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arXiv:quant-ph/0506061 · Physical Review A

Family of generalized “pretty good” measurements and the minimal-error pure-state discrimination problems for which they are optimal

Carlos Mochon

Published Mar 22, 2006• 48 citations
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Abstract

Given a quantum pure state chosen from a set with some a priori probabilities, what is the optimal measurement needed to correctly guess the given state? We show that a good choice is the family of square-root or ``pretty good'' measurements, as each measurement in the family is optimal for at least one discrimination problem with the same quantum states but possibly different a priori probabilities. Furthermore, the map from measurement to discrimination problems can be explicitly described. In fact, for linearly independent states, every pair of discrimination problem and optimal measurement can be explicitly generated this way.

DOI
10.1103/physreva.73.032328

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Given a quantum pure state chosen from a set with some a priori probabilities, what is the optimal measurement needed to correctly guess the given state? We show that a good choice is the family of square-root or ``pretty good'' measurements, as each measurement in the family is optimal for at least one discrimination problem with the same quantum states but possibly different a priori probabilities. Furthermore, the map from measurement to discrimination problems can be explicitly described. In fact, for linearly independent states, every pair of discrimination problem and optimal measurement can be explicitly generated this way.
The quantum state discrimination problem has Alice sending a quantum state to Bob who wins if he correctly identifies the state. The pretty good measurement, also known as the square root measurement, performs pretty well at this task. We study the version of this problem where Bob tries to lose with the greatest probability possible (which is harder than it sounds). We define the pretty bad measurement, which performs pretty well at this task, or in other words, pretty poorly for the original task. We show that both the pretty good measurement and the pretty bad measurement are always no worse than blind guessing at their respective tasks. As an application, we apply the pretty bad measurement to the quantum state anomaly detection problem and show how to avoid pretty bad qubits.
We show that the quantum measurement known as the pretty good measurement can be used to identify an unknown quantum state picked from any set of nnn mixed states that have pairwise fidelities upper-bounded by a constant below 1, given O(log⁡n)O(\log n)O(logn) copies of the unknown state, with high success probability in the worst case. If the unknown state is promised to be pure, there is an explicit measurement strategy which solves this worst-case quantum state discrimination problem with O~(∥G∥)\widetilde{O}(\|G\|)O(∥G∥) copies, where GGG is the Gram matrix of the states.
We consider the problem of constructing measurements optimized to distinguish between a collection of possibly nonorthogonal quantum states. We consider a collection of pure states and seek a positive operator-valued measure (POVM) consisting of rank-one operators with measurement vectors closest in squared norm to the given states. We compare our results to previous measurements suggested by Peres and Wootters (1991) and Hausladen et al. (1996), where we refer to the latter as the square-root measurement (SRM). We obtain a new characterization of the SRM, and prove that it is optimal in a least-squares sense. In addition, we show that for a geometrically uniform state set the SRM minimizes the probability of a detection error. This generalizes a similar result of Ban et al. (see Int. J. Theor. Phys., vol.36, p.1269-88, 1997).
We show a geometric formulation for minimum-error discrimination of qubit states that can be applied to arbitrary sets of qubit states given with arbitrary a priori probabilities. In particular, when qubit states are given with equal a priori probabilities, we provide a systematic way of finding optimal discrimination and the complete solution in a closed form. This generally gives a bound to cases when prior probabilities are unequal. Then it is shown that the guessing probability does not depend on detailed relations among the given states, such as the angles between them, but on a property that can be assigned by the set of given states itself. This also shows how a set of quantum states can be modified such that the guessing probability remains the same. Optimal measurements are also characterized accordingly, and a general method of finding them is provided.
Distinguishing assigned quantum states with assigned probabilities via quantum measurements is a crucial problem for the transmission of classical information through quantum channels. Measurement operators maximizing the probability of correct discrimination have been characterized by Helstrom, Holevo, and Yuen since the 1970s. On the other hand, closed-form solutions are available only for particular situations enjoying high degrees of symmetry. As a suboptimal solution to the problem, measurement operators, directly determined from states and probabilities and known as square-root measurements (SRMs), were introduced by Hausladen and Wootters. These operators were also recognized to be optimal for pure states equipped with geometrical uniform symmetry (GUS). In this paper we discuss the optimality of the SRM and find necessary and sufficient conditions in order that SRM maximize the correct decision probabilities for the set of states formed by several constellations of GUS states. The results are applied to some specific examples concerning double constellations of quantum phase-shift keying and pulse-position modulation states, with possible applications to practical systems of quantum communications.
The Barnum-Knill theorem states that the optimal success probability in the multiple state discrimination task is not more than the square root of the success probability when the pretty good or square-root measurement is used for this task. An assumption of the theorem is that the underlying ensemble consists of finitely many quantum states over a finite-dimensional quantum system. Motivated in part by the fact that the success probability is not a relevant metric for continuous ensembles, in this paper we provide a generalization of the notion of pretty good measurement and the Barnum-Knill theorem for general quantum ensembles, including those described by a continuous parameter space and an infinite-dimensional Hilbert space. To achieve this, we also design a general metric of performance for quantum measurements that generalizes the success probability, namely, the expected gain of the measurement with respect to a positive score function. A notable consequence of the main result is that, in a Bayesian estimation task, the mean square error of the generalized pretty good measurement does not exceed twice the optimal mean square error.