Distinguishing ensembles of quantum systems with identical density operators

Abstract

A method for distinguishing ensembles of quantum systems that includes receiving a first ensemble of quantum systems characterized by a first density operator, transforming each quantum system of the first ensemble, receiving the transformed first ensemble at a non-quadratic measurement device, and determining a first measurement outcome as an aggregation of detection events associated with the first ensemble. The method includes receiving a second ensemble characterized by a second density operator which is the same as the first density operator, and the quantum systems of the second ensemble are different from those of the first ensemble. The method includes transforming each quantum system of the second ensemble, receiving the transformed second ensemble at the non-quadratic measurement device, and determining a second measurement outcome as an aggregation of detection events associated with the second ensemble, in which the second measurement outcome is different from the first measurement outcome.

Description

Attorney Docket No.59456-0002WO1 DISTINGUISHING ENSEMBLES OF QUANTUM SYSTEMS WITH IDENTICAL DENSITY OPERATORS   CLAIM OF PRIORITY

[0001] This application claims priority under 35 USC §119(e) to U.S. Patent Application Serial No.63 / 562,251, filed on March 6, 2024, the entire contents of which are hereby incorporated by reference. BACKGROUND

[0002] Ensembles of quantum systems, which consist of multiple quantum systems in a statistical mixture of quantum states, play a crucial role in the advancement of quantum communication and computing. Analyzing these ensembles is essential for the development of quantum technologies, as it enables the precise manipulation and measurement of quantum information. Quantum communication relies on the secure transmission of data using quantum states, while quantum computing leverages the unique properties of quantum bits (qubits) to perform complex calculations at speeds that exceed methods of traditional computing. By improving the detection of quantum ensembles, we can enhance the reliability and efficiency of quantum systems.   SUMMARY

[0003] The systems and techniques described here relate to discriminating between ensembles of quantum systems, in which the ensembles are represented by an identical density operator. Under assumptions of orthodox quantum measurement, quantum ensembles described by the same density operator share the same statistical outcomes and are thus indistinguishable. However, the present specification is related to methods and systems for distinguishing between quantum ensembles described by identical density operators by observing non-quadratic observables of the quantum ensembles (e.g., measurements of time or measurements correlated with time).

[0004] The methods describe in the present specification relate to ensemble-level quantum tomography, which encompasses techniques for extracting information about a collection of quantum systems (e.g., a collection of photons or a collection of electrons). Ensemble-level quantum tomography has applications in both quantum communication protocols and quantum 1   Attorney Docket No.59456-0002WO1 computing, which both rely on extracting information from ensembles of quantum systems (e.g., photons in the case of quantum communication and electrons, photons, or other manifestations of quantum bits in the case of quantum computing).

[0005] The subject matter described in this specification can be implemented in particular embodiments to realize one or more of the following advantages. Techniques are described for discriminating between ensembles of quantum systems by providing access to previously inaccessible information about ensembles of quantum systems. The access to previously inaccessible information about ensembles of quantum systems is representative of an extension of a range of possible measurements, thereby augmenting available techniques in a variety of applications that involve manipulating and measuring quantum systems. The present specification is directed towards techniques for measuring characteristics of ensembles of quantum systems.

[0006] Applications that benefit from access to the previously inaccessible information include quantum communication protocols, in which access to the information provides a scenario in which a delay time between a transmission of a communication signal from a sender to a reception of the signal at a receiver is independent of a distance between the sender and receiver. As such, faster communication protocols are available in comparison with classical quantum communication protocols that require both a classical and a quantum channel, in which the protocols are limited by the speed of the classical channel. In addition to quantum communication protocols, the techniques described here enable new quantum computing algorithms based on measuring the previously inaccessible information of quantum systems.

[0007] In a first aspect, a method for distinguishing ensembles of quantum systems includes receiving, at a measurement apparatus, a first ensemble of quantum systems characterized by a first density operator. Each quantum system of the first ensemble of quantum systems is prepared in one state of a first set of multiple quantum states with an associated probability. The method includes transforming the state of each quantum system of the first ensemble of quantum systems. Each transformed state is described by an input mode correlated with a particular state of the first set of multiple quantum states. The input mode is suitable for analysis by a non- quadratic measurement. The method includes receiving each quantum system of the first ensemble of quantum systems described by the respective transformed state at a detection device. The detection device is operable to detect a non-quadratic observable of each received quantum 2   Attorney Docket No.59456-0002WO1 system. The method includes determining, by the detection device, a first measurement outcome, in which the first measurement outcome is an aggregation of detection events of the first ensemble of quantum systems over a first time interval. The method further includes receiving, at the measurement apparatus, a second ensemble of quantum systems characterized by a second density operator. Each quantum system of the second ensemble of quantum systems is prepared in one state of a second set of multiple quantum states with an associated probability, in which (i) the second density operator is the same as the first density operator and (ii) the second set of multiple quantum states is different from the first set of multiple quantum states. The method includes transforming the state of each quantum system of the second ensemble of quantum systems. Each transformed state is described by a spatial mode correlated with a particular state of the second set of multiple quantum states. The method includes receiving each quantum system of the second ensemble of quantum systems described by the respective transformed state at the detection device and determining, by the detection device, a second measurement outcome, wherein the second measurement outcome is an aggregation of detection events of the second ensemble of quantum systems over a second time interval, in which the second measurement outcome is different from the first measurement outcome.

[0008] In some implementations, the first set of multiple quantum states includes (i) a first basis state and (ii) a second basis state, and the second set of multiple quantum states includes (i) a normalized in-phase superposition of the first basis state and the second basis state and (ii) a normalized out-of-phase superposition of the first basis state and the second basis state.

[0009] In some implementations, the non-quadratic observable includes at least one of a total detection probability, a position of detection, or a time of detection.

[0010] In some implementations, the detection device is operable to detect the non-quadratic observable of a transformed quantum state by measuring a quadratic observable of the transformed quantum state that is correlated with the non-quadratic observable.

[0011] In some implementations, the non-quadratic observable corresponds to a temporal measurement.

[0012] In some implementations, the detection devices includes one of a spatially-sensitivedetection device, in which a measurement outcome is indicative of a measurement position of a quantum system, the measurement position correlated with a temporal duration of a quantum process. 3   Attorney Docket No.59456-0002WO1

[0013] In some implementations, the method further includes transforming the state of each quantum system of the first ensemble of quantum systems to erase at least one distinguishing attribute of a quantum system of the first ensemble of quantum systems.

[0014] In some implementations, the method further includes transforming the state of each quantum system of the second ensemble of quantum systems to erase at least one distinguishing attribute of a quantum system of the second ensemble of quantum systems.

[0015] In some implementations, the detection device includes a point detector, in which the point detector is operable to aggregate detection events over a time interval.

[0016] In some implementations, the measurement apparatus includes one or more recombination elements, in which the recombination elements are operable to spatially overlap quantum systems that propagate over distinct spatial modes.

[0017] In some implementations, the aggregation of detection events comprises an analysis of ahistogram, wherein the histogram comprises a plurality of time-bins, each time-bin associated with a number of detection events and a space-time coordinate.

[0018] In another aspect, a system for distinguishing ensembles of quantum system includes astate transformer. The state transformer is operable to (i) receive quantum systems of a first ensemble of quantum systems and (ii) split quantum systems of the first ensemble of quantum systems into two propagation paths depending on a quantum state of each respective quantum system, and (iii) transform a quantum state of each quantum system such that a quantum system of a first propagation can interfere with a quantum system of a second propagation path. The system includes a detection system, in which the detection includes (i) a propagation medium, in which the propagation medium is operable to receive quantum systems from the recombination element and (ii) a non-quadratic detection element, wherein the non-quadratic detection element is operable to measure a temporal characteristic of the ensemble of quantum systems.

