A method for realizing noiseless linear amplification of spatial w state using high-efficiency quantum scissors

Abstract

A method of noiseless linear amplification of spatial W state using efficient quantum scissors is proposed. The input state is a mixed state of single-photon entangled state in three different spatial modes and vacuum state. Efficient quantum scissors based on local compression are needed in each path. The total scheme of noiseless linear amplification of single-photon spatial W state is successful only when all the three quantum scissors are successful. By adjusting the conditional compression coefficient and the transmissivity of the variable beam splitter in each quantum scissors, the fidelity of the target single-photon spatial W state in the output state can be improved. The noiseless linear amplification of single-photon spatial W state is realized. Only some common linear optical devices are used in this method, which can be easily realized under the existing experimental conditions. Compared with the existing methods of noiseless linear amplification of single-photon spatial W state, the success probability of this method is higher. In summary, this method has strong application in the field of long-distance quantum communication.

Description

Technical Field

[0001] This invention belongs to the fields of quantum communication technology and noiseless linear amplification technology, specifically relating to a noiseless linear amplification method for realizing spatial W-states using efficient quantum scissors. Background Technology

[0002] Quantum communication is an interdisciplinary field combining quantum mechanics and communication science. It refers to the secure transmission of information using fundamental principles of quantum mechanics, such as the uncertainty principle, the measurement collapse principle, and the no-cloning principle. Quantum communication can detect eavesdropping and possesses absolute security, which is its greatest advantage over classical communication.

[0003] Single-photon spatial entanglement is a crucial resource for quantum communication, widely applied in quantum teleportation, quantum secure direct communication, quantum repeaters, and quantum key distribution. The single-photon spatial W-state is an important single-photon multimode spatially entangled state with significant applications in multi-party quantum communication. However, when distributing entanglement in imperfect quantum channels, transmission losses caused by environmental noise lead to an exponential decrease in photon propagation in optical fibers with increasing channel length. Due to the quantum no-cloning theorem, we cannot accurately replicate quantum states, limiting photon transmission distance. Furthermore, even within this limit, photons have a probability of being lost, leading to a chance of single-photon spatially entangled states mixing with empty states, making it impossible to establish high-quality quantum entangled channels. This significantly reduces communication efficiency and even threatens the security of quantum communication. Quantum indexed amplification, also known as noiseless linear amplification (NLA), is an effective method to address photon transmission loss. The noiseless linear amplification (NLA) protocol was first proposed by Ralph and Lund in 2009. Subsequently, numerous NLA protocols targeting single photons and photon entangled states were proposed. The quantum scissors scheme is an important NLA scheme, effectively improving the fidelity of the target state in the output state and achieving amplification of the target state. Although quantum scissors have been widely discussed, existing quantum scissors schemes generally suffer from low success rates, and the success rate gradually approaches zero as the amplification factor increases. Summary of the Invention

[0004] To address the photon loss problem during imperfect channel transmission, a noiseless quantum linear amplification scheme is proposed. During photon transmission, the channel's transmission efficiency decreases exponentially with increasing distance. To improve the fidelity of single-photon states in the output state, this invention proposes a noiseless linear amplification scheme for spatial W-states using a highly efficient quantum scissors scheme based on local compression.

[0005] This invention proposes a noiseless linear amplification method for single-photon spatial W-states using efficient quantum scissors, comprising the following steps:

[0006] Step 1: The light source S0 generates a target spatial W state and sends it to three distant receivers, namely Receiver 1, Receiver 2, and Receiver 3, through a noisy channel, where 0 and 1 represent no photon and a single photon respectively, and b1, b2, b3 represent the spatial paths connected to Receiver 1, Receiver 2, and Receiver 3 respectively; due to the existence of channel noise, photons may be lost during transmission, so its quantum state degenerates to ρ in =η|φx><φ1|+(1-η)|vac><vac|; where |vac> represents that the photon is lost during photon transmission, resulting in the quantum state of the spatial mode being a vacuum state; ρ in serves as the input state of the entire quantum scissors amplification scheme;

