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EXTREMAL COVARIANT MEASUREMENTS

Giulio Chiribella, Giacomo

Published Jan 1, 2006• 5 citations
OpenAlex

Abstract

Abstract. We characterize the extremal points of the convex set of quantum measurements that are covariant under a finite-dimensional projective representation of a compact group, with action of the group on the measurement probability space which is generally non-transitive. In this case the POVM density is made of multiple orbits of positive operators, and, in the case of extremal measurements, we provide a bound for the number of orbits and for the rank of POVM elements. Two relevant applications are considered, concerning state discrimination with mutually unbiased bases and the maximization of the mutual information. 1. introduction A fundamental issue in the theory of quantum information [1] is the investigation of the ultimate precision limits for extracting classical information from a quantum system. Indeed, when the information is encoded on quantum states, its read-out suffers the intrinsically quantum limitation of discriminating among nonorthogonal states. One then need to optimize the discrimination with respect to a given optimality

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Abstract. We characterize the extremal points of the convex set of quantum measurements that are covariant under a finite-dimensional projective representation of a compact group, with action of the group on the measurement probability space which is generally non-transitive. In this case the POVM density is made of multiple orbits of positive operators, and, in the case of extremal measurements, we provide a bound for the number of orbits and for the rank of POVM elements. Two relevant applications are considered, concerning state discrimination with mutually unbiased bases and the maximization of the mutual information. 1. introduction A fundamental issue in the theory of quantum information [1] is the investigation of the ultimate precision limits for extracting classical information from a quantum system. Indeed, when the information is encoded on quantum states, its read-out suffers the intrinsically quantum limitation of discriminating among nonorthogonal states. One then need to optimize the discrimination with respect to a given optimality
We characterize the extremal points of the convex set of quantum measurements that are covariant under a finite-dimensional projective representation of a compact group, with action of the group on the measurement probability space which is generally nontransitive. In this case the POVM density is made of multiple orbits of positive operators, and, in the case of extremal measurements, we provide a bound for the number of orbits and for the rank of POVM elements. Two relevant applications are considered, concerning state discrimination with mutually unbiased bases and the maximization of the mutual information.
Quantum detectors provide information about the microscopic properties of quantum systems by establishing correlations between those properties and a set of macroscopically distinct events that we observe. The question of how much information a quantum detector can extract from a system is therefore of fundamental significance. In this paper, we address this question within a precise framework: given a measurement apparatus implementing a specific POVM measurement, what is the optimal performance achievable with it for a specific information readout task and what is the optimal way to encode information in the quantum system in order to achieve this performance? We consider some of the most common information transmission tasks—the Bayes cost problem, unambiguous message discrimination and the maximal mutual information. We provide general solutions to the Bayesian and unambiguous discrimination problems. We also show that the maximal mutual information is equal to the classical capacity of the quantum-to-classical channel describing the measurement, and study its properties in certain special cases. For a group covariant measurement, we show that the problem is equivalent to the problem of accessible information of a group covariant ensemble of states. We give analytical proofs of optimality in some relevant cases. The framework presented here provides a natural way to characterize generalized quantum measurements in terms of their information readout capabilities.