[0019] In some implementations, the detection system includes one or more recombinationelements, in which the recombination elements are operable to overlap the quantum system of the first propagation path with the quantum system of the second propagation path.

[0020] In some implementations, the temporal characteristic of the ensemble of quantumsystems is analyzed by measuring a quadratic observable of the ensemble of quantum systems, wherein the temporal characteristic is correlated with the quadratic observable. 4   Attorney Docket No.59456-0002WO1

[0021] In some implementations, the non-quadratic detection element includes one or more detection screens. In some implementations, the non-quadratic detection element includes one or more point detectors.

[0022] In some implementations, the system further includes an input terminal. The input terminal is operable to receive an ensemble of quantum systems from at least one external system, the external system generating and transmitting the ensemble of quantum systems to the input terminal, the input terminal operable to transmit the received ensemble of quantum systems to the state transformer.

[0023] In some implementations, the quantum systems of the first ensemble are photons, wherein the quantum systems of the second ensemble are photons, wherein the state transformer includes a polarization beam splitter operable to split the photons of each ensemble into two propagation paths depending on a polarization state of each respective photon.

[0024] In some implementations, the quantum system of the first propagation path can interfere with the quantum system of the second propagation path by removing at least one distinguishing attribute of the quantum systems propagating along each path.

[0025] The details of one or more embodiments of the invention are set forth in the accompanying drawings and the description below. Other features, objects, and advantages of the invention will be apparent from the description and drawings, and from the claims. BRIEF DESCRIPTION OF THE DRAWINGS

[0026] FIG.1 is an example system for discriminating between ensembles of quantum systems described by an identical density operator.

[0027] FIG.2 is a flow diagram of an example process for discriminating between ensembles of quantum systems described by an identical density operator.

[0028] FIG. 3 is an example system for discriminating between ensembles of quantum systemsdescribed by an identical density operator.

[0029] FIG.4 is an example system for discriminating between ensembles of quantum systems described by an identical density operator.

[0030] FIG.5 is an example system for discriminating between ensembles of quantum systems described by an identical density operator. 5   Attorney Docket No.59456-0002WO1

[0031] FIG.6 is an example system for discriminating between ensembles of quantum systems described by an identical density operator.

[0032] FIG.7 is an example system for discriminating between ensembles of quantum systems described by an identical density operator.

[0033] FIG. 8 is an example system for discriminating between ensembles of quantum systemsdescribed by an identical density operator.

[0034] FIG.9 is an example implementation of a measurement system for discriminating between ensembles of quantum systems described by an identical density operator.

[0035] FIG.10 is an example implementation of a measurement system for discriminating between ensembles of quantum systems described by an identical density operator.

[0036] Like reference numbers and designations in the various drawings indicate like elements. DETAILED DESCRIPTION

[0037] The systems and techniques described here relate to discriminating between ensembles of quantum systems, in which each ensemble is described by an identical density operator (DEIDO, i.e., discriminating between ensembles with identical density operators). In particular, the present disclosure relates to implementations of quantum ensemble preparation and quantum ensemble measurement that provide a mechanism for methods of DEIDO.

[0038] The present specification is directed towards methods of ensemble-level quantum tomography, which encompasses techniques for extracting information about a collection of quantum systems (e.g., a collection of photons or a collection of electrons). Ensemble-level quantum tomography has applications in both quantum communication protocols and quantum computing, which both rely on extracting information from ensembles of quantum systems (e.g., photons in the case of quantum communication and electrons, photons, or other manifestations of quantum bits in the case of quantum computing).

[0039] An ensemble of quantum systems (e.g., an ensemble of quantum particles like photons or electrons) is a collection of quantum systems (e.g., a collection of photons), in which each quantum system is prepared in a particular quantum state (or superposition of quantum states) with a particular probability. In other words, the ensemble of quantum systems is a collection of quantum systems, each described by a pure quantum state. In some cases, an analysis of the ensemble consists of repeated measurements on individual quantum systems within the 6   Attorney Docket No.59456-0002WO1 ensemble, allowing for a determination of statistical properties of the ensemble. By measuring many quantum systems of the ensemble, a measurement system can determine a probability distribution of quantum states (or superposition of quantum states) within the ensemble of quantum systems.

[0040] In a general sense and in the context of standard quantum theory, information that can belearned about a quantum system is fundamentally limited by Heisenberg’s uncertainty principle. The principle states that determining a position of a quantum system with high precision necessarily leads to a large uncertainty in an analysis of the momentum of the quantum system. In a broader sense, standard quantum theory describes the information to be learned about a quantum system in terms of observables. In some cases, observables are represented as mathematical constructs known as self-adjoint operators. Self-adjoint operators can be represented as sums of orthogonal operators associated with corresponding eigenvalues (i.e., observables of a quantum system like position or momentum).

[0041] To expand the standard quantum theory that describes observables in terms of self-adjoint operators, a more generalized formulation of quantum theory is provided by a set of mathematical structures known as positive operator-valued measures (POVM). A POVM framework provides a generalized formulation that does not require self-adjoint operators and describes measurement scenarios that often include interactions between quantum systems and their environments (e.g., some implementations of quantum communication protocols).

[0042] The present specification is directed towards a framework of quantum measurement that extends the POVM framework. In some cases, the POVM framework fails at describing classes of quantum systems and associated measurements. The POVM framework implies that certain ensembles of quantum systems are indistinguishable by any measurement device describable within the framework. In particular, ensembles of quantum systems described by identical density operators, as described in detail below, are indistinguishable under the assumptions of the POVM framework. The present specification is directed towards systems and techniques that extend the POVM framework to include measurements that distinguish between ensembles of quantum systems described by identical density operators.

[0043] Example scenarios that lack explanation by the POVM framework include measurements of temporal characteristics of quantum systems. For example, measurements related to arrival time of quantum systems like dwell time, energy level transition time, energy level decay time, 7   Attorney Docket No.59456-0002WO1 among others, are commonly evaluated experimentally, but time measurements do not correspond to observables in the standard quantum theory. As such, in some cases, time measurements performed on ensembles of quantum systems can provide a mechanism for distinguishing between ensembles of quantum systems described by identical density operators.

[0044] In particular, the present specification is directed towards non-quadratic measurementdevices to implement methods for DEIDO. To establish a frame of reference for describing the methods and techniques of the present disclosure, and to expand upon the frameworks of quantum theory described above, it is noted that measurements of quantum systems can be described as belonging to one of three categories: (i) orthodox measurements (e.g., measurement of observables associated with self-adjoint operators), (ii) quadratic measurements (e.g., measurement of observables associated with positive operator-valued measures (POVM)), and (iii) non-quadratic measurements (e.g., measurement of observables that fit neither in a POVM framework or an orthodox framework, including measurements of temporal characteristics of quantum systems).

[0045] In relation to (i), orthodox measurements include extracting information from a quantumsystem by evaluating observables described by self-adjoint operators. Example observables evaluated with orthodox measurements include position, momentum, and polarization.

[0046] In relation to (ii), POVM measurements offer a generalization of orthodox measurements to observables that are described by operators that do not exhibit properties of self-adjoint operators. In particular, POVM measurements are described by a set of operators, in which the sum of the operators is equal to the identity operator, and each operator of the set is positive. POVM measurements allow for non-orthogonal operators, which is not allowed when considering orthodox measurements. In some cases, POVM measurements are associated with probabilistic and noisy quantum systems, and systems that interact with an environment.

[0047] In relation to (iii), non-POVM measurements (i.e., non-quadratic measurements) enableDEIDO and can be achieved with a variety of physical embodiments. To achieve a non-quadratic measurement, a device can facilitate a coupling between a first quantum system and a second quantum system, such that a duration of a process of the first quantum system determines a measurement timing of the second quantum system. As such, in some cases, as long as the second quantum system propagates predictably, its detection timing is indicative of a duration of a quantum process of the first system due to the coupling of the first and second systems. By 8   Attorney Docket No.59456-0002WO1 performing a traditional measurement (e.g., position measurement) on the second system, information is revealed about the duration of the quantum process of the first system. In other words, the timing information of the quantum process of the first system is inferred from a measurable observable of the second system. Therefore, DEIDO is achieved without a need for direct measurements of a time variable of the quantum process.

