Total variance and invariant information in complementary measurements

TL;DR

This paper explores the relationship between total variance in quantum states and the Brukner-Zeilinger invariant information, establishing an operational link through the summation of variances over specific measurement sets.

Contribution

It demonstrates that the Brukner-Zeilinger invariant information equals the difference between maximal and total variances for mutually unbiased and symmetric informationally complete measurements.

Findings

Brukner-Zeilinger invariant information equals the difference between maximal and total variances.

Sum of variances over mutually unbiased measurements relates to invariant information.

Operational interpretation of invariant information in quantum measurement contexts.

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.