[0007] Step 2: In the quantum scissors amplification device, Receivers 1, 2 and 3 use auxiliary light sources S1, S2 and S3 to generate one auxiliary single photon in spatial modes a2, a4 and a6 respectively, and three local squeezing operators act on spatial modes a1, a3 and a5 respectively; the three auxiliary photons respectively pass through a variable beam splitter 1, a variable beam splitter 2 and a variable beam splitter 3 with transmittance t to obtain quantum states and where 0≤t≤1; Receivers 1, 2 and 3 perform local squeezing operations on the photons in the a1, a3 and a5 modes at the reflection ends of the three VBSs to obtain auxiliary states |ψ'> a1a2 and |ψ'> a3a4 and |ψ'> a5a6 ;

[0008] Step 3: Receivers 1, 2 and 3 make the photons in the two spatial modes b1 and a1 simultaneously pass through a 50:50 beam splitter 1, make the photons in the two spatial modes b2 and a3 simultaneously pass through a 50:50 beam splitter 2, and make the photons in the two spatial modes b3 and a5 simultaneously pass through a 50:50 beam splitter 3; a single-photon detector is arranged at each of the two output ports of each beam splitter to detect the output photons; the scheme runs successfully and proceeds to the next step only when exactly one of Detector 1 and Detector 2 responds, exactly one of Detector 3 and Detector 4 responds, and exactly one of Detector 5 and Detector 6 responds; if other detector response situations are obtained, the scheme runs in failure;

[0009] Step 4: Under the condition that the scheme runs successfully, the output state obtained by Receivers 1, 2 and 3 at the output ends of the three high-efficiency quantum scissors is

[0010] Step 5: When the incident photon is lost, the input state of the entire quantum scissors device is a vacuum state; at this time, only the auxiliary photon enters the system through the three 50:50 beam splitters, eventually causing the single-photon detector to respond; when the system obtains the above successful detector response, the output state is a vacuum state.

[0011] Step 6: Considering both of the above situations, the output state of the quantum scissors amplification device in this invention is: Where η' is the fidelity of the target single-photon state in the output state; when η'>η, the amplification is successfully achieved in this scheme.

[0012] Beneficial effects: (1) Applying noiseless linear amplification to quantum scissors can amplify the single-photon spatial W state. (2) Introducing local compression into the quantum scissors scheme can effectively improve the overall success probability of the method; compared with existing NLA schemes for single-photon spatial W states, this method has a higher success probability. (3) The amplification devices used in this method are all commonly used optical components, which can be realized under current experimental conditions, and have stronger practicality and experimental operability. (4) This method plays an important role in resisting photon transmission loss during entanglement distribution and improving the efficiency and security of quantum communication. Attached Figure Description

[0013] Figure 1 This is a schematic diagram of the quantum state amplification process in an embodiment of the present invention.

[0014] Figure 2 This is a schematic diagram of the principle of a single high-efficiency quantum scissors in an embodiment of the present invention, where BS represents a 50:50 beam splitter and VBS represents a variable beam splitter with a transmittance of t.

[0015] Figure 3 This is a schematic diagram of the spatial W-state amplification principle in an embodiment of the present invention. Detailed Implementation

[0016] The technical solution of the present invention will be further described in detail below with reference to the accompanying drawings.

[0017] This embodiment provides a highly efficient quantum scissors scheme based on local compression to achieve a noiseless linear amplification of spatial W-states. For example... Figure 3 As shown, light source S0 generates the target space W state. The signal is transmitted through a noisy channel to three distant receivers (1, 2, and 3), where 0 and 1 represent no photon and single photon transmission, respectively, and b1, b2, and b3 represent the spatial paths connecting receivers 1, 2, and 3. Due to channel noise, photons may be lost during transmission, thus their quantum state degenerates to ρ. in=η|φ1><φ1|+(1-η)|vac><vac|。Here, |vac> represents the vacuum state, the photon has a probability η of not being lost, and a probability 1-η of being lost. η is the transmission efficiency of the photon in the channel, η=e -αL / 10 , where α is the attenuation coefficient, and in standard optical fibers under current experimental conditions, α=0.2, and L is the transmission distance.