[0048] Non-quadratic measurement devices can implement time measurements of quantum mechanical systems as a source of non-quadratic behavior. For example, decaying particles provide non-quadratic behavior in which the temporal duration of an excited or unstable state can be correlated with an orthodox observable like position. Decaying particles can be represented by an unstable or radioactive particle that decays into multiple particles or an atom in an excited state that emits a photon when decaying to a lower energy state. A system can measure the position of the decay products (e.g., photons or atoms) to infer information pertaining to the temporal duration of the excited state. The measurement of position as a proxy for timing measurement of a quantum mechanical process is an example of a non-quadratic measurement, and thus not described by the POVM framework.

[0049] In a general sense, a non-quadratic observable is any experimental outcome O, the probability of which depends on an input state “non-quadratically.” In more detail, O can dependon the input state quadratically if given any collection of orthogonal states ^^^, ^^ଶ, …^^^ for anynormalized state of the form ^^^^^^, … , ^^^^ ൌ ^^^^^^ ^ ^^ଶ^^ଶ ^ ⋯^ ^^^^^^, where the probability ofobserving the experimental outcome O given the collection of orthogonal states^^^^^, ^^^^^^, … , ^^^^^ is a homogeneous quadratic polynomial in the variables ^^^, ^^ଶ, … , ^^^. It isnoted that instantaneous measurements of standard observables or POVMs arequadratic. Furthermore, quadraticity is expected to fail for measurements of parameters like time that fall outside of the standard measurement frameworks.

[0050] To understand the statistical nature of an ensemble of quantum systems (e.g., to understand which states are represented in the ensemble with which probabilities), a density operator (e.g., a density matrix) encodes all statistical properties of the ensemble of quantum systems that are described by self-adjoint operators or POVMs. The density operator provides a means of computing an expectation value of an associated observable (e.g., position, momentum, energy, etc.). 9   Attorney Docket No.59456-0002WO1

[0051] The present specification is particularly related to a description of ensembles of quantum systems. An ensemble of quantum systems is represented as a collection of pure states, e.g.,^^^^^,^^^^, ^^^ଶ, ^^ଶ^, … ^, in which an associated ensemble density operator ^^^ can be written as,^^^ ൌ ^^^|^^^^^^^^| ^ ^^ଶ|^^ଶ^^^^ଶ| ^ ⋯,where ^^^is a probability that a quantum system of the ensemble described by^^^is described by the quantum state |^^^^. Each quantum system of the ensemble has a particular probability ^^^of being represented by the quantum state |^^^^. The density operator ^^^ is a lossy mathematicalrepresentation of the ensemble of quantum states ^^^^^,^^^^, ^^^ଶ, ^^ଶ^, … ^, and as such, the presentspecification is directed towards detecting information that is not described by the density operator representation.

[0052] The traditional formulation of quantum mechanics expresses an expectation value of an observable quantity of a quantum system as 〈^^^〉 ൌ ^^^^൫^^^ ^^^൯,where ^^^^ is the trace operator, ^^^ represents the observable quantity, and 〈^^^〉 represents anexpectation value of the observable quantity. Therefore, the expectation value of the observable quantity (i.e., a result of an ideal measurement, e.g., an instantaneous measurement of position) depends on the density operator ^^^. As such, if two ensembles can be described by an identical density operator, the traditional formulation of quantum mechanics yields that statistics of measurement outcomes associated with standard observables do not discriminate between the two ensembles.

[0053] In some cases, a first ensemble of quantum systems is described by a particular density operator, and a second ensemble of quantum systems is described by an identical particular density operator. However, the probabilities of quantum states of the first ensemble need not be the same as the probabilities of quantum states of the second ensemble, even if the statistics of all POVM measurement outcomes are the same for both ensembles. Furthermore, the particular quantum states |^^^^ of the first and second ensembles need not be the same. For example, consider a first ensemble of quantum bits (qubits) in which each qubit of the first ensemble ofqubits can be described by a first state|0^ (i.e., ൫^^൯) or a second state|1^ (i.e., ൫^ ^൯) with equal probability. The corresponding density operator for the first ensemble of qubits is written as ^^^ ^^ ^^ ^ ^1 0 Attorney Docket No.59456-0002WO1 Similarly, consider a second ensemble of qubits in which each qubit of the second ensemble of ^ qubits can be described by a first state √ଶ^|0^ ^ |1^^ (i.e.,^^^ √ଶ൫^൯) or a second state√ଶ^|0^ െ |1^^^ (i.e., √ଶ൫ ^ି^൯) with equal probability. The corresponding density operator for the secondis written as4^| ^^ | |1 ^^^ଶൌ0 0 ^ 0^^1| ^ |1^^0| ^ |1^^1|^ ^4^|0^^0| െ |0^^1| െ |1^^0| ^ |1^^1|^ ൌtothe to the standard quantum theory as well as the POVM framework, a measurement device will yield a measurement of an observable (e.g., ^^^, corresponding to polarization, momentum, etc.) with the same probability distribution of output values (e.g., a data value indicative of an instantaneous position measurement) regardless of whether the measurement device is sampling qubits prepared in the first ensemble of qubits or the second ensemble of qubits.

[0055] The present disclosure describes methods and systems for distinguishing the first ensemble of qubits (e.g., a collection of qubits described by one of two basis states) from the second ensemble of qubits (e.g., a collection of qubits described by one of two superpositions of the two basis states), despite the ensembles of qubits being described by the same density operator (e.g., ^^^^ ൌ ^^^ଶ). The density operator merely describes the statistical properties of theensembles and does not describe the preparation of the states that make up the ensembles.

[0056] In particular, given a particular non-quadratic measurement device that is sensitive to a non-quadratic observable, DEIDO is achievable if a statistical property of detection outputs from the non-quadratic measurement device fails a generalized parallelogram law in response to an analysis of ensembles of quantum systems. For instance, for a function^^^|^^^^that describes aprobability that a quantum system (e.g., a photon) described by a state |^^^is detected within a particular time interval, the function fails the generalized parallelogram law if there exists orthogonal states |^^^ and |^^^ such that ^^^|^^^^ ^ ^^^|^^^^ ് ^^^| ^^^ ^ ^^^| െ^^, where |^^^ and|^^ can be theprevious example and | ^^ and | െ^ are given by^ √ଶ^|^^^ ^ |^^^^ and^|^^^ െ |^^^^Attorney Docket No.59456-0002WO1

[0057] In a general sense, the existence of any pair of orthogonal states (e.g., the basis states of the first ensemble of qubits) that leads to a failure of the generalized parallelogram law provides an opportunity for DEIDO using a non-quadratic measurement device. In some implementations, variations of a generalized parallelogram law are implemented to determine a set of states that provide a means for implementing DEIDO (e.g., basis states that need not be orthogonal, uneven mixtures of basis states, and ensembles that include more than one basis state, among others).

[0058] As described below in relation to the description of the embodiments of FIGS.3-8, a coupling between two (or more) parameters of a quantum system facilitates a measurement of a standard quantum observable to be correlated with a temporal measurement, thus providing a mechanism to achieve DEIDO.

