Quantum measurements on finite dimensional systems: relabeling and mixing

TL;DR

This paper shows that any finite outcome POVM can be constructed from extremal rank-1 POVMs using mixing and relabeling, clarifying the structure of extremal POVMs with higher rank.

Contribution

It demonstrates that extremal rank-1 POVMs, combined with mixing and relabeling, suffice to generate all finite outcome POVMs, simplifying their analysis.

Findings

Any finite outcome POVM can be derived from extremal rank-1 POVMs.

Extremal POVMs with higher rank are relabelings of rank-1 POVMs.

The structure of extremal POVMs is clarified through these procedures.

Abstract

Concentrating on finite dimensional systems, we show that one can limit to extremal rank-1 POVMs if two simple procedures of mixing and relabeling are permitted. We demonstrate that any finite outcome POVM can be obtained from extremal rank-1 POVMs with these two procedures. In particular, extremal POVMs with higher rank are just relabelings of extremal rank-1 POVMs and their structure is therefore clarified.