[0018] First consider the case where the incident photon is not lost. The input state of the light source S0 is respectively connected to three high-efficiency quantum scissors, and the amplification of the spatial W state is realized through the three high-efficiency quantum scissors.

[0019] In the quantum scissors amplification device, receivers 1, 2, and 3 use auxiliary light sources S1, S2, and S3 to generate one auxiliary single photon in spatial modes a2, a4, and a6 respectively, and three local squeezing operators are respectively on spatial modes a1, a3, and a5. The three auxiliary photons respectively pass through a variable beam splitter 1, a variable beam splitter 2, and a variable beam splitter 3 with a transmittance of t to obtain a quantum state:

[0020]

[0021]

[0022]

[0023] Receivers 1, 2, and 3 apply local squeezing operations to the photons in the reflection end a1, a3, and a5 modes of the three VBSs to obtain an auxiliary state:

[0024]

[0025]

[0026]

[0027] After that, a quantum state of the system is obtained:

[0028]

[0029] Because the single quantum scissors scheme can only obtain a successful detector response situation when only one of the two input ports of the 50:50 beam splitter has one photon, the total quantum state of the system collapses to:

[0030]

[0031] After that, the quantum state under this system enters a 50:50 beam splitter (BS) and is distributed to the detector path. The function of the 50:50 beam splitter is:

[0032]

[0033]

[0034]

[0035] After passing through three 50:50 beam splitters, the quantum state |φ3> evolves as follows:

[0036]

[0037]

[0038]

[0039]

[0040] Next, Alice set up single-photon detectors D1, D2, D3, D4, D5, and D6 for the d1, d2, d3, d4, d5, and d6 spatial modes respectively and detected the emitted photons. All successful detection responses are categorized into eight types:

[0041] The first type is where only D1, D3, and D5 each detect one photon, while D2, D4, and D6 do not detect any photons, i.e., |111> d1d3d5 ;

[0042] The second type is where only D1, D3, and D6 each detect one photon, while D2, D4, and D5 do not detect any photons, i.e., |111> d1d3d6 ;

[0043] The third type is where only D1, D4, and D5 each detect one photon, while D2, D3, and D6 do not detect any photons, i.e., |111> d1d4d5 ;

[0044] The fourth type is where only D1, D4, and D6 each detect one photon, while D2, D3, and D5 do not detect any photons, i.e., |111> d1d4d6 ;

[0045] The fifth type is where only D2, D3, and D5 each detect one photon, while D1, D4, and D6 do not detect any photons, i.e., |111> d2d3d5 ;

[0046] The sixth type is where only D2, D3, and D6 each detect one photon, while D1, D4, and D5 do not detect any photons, i.e., |111> d2d3d6 ;

[0047] The seventh type is where only D2, D4, and D5 each detect one photon, while D1, D3, and D6 do not detect any photons, i.e., |111> d2d4d5 ;

[0048] The eighth category is where only D2, D4, and D6 each detect one photon, while D1, D3, and D5 do not detect any photons, i.e., |111> d2d4d6 .

[0049] Taking the example where detectors D1, D3, and D5 successfully detect one photon, while detectors D2, D4, and D6 do not detect any photons, |φ4> collapses to:

[0050]

[0051] If other successful detector responses are obtained, the output state |φ5> can also be obtained in the end. |φ5> is still the W state of the three spatial modes.

[0052] Based on the above description, the probability of the output state |φ5> can be obtained as:

[0053]

[0054] Secondly, considering that the initial photon is lost during transmission with a probability of 1-η, the only states entering the amplified system are auxiliary states. Therefore, the total number of sub-states of the system can be expressed as:

[0055]

[0056] Because the quantum scissors scheme can only achieve a successful detector response when only one of the two incident ports of the 50:50 beam splitter contains one photon, the total number of quantum states in the system collapses as follows:

[0057]

[0058] The quantum states of this system are then distributed onto the detector path by three 50:50 beam splitters (BSs), and the total number of quantum states of the system can be expressed as:

[0059]

[0060] If any of the aforementioned detectors successfully respond, the system's output state will collapse into an empty state.

