurement of MUB by a finite field of dimension d = p N (where p is a prime number).

[0155] FIG. 20 is a diagram showing the configuration of a two-particle p-dimensional Fourier transform state measurement unit using a delay line. The Fourier transform state measurement unit 520 in FIG. 20 corresponds to Configuration Example 2 of the Hadamard transform state measurement unit 90 using a delay line in the measurement of the MUB state by the finite field of d = 2 in FIG. 10. The Fourier transform state measurement unit 520 has two multi-arm delay interferometers 521 and 523 connected in cascade. That is, a multi-arm delay interferometer 521 including p arm waveguides in the first layer and a multi-arm delay interferometer 523 including p arm waveguides in the second layer are connected in cascade via a delay line section 522. The first-layer multi-arm delay interferometer 521 is composed of an input-side coupler 525 with p inputs and p outputs, p arm waveguides 526 with different lengths, and an output-side coupler 527 with p inputs and p outputs. The second-layer multi-arm delay interferometer 530 also has the same configuration as the first layer, and is composed of an input-side coupler 529 with p inputs and p outputs, p arm waveguides 530 with different lengths, and an output-side coupler 531 with p inputs and p outputs. The p arm waveguides of the first-layer multi-arm delay interferometer 521 have delay times of (p - 1)pτ ··· 2pτ, pτ, 0. The p arm waveguides of the second-layer multi-arm delay interferometer 530 have delay times of (p - 1)τ ··· 2τ, τ, 0. N This corresponds to Configuration Example 2 of the Hadamard transform state measurement unit 90 using a delay line in the measurement of the MUB state by the finite field of d = 2 in FIG. 10. The Fourier transform state measurement unit 520 has two multi-arm delay interferometers 521 and 523 connected in cascade. That is, a multi-arm delay interferometer 521 including p arm waveguides in the first layer and a multi-arm delay interferometer 523 including p arm waveguides in the second layer are connected in cascade via a delay line section 522. The first-layer multi-arm delay interferometer 521 is composed of an input-side coupler 525 with p inputs and p outputs, p arm waveguides 526 with different lengths, and an output-side coupler 527 with p inputs and p outputs. The second-layer multi-arm delay interferometer 530 also has the same configuration as the first layer, and is composed of an input-side coupler 529 with p inputs and p outputs, p arm waveguides 530 with different lengths, and an output-side coupler 531 with p inputs and p outputs. The p arm waveguides of the first-layer multi-arm delay interferometer 521 have delay times of (p - 1)pτ ··· 2pτ, pτ, 0. The p arm waveguides of the second-layer multi-arm delay interferometer 530 have delay times of (p - 1)τ ··· 2τ, τ, 0.

[0156] Between the two-layer multi-arm delay interferometers, there is a delay line section 522 including p delay lines having different delay times. For p - 1 delay lines, delay times of τ', τ'' ··· τ''' different from the time interval τ of the input continuous pulses are set.

[0157] In the Fourier transform state measurement unit 520 of FIG. 20, with the configuration including the above-mentioned delay line, at a predetermined observation time, a superposition state of d = p 2 input continuous pulses is generated. Similar to the explanation in FIG. 10(b), by the combination of the information on which photon detector among the p photon detectors detected a photon and the observation timing among the p, it is possible to determine which state projection measurement in the p 2 dimensional quantum state was performed. For example, if p = 3 in FIG. 20, three photon detectors are provided at the three outputs of the second-layer multi-arm delay interferometer, and d = 9 (= 32 ) Photons will be detected at any one of three timings when the overlapping state of the input continuous pulses occurs.

[0158] By combining the Fourier transform state measurement unit described in FIGS. 19 and 20 with a phase modulation unit arranged on the front stage side, a state measurement apparatus having the same configuration as that described in FIG. 6(b) can be realized. That is, similar to the measurement apparatus 50-2 in FIG. 6(b), a phase modulation unit 51 which is the first unit on the front stage side, and a measurement unit 52 which is the second unit corresponding to B mn (0) can be realized. However, the measurement unit 52 corresponding to B mn (0) corresponds to Equation (14), and for the received d-dimensional quantum state, projective measurements are performed on the N-particle tensor product of p states |f0>, |f1>, ··· |f p-1 >. Here, the basis state of the received d-dimensional quantum state is represented by the tensor product of any N of the p states |f0>, |f1>, ··· |f p-1 >. The second unit in FIG. 6(b) performs projective measurements on the state obtained by performing a d = 2 N -dimensional Hadamard transform on the computational basis, while using the Fourier transform state measurement units in FIGS. 19 and 20, projective measurements are performed on the N-particle tensor product state of the p-dimensional Fourier transform basis in d = p N dimensions (p is a prime number).

[0159] In the generation and measurement of states of high-dimensional MUBs using the Fourier transform basis |f n > shown in Equation (1), the problem was that a high phase resolution was required as the dimension d increased. Therefore, in the state measurement of MUBs using a finite field of d = p N dimensions (p is a prime number), it may seem strange that the Fourier transform state measurement units in FIGS. 19 and 20 are used. However, if the following points are noted, the advantages of using the Fourier transform state measurement unit as the second unit of the state measurement apparatus will be understood.

