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IEEE Transactions on Information Theory

Square-root measurement for quantum symmetric mixed state signals

Kentaro Kato, Osamu Hirota

Published Dec 1, 2003• 29 citations
Publisher

Abstract

A certain class of mixed states artificially constructed that is used in quantum cryptography is crucial for representation of signals. This correspondence shows that the square-root measurement gives the optimal measurement for such a class of mixed states.
DOI
10.1109/tit.2003.820050

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To derive an optimal quantum detection that minimizes the average probability of detection error is an important problem in quantum information theory. Especially, a study of mixed-state signals is important not only for reliable communications but also for secure communications. To date, the Square-Root Measurement (SRM) has been received much attention because it is the optimal or asymptotically-optimal detection for quantum state signals. The main purpose of this correspondence is to consider the Pseudo-Classicality of the SRM.
The optimum quantum measurements that minimize the error probability for mixed‐state signals are considered. The square‐root measurement (SRM) is known to be the optimal measurement for many pure‐state signals, e.g., symmetric states, linear codes, etc. However, when the states are mixed, the SRM is no longer guaranteed to be optimal even if the states are symmetric. Here, we define the ‘pure’ square‐root measurement (pure‐SRM) which is the SRM for pure states and its optimality for mixed states. It is shown that the pure‐SRM is satisfied the necessary and sufficient conditions of the quantum Bayes strategy when the signals are mixed by transmission errors, and therefore the pure‐SRM is the optimal measurement.
We develop a sufficient condition for the least-squares measurement (LSM), or the square-root measurement, to minimize the probability of a detection error when distinguishing between a collection of mixed quantum states. Using this condition we derive the optimal measurement for state sets with a broad class of symmetries. We first consider geometrically uniform (GU) state sets with a possibly non-Abelian generating group, and show that if the generator satisfies a weighted norm constraint, then the LSM is optimal. In particular, for pure-state GU ensembles, the LSM is shown to be optimal. For arbitrary GU state sets we show that the optimal measurement operators are GU with generator that can be computed very efficiently in polynomial time, within any desired accuracy. We then consider compound GU (CGU) state sets which consist of subsets that are GU. When the generators satisfy a certain constraint, the LSM is again optimal. For arbitrary CGU state sets, the optimal measurement operators are shown to be CGU with generators that can be computed efficiently in polynomial time.
Knowing about optimal quantum measurements is important for many applications in quantum information and quantum communication. However, deriving optimal quantum measurements is often difficult. We present a collection of results for minimum-cost quantum measurements, and give examples of how they can be used. Among other results, we show that a minimum-cost measurement for a set of given pure states is formally equivalent to a minimum-error measurement for mixed states of those same pure states. For pure symmetric states it turns out that for a certain class of cost matrices, the minimum-cost measurement is the square-root measurement. That is, the optimal minimum-cost measurement is in this case the same as the minimum-error measurement. Finally, we consider sequences of individual ``local" systems, and examine when the global minimum-cost measurement is a sequence of optimal local measurements. We also consider an example where the global minimum-cost measurement is, perhaps counter-intuitively, not a sequence of local measurements, and discuss how this is related to related to the Pusey-Barrett-Rudolph argument for the nature of the wave function.
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).
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.