[0059] FIG.1 illustrates an example system 100 for discriminating between two prepared ensembles of quantum systems that are described by identical density operators. The system 100 includes an input terminal 102 that is operable to receive an ensemble of quantum systems (e.g., photons or electrons). A quantum system can be a particle with a particular set of properties (e.g., polarization, frequency, momentum, etc.). The system 100 illustrates two cases, in which the input terminal 102 is operable to receive a first ensemble of quantum systems 104a and a second ensemble of quantum systems 104b. In a general sense, the input terminal 102 outputs the first ensemble of quantum systems 104a for a first period of time and the ensemble of quantum systems 104b for a second period of time. The system 100 is operable to generate respective measurement outputs 114a and 114b indicative of the prepared ensemble by the input terminal 102. In some implementations, the input terminal 102 receives an ensemble of quantum states from a separate quantum system (e.g., a source entangled photons).

[0060] The system 100 includes a state transformer 106 that transforms a quantum state of each quantum system of the ensembles of quantum systems 104a-b received by the input terminal 102. In some implementations, the state transformer 106 includes optical elements in the case that the input terminal 102 receives photonic quantum systems. For example, the optical elements can include one or more of a beam splitter, mirror, waveplate, diffraction grating, lens, etc. The system 100 includes a measurement apparatus 110 that is operable to receive the ensembles of quantum systems 104a-b with transformed quantum states from the state transformer 106 and to discriminate a class of quantum states that are described by an identical density operator, as described in the present specification above. 12   Attorney Docket No.59456-0002WO1

[0061] The state transformer 106 receives the example ensembles of quantum systems 104a-b. In some implementations, the transformer 106 transforms one or more physical parameters related to each received quantum system (e.g., polarization, momentum, spatial mode, etc.).

[0062] The measurement apparatus 110 receives each transformed ensemble of quantum systems 108a-b. In the example system 100, the measurement apparatus 110 receives the first transformed ensemble of quantum systems 108a and a second transformed ensemble of quantum systems 108b. The measurement apparatus 110 generates a first measurement output (^^^^ and a second measurement output (^^^), such that the statistics of the first and second measurementoutputs are different ^^^^ ് ^^^^. In some implementations, a certain number of measurements ofquantum systems of a particular ensemble of quantum systems are measured to determine astatistical difference between^^^and^^^.For example, a difference between^^^and^^^may beobserved over a large number of measurements (e.g., 1 in 1,000 measurements of quantum systems of an ensemble may be different, leading to a discrimination between two associated ensembles of quantum systems described by identical density operators).

[0063] The measurement apparatus 100 is operative to perform a non-quadratic measurement. The non-quadratic measurement, as described throughout the present specification, is performed by a measurement device sensitive to a parameter of a quantum system that exhibits at least one non-quadratic property such as a quantum property indicative of a temporal duration of a quantum process (e.g., decay time of an excited quantum state). The measurement apparatus 100 is operable to correlate a measurement of a non-temporal measurement (e.g., position) with a temporal measurement.

[0064] In some implementations, the measurement apparatus 100 performs measurementsrepeatedly over a period of time. As such, the measurement apparatus 100 samples the prepared quantum systems of ensembles prepared by the input terminal 102. The measurement apparatus 100 can determine a statistical distribution over space and / or time of the transformed ensembles received by the measurement apparatus 110. In some implementations, the measurement apparatus 100 determines a total probability distribution of detection events over a time interval (e.g., determining if a quantum state eventually is detected at particular spatial coordinate within the measurement apparatus 110).

[0065] FIG.2 is a flow diagram of an example process 200 for discriminating between two quantum states described by an identical density operator. The process 200 can be implemented 13   Attorney Docket No.59456-0002WO1 by a system that includes an ensemble input terminal, state transformer, and measurement apparatus, as described in relation to FIGS.3-8.

[0066] The system receives (202), at a measurement apparatus, a first ensemble of quantum systems characterized by a first density operator. Each quantum system of the first ensemble of quantum systems is prepared in one state of a first set of multiple quantum states with an associated probability.

[0067] The system transforms (204) the state of each quantum system of the first ensemble of quantum systems. Each transformed state is described by an input mode correlated with a particular state of the first set of multiple quantum states. The input mode is suitable for analysis by a non-quadratic measurement.

[0068] The system receives (206) each quantum system of the first ensemble of quantum systems described by the respective transformed state at a detection device. The detection device is operable to detect a non-quadratic observable of each received quantum system.

[0069] The system determines (208), by the detection device, a first measurement outcome, in which the first measurement outcome is an aggregation of detection events of the first ensemble of quantum systems over a first time interval.

[0070] The system receives (210), at the measurement apparatus, a second ensemble of quantum systems characterized by a second density operator. Each quantum system of the second ensemble of quantum systems is prepared in one state of a second set of multiple quantum states with an associated probability, in which (i) the second density operator is the same as the first density operator and (ii) the second set of multiple quantum states is different from the first set of multiple quantum states.

[0071] The system transforms (212) the state of each quantum system of the second ensemble of quantum systems. Each transformed state is described by a spatial mode correlated with a particular state of the second set of multiple quantum states.

[0072] The system receives (214) each quantum system of the second ensemble of quantum systems described by the respective transformed state at the detection device.

[0073] The system determines (216), by the detection device, a second measurement outcome,wherein the second measurement outcome is an aggregation of detection events of the second ensemble of quantum systems over a second time interval, in which the second measurement outcome is different from the first measurement outcome. 14   Attorney Docket No.59456-0002WO1

[0074] FIG.3 is an example system 300 for discriminating between two ensembles of quantum systems described by an identical density operator. The system 300 depicts a configuration that receives, transforms, and measures ensembles of quantum systems (e.g., an ensemble of photons). In a general sense, embodiments of the methods described in the present specification can be configured to receive, transform, and measure photonic systems, electronic systems, or systems of any other quantum particle or composite system. For example, an optical component that maps a spin state (e.g., polarization state of a photon or a spin state of an electron) to distinct spatial modes can be implemented using a polarizing beam splitter for photons and a Stern- Gerlach apparatus for an electron. For illustrative purposes, the methods of the present specification are described in relation to photonic quantum systems but can be generalized for other quantum systems.

[0075] The system 300 includes an input terminal 302 that receives an ensemble of photons 304described by a particular collection of polarization states with associated probabilities. For example, the ensemble of photons 304 can be described by a set of polarization states (qubits)with associated probabilities, ^^|0^, ^ଶ^ , ^|1^, ^ଶ^^ or ^^|^^, ^ଶ^ , ^|െ^, ^ଶ ^^. In some cases, the input terminal 302 receives a first receives a secondensemble of photons for a second period of time, where the polarization states of the photons of the first and second ensembles are different.

[0076] A state transformer 306 receives the ensemble of photons received by the input terminal 302. The state transformer 306 includes a polarizing beam splitter 308 that splits a first polarization (e.g., horizontal, |0^^ into a top path 314a and a second polarization (e.g., vertical, |1^^ into a bottom path 314b. The state transformer 306 includes a wave plate 316a associated with the top path 314a and a wave plate 316b associated with the bottom path 314b. The waveplates 316a-b are operable to rotate the polarization of the photons such that the polarization of any photon exiting the respective waveplate is the same. In other words, both horizontally polarized photons and vertically polarized photons are rotated into aligned polarization states. The waveplates 316a-b provide polarization indistinguishability between the photons propagating along the top path 314a and the bottom path 314b.

[0077] The state transformer 306 includes, for both the top path 314a and the bottom path 314b, a respective spatial recombination element 318a and 318b. In some implementations, the recombination elements 318a-b are lenses, diffraction gratings, or any element that is operable to 15   Attorney Docket No.59456-0002WO1 recombine photons that propagate along the top path 314a with photons that propagate along the bottom path 314b. In some implementations of the state transformer 306, the spatial recombination elements 318a and 318b are not included. In these implementations, transverse spatial dispersion creates an overlap between the photons that propagate along the top path 314a with the photons that propagate along the bottom path 314b.

