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Physical Review A

Iterative methods for finding optimal quantum measurements under minimum-error and minimax criteria

Kenji Nakahira, Kentaro Kato, Tsuyoshi Sasaki Usuda

Published Jan 14, 2015• 10 citations
Publisher

Abstract

We investigate the problem of computing optimal quantum measurements in both minimal measuring and minimax strategies. A Belavkin weighted square-root measurement (BWSRM) with appropriate weights can represent the measurement that maximizes the correct probability for any given prior probabilities of quantum states. Using this fact, we propose methods for computing optimal solutions by optimizing the weights of the BWSRM. First, we explain the conditions for the BWSRM to be optimal. In particular, we argue that if a BWSRM with certain weights is a minimax measurement, then the minimax probabilities can be immediately obtained. Next, we propose an extension of the iterative algorithm developed by Je\ifmmode \check{z}\else \v{z}\fi{}ek et al. [Phys. Rev. A 65, 060301 (2002)] for maximizing the correct probability. We prove that, for a linearly independent pure state set, Je\ifmmode \check{z}\else \v{z}\fi{}ek et al.'s algorithm converges to an optimal measurement. We also propose an iterative algorithm for a minimax solution and prove that, for a pure state set, our algorithm monotonically decreases the difference between estimated and true minimax values. Finally the performance of our algorithms is evaluated through numerical experiments.

DOI
10.1103/physreva.91.012318

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We investigate the problem of computing optimal quantum measurements in both minimal measuring and minimax strategies. A Belavkin weighted square-root measurement (BWSRM) with appropriate weights can represent the measurement that maximizes the correct probability for any given prior probabilities of quantum states. Using this fact, we propose methods for computing optimal solutions by optimizing the weights of the BWSRM. First, we explain the conditions for the BWSRM to be optimal. In particular, we argue that if a BWSRM with certain weights is a minimax measurement, then the minimax probabilities can be immediately obtained. Next, we propose an extension of the iterative algorithm developed by Je\ifmmode \check{z}\else \v{z}\fi{}ek et al. [Phys. Rev. A 65, 060301 (2002)] for maximizing the correct probability. We prove that, for a linearly independent pure state set, Je\ifmmode \check{z}\else \v{z}\fi{}ek et al.'s algorithm converges to an optimal measurement. We also propose an iterative algorithm for a minimax solution and prove that, for a pure state set, our algorithm monotonically decreases the difference between estimated and true minimax values. Finally the performance of our algorithms is evaluated through numerical experiments.
We compare several instances of pure-state Belavkin weighted square-root measurements from the standpoint of minimum-error discrimination of quantum states. The quadratically weighted measurement is proven superior to the so-called ``pretty good measurement'' (PGM) in a number of respects: (1) Holevo's quadratic weighting unconditionally outperforms the PGM in the case of two-state ensembles, with equality only in trivial cases. (2) A converse of a theorem of Holevo is proven, showing that a weighted measurement is asymptotically optimal only if it is quadratically weighted. Counterexamples for three states are constructed. The cube-weighted measurement of Ballester, Wehner, and Winter is also considered. Sufficient optimality conditions for various weights are compared.
K. Nakahira et al. introduced iterative algorithms to determine the optimal quantum detection for general quantum state signals under the minimum-error and minimax criteria. In this paper, we aim to modify these algorithms, assuming that the signal states are pure and linearly independent. We demonstrate that the algorithms can be modified using the properties of the Gram matrix of linearly independent pure states. This modification eliminates the need to compute the inverse of the square root of the weighted Gram matrix, which is required in the original algorithms, thereby simplifying their implementation.
We address a broad class of optimization problems of finding quantum measurements, which includes the problems of finding an optimal measurement in the Bayes criterion and a measurement maximizing the average correct probability with a fixed rate of inconclusive results. Our approach can deal with any problem in which each of the objective and constraint functions is formulated by the sum of the traces of the multiplication of a Hermitian operator and a detection operator. We first derive dual problems and necessary and sufficient conditions for an optimal measurement. We also consider the minimax version of these problems and provide necessary and sufficient conditions for a minimax solution. Finally, for optimization problem having a certain symmetry, there exists an optimal solution with the same symmetry. Examples are shown to illustrate how our results can be used.
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.
We consider the application of the minimax strategy to discrimination between quantum states with a certain fraction of inconclusive results. An inconclusive minimax solution is defined as the pair of a collection of prior probabilities and a quantum measurement that achieves the minimax probability of a detection error subject to the constraint that the highest probability of an inconclusive result does not exceed a given value. We first show a necessary and sufficient condition for an inconclusive minimax solution, and derive that the problem of obtaining an inconclusive minimax measurement can be expressed as a semidefinite programming problem. We then present a condition under which there exists a solution such that at least one of the prior probabilities is a zero. We finally show that for any state set with a certain symmetry, an inconclusive minimax solution exists having the same symmetry.
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.