Heralded orthogonalisation of coherent states and their conversion to discrete-variable superpositions

Unambiguous quantum state discrimination is a strategy where the conclusive result can always be trusted. This strategy is very important, since it can be used for various quantum information protocols, including quantum key distribution. However, in the view of quantumness, it is not clear what is going on in performing unambiguous quantum state discrimination. To answer the question, we investigate coherence distribution when unambiguous discrimination is performed by generalized measurement. Specially, we study coherence distribution in three cases, which consist of unambiguous quantum state discrimination, sequential quantum state discrimination, and assisted optimal discrimination, which are considered to be a family of unambiguous quantum state discrimination. In this investigation, we show that the structure of generalized measurements performing various types of unambiguous quantum state discrimination can be understood in terms of coherence distribution. Our result is not limited to the discrimination of two pure quantum states, but it is extended to the discrimination of two mixed states.

Coherent states of the quantum electromagnetic field, the quantum description of ideal laser light, are prime candidates as information carriers for optical communications. A large body of literature exists on their quantum-limited estimation and discrimination. However, very little is known about the practical realizations of receivers for unambiguous state discrimination (USD) of coherent states. Here we fill this gap and outline a theory of USD with receivers that are allowed to employ: passive multimode linear optics, phase-space displacements, auxiliary vacuum modes, and on-off photon detection. Our results indicate that, in some regimes, these currently-available optical components are typically sufficient to achieve near-optimal unambiguous discrimination of multiple, multimode coherent states.

We prove that the ultimate channel capacity, the Holevo bound, for sending classical data on a quantum channel (the so-called classical-quantum, or cq channel) can be achieved for a pure-state cq channel by decoding codewords via a collective quantum measurement based on unambiguous state discrimination (USD). In cq communication theory, the channel decoder acts directly on the modulated codeword waveform in the quantum (viz., electromagnetic or optical) domain, and it is known that collective measurements on long codeword blocks are needed to attain the Holevo capacity, which is strictly larger than the Shannon capacity of the classical channel induced by any specific measurement choice on each channel use. The USD measurement based channel decoder we propose, can distinguish finite blocklength codeword quantum states unambiguously (i.e., an incorrect codeword is never chosen) provided one allows for a finite probability of obtaining an inconclusive (erasure) outcome. We show that the probability of the inconclusive outcome goes to zero for asymptotically long codewords whenever the code rate is below the Holevo bound. The USD channel decoder is an addition to a small list of other collective measurements known to achieve the Holevo capacity (such as, the square root measurement, the minimum probability of error measurement, the sequential decoding measurement, and the quantum successive cancellation decoder for the cq polar code). A structured optical receiver design is not known yet for any of these decoders. What makes the USD decoder special is that there is no classical analogue to truly unambiguous discrimination (say, of samples drawn from a set of probability distributions). Secondly, the erasures-only decoding of USD is likely to result in a better channel reliability function. Finally, the USD measurement seems more likely to lead naturally to a structured optical receiver design and implementation.

Unambiguous discrimination and exact cloning reduce the square-overlap between quantum states, exemplifying the more general type of procedure we term state separation. We obtain the maximum probability with which two equiprobable quantum states can be separated by an arbitrary degree, and find that the established bounds on the success probabilities for discrimination and cloning are special cases of this general bound. The latter also gives the maximum probability of successfully producing N exact copies of a quantum system whose state is chosen secretly from a known pair, given M initial realizations of the state, where N > M. We also discuss the relationship between this bound and that on unambiguous state discrimination.

Unambiguous discrimination of two squeezed states using probabilistic quantum cloning

Unambiguous (non-orthogonal) state discrimination (USD) has a fundamental importance in quantum information and quantum cryptography. Various aspects of two-state and multiple-state USD are studied here using singular value decomposition of the evolution operator that describes a given state discriminating system. In particular, we relate the minimal angle between states to the ratio of the minimal and maximal singular values. This is supported by a simple geometrical picture in two-state USD. Furthermore, by studying the singular vectors population we find that the minimal angle between input vectors in multiple-state USD is always larger than the minimal angle in two-state USD in the same system. As an example we study what pure states can be probabilistically transformed into maximally entangled pure states in a given system .

Unambiguous discrimination of mixed quantum states

Generalized quantum measurements implemented to allow for measurement outcomes termed inconclusive can perform perfect discrimination of non-orthogonal states, a task which is impossible using only measurements with definitive outcomes. Here we demonstrate such generalized quantum measurements for unambiguous discrimination of four non-orthogonal coherent states and obtain their quantum mechanical description, the positive-operator valued measure. For practical realizations of this positive-operator valued measure, where noise and realistic imperfections prevent perfect unambiguous discrimination, we show that our experimental implementation outperforms any ideal standard-quantum-limited measurement performing the same non-ideal unambiguous state discrimination task for coherent states with low mean photon numbers.

