Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Physical Review A

Martingale dynamics in quantum discrimination

Michael R. Frey

Published Jul 14, 2008• 0 citations
Publisher

Abstract

Conclusive discrimination of quantum states is expressible as a sequence of optimal weak measurements. The sequence of probabilities of successful discrimination realized in this process is shown to be a martingale, a model of a fair game. For discriminating between two pure states, this martingale is constant; for at least one strictly mixed state or more than two pure states, it is constant just on average. In all these latter cases, therefore, the realized discrimination probability conditioned on prior measurements may be greater than the optimal success probability of the original ensemble. So in these cases superoptimal discrimination is conditionally achievable. This phenomenon is demonstrated by examples.

DOI
10.1103/physreva.78.012324

Related papers

A quantum state discrimination martingale

Michael R. Frey

A martingale appears in conclusive Bayesian discrimination of two quantum states subjected to a sequence of optimal weak measurements. This martingale is solely a function of the two system states and their respective probabilities at the time of each measurement, and it directly determines the evolving probability of discrimination error. Also, it is constant if and only if the states are pure. For strictly mixed quantum states the martingle is constant just on average, with the consequence that, with some probability, the realized discrimination error probability conditioned on prior measurements may be less (better) than the optimal Helstrom error probability. So for mixed states conditionally superoptimal discrimination is possible. This phenomenon is demonstrated numerically in an example.

Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE•Apr 27, 2007•0 citations

Quantum-state comparison and discrimination

A. Hayashi, T. Hashimoto, M. Horibe

We investigate the performance of discrimination strategy in the comparison task of known quantum states. In the discrimination strategy, one infers whether or not two quantum systems are in the same state on the basis of the outcomes of separate discrimination measurements on each system. In some cases with more than two possible states, the optimal strategy in minimum-error comparison is that one should infer the two systems are in different states without any measurement, implying that the discrimination strategy performs worse than the trivial ``no-measurement'' strategy. We present a sufficient condition for this phenomenon to happen. For two pure states with equal prior probabilities, we determine the optimal comparison success probability with an error margin, which interpolates the minimum-error and unambiguous comparison. We find that the discrimination strategy is not optimal except for the minimum-error case.

arXiv:1803.09030•Physical Review A•May 21, 2018•3 citations

Sequential Discrimination of Mixed Quantum States

Jinhua Zhang, Fulin Zhang, Yan Gao +2 more

Classical mixtures of quantum states often give rise to decoherence and are generally considered detrimental to quantum processing. However, in the framework of sequential measurement, such mixtures can be beneficial for state discrimination. We investigate the sequential discrimination of mixed states and compare the results with those of pure states under the condition of equal fidelity. It is found that the successful probability of the mixed-state protocol is superior to the pure one under the equal-fidelity condition. It is shown that the difference between the sequential discrimination of pure and mixed states is more reliable under the equal-fidelity condition than under single-shot discrimination, and this difference increases with the mixability of the initial mixed states. For scenarios in which classical communication is allowed, the optimal successful probability of pure-state discriminations is larger than that for mixed states on the contrary. We also show that the classical mixture of basic vectors from quantum decoherence has a subtle impact on the communication channel induced by the coincidence of the maximal mutual information and optimal successful probability of sequential discrimination for pure states.

Entropy•Feb 27, 2025•0 citations

Optimal state discrimination with a fixed rate of inconclusive results: Analytical solutions and relation to state discrimination with a fixed error rate

Ulrike Herzog

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 $N$ 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.

arXiv:1206.4412•Physical Review A•Sep 14, 2012•45 citations

Distinguishing mixed quantum states: Minimum-error discrimination versus optimum unambiguous discrimination

Ulrike Herzog, János A. Bergou

We consider two different optimized measurement strategies for the discrimination of nonorthogonal quantum states. The first is ambiguous discrimination with a minimum probability of inferring an erroneous result, and the second is unambiguous, i.e., error-free, discrimination with a minimum probability of getting an inconclusive outcome, where the measurement fails to give a definite answer. For distinguishing between two mixed quantum states, we investigate the relation between the minimum-error probability achievable in ambiguous discrimination, and the minimum failure probability that can be reached in unambiguous discrimination of the same two states. The latter turns out to be at least twice as large as the former for any two given states. As an example, we treat the case where the state of the quantum system is known to be, with arbitrary prior probability, either a given pure state, or a uniform statistical mixture of any number of mutually orthogonal states. For this case we derive an analytical result for the minimum probability of error and perform a quantitative comparison with the minimum failure probability.

arXiv:quant-ph/0403124•Physical Review A•Aug 5, 2004•64 citations

Optimal unambiguous filtering of a quantum state: An instance in mixed state discrimination

János A. Bergou, Ulrike Herzog, Mark S. Hillery

Deterministic discrimination of nonorthogonal states is forbidden by quantum measurement theory. However, if we do not want to succeed all the time, i.e., allow for inconclusive outcomes to occur, then unambiguous discrimination becomes possible with a certain probability of success. A variant of the problem is set discrimination: the states are grouped in sets and we want to determine to which particular set a given pure input state belongs. We consider here the simplest case, termed quantum state filtering, when the $N$ given nonorthogonal states ${\ensuremath{\mid}{\ensuremath{\psi}}_{1}⟩,\dots{},\ensuremath{\mid}{\ensuremath{\psi}}_{N}⟩}$ are divided into two sets and the first set consists of one state only while the second consists of all of the remaining states. We present the derivation of the optimal measurement strategy, in terms of a generalized measurement (positive-operator-valued measure), to distinguish $\ensuremath{\mid}{\ensuremath{\psi}}_{1}⟩$ from the set ${\ensuremath{\mid}{\ensuremath{\psi}}_{2}⟩,\dots{},\ensuremath{\mid}{\ensuremath{\psi}}_{N}⟩}$ and the corresponding optimal success and failure probabilities. The results, but not the complete derivation, were presented previously [Phys. Rev. Lett. 90, 25901 (2003)] as the emphasis there was on appplication of the results to probabilistic quantum algorithms. We also show that the problem is equivalent to the discrimination of a pure state and an arbitrary mixed state.

