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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE

A quantum state discrimination martingale

Michael R. Frey

Published Apr 27, 2007• 0 citations
Publisher

Abstract

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.

DOI
10.1117/12.717679

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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.
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.
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.
A Gaussian statistic, or discriminator, is proposed for conclusively discriminating between two pure quantum states. The discriminator is based on registrations from a limit of an increasing number of weak measurements on a single system. The average error of the discriminator is near that of the optimal conclusive Helstrom discriminator and, in fact, the Gaussian discriminator is the limit point of a class of conclusive discriminators which includes the Helstrom discriminator. The Gaussian discriminator always leaves some post-measurement state (PMS) separation; by contrast, the PMS separation with the Helstrom discriminator is always zero. Simple, closed-form expressions are given for the distribution and error performance of the Gaussian discriminator.
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.
Abstract Consider a quantum system prepared in a state chosen from a known set, the aim of quantum state discrimination is to perform a measurement on the system which can determine that a subset of the possible preparation procedures have been prepared. However, the aim of quantum state exclusion is to perform a measurement on the system which can conclusively rule that a subset of the possible preparation procedures have not taken place. For classical states, probabilities of error in discrimination and exclusion are always equal. For quantum states, we point out that probabilities of error in discrimination and exclusion are still equal in two-state ensembles, however they are no longer equal in multiple-state ensembles. More specifically, the error-probability of discrimination is not less than the error-probability of exclusion. We also show that the error-probability of exclusion is continuous which is similar with the error-probability of discrimination.