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arXiv:1502.05785 · AIP conference proceedings

How much a quantum measurement is informative?

Michele Dall'Arno, Giacomo Mauro D’Ariano, Massimiliano Federico Sacchi

Published Jan 1, 2014• 2 citations
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Abstract

The informational power of a quantum measurement is the maximum amount of classical information that the measurement can extract from any ensemble of quantum states. We discuss its main properties. Informational power is an additive quantity, being equivalent to the classical capacity of a quantum-classical channel. The informational power of a quantum measurement is the maximum of the accessible information of a quantum ensemble that depends on the measurement. We present some examples where the symmetry of the measurement allows to analytically derive its informational power.

DOI
10.1063/1.4903120

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The informational power of a quantum measurement is the maximum amount of classical information that the measurement can extract from any ensemble of quantum states. We discuss its main properties. Informational power is an additive quantity, being equivalent to the classical capacity of a quantum-classical channel. The informational power of a quantum measurement is the maximum of the accessible information of a quantum ensemble that depends on the measurement. We present some examples where the symmetry of the measurement allows to analytically derive its informational power.
We introduce the informational power of a quantum measurement as the maximum amount of classical information that the measurement can extract from any ensemble of quantum states. We prove the additivity by showing that the informational power corresponds to the classical capacity of a quantum-classical channel. We restate the problem of evaluating the informational power as the maximization of the accessible information of a suitable ensemble. We provide a numerical algorithm to find an optimal ensemble and quantify the informational power.
The accessible information quantifies the amount of classical information that can be extracted from an ensemble of quantum states. Analogously, the informational power quantifies the amount of classical information that can be extracted by a quantum measurement. For both quantities, we provide upper and lower bounds that depend only on the dimension of the system, and we prove their tightness. In the case of symmetric informationally complete (SIC) ensembles and measurements, stronger bounds are provided and their tightness proved for qubits and qutrits. From our upper bounds, we notice, perhaps surprisingly, that the statistics generated by SIC ensembles or measurements in arbitrary dimension, though optimal for tomographic purposes, in fact never contain more than just one bit of information, the rest being constituted by completely random bits. On the other hand, from our lower bounds, we obtain an explicit strategy beating the so-called pretty-good one for the extraction of mutual information in the case of SIC ensembles and measurements.
In this paper we consider the classical capacities of quantum-classical channels corresponding to measurement of observables. Special attention is paid to the case of continuous observables. We give formulas for unassisted and entanglement-assisted classical capacities C and C ea and consider some explicitly solvable cases, which give simple examples of entanglement-breaking channels with C < C ea. We also elaborate on the ensemble-observable duality to show that C ea for the measurement channel is related to the χ-quantity for the dual ensemble in the same way as C is related to the accessible information. This provides both accessible information and the χ-quantity for quantum ensembles dual to our examples.
The accessible information and the informational power quantify the maximum amount of information that can be extracted from a quantum ensemble and by a quantum measurement, respectively. Here, we investigate the tradeoff between the accessible information (informational power, respectively) and the purity of the states of the ensemble (the elements of the measurement, respectively). Under any given lower bound on the purity, 1) we compute the minimum informational power and show that it is attained by the depolarized uniformly-distributed measurement; and 2) we give a lower bound on the accessible information. Under any given upper bound on the purity, 1) we compute the maximum accessible information and show that it is attained by an ensemble of pairwise commuting states with at most two distinct non-null eigenvalues; and 2) we give a lower bound on the maximum informational power. The present results provide, as a corollary, novel sufficient conditions for the tightness of the Jozsa-Robb-Wootters lower bound to the accessible information.
The accessible information and the informational power quantify the amount of information extractable from a quantum ensemble and by a quantum measurement, respectively. So-called spherical quantum 2-designs constitute a class of ensembles and measurements relevant in testing entropic uncertainty relations, quantum cryptography, and quantum tomography. We provide lower bounds on the entropy of 2-design ensembles and measurements, from which upper bounds on their accessible information and informational power follow, as a function of the dimension only. We show that the statistics generated by 2-designs, although optimal for the above-mentioned protocols, never contains more than 1 bit of information. Finally, we specialize our results to the relevant cases of symmetric informationally complete sets and maximal sets of mutually unbiased bases, and we generalize them to the arbitrary-rank case.
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.