Unambiguous discrimination between nonorthogonal but linearly independent quantum states is a challenging problem in quantum information processing. In this study, using the connection between Lewenstein-Sanpera decomposition (LSD) and optimal unambiguous state discrimination (OPUSD), an analytic relation for the feasible region of $N$ linearly independent quantum states is presented in terms of inner product of reciprocal states. Then, for the real inner product of states, an exact analytic solution for the OPUSD problem involving an arbitrary number of pure linearly independent quantum states is presented using the Karush-Kuhn-Tucker convex optimization method. In another approach, an analytic relation for the feasible region for an arbitrary number of pure linearly independent quantum states is presented in terms of the inner product of states. To this end, the relevant semidefinite programming task is reduced to a linear programming (LP) one with a feasible region of polygon type which can be solved via simplex method. Moreover, using the obtained feasible region, an exact analytic solution to an OPUSD problem involving two and three pure linearly independent quantum states is provided.

In this report, we present a framework for implementing an arbitrary n-outcome generalized quantum measurement (POVM) on an m-qubit register as a sequence of two-outcome measurements requiring only single ancillary qubit. Our procedure offers a particular construction for the two-outcome partial measurements which can be composed into a full implementation of the measurement on any gate architecture. This implementation in general requires classical feedback; we present specific cases when this is not the case. We apply this framework on the unambiguous state discrimination and analyze possible strategies. In the simplest case, it gives the same construction as is known, if we opt for performing conclusiveness measurement first. However, it also offers possibility of performing measurement for one of the state outcomes first, leaving conclusiveness measurement for later. This shows flexibility of presented framework and opens possibilities for further optimization. We present discussion also on biased qubit case as well as general case of unambiguous quantum state discrimination in higher dimension.

Considering quantum detection, there are two different ways to make a measurement on signals set. One is minimum error discrimination and another is unambiguous states discrimination. In this work, we study the unambiguous state discrimination of coherent states. The quantum measurement model investigated is based on binary signals. All necessary positive-operator valued measurements are established to implement the quantum measurements in non-ambiguity way. The conclusive probability and inconclusive probability for the unambiguous discrimination of both on-off keying and binary-phase-shifting keying modulations are derived rigorously to show the measurement performance of proposed detection method theoretically.

Considering quantum detection, there are two different ways to make a measurement on signals set. One is minimum error discrimination and another is unambiguous states discrimination. In this work, we study the unambiguous state discrimination of coherent states. The quantum measurement model investigated is based on binary signals. All necessary positive-operator valued measurements are established to implement the quantum measurements in non-ambiguity way. The conclusive probability and inconclusive probability for the unambiguous discrimination of both on-off keying and binary-phase-shifting keying modulations are derived rigorously to show the measurement performance of proposed detection method theoretically.

We present the solution to the problem of optimally discriminating among quantum states, i.e., identifying the states with maximum probability of success when a certain fixed rate of inconclusive answers is allowed. By varying the inconclusive rate, the scheme optimally interpolates between unambiguous and minimum error discrimination, the two standard approaches to quantum state discrimination. We introduce a very general method that enables us to obtain the solution in a wide range of cases and give a complete characterization of the minimum discrimination error as a function of the rate of inconclusive answers. A critical value of this rate is identified that coincides with the minimum failure probability in the cases where unambiguous discrimination is possible and provides a natural generalization of it when states cannot be unambiguously discriminated. The method is illustrated on two explicit examples: discrimination of two pure states with arbitrary prior probabilities and discrimination of trine states.

Unambiguous discrimination and exact cloning reduce the square-overlap between quantum states, exemplifying the more general type of procedure we term state separation. We obtain the maximum probability with which two equiprobable quantum states can be separated by an arbitrary degree, and find that the established bounds on the success probabilities for discrimination and cloning are special cases of this general bound. The latter also gives the maximum probability of successfully producing N exact copies of a quantum system whose state is chosen secretly from a known pair, given M initial realizations of the state, where N > M. We also discuss the relationship between this bound and that on unambiguous state discrimination.

Quantum-enhanced measurement technologies can unambiguously discriminate coherent states with accuracy beyond the classical heterodyne measurement. However, typical quantum-enhanced measurement scheme is vulnerable to the thermal noise, which will change the photon counting statistics of the coherent state. This paper presents a threshold-switching strategy that can discriminate quadrature phase-shift-keying coherent states with performance surpassing the typical quantum-enhanced scheme. In our scheme, photon number resolving detectors are used to switch the value of the threshold, which can mitigate the influence of thermal noise and other imperfections. Simulation results show that our scheme unambiguously discriminates the signal states with higher correct probability and the same error ratio compared with the typical scheme. Besides, this scheme can reduce the error ratio simultaneously for thermal noise N ≤ 0.2. The paper demonstrations that quantum-enhanced measurement with the threshold-switching strategy can adapt to different thermal noises by switching the value of the threshold under situations of different thermal noises and signal states.