[0160] As a conventional technique, in the high-dimensional MUB using the Fourier transform basis |f shown in Equation (1), as can be seen from the term of e in Equation (1), it has a phase proportional to 1 / d with respect to the dimension d. For the generation of MUB states, a very high-resolution phase modulation is required as the dimension d increases, and the same applies to the state measurement device. n On the other hand, the state measurement device of the present disclosure decomposes into units (sub-systems) corresponding to each of the two matrices, and the measurement unit 52 corresponds to the probability amplitude of Equation (14), and performs a projective measurement onto the tensor product of p-dimensional Fourier transforms smaller than the d dimension. In the example of p = 3, the three states of the 3D Fourier transform basis used for the projective measurement consist of |f0>, |f1>, |f2> according to Equations (15-1) to (15-3) obtained from Equation (14). As can be seen from the term of e in Equation (14), as the dimension d = p

[0161] increases, the phase resolution is required according to the base number p rather than depending on the dimension d. For example, in the Fourier transform state measurement unit of the present disclosure, when the base number p = 5 and the dimension d = 5 N = 25, the required phase resolution is 1 / 5 of 2π. On the other hand, in the state measurement of the high-dimensional MUB using the conventional Fourier transform basis |f 2 shown in Equation (1), the required phase resolution is 1 / 25 of 2π. In the case of p = 11, the required phase resolution is an extreme difference of 1 / 11 and 1 / 121 of 2π, and there is a large difference in the required phase resolution between the conventional state measurement device and the state measurement device using the Fourier transform state measurement unit of the present disclosure. n >

[0162] Such an advantage of the state measurement device using the Fourier transform state measurement unit of the present disclosure is attributed to performing a projective measurement onto the tensor product state of p states |f0>, |f1>, ··· |f N as a method for measuring a d-dimensional quantum state of the measurement target with the dimension d = p p-1 >. Here, the base state of the d-dimensional quantum state is the p states |f0>, |f1>, ··· |f p-1> It is represented by any N tensor products among them.

[0163] Similarly, the advantages of the state measurement apparatus that uses the Hadamard transform state measurement unit of the present disclosure described above also apply when the dimension d = 2 of the measurement target N As a method for measuring a d-dimensional quantum state, it comes down to performing a projective measurement on a tensor product state of two-dimensional basis states (|+>, |->). Here, the basis states of the received d-dimensional quantum state are represented by the tensor product of two basis states (|+>, |->).

[0164] The descriptions from FIG. 4 to FIG. 20 have been about the state generation and state measurement of MUB using a finite field, by utilizing the time-bin quantum state of light which is an orthogonal mode of time as the computational basis. However, even when using a mode of light different from the time-bin quantum state, for d = 2 in equations (2) to (4) N dimensional states and probability amplitudes B mn (r) Replacing with equation (13), an MUB state of d = p N dimensions (p is an odd prime number) can be generated. Also, even when using different modes of light, as shown in equation (6), the probability amplitude B mn (r) is decomposed into two matrix components, and a state measurement apparatus for MUB can be realized with the corresponding two measurement units shown in FIG. 6(b). The two measurement units are a phase modulation unit and a Hadamard transform state measurement unit (d = 2 N dimensions) or a Fourier transform state measurement unit (d = p N dimensions, p is an odd prime number).

[0165] The term "mode of light" refers to a state of light that is physically orthogonal, regardless of the type of degree of freedom such as time, frequency, space, etc. When implementing a quantum state into physical elements, for example, there are modes of light in the following states. (a) Time-bin quantum state: A state of light that can be distinguished temporally as a pulse (b) Frequency-bin quantum state: A state of light that can be distinguished frequency-wise (c) Quantum state using spatial mode Regarding the quantum state using the spatial mode in (c) above, for example, the following exist.

[0166] - Polarization: State of light based on two orthogonal polarization states such as vertical polarization / horizontal polarization - Orbital angular momentum: State distinguishable orthogonally by intensity distribution / phase distribution in the beam cross-section - Propagation mode in fiber: State using orthogonal propagation modes in fiber such as TE / TM modes, etc. - Using optical path information: State of light distinguishable by information such as which core of a multi-core fiber to propagate through or which optical path of an optical circuit to propagate through The state generation device and state measurement device of the present disclosure are not limited to the types of optical modes that are physical elements for implementing the formulas (2) to (5) and formula (13) that define MUB. Further, in the measurement device, the implementation of the two measurement units by formulas (6), (7), and (13) that define the matrix decomposed into two is not limited to the types of optical modes that are physical elements. The formulas (2) to (4) and formula (13) that define MUB do not describe physical entities such as electric fields and magnetic fields. The descriptions from FIG. 4 to FIG. 20 are the cases where quantum states such as |0>, |1>, |2>, ··· |d - 1> in the above various formulas are mapped to time-bin quantum states which are one of the physical optical modes. Therefore, the above-mentioned formulas used for state generation and state measurement are completely common regardless of the types of optical modes. As an example of the use of optical modes other than time-bin quantum states, next, an example using frequency-bin quantum states for the state of high-dimensional MUB using a finite field is shown.

