本文研究线性无关纯量子态的无模糊判别问题。我们表明，设计的最佳测量，最大限度地减少了不确定的结果的概率，可以制定为一个半定规划问题。基于这个公式，我们发展了一组最佳量子测量的充分必要条件。我们表明，最佳的测量可以非常有效地计算在多项式时间内利用许多著名的算法来解决半定方案，这是保证收敛到全局最优。使用最优性的一般条件，我们得到必要和充分条件，使测量结果在一个不确定的结果为每一个量子态的概率相等是最佳的。我们把这种测量称为等概率测量（Equal Probability Measurement，简称EMT）。然后，我们表明，对于任何状态集，状态的先验概率可以选择，使得最优的。最后，我们考虑具有强对称性和等先验概率的状态集，对于这些状态集，最优解是最优的。我们首先考虑定义在一组酉矩阵上的几何一致状态集，它们由单个生成向量生成。然后，我们考虑复合几何均匀状态集所产生的一组酉矩阵使用多个生成向量，其中的生成向量满足一定的（加权）范数约束。
英文摘要：
In this paper we consider the problem of unambiguous discrimination between a set of linearly independent pure quantum states. We show that the design of the optimal measurement that minimizes the probability of an inconclusive result can be formulated as a semidefinite programming problem. Based on this formulation, we develop a set of necessary and sufficient conditions for an optimal quantum measurement. We show that the optimal measurement can be computed very efficiently in polynomial time by exploiting the many well-known algorithms for solving semidefinite programs, which are guaranteed to converge to the global optimum.
Using the general conditions for optimality, we derive necessary and sufficient conditions so that the measurement that results in an equal probability of an inconclusive result for each one of the quantum states is optimal. We refer to this measurement as the equal-probability measurement (EPM). We then show that for any state set, the prior probabilities of the states can be chosen such that the EPM is optimal.
Finally, we consider state sets with strong symmetry properties and equal prior probabilities for which the EPM is optimal. We first consider geometrically uniform state sets that are defined over a group of unitary matrices and are generated by a single generating vector. We then consider compound geometrically uniform state sets which are generated by a group of unitary matrices using multiple generating vectors, where the generating vectors satisfy a certain (weighted) norm constraint.
Roa et al. showed that quantum state discrimination between two nonorthogonal quantum states does not require quantum entanglement but quantum dissonance only. We find that quantum coherence can also be utilized for unambiguous quantum state discrimination. We present a protocol and quantify the required coherence for this task. We discuss the optimal unambiguous quantum state discrimination strategy in some cases. In particular, our work illustrates an avenue to find the optimal strategy for discriminating two nonorthogonal quantum states by measuring quantum coherence.
The paper studies unambiguous discrimination of Fermionic states through local operations and classical communication (LOCC). In the task of unambiguous discrimination, no error is tolerated but an inconclusive result is allowed. We show that contrary to the quantum case, it is not always possible to distinguish two Fermionic states through LOCC unambiguously with the same success probability as if global measurements were allowed. Furthermore, we prove that we can overcome such a limit through an ancillary system made of two Fermionic modes, independently of the dimension of the system, prepa…
Quantum hypothesis testing is an important tool for quantum information processing. Two main strategies have been widely adopted: in a minimum error discrimination strategy, the average error probability is minimized; while in an unambiguous discrimination strategy, an inconclusive decision (abstention) is allowed to vanish any possibility of errors when a conclusive result is obtained. In both scenarios, the testing between quantum states are relatively well-understood, for example, the ultimate limits of the performance are established decades ago; however, the testing between quantum channe…
We present an in-depth study of the problem of multiple-shot discrimination of von Neumann measurements in finite-dimensional Hilbert spaces. Specifically, we consider two scenarios: minimum error and unambiguous discrimination. In the case of minimum error discrimination, we focus on discrimination of measurements with the assistance of entanglement. We provide an alternative proof of the fact that all pairs of distinct von Neumann measurements can be distinguished perfectly (i.e. with the unit success probability) using only a finite number of queries. Moreover, we analytically find the mini…
Nonlocality without entanglement(NLWE) is a nonlocal phenomenon that occurs in quantum state discrimination of multipartite separable states. In the discrimination of orthogonal separable states, the term NLWE is used when the quantum states cannot be discriminated perfectly by local operations and classical communication. In this case, the occurrence of NLWE is independent of nonzero prior probabilities of quantum states being prepared. Recently, it has been found that the occurrence of NLWE can depend on nonzero prior probabilities in minimum-error discrimination of nonorthogonal separable s…
Generalized quantum measurements identifying non-orthogonal states without ambiguity often play an indispensable role in various quantum applications. For such unambiguous state discrimination scenario, we have a finite probability of obtaining inconclusive results and minimizing the probability of inconclusive results is of particular importance. In this paper, we experimentally demonstrate an adaptive generalized measurement that can unambiguously discriminate the quaternary phase-shift-keying coherent states with a near-optimal performance. Our scheme is composed of displacement operations,…
We investigate the discrimination of pure-mixed (quantum filtering) and mixed-mixed states and compare their optimal success probability with the one for discriminating other pairs of pure states superposed by the vectors included in the mixed states. We prove that under the equal-fidelity condition, the pure-pure state discrimination scheme is superior to the pure-mixed (mixed-mixed) one. With respect to quantum filtering, the coherence exists only in one pure state and is detrimental to the state discrimination for lower dimensional systems; while it is the opposite for the mixed-mixed case …