[0078] A detection system 326 receives the transformed ensemble of photons from the state transformer 306, in which the propagation paths of the photons of the top path 314a overlap with the photons of the bottom path 314b (via overlap propagation path 324a and overlap propagation path 324b). The detection system 326 includes a propagation medium 322. In some implementations, the propagation medium 322 includes dispersive optical elements (e.g., glass) or vacuum. The propagation medium 322 includes a detection element 320 positioned a non-zero distance from the output of the state transformer 306. One or more geometrics can be implemented to position the propagation medium 322 between the state transformer 308 and the detection element 320. The detection element 320 of the detection system 326 is a scintillating screen, detector array, or any device that is sensitive to detecting individual photons that arrive at the detection element 320. For example, the detection element 320 can include an array of single photon detectors that generate output voltage signals in response to an arrival of single photons.

[0079] In some implementations, photons of the ensemble of photons received by the input terminal 302 enter the state transformer 306 at a particular known time^^^. Due to a potential difference in path lengths between the top path 314a and the bottom path 314b, delay elements can be included in either path to provide temporal indistinguishability between the two paths (e.g., photons that propagate along the top path 314a should be temporally aligned with photons that propagate along the bottom path 314b when entering the detection system 326). In some implementations, delay lines including additional dispersive material, fiber optic spans, or additional free space propagation paths can be included in either the top path 314a or the bottom path 314b to ensure temporal indistinguishability between the two paths.

[0080] The detection element 320 is operable to detect single photons at a time ^^^at position ^^,along the length of the detection element 320. Joint statistics of the time difference ^^^ െ ^^^ andposition ^^ across a particular time range for an incoming ensemble of photons provide a mechanism to perform DEIDO. In other words, an observation of the joint statistics between arrival time and position reveals a method of discrimination between a first ensemble of photons 16   Attorney Docket No.59456-0002WO1 received by the input terminal 302 and a second ensemble of photons received by the input terminal 302, in which the first and second ensembles are described by an identical density operator.

[0081] In some implementations, the detection element 320 is coupled with a system that includes a processor that is operable to execute instructions for computing one or more statistical analysis of associated detection events. In some cases, a time-based statistical analysis is accomplished by first assigning each detector of the n detectors of the detection element 320 with an integer between 1 and n, and assign evenly spaced time attributes to each detector (e.g.,^^^^, ^^^ ^ ^^, ^^^ ^ 2^^, … ^^. For a single detection event by a detector of the detection element 320,an outcome corresponding to a detection event at the k-th detector is triggered at a time t (e.g., a detection event corresponding to space-time coordinate of (k,t)). For each detection event, the system assigns the space-time coordinate (k,t) to a particular time bin, according to the assignedevenly spaced time attributes of each detector (e.g., determine which time bin ^^^^, ^^^ ^ ^^, ^^^ ^2^^, … ^^ contains the time t. Over many detection events, the system can generate a histogram, inwhich each time bin is associated with a number of detection events, corresponding to detection events of each detector of the detection element 320.

[0082] To determine that two ensembles of quantum systems are distinguishable via a non-quadratic measurement by the detection element 320, the system can compare a first histogram associated with a first ensemble with a second histogram associated with a second ensemble, in which the first and second ensembles are described by an identical density operator.

[0083] FIG.4 is an example system 400 for discriminating between two ensembles of quantum systems described by an identical density operator. The system 400 depicts a configuration that receives, transforms, and measures ensembles of quantum systems (e.g., an ensemble of photons). The system 400 is a variation of the system 300. The system 400 includes a detection system 426 a detection element 420 positioned within a propagation medium 422.

[0084] The detection element 420 is a curved surface of detector screen. In some implementations, the detection element 420 is circular. In some implementations, the detection element 420 is composed of detectors equivalent to the detectors included in the detection element 320 of FIG.3. The detection element 420 is positioned at a point of intersection between a top propagation path 424a and a bottom propagation path 424b, in which the propagation paths 17   Attorney Docket No.59456-0002WO1 424a-b correspond to a propagation of photons from a state transformer 406 into the detection system 426.

[0085] The state transformer 406 is identical to the state transformer 306 of FIG.3, including a polarizing beam splitter 408 that receives ensembles of photons 404 from an input terminal 402. In addition, the state transformer 406 includes a top path 414a and a bottom path 414b, in which photons that propagate along each path correspond to photons described by a particular polarization state before the polarizing beam splitter 408. The state transformer includes a waveplate 416a and a waveplate 416b in the top path 414a and bottom path 414b respectively. Furthermore, the state transformer includes a top recombination element 418a and a bottom recombination element 418b in the top path 414a and bottom path 414b respectively, each operable to spatially disperse the transformed photons to enable overlap of states at a position in a vicinity of the curved surface detection element 420.

[0086] FIG.5 is an example system 500 for discriminating between two ensembles of quantum systems described by an identical density operator. The system 500 depicts a configuration that receives, transforms, and measures ensembles of quantum systems (e.g., an ensemble of photons). The system 500 is a variation of the system 300. The system 500 includes a detection system 526 with a first detection element 520a and a second detection element 520b. The detection elements 520a-b are perpendicular to the detection element 320 of the detection system 326 of FIG.3. Similar to the detection element 320, the first detection element 520a and the second detection element 520b are operable to detect single photon detection events at various longitudinal positions ^^, in which ^^ is colinear with the propagation paths of a top path 514a and a bottom path 514b of a state transformer 506.

[0087] The state transformer 506 operates similar to the state transformer 306. The statetransformer 506 receives ensembles of photons 504 from an input terminal 502. The state transformer 506 includes a polarization beam splitter 508 that splits the ensemble of photons 504 into the top path 514a and the bottom path 514b, each path including a respective waveplate 516a-b and spatial recombination elements 518a-b.

[0088] The detection system 526 includes a propagation medium 522 and the two detection systems 520a-b that are operable to receive photons that propagate along paths 524a-b. Due to the spatial recombination elements 518a-b, each detection system receives one or more photons from the ensemble of photons that propagate along the top path 514a and the bottom path 514b. 18   Attorney Docket No.59456-0002WO1 The spatial recombination elements 518a-b (e.g., lens or diffraction gratings) introduce a transverse spread of momentum through the propagation medium 522 providing spatial overlap between photons from the top path 514a and photons from the bottom path 514b.

[0089] Although the system 500 contains components similar to the system 300, a difference in orientation between the detection element 320 and the detection elements 520a-b provide a different measurement approach to achieving DEIDO. Precise timing of photons entering the state transformer 506 from the input terminal 502 and precise timing of detection events that the detection elements 520a-b of the detection system 526 are not necessary to achieve DEIDO, unlike the configuration of FIG.3.

[0090] A position of a detection event (e.g., a single photon detection) along the^^direction of each detection element 520a-b is indicative of a time-of-flight of a respective photon. For example, a detection event that happens at a first position along the detection element 520a is indicative of a photon propagating for a time period through the propagation medium 522 greater than a time period associated with a photon detected by the detection element 520a at a position further from the entrance of the detection system 526. As such, a position of a detection event isindicative of a time difference ^^^ െ ^^^, which relates to a time difference between when a photonof an ensemble of photons enters the state transformer 506 and is detected by a detection element of the detection system 526. Joint statistics between a detection time and a position is typically not needed, and statistics related to spatial distribution of detection events along detection elements 520a-b reveal information that enables DEIDO.

[0091] FIG.6 is an example system 600 for discriminating between two ensembles of quantum systems described by an identical density operator. The system 600 depicts a configuration that receives, transforms, and measures ensembles of quantum systems (e.g., an ensemble of photons). The system 600 is a variation of the system 300. The system 600 includes a detection system 626 with a reflective chamber 622 with an internal partially reflective barrier 628 and a detection element 620 positioned behind the partially reflective barrier 628.

