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

A Gaussian quantum state discriminator

Michael R. Frey, Jaclyn D. Szymonifka

Published May 25, 2005• 1 citation
Publisher

Abstract

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.

DOI
10.1117/12.604340

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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.
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.
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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.
Any measurement strategy designed to discriminate between nonorthogonal quantum states will have a finite probability of yielding erroneous or inconclusive results. The lower bounds on these probabilities for conclusive and error-free strategies are given by the Helstrom and Ivanovic–Feres limits respectively. We consider intermediate measurements which interpolate between these extremes and derive a general lower bound on the combination of errors and inconclusive results. We also describe a family of measurement strategies which attains this limit.
The problem of discriminating with minimum error between two mixed quantum states is reviewed, with emphasis on the detection operators necessary for performing the measurement. An analytical result is derived for the minimum probability of errors in deciding whether the state of a quantum system is either a given pure state or a uniform statistical mixture of any number of mutually orthogonal states. The result is applied to two-qubit states, and the minimum error probabilities achievable by collective and local measurements on the qubits are compared.
NNN-sequential Conclusive State Discrimination in a General Case
In the present article, we develop a general framework for the description of an NNN-sequential state discrimination, where each of NNN receivers always obtains a conclusive result. For this new state discrimination scenario, we derive two mutually equivalent general representations of the success probability and prove that if one of two states, pure or mixed, is prepared by a sender, then the optimal success probability is given by the Helstrom bound for any number NNN of sequential receivers. Furthermore, we specify receivers' indirect measurements resulting in the optimal NNN-sequential conclusive state discrimination protocol. The developed framework is true for any number NNN of sequential receivers, any number of arbitrary quantum states, pure or mixed, to be discriminated, and all types of receivers' quantum measurements. The new general results derived within the developed framework are important both from the theoretical point of view and for a successful multipartite quantum communication even in the presence of a quantum noise.