Quantum measurements on finite dimensional systems: relabeling and mixing

Understand

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.