[0167] [Implementation of State Generation Device by Frequency-Bin State] So far, in the MUB quantum state generation and measurement apparatus using the finite field of the present disclosure, in the case of the time-bin quantum state using the orthogonal mode of time as the computational basis, it has been described. As described above, there is no limitation on the type of optical mode, which is a physical element for implementing the generation of the quantum states of the formulas (2) to (5) and (13) that define the MUB. In the measurement apparatus, the implementation by the two measurement units based on the formulas (6), (7), and (13) similarly does not depend on the optical mode.

[0168] As a computational basis, there is a frequency-bin quantum state that uses the orthogonal mode of the optical frequency as the computational basis in another optical mode different from the time-bin quantum state. The frequency-bin quantum state uses a plurality of lights with different frequencies existing within a predetermined time period as the computational basis. Therefore, in the light of the d-dimensional frequency-bin quantum state, instead of the d pulse positions arranged on the time axis obtained by the time-bin quantum state generation apparatus of FIG. 4, the light at any of the d frequency positions arranged on the frequency axis is treated as the computational basis. As long as the frequency width of each light is sufficiently narrower than the frequency interval Δf, the d lights can be distinguished, so the plurality of lights do not need to have an equal frequency interval. For simplicity, in the following description, the light of the computational basis is at any position with an equal frequency interval, and the high-dimensional quantum state generated by the formulas (2) to (5) is represented as a superposition of the states of the computational basis.

[0169] FIG. 21 is a diagram showing the configuration of a state generation apparatus using the frequency-bin quantum state of the MUB using the finite field of the present disclosure. The state generation apparatus 600 is an MUB using a finite field with a digit d = 2 N and generates the states defined by the formulas (3) to (5). When compared with the state generation apparatus 30 using the time-bin quantum state shown in FIG. 4, it is also common in that it includes a phase modulator 602. The difference from the case of the state generation apparatus of the time-bin quantum state shown in FIG. 4 is, firstly, that the phase modulator 602 has the ability to independently modulate lights with different frequencies instead of different times. Secondly, in order to generate the state of the computational basis of the formula (5) with the frequency-bin quantum state, instead of the intensity modulator 31 in FIG. 4, it includes a variable frequency filter 601.

[0170] To the state generator 600, light 604 of a plurality of frequencies including at least all the frequencies of the computational basis light is input. The input light 604 can be controlled as continuous light so that the variable frequency filter 601 or the phase modulator 602 outputs only for a certain time period. Also, the input light 604 may be input only for a certain time period corresponding to the frequency bin state, and in synchronization therewith, the variable frequency filter 601 and the phase modulator 602 may be operated synchronously. For the input light 604 including d different frequencies, by applying a phase modulation different for each frequency by the phase modulator 602, a quantum state 607 of a predetermined MUB can be obtained. The modulation signal generator 603 generates a modulation signal (control signal) 606 for applying a phase modulation according to the probability amplitude of Equation (4) to the input light 605 including d different frequencies based on the information r of the basis and the information n of the label of the state in the basis.

[0171] By the generator 600 having the configuration of FIG. 21, a high-dimensional quantum state of an MUB using a finite field can be generated, in which the relative phase based on the light located at one end on the frequency axis in a plurality of lights of d different frequencies has only four values. Dimension d = 2 N Even when it increases, since each light can have at most four phases, the required conditions for the phase resolution can be significantly relaxed, just like the time-bin quantum state.

[0172] In the state generator 600 of FIG. 21, the variable frequency filter 601 is used to select only the light of a desired one frequency among the input lights 605 and generate the state of the computational basis. Here, if the phase modulator 602 also has an amplitude modulation function, the variable frequency filter 601 can be omitted. That is, the state generator 600 can be realized only by an optical modulator that can modulate the phase and amplitude independently for each frequency with respect to a plurality of lights having different frequencies. In this case, the optical IQ modulator 41 in FIG. 5 in the time-bin quantum state may be replaced with such an optical modulator that can modulate the phase and amplitude independently for each frequency.

[0173] The optical modulator capable of modulating the above-described phase and amplitude can be realized by, for example, combining a spatial optical component and an LCOS (Liquid Crystal on Silicon) as described in Non-Patent Document 9. That is, the input light from the input fiber is separated in the x direction by a diffraction grating, further input to the element formation surface of the LCOS, phase modulation is applied in the x direction, and it is returned to the output fiber to give phase modulation for each frequency. The amplitude modulation can be realized, for example, by changing the coupling ratio between the modulated light and the output fiber by some means. In the MUB state generation device using the frequency bin state of the present disclosure, as long as the phase and amplitude can be modulated independently for each frequency, the realization method and configuration of the optical modulator are not limited.

[0174] Dimension d = p N (p: odd prime number), the configuration of the above-described state generation device 600 can be applied in exactly the same manner even in the case of the MUB state generation device using a finite field. Since the set phase to the phase modulator 602 is different, the modulation signal 606 applied from the modulation signal generator 603 is different only from the case where the dimension d = 2 N is different.

