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Journal of Modern Optics

Unambiguous correct discrimination of real quantum states

Zhang Shi-jun, Wen‐Hai Zhang

Published May 21, 2023• 0 citations
Publisher

Abstract

We investigate the unambiguous correct discrimination strategy of a set of linearly independent equidistant real quantum states and three sets of three linearly independent states. The measurement operators include the success and correct operators, and can gain more information about the input states than by using the conventional unambiguous discrimination strategy. Five examples are given to demonstrate the efficiency of the unambiguous correct discrimination strategy.

DOI
10.1080/09500340.2023.2294435

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We investigate the unambiguous correct discrimination strategy of a set of linearly independent equidistant real quantum states and three sets of three linearly independent states. The measurement operators include the success and correct operators, and can gain more information about the input states than by using the conventional unambiguous discrimination strategy. Five examples are given to demonstrate the efficiency of the unambiguous correct discrimination strategy.
It is known that unambiguous discrimination among nonorthogonal but linearly independent quantum states is possible with a certain probability of success. Here, we consider a variant of that problem. Instead of discriminating among all of the different states, we shall only discriminate between two subsets of them. In particular, for the case of three nonorthogonal states, ${|{\ensuremath{\psi}}_{1}〉,|{\ensuremath{\psi}}_{2}〉,|{\ensuremath{\psi}}_{3}〉},$ we show that the optimal strategy to distinguish $|{\ensuremath{\psi}}_{1}〉$ from the set ${|{\ensuremath{\psi}}_{2}〉,|{\ensuremath{\psi}}_{3}〉}$ has a higher success rate than if we wish to discriminate among all three states. Somewhat surprisingly, for unambiguous discrimination the subsets need not be linearly independent. A fully analytical solution is presented, and we also show how to construct generalized interferometers (multiport) which provide an optical implementation of the optimal strategy.
Given nnn linearly independent pure states and their prior probabilities, we study the optimum unambiguous state discrimination problem. We derive the conditions for the optimum measurement strategy to achieve the maximum average success probability and establish two sets of equations that must be satisfied by the optimum solution in different situations. We also provide the detailed steps to find the optimum measurement strategy. The method and results we obtain are given a geometrical illustration with a numerical example. Furthermore, using these equations, we derive a formula which shows a clear analytical relation between the optimum solution and the nnn states to be discriminated. We also solve a generalized equal-probability measurement problem analytically. Finally, as another application of our result, the unambiguous discrimination problem of three pure states is studied in detail and analytical solutions are obtained for some interesting cases.
The quantum state discrimination problem is ubiquitous in quantum information and serves as the foundation for many quantum communication schemes proposed so far. This paper investigates the optimal discrimination of three symmetric quantum states with equal prior probabilities. To obtain completely accessible information about the input pure states, this strategy makes improvements to the two standard strategies of quantum state discrimination, namely the minimum error discrimination and unambiguous discrimination. The new discrimination strategy proposed in this paper can be applied to both linearly independent and dependent quantum states.
We address the problem of unambiguous discrimination between a given set of quantum operations. The necessary and sufficient condition for them to be unambiguously distinguishable is derived in the cases of single use and multiple uses respectively. For the latter case we explicitly construct the input states and corresponding measurements that accomplish the task. It is found that the introduction of entanglement can improve the discrimination. The advantage of our strategy over minimum-error discrimination is also discussed.
This paper presents a complete discrimination strategy of pure quantum states, by which the accessible information about the input pure states can be obtained completely. The complete discrimination is showed by taking the example of discriminating symmetric pure states. The measurement operators of complete discrimination can be represented by the multi-order correct operators, and the m-order relative average correct probability in the figure of merit is introduced. Since the rank of an inconclusive operator is one and an inconclusive operator provides the minimum relative average correct probability, no more information about the input state can be extracted. Our strategy involves two standard discrimination strategies, namely minimum-error discrimination and unambiguous discrimination, while overcoming their deficiencies. It is promising for complete discrimination and may find important applications in quantum information science.
Unambiguous discrimination among non-orthogonal but linearly independent quantum states is possible with a certain probability of success. Here we consider a variant of that problem; rather than discriminating among all of the different states, we shall discriminate between two subsets of them. In particular, for the case of 3 states, we find the optimal strategy to distinguish one from the other two. This procedure has a higher probability of success than distinguishing all 3 states. We show how to construct generalized interferometers that provide an implementation of the optimal strategy.