arXiv:quant-ph/0504138•Physical Review A•Apr 11, 2005•29 citations
Conclusive discrimination of quantum states is expressible as a sequence of optimal weak measurements. The sequence of probabilities of successful discrimination realized in this process is shown to be a martingale, a model of a fair game. For discriminating between two pure states, this martingale is constant; for at least one strictly mixed state or more than two pure states, it is constant just on average. In all these latter cases, therefore, the realized discrimination probability conditioned on prior measurements may be greater than the optimal success probability of the original ensemble. So in these cases superoptimal discrimination is conditionally achievable. This phenomenon is demonstrated by examples.
A martingale appears in conclusive Bayesian discrimination of two quantum states subjected to a sequence of optimal weak measurements. This martingale is solely a function of the two system states and their respective probabilities at the time of each measurement, and it directly determines the evolving probability of discrimination error. Also, it is constant if and only if the states are pure. For strictly mixed quantum states the martingle is constant just on average, with the consequence that, with some probability, the realized discrimination error probability conditioned on prior measurements may be less (better) than the optimal Helstrom error probability. So for mixed states conditionally superoptimal discrimination is possible. This phenomenon is demonstrated numerically in an example.
We investigate the performance of discrimination strategy in the comparison task of known quantum states. In the discrimination strategy, one infers whether or not two quantum systems are in the same state on the basis of the outcomes of separate discrimination measurements on each system. In some cases with more than two possible states, the optimal strategy in minimum-error comparison is that one should infer the two systems are in different states without any measurement, implying that the discrimination strategy performs worse than the trivial ``no-measurement'' strategy. We present a sufficient condition for this phenomenon to happen. For two pure states with equal prior probabilities, we determine the optimal comparison success probability with an error margin, which interpolates the minimum-error and unambiguous comparison. We find that the discrimination strategy is not optimal except for the minimum-error case.
Classical mixtures of quantum states often give rise to decoherence and are generally considered detrimental to quantum processing. However, in the framework of sequential measurement, such mixtures can be beneficial for state discrimination. We investigate the sequential discrimination of mixed states and compare the results with those of pure states under the condition of equal fidelity. It is found that the successful probability of the mixed-state protocol is superior to the pure one under the equal-fidelity condition. It is shown that the difference between the sequential discrimination of pure and mixed states is more reliable under the equal-fidelity condition than under single-shot discrimination, and this difference increases with the mixability of the initial mixed states. For scenarios in which classical communication is allowed, the optimal successful probability of pure-state discriminations is larger than that for mixed states on the contrary. We also show that the classical mixture of basic vectors from quantum decoherence has a subtle impact on the communication channel induced by the coincidence of the maximal mutual information and optimal successful probability of sequential discrimination for pure 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.
We consider two different optimized measurement strategies for the discrimination of nonorthogonal quantum states. The first is ambiguous discrimination with a minimum probability of inferring an erroneous result, and the second is unambiguous, i.e., error-free, discrimination with a minimum probability of getting an inconclusive outcome, where the measurement fails to give a definite answer. For distinguishing between two mixed quantum states, we investigate the relation between the minimum-error probability achievable in ambiguous discrimination, and the minimum failure probability that can be reached in unambiguous discrimination of the same two states. The latter turns out to be at least twice as large as the former for any two given states. As an example, we treat the case where the state of the quantum system is known to be, with arbitrary prior probability, either a given pure state, or a uniform statistical mixture of any number of mutually orthogonal states. For this case we derive an analytical result for the minimum probability of error and perform a quantitative comparison with the minimum failure probability.
Deterministic discrimination of nonorthogonal states is forbidden by quantum measurement theory. However, if we do not want to succeed all the time, i.e., allow for inconclusive outcomes to occur, then unambiguous discrimination becomes possible with a certain probability of success. A variant of the problem is set discrimination: the states are grouped in sets and we want to determine to which particular set a given pure input state belongs. We consider here the simplest case, termed quantum state filtering, when the NNN given nonorthogonal states ${\ensuremath{\mid}{\ensuremath{\psi}}_{1}⟩,\dots{},\ensuremath{\mid}{\ensuremath{\psi}}_{N}⟩}$ are divided into two sets and the first set consists of one state only while the second consists of all of the remaining states. We present the derivation of the optimal measurement strategy, in terms of a generalized measurement (positive-operator-valued measure), to distinguish $\ensuremath{\mid}{\ensuremath{\psi}}_{1}⟩$ from the set ${\ensuremath{\mid}{\ensuremath{\psi}}_{2}⟩,\dots{},\ensuremath{\mid}{\ensuremath{\psi}}_{N}⟩}$ and the corresponding optimal success and failure probabilities. The results, but not the complete derivation, were presented previously [Phys. Rev. Lett. 90, 25901 (2003)] as the emphasis there was on appplication of the results to probabilistic quantum algorithms. We also show that the problem is equivalent to the discrimination of a pure state and an arbitrary mixed state.