[0092] Similar to the detection element 320, the detection element 620 is operable to detect single photon detection events at various longitudinal positions ^^, in which ^^ is colinear with the propagation paths of a top path 614a and a bottom path 614b of a state transformer 606. However, in some other implementations, the orientation of the partially reflective barrier 628 and the detection element 620 can be different because the reflective chamber 622 allows for 19   Attorney Docket No.59456-0002WO1 interference between different input states of the ensemble of photons that enter the detection system 626.

[0093] The detection system 626 receives photons from the state transformer 606 through an opening in the reflective chamber 622 coupled to each of the top path 614a and the bottom path 614b. Similar to the state transformer 306, the state transformer 606 includes a polarizing beam splitter that splits an incoming ensemble of photons 604 received by an input terminal 602 into the top path 614a and the bottom path 614b depending on the polarization state of the incoming ensemble. To erase polarization distinguishability between the two paths, each of the top path 614a and the bottom path 614b include a respective waveplate 616a and waveplate 616b to align the polarizations of the respective photons.

[0094] The reflective chamber 622, with multiple reflective walls and at least one partially reflective barrier, confines photons that enter via a top propagation path 624a through a top opening and photons that enter via a bottom propagation path 624b through a bottom opening for a particular amount of time before they propagate through the partially reflective barrier 628 and are detected by the detection element 620. A state-dependent delay in detection time by the detection element 620 enables DEIDO. Because the top propagation path 624a is closer to the detection element 620 than the bottom propagation path 624b, it is expected that photons that propagate along the top propagation path 624a will reach the detection element 620 on average faster than the photons that propagate along the bottom propagation path 624b.

[0095] FIG.7 is an example system 700 for discriminating between two ensembles of quantum systems described by an identical density operator. The system 700 depicts a configuration that receives, transforms, and measures ensembles of quantum systems (e.g., an ensemble of photons). The system 700 is a variation of the system 600.

[0096] The system 700 includes a detection system 726 with a reflective chamber 722. The reflective chamber 722 contains two regions of absorptive media. A first region of absorptive media 728a is positioned closer to a first opening in the reflective chamber 722 aligned with a top path 714a of a state transformer 706. A second region of absorptive media 728b is positioned closer to a second opening in the reflective chamber 722 aligned with a bottom path 714b of the state transformer 706. The state transformer 706 includes a polarization beam splitter 708, waveplates 716a-b and is operative similar to the state transformer 606 described in relation to FIG.6. The detection system 726 includes detection element 720 positioned within the reflective 20   Attorney Docket No.59456-0002WO1 chamber 722 but external to both the first region of absorptive media 728a and the second region of absorptive media 728b.

[0097] In some implementations, material (or materials) within the regions of absorptive media 728a-b is chosen such that a quantum process of energy level decay occurs upon excitation by a photon that enters the reflective chamber 722 from either an input path 724a or an output path 724b. Upon interaction with a photon, atoms within a region of absorptive material enters an excited energy state and then releases the energy via a decay process and emits a photon in turn. The lifetime of the excited state is related to the particular material within a region of absorptive material. The resulting emitted photon is emitted back into the reflective chamber 722 and ultimately detected by the detection element 720.

[0098] The detection element 720 records detection events and the statistics of the detection timing reveals information about the decay time of the material within the regions of absorptive material 728a-728b. In a general sense, the detection element 720 can be positioned anywhere in or out of the reflective chamber such that detection events are efficiently captured. However, like the reflective chamber 622 of FIG.6, the regions of absorptive media 728a-b erase any information encoded in the position or momentum of incoming photons.

[0099] FIG.8 illustrates a first implementation 800 of an example system and a second implementation 850 of the example system for discriminating between two ensembles of quantum systems described by an identical density operator. The first implementation 800 and the second implementation 850 each depict a three-dimensional apparatus operable to enable a computation of conditional probabilities that allow for DEIDO. The second implementation 850 illustrates an internal cutaway of the system that includes a sparse detector array 852 aligned along an x-z plane of a coordinate system 801. The first implementation 800 illustrates an external view of the example system.

[0100] The first implementation 800 includes an input terminal 802. In some implementations,the input terminal 802 is an optical fiber. The first implementation 800 includes a polarizing beam splitter 804 coupled to the input terminal 802. The polarizing beam splitter 804 is operable to split photons of an ensemble of photons received from the input terminal 802 into a first optical path 806a and a second optical path 806b, depending on a polarization state of an incoming photon. In some implementations, the first and second optical paths 806a-b are implemented as optical fibers. In some implementations, the polarizing beam splitter 804 is 21   Attorney Docket No.59456-0002WO1 configured such that a plane of polarization is oriented along a diagonal between horizontal and vertical axes of the coordinate system 801.

[0101] Each optical path 806a-b include a waveplate or other polarization rotating component. For example, the first optical path 806a includes waveplate 808a and the second optical path 806b includes waveplate 808b. The waveplates 808a-b are operable to align the polarizations of incoming photons between the first optical path 806a and second optical path 806b.

[0102] A cavity housing 810 receives photons of an ensemble of photons from the first optical path 806a and the second optical path 806b, after the polarization states of photons propagating through each pass is rotated to be indistinguishable between paths. As indicated by a square outline 812 illustrated on an entrance surface of the cavity housing 810, the entrance point of the first optical path 806a is offset in both the x and z directions in relation to the entrance point of the second optical path 806b.

[0103] The second implementation 850, which is an internal cutaway of the first implementation 800, depicts the sparse detector array 852 disposed within a cavity housing 860. Similar to the first implementation 800, the second implementation 850 depicts an optical input 852, polarizing beam splitter 854, and entrance points 858a-b to the cavity that are offset in both the x and y directions with respect to the coordinate system 801.

[0104] The sparse detector array 852 generates output detection events (e.g., voltage signals in response to a detection of a single photon received through the entrance point 858a or the entrance point 858b to the cavity housing 860). In some implementations, the sparse detector array 852 includes a semi-transparent scintillating screen together with optical detectors capable of pinpointing locations of any scintillations that occur. The configuration allows for a joint probability of x and z arrival positions to be computed, which in turn allows for conditional probabilities P(x|z), the probability of x given z, and P(z|x), the probability of z given x, to be computed. Since in this arrangement, the z direction corresponds to depth and is correlated with time-of-flight within the cavity housing 860, the observation of joint statistics of x-z probabilities enables DEIDO.

[0105] In some implementations, the sparse detector array 852 includes a single detector. Insome implementations, the sparse detector array 852 includes multiple detectors arranged in an ordered array. In a general sense, detectors of the sparse detector array 852 can be positioned in any geometric pattern. In some cases, a particular geometry of detectors of the sparse detector 22   Attorney Docket No.59456-0002WO1 array 852 may be advantageous over another depending on the position of the nodes of the interference pattern within the cavity housing 860.

[0106] FIG.9 is an example system 900 for discriminating between two ensembles of quantum systems described by an identical density operator as it relates to an implementation of a standard quantum communication protocol. The system 900 depicts a configuration that generates, transforms, and measures ensembles of quantum systems (e.g., an ensemble of photons).

[0107] The system 900 includes a source of entangled quantum systems 902. Entangled quantum systems are two-particle systems (e.g., two photons), in which the particles have correlated state values in addition to being described by a superposition of state values. For example, an entangled quantum system embodied as a polarization photonic state includes two photons. Each of the two photons are described by a superposition of two basis polarization states (e.g., each photon can be described as having a polarization state of a superposition of horizontal and vertical polarizations), and the polarization state of each photon of the two photons is correlated with the other (e.g., if a first photon is measured to have horizontal polarization, a second photon of the entangled pair of photons will be measured to have vertical polarization with certainty, despite the second photon being in a superposition of horizontal and vertical polarizations).