[0175] [Implementation of the state measurement device using the frequency bin state] The MUB state measurement device using a finite field in the frequency bin state can also be realized by replacing some of the components of the configuration of the time bin quantum state described in FIGS. 6 to 7. Also, various configuration variations in the case of the time bin quantum state shown in FIGS. 8 to 14 can be applied in the same manner.

[0176] Similar to what was described for the state measurement device of the time bin quantum state, also in the state measurement device of the frequency bin state, the expression (4) representing the probability amplitude B mn (r) of the MUB using a finite field is common. Therefore, D mm (r) in Expression (6) becomes a diagonal unitary matrix, and in the case of the frequency bin quantum state, D mm (r) corresponds to the phase modulation for each of a plurality of lights of different frequencies. D in Expression (6) mm (r)The first unit that performs an operation corresponding to a diagonal unitary matrix can be realized as a phase modulator for the computational basis of the frequency bin state.

[0177] Furthermore, the operation corresponding to B in Equation (6) mn (0) can be realized as a second unit that performs a projective measurement onto the Hadamard transform basis. The state measurement device of the MUB in the frequency bin state has a configuration that includes a unit that performs phase modulation according to Equation (7) on the front stage side of the measurement unit that performs the operation of the Hadamard transform matrix. The probability amplitude B of the MUB mn (r) The measurement to the basis (MUB of label r) having the probability amplitude B of Equation (4) can be realized by units corresponding to the two matrices decomposed from mn (r) Here, the operation of the Hadamard transform unit in the frequency bin state will be described in comparison with the operation in the time bin quantum state of FIG. 7.

[0178] FIG. 22 is a diagram conceptually explaining the operation of the Hadamard transform unit in the frequency bin quantum state. (a) of FIG. 22 explains the replacement of the frequency bin quantum state from a d-dimensional quantum state to a state of a plurality of equivalent particles. Consider the case where the dimension d = 4, that is, a 4-dimensional quantum state 611. Each position of the light arranged at a frequency interval Δf on the frequency axis corresponds to the four quantum states |0>, |1>, |2>, |3> of the computational basis. Here, this 4-dimensional quantum state is considered as an N-particle 2-dimensional quantum state having an equivalent dimension. Similar to the case of the time bin quantum state, the four quantum states 611 can be replaced and considered as four quantum states 612 of |00>, |01>, |10>, |11> as a bit representation of 2 bits (N = 2).

[0179] (b) of FIG. 22 illustrates the operation of decomposing the N-particle two-dimensional quantum state bit by bit at the next stage. The four quantum states 612, which are two-particle two-dimensional states, can be divided into two blocks with different frequency differences Δf when decomposed into two states for each digit by focusing on the bits of each digit. Focusing on the first digit of the quantum state 612, it is divided into the block 613-1 of state |0> and the block 613-2 of state |1>, and the frequency difference between the two blocks is Δf. Also, focusing on the second digit of the quantum state 612, it is divided into the block 614-1 of state |0> and the block 614-2 of state |1>, and the frequency difference between the two blocks is 2Δf. The conversion of the second unit B in the frequency bin quantum state is an operation of performing a two-dimensional Hadamard transform on all of the equivalent qubits obtained by decomposing the N-particle two-dimensional quantum state equivalent to the quantum state to be measured shown in (b) of FIG. 22 bit by bit. There is no difference in this regard between the case of the time bin quantum state and the case of the frequency bin quantum state. The difference from the time bin quantum state lies in the configuration of the interference structure for realizing the superposition state of all the states of a plurality of lights with different frequencies. mn (0) The conversion of B is an operation of performing a two-dimensional Hadamard transform on all of the equivalent qubits obtained by decomposing the N-particle two-dimensional quantum state equivalent to the quantum state to be measured shown in (b) of FIG. 22 bit by bit, that is, for all of the equivalent qubits. There is no difference in this regard between the case of the time bin quantum state and the case of the frequency bin quantum state. The difference from the time bin quantum state lies in the configuration of the interference structure for realizing the superposition state of all the states of a plurality of lights with different frequencies.

[0180] Therefore, the Hadamard transform unit in the frequency bin quantum state can be realized with a configuration adapted to the frequency bin quantum state, and it is only necessary to associate the position (photon detection position) that outputs different interference states with the projective measurement to the corresponding superposition state. The configuration of the Hadamard transform unit can be easily realized by replacing a part of the configuration in the time bin quantum state with a configuration adapted to the frequency bin quantum state. In order to superpose a plurality of lights with different frequencies, it is only necessary to replace the configuration for causing a time delay in the time bin quantum state with a configuration for causing a frequency shift. Also, the configuration for causing different interference states can directly apply the arrangement variations of the interference structure in Configuration Examples 1 to 4 described for the time bin quantum state.

