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arXiv:1506.07617 · Proceedings of the Royal Society A Mathematical Physical and Engineering Sciences

On the Brukner–Zeilinger approach to information in quantum measurements

Alexey E. Rastegin

Published Nov 1, 2015• 18 citations

Abstract

We address the problem of properly quantifying information in quantum theory. Brukner and Zeilinger proposed the concept of an operationally invariant measure based on measurement statistics. Their measure of information is calculated with probabilities generated in a complete set of mutually complementary observations. This approach was later criticized for several reasons. We show that some critical points can be overcome by means of natural extension or reformulation of the Brukner–Zeilinger approach. In particular, this approach is connected with symmetric informationally complete measurements. The ‘total information’ of Brukner and Zeilinger can further be treated in the context of mutually unbiased measurements as well as general symmetric informationally complete measurements. The Brukner–Zeilinger measure of information is also examined in the case of detection inefficiencies. It is shown to be decreasing under the action of bistochastic maps. The Brukner–Zeilinger total information can be used for estimating the map norm of quantum operations.

DOI: 10.1098/rspa.2015.0435

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We address the problem of properly quantifying information in quantum theory. Brukner and Zeilinger proposed the concept of an operationally invariant measure based on measurement statistics. Their measure of information is calculated with probabilities generated in a complete set of mutually complementary observations. This approach was later criticized for several reasons. We show that some critical points can be overcome by means of natural extension or reformulation of the Brukner–Zeilinger approach. In particular, this approach is connected with symmetric informationally complete measurements. The ‘total information’ of Brukner and Zeilinger can further be treated in the context of mutually unbiased measurements as well as general symmetric informationally complete measurements. The Brukner–Zeilinger measure of information is also examined in the case of detection inefficiencies. It is shown to be decreasing under the action of bistochastic maps. The Brukner–Zeilinger total information can be used for estimating the map norm of quantum operations.

Abstract We investigate the total variance of a quantum state with respect to a complete set of mutually complementary measurements and its relation to the Brukner–Zeilinger invariant information. By summing the variances over any complete set of mutually unbiased measurements and general symmetric informationally complete measurements respectively, we show that the Brukner–Zeilinger invariant information associated with such types of quantum measurements is equal to the difference between the maximal variance and the total variance obtained. These results provide an operational link between the previous interpretations of the Brukner–Zeilinger invariant information.

Abstract Brukner–Zeilinger invariants, including invariant information and invariant uncertainty, play a remarkable role in quantum information theory. Recently, a broad class of symmetric informationally complete measurements is presented, which contains a general symmetrical information complete POVMs and mutually unbiased measurements (2022, Phys. Rev. A 105, 042 209). Based on this important measurement and variance, we further analyze the internal relationship between Brukner–Zeilinger invariants in detail. The results maybe provide a method to explain the importance of Brukner–Zeilinger invariants.

To quantify the intrinsic information content in a quantum state, Brukner and Zeilinger introduced the concept of operationally invariant information in terms of the outcome probabilities of measuring a complete set of mutually complementary observables [\ifmmode \check{C}\else \v{C}\fi{}. Brukner and A. Zeilinger, Phys. Rev. Lett. 83, 3354 (1999)]. This information quantity has basic significance and implications, and the present work is devoted to some further studies of it. We first introduce the Brukner-Zeilinger invariant information in the presence of conjugate symmetry or antisymmetry, which are motivated by considerations of fundamental issues concerning conjugate symmetry in quantum mechanics. Then we prove that both the Brukner-Zeilinger invariant information with conjugate symmetry and that with conjugate antisymmetry are convex in the quantum state, and we show that they constitute a natural decomposition of the Brukner-Zeilinger invariant information. We further relate them to the imaginarity (i.e., the usage of a complex number field) of quantum mechanics and evaluate their extreme values.

The invariant information, introduced by C. Brukner and A. Zeilinger [Phys. Rev. Lett. 83, 3354 (1999)], is reconsidered from the point of view of quantum state estimation. We show that this quantity is directly related to the mean error of the standard reconstruction from the measurement of a complete set of mutually complementary observables. We give its generalization in terms of the Fisher information. Provided that the optimum reconstruction is adopted, the information loses its invariant character.

Abstract In order to measure the inherent information content in a quantum state, Brukner and Zeilinger developed the idea of operationally invariant information in terms of the outcome probabilities of measuring a whole collection of mutually complementary observables. In this work, by partitioning the orthonormal operator basis of the associated operator Hilbert space into two complementary subsets, we establish a new decomposition of the Brukner-Zeilinger information. We further demonstrate that the two complementary components of the Brukner-Zeilinger information capture nicely the wave and particle features encoded in the quantum state after an n -path interferometer. Thus, we establish a universal complementarity relation involving the interference (wave feature) and the predictability (particle feature) via the unified concept of the Brukner-Zeilinger invariant information. These findings provide new characterizations of the wave-particle duality via the fundamental Brukner-Zeilinger invariant information.