Minimal sufficient positive-operator valued measure on a separable Hilbert space

TL;DR

This paper introduces the concept of a minimal sufficient POVM in quantum measurement theory, establishing its existence, uniqueness, and construction for separable Hilbert spaces, and explores its applications to discrete POVMs.

Contribution

It defines and proves the existence and uniqueness of minimal sufficient POVMs, providing a foundational tool for quantum measurement analysis.

Findings

Every POVM has an equivalent minimal sufficient POVM.

Minimal sufficient POVMs are unique up to relabeling.

The concept applies to discrete POVMs and information conservation.

Abstract

We introduce a concept of a minimal sufficient positive-operator valued measure (POVM), which is the least redundant POVM among the POVMs that have the equivalent information about the measured quantum system. Assuming the system Hilbert space to be separable, we show that for a given POVM a sufficient statistic called a Lehmann-Scheff\'{e}-Bahadur statistic induces a minimal sufficient POVM. We also show that every POVM has an equivalent minimal sufficient POVM and that such a minimal sufficient POVM is unique up to relabeling neglecting null sets. We apply these results to discrete POVMs and information conservation conditions proposed by the author.