[0181] FIG. 23 is a diagram showing the configuration of the MUB state measurement apparatus in the frequency bin quantum state. The state measurement apparatus 700-1 in FIG. 23(a) has the same configuration as the state measurement apparatus 50-1 for the time bin quantum state shown in FIG. 6(a), and measures the probability amplitude B represented by Equation (4). mn (r) It only differs in that the input light to be measured is the light 705 arranged at frequency intervals Δf on the frequency axis. The quantum state measurement apparatus 700-1 can be understood as a measurement apparatus 50-1 that identifies the label n of the quantum state |ψ> having the probability amplitude B represented by Equation (4) for the input light 705 to be measured, for example, in the frequency bin quantum state. mn (r) n (r) >

[0182] The state measurement apparatus 700-2 in FIG. 23(b) shows a state measurement apparatus 700-2 having a configuration combining the above two units. The measurement apparatus 700-2 includes an optical modulation unit 701 corresponding to D, which is the first unit on the front stage side, and a measurement unit 702 corresponding to B, which is the second unit. The frequency bin quantum state measurement apparatus 700-2 is common to the state measurement apparatus for the time bin quantum state in FIG. 6(b) in that it consists of two units corresponding to the diagonal unitary matrix and the Hadamard transformation matrix of Equation (6). mm (r) mn (0)

[0183] ​​​When comparing the state measurement device 700-2 with the state measurement device 50-2 based on the time-bin quantum state shown in Fig. 6(b), they differ in that the optical modulator 704, which is the first unit 701, has the ability to independently modulate lights of different frequencies. Therefore, the optical modulator 704 of the state measurement device 700-2 can use the same one as the phase modulator 602 in the state generator 600 of the MUB of the frequency-bin quantum state. The modulation signal generator 703 supplies a modulation signal to the phase modulator 704. Similar to the case of the time-bin quantum state, in the optical modulation unit 704, instead of the conversion from the 0th basis (label 0) to the rth basis (label r), the conversion from the rth basis to the 0th basis is required. For this reason, in the measurement device, the inverse conversion of the conversion represented by Equation (6), that is, the phase modulation that is the complex conjugate of D mm (r) is performed.

[0184] The measurement corresponding to matrix B, which is the second unit of the state measurement device 700-2 mn (0) can be implemented as a projective measurement on the state obtained by decomposing the frequency-bin quantum state to be measured into equivalent lower-dimensional two-dimensional quantum states (qubits) and performing a two-dimensional Hadamard transformation on these states.

[0185] FIG. 24 is a diagram showing the configuration of the Hadamard transform state measurement unit in the frequency bin quantum state. The Hadamard transform state measurement unit 800 in FIG. 24 corresponds to the Hadamard transform state measurement unit 100 of Configuration Example 3 in the time bin quantum state. The Hadamard transform state measurement unit 800 is similar to the Hadamard transform state measurement unit 100 of Configuration Example 3 in that three interference structures 801-1 to 801-3 are cascade-connected and are composed of paths connecting in parallel between the interference structures (between layers). However, in order to realize the superposition state of a plurality of lights for a plurality of lights 806 (d = 8) having different frequencies of the target frequency bin quantum state, it has the following-described configuration different from the case of the time bin quantum state. The "interference structure" in the following description corresponds to the interferometer (MZI) in the time bin quantum state, and all of them include an optical coupler, and interference can be generated by appropriately operating the branched input light.

[0186] The interference structure 801-1 in the first layer includes a wavelength separation filter 802-1, two branch paths a and b, and an optical coupler 804-1, and a frequency shifter 803-1 is provided in one of the branch paths a. The wavelength separation filter 802-1 separates the four lights on the low-frequency side into the branch path a and the four lights on the high-frequency side into the branch path b for eight lights 806 having different frequencies arranged at a frequency interval Δf in the frequency bin state. The frequency shifter 803-1 in the branch path a gives a frequency shift of 4Δf to the four lights on the low-frequency side. At this time, at the output of the optical coupler 804-1, the phase difference between the plurality of lights propagating through the branch path a and the plurality of lights propagating through the branch path b is set to 0.

[0187] The adjacent layers of the interference structure are connected by two paths with different frequency shift amounts. For example, the first-layer interference structure 801-1 is connected to the second-layer interference structure 801-2 by a branch path c and a branch path d that connect the layers. In the branch path d, a frequency shift of 4Δf, which is the same as the frequency shift amount on the previous layer side, is given by the frequency shifter 805-1. Also, between the second and third layers and between the third layer and the wavelength separation filter 802-4 in the final stage, frequency shifts of 2Δf and Δf, which are the same as those on the previous layer side, are given in one of the branch paths, respectively. Note that the Hadamard transform state measurement unit 800 in Fig. 24(a) is drawn so that the configuration in the case of the time-bin quantum state in Fig. 11(a) and the correspondence relationship of the components and their connections can be understood. The configuration including the branch paths in the three interference structures is drawn as if they were waveguides with different lengths, but it should be noted that there is no time delay difference between the two branch paths a and b. Similarly, there is no time delay difference between the branch paths c and d that connect the two interference structures. The above-mentioned wavelength separation filter can use, for example, a WDM filter.