[0108] In some implementations, the source of entangled quantum systems 902 generates polarization entangled photons. In some implementations, the source of polarization entangled photons includes a non-linear crystal that exhibits spontaneous parametric down conversion in response to an excitation of the non-linear crystal by a laser. Photons within the exciting laser spontaneously decompose into a pair of lower-energy photon pairs while conserving energy and momentum, and with polarization states dependent on a particular orientation of the non-linear crystal. The non-linear crystal of the source of entangled quantum systems 902 can be configured to generate pairs of polarization entangled photons along pre-determined axes.

[0109] The source of entangled quantum systems 902 emits one quantum system of the pair of quantum systems along a first path 904a to be received by a first measurement device 906. The source of entangled quantum systems 902 emits the other quantum system of the pair of quantum systems along a second path 904b to be received by a second measurement device 908.

[0110] The first measurement device 906 is operable to evaluate an entangled degree of freedom of the incoming photon from the pair of entangled photons generated by the source of entangled 23   Attorney Docket No.59456-0002WO1 quantum systems 902. For example, the first measurement device 906 can include a polarizer 910, one or more waveplates, and a detection screen 912. According to standard quantum theory, a measurement of a photonic state of an ensemble of incoming photons at the first measurement device 906 (e.g., a polarization measurement), instantly projects a counterpart ensemble of states propagating along the second path 904 in a particular ensemble state, depending on the configuration of the first measurement device 906.

[0111] The second measurement device 908 can include a DEIDO-capable device, similar to the devices described in relation to FIGS.3-8. The second measurement device 908 can distinguish between ensembles of quantum systems received by the source of entangled quantum systems 902 by evaluating statistical properties of the incoming ensemble, as described in detail in relation to FIGS.3-8. By distinguishing between ensembles of quantum systems that are represented in different polarization bases but described by identical density operators, an output of the second measurement device 908 is indicative of a configuration of the first measurement device 906. Because the output of the second measurement device 908, which is a DEIDO- enabled device, is indicative of an orientation of the first measurement device, the particular quantum communication protocol does not require a classical channel between the first measurement device 906 and the second measurement device 908 to communicate the particular orientation of the first measurement device 906 (i.e., no classical channel is required to communicate a measurement basis used at the first measurement device 906).

[0112] In some implementations, the first path 904a and the second path 904b are adjusted such that photons that belong to a particular pair of entangled photons from the source of entangled quantum systems 902 arrive at the respective measurement devices at the same time.

[0113] FIG.10 is an example process 1000 for performing quantum computations that includes discriminating between two quantum ensembles described by an identical density operator. The example process 1000 includes a (i) state preparation stage 1002, (ii) a mid-circuit measurement stage 1004, (iii) a state manipulation stage 1006, and (iv) a terminal measurement stage 1008.

[0114] The process 1000 can be implemented by a quantum computer that process multiple inputquantum systems 1010 prepared in respective quantum states ^^^ … ^^^. The system can perform avariety of operations on each quantum system of theinput quantum system 1010 that include entanglement generation, measurement, and state transformation. The process 1000 includes a first DEIDO measurement 1012 during the mid-circuit measurement stage 1004 and a 24   Attorney Docket No.59456-0002WO1 second DEIDO measurement 1014 during the terminal measurement stage 1008. In addition, the process 1000 includes multiple standard quantum computing measurements. DEIDO-enabled measurement devices within the quantum computing system allow for new forms of ensemble- level state tomography to be employed by the quantum computing system.

[0115] In addition to the embodiments described above, the following embodiments are alsoinnovative:

[0116] Embodiment 1 is a method for distinguishing ensembles of quantum systems, the method comprising:

[0117] receiving, at a measurement apparatus, a first ensemble of quantum systems characterized by a first density operator, wherein each quantum system of the first ensemble of quantum systems is prepared in one state of a first plurality of quantum states with an associated probability;

[0118] transforming the state of each quantum system of the first ensemble of quantum systems, wherein each transformed state is described by an input mode correlated with a particular state of the first plurality of quantum states, the input mode suitable for analysis by a non-quadratic measurement;

[0119] receiving each quantum system of the first ensemble of quantum systems described by the respective transformed state at a detection device, the detection device operable to detect a non-quadratic observable of each received quantum system;

[0120] determining, by the detection device, a first measurement outcome, wherein the first measurement outcome is an aggregation of detection events of the first ensemble of quantum systems over a first time interval;

[0121] receiving, at the measurement apparatus, a second ensemble of quantum systems characterized by a second density operator, wherein each quantum system of the second ensemble of quantum systems is prepared in one state of a second plurality of quantum states with an associated probability, wherein (i) the second density operator is the same as the first density operator and (ii) the second plurality of quantum states is different from the first plurality of quantum states;

[0122] transforming the state of each quantum system of the second ensemble of quantum systems, wherein each transformed state is described by a spatial mode correlated with a particular state of the second plurality of quantum states; 25   Attorney Docket No.59456-0002WO1

[0123] receiving each quantum system of the second ensemble of quantum systems described by the respective transformed state at the detection device; and

[0124] determining, by the detection device, a second measurement outcome, wherein the second measurement outcome is an aggregation of detection events of the second ensemble of quantum systems over a second time interval, wherein the second measurement outcome is different from the first measurement outcome.

[0125] Embodiment 2 is the method of embodiment 1, wherein the first plurality of quantum states comprises (i) a first basis state and (ii) a second basis state, and the second plurality of quantum states comprises (i) a normalized in-phase superposition of the first basis state and the second basis state and (ii) a normalized out-of-phase superposition of the first basis state and the second basis state.

[0126] Embodiment 3 is the method of any of embodiments 1-2, wherein the non-quadraticobservable comprises at least one of a total detection probability, a position of detection, or a time of detection.

[0127] Embodiment 4 is the method of any of embodiments 1-3, wherein the detection device isoperable to detect the non-quadratic observable of a transformed quantum state by measuring a quadratic observable of the transformed quantum state that is correlated with the non-quadratic observable.

[0128] Embodiment 5 is the method of any of embodiments 1-4, wherein the non-quadratic observable corresponds to a temporal measurement.

[0129] Embodiment 6 is the method of any of embodiments 1-5, wherein the detection devices comprises one of a spatially-sensitive detection device, wherein a measurement outcome is indicative of a measurement position of a quantum system, the measurement position correlated with a temporal duration of a quantum process.

[0130] Embodiment 7 is the method of any of embodiments 1-6, further comprisingtransforming the state of each quantum system of the first ensemble of quantum systems to erase at least one distinguishing attribute of a quantum system of the first ensemble of quantum systems.

[0131] Embodiment 8 is the method of any of embodiments 1-7, further comprising transforming the state of each quantum system of the second ensemble of quantum systems to 26   Attorney Docket No.59456-0002WO1 erase at least one distinguishing attribute of a quantum system of the second ensemble of quantum systems.

[0132] Embodiment 9 is the method of any of embodiments 1-8, wherein the detection device comprises a point detector, wherein the point detector is operable to aggregate detection events over a time interval.

[0133] Embodiment 10 is the method of any of embodiments 1-9, wherein the measurement apparatus comprises one or more recombination elements, wherein the recombination elements are operable to spatially overlap quantum systems that propagate over distinct spatial modes.

[0134] Embodiment 11 is the method of any of embodiments 1-10, wherein the aggregation of detection events comprises an analysis of a histogram, wherein the histogram comprises a plurality of time-bins, each time-bin associated with a number of detection events and a space- time coordinate.

[0135] Embodiment 12 is a system for distinguishing ensembles of quantum systems, the system comprising:

[0136] a state transformer, wherein the state transformer is operable to (i) receive quantumsystems of a first ensemble of quantum systems and (ii) split quantum systems of the first ensemble of quantum systems into two propagation paths depending on a quantum state of each respective quantum system, and (iii) transform a quantum state of each quantum system such that a quantum system of a first propagation path can interfere with a quantum system of a second propagation path;

[0137] a detection system, wherein the detection system comprises:

[0138] a propagation medium, wherein the propagation medium is operable to receive quantum systems from the recombination element; and

[0139] a non-quadratic detection element, wherein the non-quadratic detection element is operable to analyze a temporal characteristic of the ensemble of quantum systems.

