Entropy reduction of quantum measurements

TL;DR

This paper explores the entropy reduction in quantum measurements, linking it to quantum mutual information, and extends its definition and properties to a broader class of states, providing new analytical insights.

Contribution

It introduces a generalized approach to entropy reduction in quantum measurements using quantum mutual information, applicable to arbitrary states and analyzing its mathematical properties.

Findings

Entropy reduction equals quantum mutual information for efficient measurements.

Entropy reduction is a nonnegative, lower semicontinuous, concave function.

Monotonicity and subadditivity of entropy reduction are established.

Abstract

It is observed that the entropy reduction (the information gain in the initial terminology) of an efficient (ideal or pure) quantum measurement coincides with the generalized quantum mutual information of a q-c channel mapping an a priori state to the corresponding posteriori probability distribution of the outcomes of the measurement. This observation makes it possible to define the entropy reduction for arbitrary a priori states (not only for states with finite von Neumann entropy) and to study its analytical properties by using general properties of the quantum mutual information. By using this approach one can show that the entropy reduction of an efficient quantum measurement is a nonnegative lower semicontinuous concave function on the set of all a priori states having continuous restrictions to subsets on which the von Neumann entropy is continuous. Monotonicity and subadditivity of the entropy reduction are also easily proved by this method. A simple continuity condition for the entropy reduction and for the mean posteriori entropy considered as functions of a pair (a priori state, measurement) is obtained. A characterization of an irreducible measurement (in the Ozawa sense) which is not efficient is considered in the Appendix.