[0061] The probability of obtaining an empty state in this case is:

[0062]

[0063] In summary, the overall probability of success for the proposed solution is:

[0064]

[0065] When the scheme succeeds, a new mixture state can be obtained as follows:

[0066] ρ out =η'|φ> out out <φ|+(1-η')|vac> out out <vac|

[0067] Where the target state |φ> out The fidelity is:

[0068]

[0069] The magnification factor is:

[0070]

[0071] To achieve high fidelity amplification, G must be greater than 1. Calculations show that when... When η' > η, then the amplification of the original incident state can be achieved. For example, when η = 0.8 and cosh(ξ) = 0.5, amplification can be achieved when 0.8 ≤ t ≤ 1, and the total success probability is...

[0072] This amplification scheme is highly practical because it uses optical components commonly found under current experimental conditions.

[0073] The above description is only a preferred embodiment of the present invention. The scope of protection of the present invention is not limited to the above embodiments. Any equivalent modifications or changes made by those skilled in the art based on the content disclosed in the present invention should be included within the scope of protection set forth in the claims.

Claims

1. A noiseless linear amplification method for spatial W-states using efficient quantum scissors, characterized in that: The method includes the following steps: Step 1: The light source S0 generates the target spatial W state and transmits it to the distant receiver through a noisy channel, where there are three receivers 1, 2, 3, 0 and 1 respectively represent no photon and single photon, and b1, b2, b3 respectively represent the spatial paths connected to receivers 1, 2, 3; the photon quantum state is ρ in =η|φ1><φ1|+(1-η)|vac><vac|; where η is the transmission efficiency of photons in the channel, |υ ac > represents that the photon is lost during the photon transmission process, resulting in the quantum state of the spatial mode being a vacuum state; ρ in serves as the input state of the entire quantum scissors amplification scheme; Step 2: In the quantum scissors amplification device, receivers 1, 2, and 3 use auxiliary light sources S1, S2, and S3 to generate an auxiliary single photon in spatial modes a2, a4, and a6, respectively, and three local compression operators. In spatial modes a1, a3, and a5, three auxiliary photons respectively pass through a variable beam splitter 1, variable beam splitter 2, and variable beam splitter 3 with a transmittance of t to obtain quantum states. and Where 0 ≤ t ≤ 1; receivers 1, 2, and 3 apply a local compression operation S(-ξ) to the photons of modes a1, a3, and a5 at the three VBS reflectors to obtain the auxiliary state |ψ'> a1a2 and |ψ'> a3a4 and |ψ'> a5a6 ; Step 3: Receivers 1, 2, and 3 simultaneously pass photons in spatial modes b1 and a1 through 50:50 beamsplitter 1, photons in spatial modes b2 and a3 through 50:50 beamsplitter 2, and photons in spatial modes b3 and a5 through 50:50 beamsplitter 3. A single-photon detector is placed at each of the two output ports of each beamsplitter to detect the emitted photons. The process is successful and proceeds to the next step only when only one detector in detectors 1 and 2, only one detector in detectors 3 and 4, and only one detector in detectors 5 and 6 responds. If other detectors respond, the process ends. Step 4: If the scheme runs successfully, the output states obtained by receivers 1, 2, and 3 at the output terminals of the three efficient quantum scissors are as follows: Step 5: When the incident photon is lost, the input state of the entire quantum scissors device is a vacuum state; at this time, only the auxiliary photon enters the system through the three 50:50 beam splitters, eventually causing the single-photon detector to respond; when the system obtains the above successful detector response, the output state is a vacuum state. Step 6: Considering both scenarios, the output state of the quantum scissors amplifier is: Where η' is the fidelity of the target single-photon state in the output state; when η'>η, amplification is successfully achieved.

