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AIP Advances

Complete discrimination of pure quantum states

Dawei Guo, Lan-Lan Li, Shouya Wang, Wen‐Hai Zhang

Published Jul 1, 2025• 0 citations
Publisher

Abstract

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.

DOI
10.1063/5.0271783

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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.
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 investigate the optimum strategy for state discrimination of two pure states when the probability of inconclusive answers is fixed. By varying a given rate of inconclusive probabilities, the strategy optimally interpolates between Unambiguous and Minimum-Error discrimination, the two standard approaches to quantum state discrimination. We derive the explicit expressions of all the probabilities using an ancillary system and introducing unitary transformations which act on the input states and produce the output measured on the ancilla. By exploring physically accessible transformation acting on the input states, we present an optical scheme to implement the state discrimination.
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.
We study an optimum measurement for quantum state discrimination, which maximizes the probability of correct results when the probability of inconclusive results is fixed at a given value. The measurement describes minimum-error discrimination if this value is zero, while under certain conditions it corresponds to optimized maximum-confidence discrimination, or to optimum unambiguous discrimination, respectively, when the fixed value reaches a definite minimum. Using operator conditions that determine the optimum measurement, we derive analytical solutions for the discrimination of two mixed qubit states, including the case of two pure states occurring with arbitrary prior probabilities, and for the discrimination of NNN symmetric states, both pure and mixed. We also consider a case where the given density operators resolve the identity operator, and we specify the optimality conditions for partially symmetric states. Moreover, we show that from the complete solution for arbitrary values of the fixed rate of inconclusive results one can always obtain the optimum measurement in another strategy where the error rate is fixed, and vice versa.
The discrimination of two mixed quantum states is a fundamental task in quantum state estimation and quantum information theory. In quantum state discrimination a quantum system is assumed to be in one of two possible – in general mixed – non-orthogonal quantum states. The discrimination then consists of a measurement strategy that allows to decide in which state the system was before the measurement. In unambiguous state discrimination the aim is to make this decision without errors, but it is allowed to give an inconclusive answer. Especially interesting are measurement strategies that minimize the probability of an inconclusive answer. A starting point for the analysis of this optimization problem was a result by Eldar et al. [Phys. Rev. A 69, 062318 (2004)], which provides non-operational necessary and sufficient conditions for a given measurement strategy to be optimal. These conditions are reconsidered and simplified in such a way that they become operational. The simplified conditions are the basis for further central results: It is shown that the optimal measurement strategy is unique, a statement that is e.g. of importance for the complexity analysis of optimal measurement devices. The optimal measurement strategy is derived for the case, where one of the possible input states has at most rank two, which was an open problem for many years. Furthermore, using the optimality criterion it is shown that there always exists a threshold probability for each state, such that below this probability it is optimal to exclude this state from the discrimination strategy. If the two states subject to discrimination can be brought to a diagonal structure with (2×2)-dimensional blocks, then the unambiguous discrimination of these states can be reduced to the unambiguous discrimination of pure states. A criterion is presented that allows to identify the presence of such a structure for two self-adjoint operators. This criterion consists of the evaluation of three commutators and allows an explicit construction of the (2×2)-dimensional blocks. As an important application of unambiguous state discrimination, unambiguous state comparison, i.e., the question whether two states are identical or not, is generalized and optimal measurements for this problem are constructed. If for a certain family of states, a physical device maps the input state to an output state, such that a second device can be built that yields back the original input state, such a map is called reversible on this family. With respect to state discrimination, such reversible maps are particularly interesting, if the output states are pure. A complete characterization of all families that allow such a reversible and purifying map is provided. If the states are mapped to pure states, but the map itself is not reversible, upper and lower bounds are analyzed for the “deviation from perfect faithfulness”, a quantity which measures the deviation from a reversible mapping.
We consider the problem of discrimination between a set of quantum states with the maximum probability of correct detection for a fixed rate of inconclusive results. We show that this type of measurement can be decomposed into two steps. The first step is the discrimination of whether the measurement result is inconclusive. The second step, which is performed in the case in which a conclusive result was obtained, is minimum error discrimination of output states of the first step. Using this result, we derive a closed-form analytical expression for an optimal measurement for both a geometrically uniform pure-state set and a set of binary doubly symmetric mixed states for any probability of an inconclusive result.