On the Brukner-Zeilinger approach to information in quantum measurements

TL;DR

This paper revisits and refines the Brukner-Zeilinger measure of quantum information, addressing criticisms and connecting it with symmetric informationally complete measurements, while exploring its properties under various quantum operations.

Contribution

The authors propose extensions and reformulations of the Brukner-Zeilinger information measure, linking it to symmetric informationally complete measurements and analyzing its behavior under detection inefficiencies and quantum maps.

Findings

The measure can be connected with SIC measurements.

It decreases under bistochastic maps.

It can estimate the map norm of quantum operations.

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.