[0188] With the interference structure including the above-mentioned three-layer frequency shift elements, eight input lights with different frequencies correspond to the eight states of the computational basis (dimension d = 2 3 ). When corresponding to the equivalent three-particle two-dimensional quantum state (q2, q1, q0), the paths of the wavelength separation filters for each layer are determined according to |0> and |1> of each qubit. First, in the wavelength separation filter 802-1, the eight input lights are separated such that the light with the frequency corresponding to the equivalent q2 being |0> passes through the branch path a, and the light with the frequency corresponding to |1> passes through the branch path b. The light passing through the upper branch path a is given a frequency shift of 4Δf relative to the branch path b by the frequency shifter 803-1. The two sets of lights separated into the two branch paths interfere with a relative phase of 0 using the optical coupler 804-1. As a result of this interference in the optical coupler 804-1, whether the output is to the branch path c or the branch path d of the coupler output gives the measurement information for the equivalent qubit, that is, | +> , | -> .

[0189] Similar to the first layer, for each interference structure in the second and third layers, path separation, frequency shift, and interference are repeatedly performed. At the final interferometer output point 810, depending on which frequency of light the eight input lights 806 are observed as, it is possible to determine which superposition state of the high-dimensional Hadamard transform basis they are projected onto.

[0190] Figure 24(b) is a diagram showing the correspondence between the frequencies of the superimposed light observed at the interferometer output point 810 and the detected three-particle two-dimensional quantum state. It is the same as the correspondence in the case of Configuration Example 3 in the time-bin quantum state shown in Figure 11(b), showing the light 811 in a state where multiple lights are superimposed on one frequency and interfere. The difference from the case of the time-bin quantum state lies only in whether the fact of photon observation is identified by the detection times (t0, t1, ··· t7) of photon detection or by the detection frequencies (f0, f1, ··· f7). For example, if a photon is observed at the frequency f0 of the superposition state of the eight lights represented by the dotted line region 812, it can be determined that a projective measurement onto the |+, +, +> state has been performed. Similarly, if a photon is observed at the frequency f1 represented by the dotted line region 813, it can be determined that a projective measurement onto the |+, +, -> state has been performed.

[0191] When an 8-dimensional quantum state is qubit-represented as an equivalent three-particle two-dimensional quantum state, it is possible to determine which projective measurement onto which state of the combination of the two-dimensional quantum states |+> and |-> on the equivalent qubits has been performed as follows.

[0192] * Photon detection at frequency f0 -> |+, +, +> state * Photon detection at frequency f1 -> |+, +, -> state * Photon detection at frequency f2 -> |+, -, -> state * Photon detection at frequency f3 -> |+, -, +> state * Photon detection at frequency f4 -> | - , -, +> state * Photon detection at frequency f5 -> |-, -, -> state *Photon detection at frequency f6 -> |-,+,-> state *Photon detection at frequency f7 -> |-,+,+> state In the Hadamard transform state measurement unit 800 of FIG. 24, the frequency at which photon detection is performed by the photon detector 808 must be specified. Generally, a photon detector does not have frequency discrimination ability. However, for example, by converting frequency information into time information using a high-dispersion fiber 807, the frequency at which photons are detected can be measured. Further, at the interferometer output point 810, frequency discrimination is performed using a wavelength separation filter, and by providing a corresponding photon detector for each frequency, the final projective measurement can be easily implemented.

[0193] Therefore, the Hadamard transform state measurement unit in the frequency bin quantum state is a first configuration including a plurality of optical interference structures arranged in a tree shape in N layers, each of which is given a frequency shift corresponding to the layer position of the N layers (corresponding to Configuration Example 1 in FIG. 8), a second configuration including a plurality of optical interference structures cascade-connected in N layers, each of which is given a frequency shift corresponding to the layer position of the N layers, and one or more paths in parallel to the connection between two adjacent layers, in which a frequency shift corresponding to the frequency shift of the optical interference structure in the previous layer is set (corresponding to Configuration Examples 2 and 3 in FIGS. 10 and 11), or a third configuration including an optical interference structure connected in a loop shape, in which a variable frequency shift corresponding to the number of turns is given to the optical interference structure become( It can be implemented as including any one of (corresponding to Configuration Example 4 in FIG. 12).

[0194] In the configuration of FIG. 24 described above, it will be understood that in the interference structure of each layer, by repeatedly performing path separation, frequency shift, and interference, an operation of superimposing a plurality of lights at different frequencies onto one frequency is being carried out. And the interference output from one interference structure is associated with the |+>, |-> projective measurements for each of the qubits equivalent to the high-dimensional quantum state to be measured. Also in this regard, the operation of the Hadamard transform state measurement unit is common between the time-bin quantum state and the frequency-bin quantum state. Furthermore, the interference structure consists of an element that separates the input light into two or more paths, an element that converts or shifts the orthogonal mode for the light in at least one path, and an element that combines and interferes the separated lights. The specific configuration of these elements may be adapted according to the mode of light such as the time-bin quantum state or the frequency-bin quantum state, as shown in Table 1 below. TIFF0007705068000030.tif26160 Table 1: Comparison of elements of the interference structure

[0195] It is also commonly possible regardless of the mode of light to use the above-described interference structure in multiple layers to form a tree structure (Configuration Example 1), reuse one interference structure (Configuration Example 2), connect in cascade (Configuration Example 3), or configure in a loop (Configuration Example 4).