2. The method for noiseless linear amplification of spatial W-states using efficient quantum scissors according to claim 1, characterized in that: In step 1, This represents the probability that a photon in state W of the input space is not lost (η probability) and is lost (1-η probability). η is the transmission efficiency of photons in the channel, η = e -αL / 10 , where α is the attenuation coefficient, which is 0.2 dB / km in standard optical fiber under current experimental conditions, and L is the transmission distance.

3. The method for noiseless linear amplification of spatial W-states using efficient quantum scissors according to claim 1, characterized in that: In step 3, local compression operations are performed in space modes a1, a3, and a5. Its function is: Where ξ represents the compression coefficient, and k represents the number of photons in the photon pulse after the compression operator; Therefore, the quantum state of the auxiliary photon after the local squeezing operation is represented as: Where k1 represents the number of photons in the optical pulse after zero photons or single photons pass through the compression operator of spatial mode a1, k2 represents the number of photons in the optical pulse after zero photons or single photons pass through the compression operator of spatial mode a3, and k3 represents the number of photons in the optical pulse after zero photons or single photons pass through the compression operator of spatial mode a5.

4. The method for noiseless linear amplification of spatial W-states using efficient quantum scissors according to claim 1, characterized in that: In step 3, when photons are not lost during transmission, the total number of sub-states of the system is: Where b1, b2, and b3 represent the spatial paths connecting receivers 1, 2, and 3, respectively; The total number of substates of the system collapses as follows: After passing through three 50:50 beam splitters, the quantum state |φ3> evolves as follows: In this context, spatial modes d1, d2, d3, d4, d5, and d6 are configured with single-photon detectors D1, D2, D3, D4, D5, and D6, respectively, to detect the emitted photons. The successful detector responses are as follows: The first type is where only one photon is detected for each of D1, D3, and D5, i.e., |111> d1d3d5 ; The second type is where only one photon is detected in each of D1, D3, and D6, i.e., |111> d1d3d6 ; The third type is where only one photon is detected in each of D1, D4, and D5, i.e., |111> d1d4d5 ; The fourth type is where only D1, D4, and D6 each detect one photon, i.e., |111> d1d4d6 ; The fifth type is when only one photon is detected in each of D2, D3, and D5, i.e., |111> d2d3d5 ; The sixth type is when only one photon is detected in each of D2, D3, and D6, i.e., |111> d2d3d6 ; The seventh type is when only one photon is detected in each of D2, D4, and D5, i.e., |111> d2d4d5 ; The eighth type is when only one photon is detected in D2, D4, and D6 respectively, i.e., |111> d2d4d6 ; Taking the successful responses of detectors D1, D3, and D5 as an example, |φ4> collapses to: If other successful detector responses are obtained, the output state |φ5> will eventually be obtained, and |φ5> will still be the W state of the three spatial modes.

5. The method for noiseless linear amplification of spatial W-states using efficient quantum scissors according to claim 1, characterized in that: In step 5, the probability of obtaining the output state |φ5> using the above scheme is:

6. The method for noiseless linear amplification of spatial W-states using efficient quantum scissors according to claim 1, characterized in that: In step 5, when the input photon is lost during transmission, the input state is empty, and the overall state of the system is: The total number of substates of the system collapses as follows: The total substates of the system after three 50:50 beam splitters are written as follows: If any of the aforementioned detectors successfully respond, the system's output state will collapse into an empty state; The probability of obtaining an empty state using the above scheme is:

7. The method for noiseless linear amplification of spatial W-states using efficient quantum scissors according to claim 1, characterized in that: In step 6, assuming no photons are lost when input with probability η, the success probability of this scheme is: When the input photon is lost with a probability of 1-η, the success probability is: The overall success probability is:

8. The method for noiseless linear amplification of spatial W-states using efficient quantum scissors according to claim 7, characterized in that: In step 6, the solution is successful only if the fidelity of the output state is greater than the fidelity of the input state; the fidelity after the solution is successful is: η' = Then when When η' > η, the amplification of the original incident state is achieved; when η = 0.8 and cosh(ξ) = 0.5, the amplification is achieved when 0.8 ≤ t ≤ 1, and the total success probability is...

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