[0140] Embodiment 13 is the system of embodiment 12, wherein the detection system further comprises:

[0141] one or more recombination elements, wherein the recombination elements are operable tooverlap the quantum system of the first propagation path with the quantum system of the second propagation path. 27   Attorney Docket No.59456-0002WO1

[0142] Embodiment 14 is the system of any of embodiments 12-13, wherein the temporal characteristic of the ensemble of quantum systems is analyzed by measuring a quadratic observable of the ensemble of quantum systems, wherein the temporal characteristic is correlated with the quadratic observable.

[0143] Embodiment 15 is the system of any of embodiments 12-14, wherein the non-quadraticdetection element comprises one or more detection screens.

[0144] Embodiment 16 is the system of any of embodiments 12-15, wherein the non-quadratic detection element comprises one or more point detectors.

[0145] Embodiment 17 is the system of any of embodiments 12-16, the system further comprising:

[0146] an input terminal, wherein the input terminal is operable to receive an ensemble of quantum systems from at least one external system, the external system generating and transmitting the ensemble of quantum systems to the input terminal, the input terminal operable to transmit the received ensemble of quantum systems to the state transformer.

[0147] Embodiment 18 is the system of any of embodiments 12-17, wherein the quantumsystems of the first ensemble are photons, wherein the quantum systems of the second ensemble are photons, wherein the state transformer comprises a polarization beam splitter operable to split the photons of each ensemble into two propagation paths depending on a polarization state of each respective photon.

[0148] Embodiment 19 is the system of any of embodiments 12-18, wherein the quantum system of the first propagation path can interfere with the quantum system of the second propagation path by removing at least one distinguishing feature of the quantum systems propagating along each path.

[0149] Particular embodiments of the subject matter have been described. Other embodiments are within the scope of the following claims. For example, the actions recited in the claims can be performed in a different order and still achieve desirable results. As one example, the processes depicted in the accompanying figures do not necessarily require the particular order shown, or sequential order, to achieve desirable results. In certain implementations, multitasking and parallel processing may be advantageous. 28

Claims

Attorney Docket No.59456-0002WO1 WHAT IS CLAIMED IS 1. A method for distinguishing ensembles of quantum systems, the method comprising: receiving, at a measurement apparatus, a first ensemble of quantum systems characterized by a first density operator, wherein each quantum system of the first ensemble of quantum systems is prepared in one state of a first plurality of quantum states with an associated probability; transforming the state of each quantum system of the first ensemble of quantum systems, wherein each transformed state is described by an input mode correlated with a particular state of the first plurality of quantum states, the input mode suitable for analysis by a non-quadratic measurement; receiving each quantum system of the first ensemble of quantum systems described by the respective transformed state at a detection device, the detection device operable to detect a non-quadratic observable of each received quantum system; determining, by the detection device, a first measurement outcome, wherein the first measurement outcome is an aggregation of detection events of the first ensemble of quantum systems over a first time interval; receiving, at the measurement apparatus, a second ensemble of quantum systems characterized by a second density operator, wherein each quantum system of the second ensemble of quantum systems is prepared in one state of a second plurality of quantum states with an associated probability, wherein (i) the second density operator is the same as the first density operator and (ii) the second plurality of quantum states is different from the first plurality of quantum states; transforming the state of each quantum system of the second ensemble of quantum systems, wherein each transformed state is described by a spatial mode correlated with a particular state of the second plurality of quantum states; receiving each quantum system of the second ensemble of quantum systems described by the respective transformed state at the detection device; and determining, by the detection device, a second measurement outcome, wherein the second measurement outcome is an aggregation of detection events of the second ensemble of 29   Attorney Docket No.59456-0002WO1 quantum systems over a second time interval, wherein the second measurement outcome is different from the first measurement outcome.

2. The method of claim 1, wherein the first plurality of quantum states comprises (i) a first basis state and (ii) a second basis state, and the second plurality of quantum states comprises (i) a normalized in-phase superposition of the first basis state and the second basis state and (ii) a normalized out-of-phase superposition of the first basis state and the second basis state.

3. The method of claim 1, wherein the non-quadratic observable comprises at least one of a total detection probability, a position of detection, or a time of detection.

4. The method of claim 1, wherein the detection device is operable to detect the non- quadratic observable of a transformed quantum state by measuring a quadratic observable of the transformed quantum state that is correlated with the non-quadratic observable.

5. The method of claim 4, wherein the non-quadratic observable corresponds to a temporal measurement.

6. The method of claim 1, wherein the detection devices comprises one of a spatially- sensitive detection device, wherein a measurement outcome is indicative of a measurement position of a quantum system, the measurement position correlated with a temporal duration of a quantum process.

7. The method of claim 1, further comprising transforming the state of each quantum system of the first ensemble of quantum systems to erase at least one distinguishing attribute of a quantum system of the first ensemble of quantum systems.

8. The method of claim 1, further comprising transforming the state of each quantum system of the second ensemble of quantum systems to erase at least one distinguishing attribute of a quantum system of the second ensemble of quantum systems. 30   Attorney Docket No.59456-0002WO1 9. The method of claim 1, wherein the detection device comprises a point detector, wherein the point detector is operable to aggregate detection events over a time interval.

10. The method of claim 1, wherein the measurement apparatus comprises one or more recombination elements, wherein the recombination elements are operable to spatially overlap quantum systems that propagate over distinct spatial modes.

11. The method of claim 1, wherein the aggregation of detection events comprises an analysis of a histogram, wherein the histogram comprises a plurality of time-bins, each time-bin associated with a number of detection events and a space-time coordinate.

12. A system for distinguishing ensembles of quantum systems, the system comprising: a state transformer, wherein the state transformer is operable to (i) receive quantum systems of a first ensemble of quantum systems and (ii) split quantum systems of the first ensemble of quantum systems into two propagation paths depending on a quantum state of each respective quantum system, and (iii) transform a quantum state of each quantum system such that a quantum system of a first propagation path can interfere with a quantum system of a second propagation path; a detection system, wherein the detection system comprises:  a propagation medium, wherein the propagation medium is operable to receive quantum systems from the recombination element; and a non-quadratic detection element, wherein the non-quadratic detection element is operable to analyze a temporal characteristic of the ensemble of quantum systems.

13. The system of claim 11, wherein the detection system further comprises: one or more recombination elements, wherein the recombination elements are operable to overlap the quantum system of the first propagation path with the quantum system of the second propagation path.

14. The system of claim 11, wherein the temporal characteristic of the ensemble of quantum systems is analyzed by measuring a quadratic observable of the ensemble of quantum systems, 31   Attorney Docket No.59456-0002WO1 wherein the temporal characteristic is correlated with the quadratic observable.

15. The system of claim 11, wherein the non-quadratic detection element comprises one or more detection screens.

16. The system of claim 11, wherein the non-quadratic detection element comprises one or more point detectors.

17. The system of claim 11, the system further comprising: an input terminal, wherein the input terminal is operable to receive an ensemble of quantum systems from at least one external system, the external system generating and transmitting the ensemble of quantum systems to the input terminal, the input terminal operable to transmit the received ensemble of quantum systems to the state transformer.

18. The system of claim 11, wherein the quantum systems of the first ensemble are photons, wherein the quantum systems of the second ensemble are photons, wherein the state transformer comprises a polarization beam splitter operable to split the photons of each ensemble into two propagation paths depending on a polarization state of each respective photon.

19. The system of claim 11, wherein the quantum system of the first propagation path can interfere with the quantum system of the second propagation path by removing at least one distinguishing feature of the quantum systems propagating along each path. 32

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