[0196] As described above, the state measurement apparatus for MUB using a finite field utilizing the frequency-bin quantum state is the first unit corresponding to the diagonal unitary matrix of D in Equation (6), mm (r) and the second unit that performs projective measurement onto the Hadamard transform basis corresponding to B in Equation (6). mn (0) In that it decomposes the probability amplitude of the quantum state to be measured into a diagonal unitary transformation and a high-dimensional Hadamard transformation, and decomposes and implements them in the units of each subsystem, it has a common structural feature regardless of the mode of light. The operation corresponding to the diagonal unitary transformation is implemented as at most a four-valued phase modulation for the orthogonal modes of light used if the dimension d = 2. N The requirements for the phase resolution of the prior art can be relaxed.

[0197] The above discussion also holds for the case where the dimension d = p N (p: odd prime number), and the probability amplitude B given by Equation (13) mn (r) can be realized as the first unit corresponding to the diagonal unitary matrix of D mm (r) and the second unit corresponding to the tensor product of the Fourier transform matrix of B in Equation (14). It is obvious that the configurations of the multi-arm interferometers in FIGS. 19 and 20 can be applied to the frequency bin quantum state if the time delay is replaced by a frequency shift. If the dimension d = p mn (0) it can be implemented as a phase modulation with at most p values for the orthogonal modes of the light used, and the effect of relaxing the requirements for the phase resolution of the prior art is the same. N Needless to say, the features of the configuration of the state measurement apparatus of the present disclosure described above can also be applied to various high-dimensional quantum states using other orbital angular momenta, optical path information, spatial modes in multimode fibers, etc.

[0198] The present invention can be used for quantum information processing and quantum communication such as quantum key distribution and quantum state tomography.

Industrial Applicability

[0199] The present invention can be used for quantum information processing and quantum communication such as quantum key distribution and quantum state tomography.

Claims

1. A measuring device that performs a projective measurement on a high-dimensional quantum state defined by one computational basis {|m> | m ∈ {0, 1, ···, d−1}} of a d-dimensional quantum state consisting of states of orthogonal light, and a mutually unbiased basis of label r (an integer greater than or equal to 0) for which a quantum state of label n (0, 1, ···, d−1) is defined and which is non-orthogonal to the computational basis, d = 2 Ｎ where N is a natural number of 2 or more, wherein the quantum state of the label n is represented by the following formula Probability amplitude B ｍｎ （ｒ） When decomposed into a diagonal unitary matrix and a Hadamard transformation matrix, a phase modulation unit corresponding to the diagonal unitary matrix and applying a phase modulation to each state of the computational basis of the received d-dimensional quantum state, and a measurement unit corresponding to the Hadamard transformation matrix and determining the label n of the d-dimensional quantum state A measuring device for a high-dimensional quantum state comprising the above.

2. Let \(f\) be an element that forms a basis of a finite field of degree \(d\). ｉ Let the symmetric matrix \(A\) （ｊ） satisfy the following equation. The probability amplitude is represented by, taking only four phase states, wherein the measurement unit uses two states |+>, |−> for each of the N-particle two-dimensional quantum states equivalent to the d-dimensional quantum state, and the state constituting the basis of r = 0 is represented by the tensor product of the two states |+>, |−>, and performs a projective measurement on the tensor product. The measuring device according to claim 1.

3. A measuring device that performs a projective measurement on a high-dimensional quantum state defined by one computational basis {|m> | m ∈ {0, 1, ···, d−1}} of a d-dimensional quantum state consisting of states of orthogonal light, and a mutually unbiased basis of label r (an integer greater than or equal to 0) for which a quantum state of label n (0, 1, ···, d−1) is defined and which is non-orthogonal to the computational basis, Let \(p\) be an odd prime number and \(d = p\) Ｎ where \(N\) is a natural number wherein the quantum state of the label n is represented by the following formula Probability amplitude B ｍｎ （ｒ） when decomposed into the tensor product of a diagonal unitary matrix and a Fourier transform matrix, a phase modulation unit corresponding to the diagonal unitary matrix and applying a phase modulation to each state of the computational basis of the received d-dimensional quantum state, and a measurement unit corresponding to the Fourier transformation matrix and determining the label n of the d-dimensional quantum state A measuring device for a high-dimensional quantum state comprising the above.

4. Let \(f\) be an element that is a basis of a finite field of degree \(d\). ｉ Let the symmetric matrix \(A\) （ｊ） satisfy the following equation. The probability amplitude is represented by, taking only p phase states, The measurement unit uses p states |f ０ >、|f １ >、· · |f ｐ－１ > for each of the N-particle p-dimensional quantum states equivalent to the d-dimensional quantum state, and the state constituting the basis of r = 0 is one of the p states |f ０ >、|f １ >、· · |f ｐ－１ > is represented by any N tensor products of them, and the measurement device according to claim 3 that performs a projective measurement on the tensor product.

5. The state of the orthogonal light is a time-bin quantum state that associates each pulse of d continuous pulse trains with the state of the computational basis and uses an orthogonal mode of time, or a frequency-bin quantum state that associates light having different frequencies with each state of the computational basis and uses an orthogonal mode of frequency The measuring device according to any one of claims 1 to 4, which is any one of the above.

6. The state of the orthogonal light is a time-bin quantum state, and the measurement unit is A first configuration including a plurality of optical interferometers arranged in a tree shape in N layers, each provided with a delay time corresponding to the layer position of the N layers, A plurality of optical interferometers cascade-connected in N layers, each provided with a delay time corresponding to the layer position of the N layers, and A second configuration including one or more delay lines having a delay time set corresponding to the delay time of the optical interferometer of the previous layer, in parallel with the connection between two adjacent layers, or A third configuration including an optical interferometer connected in a loop shape, provided with a variable delay time corresponding to the number of turns The measuring device according to claim 5, including any one of them.

7. where p is an odd prime number and the dimension d = p Ｎ The measuring device according to claim 6, wherein, in the case of Ｎ , the interferometer is composed of a multi-arm interferometer including p arm waveguides having different lengths.

8. The state of the orthogonal light is a frequency bin quantum state, and The measuring unit is A first configuration including a plurality of optical interference structures arranged in a tree shape in N layers, each provided with a frequency shift corresponding to the layer position of the N layers, A plurality of optical interference structures cascade-connected in N layers, each provided with a frequency shift corresponding to the layer position of the N layers, and A second configuration including one or more paths having a frequency shift set corresponding to the frequency shift of the optical interference structure of the previous layer, in parallel with the connection between two adjacent layers, or A third configuration including an optical interference structure connected in a loop shape, provided with a variable frequency shift corresponding to the number of turns The measuring device according to claim 5, including any one of them.

9. A high-dimensional quantum state generating device defined by one computational basis {|m>|m ∈ {0, 1, ···, d - 1}} of a d-dimensional quantum state composed of states of orthogonal light, and a mutually unbiased basis of label r (an integer of 0 or more) in which a quantum state of label n (0, 1, ···, d - 1) is defined and is non-orthogonal to the computational basis, d = 2 Ｎ where N is a natural number of 2 or more, and the quantum state of the label n is represented by the following equation Let \(f\) be an element that is a basis of a finite field of degree \(d\). ｉ Let the symmetric matrix \(A\) （ｊ） satisfy the following equation. The probability amplitude is represented by, and a state generating device that takes only four phase states.

10. A high-dimensional quantum state generating device defined by one computational basis {|m>|m ∈ {0, 1, ···, d - 1}} of a d-dimensional quantum state composed of states of orthogonal light, and a mutually unbiased basis of label r (an integer of 0 or more) in which a quantum state of label n (0, 1, ···, d - 1) is defined and is non-orthogonal to the computational basis, d = p Ｎ where N is a natural number and p is an odd prime number, and the quantum state of the label n is represented by the following formula Let \(f\) be an element that is a basis of a finite field of degree \(d\). ｉ and let the symmetric matrix \(A\) （ｊ） satisfy the following equation. The probability amplitude is represented by, and a state generating device that takes only p phase states.

11. The state of the orthogonal light is a time-bin quantum state that corresponds each pulse of d consecutive pulse trains to the state of the computational basis and uses an orthogonal mode of time. An amplitude modulator that switches between the state of a single pulse that is a quantum state belonging to the computational basis and a state composed of d pulses that is a quantum state belonging to a basis non-orthogonal to the computational basis. A phase modulator that applies phase modulation to each pulse of the d consecutive pulse trains based on the information r of the basis and the information n of the label of the state in the basis. The state generation device according to claim 9 or 10, comprising:

12. The state of the orthogonal light is a frequency-bin quantum state that corresponds d lights having different frequencies on the frequency axis to each state of the computational basis and uses an orthogonal mode of frequency. A frequency selection filter that switches between the state of a single frequency that is a quantum state belonging to the computational basis and a state composed of d frequencies that is a quantum state belonging to a basis non-orthogonal to the computational basis. A phase modulator that applies phase modulation to each of the d lights having different frequencies based on the information r of the basis and the information n of the label of the state in the basis. The state generation device according to claim 9 or 10, comprising:

13. The state of the orthogonal light is a time-bin quantum state that corresponds each pulse of d consecutive pulse trains to the state of the computational basis and uses an orthogonal mode of time. A first configuration including a plurality of optical interferometers arranged in an inverted tree shape in N layers from the input side to the output side, each of which is given a delay time corresponding to the layer position of the N layers. A second configuration including a plurality of optical interferometers cascade-connected in N layers, each of which is given a delay time corresponding to the layer position of the N layers, or A third configuration including an optical interferometer connected in a loop shape, the optical interferometer being given a variable delay time corresponding to the number of turns. Any one of the configurations and A phase modulator that is connected to the last stage of any one of the configurations and applies phase modulation to each of the pulses based on the information r of the basis and the information n of the label of the state in the basis. The state generation device according to claim 9 or 10, comprising:

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