The canonical phase measurement is pure

TL;DR

This paper proves that the canonical phase measurement in quantum mechanics is a pure, extremal POVM, indicating it cannot be viewed as a noisy or mixed measurement, even though it is not a projection-valued measure.

Contribution

It establishes the extremality and purity of the canonical phase measurement, clarifying its fundamental nature in quantum measurement theory.

Findings

Canonical phase measurement is an extremal POVM.

It is pure and cannot be interpreted as noisy.

The measurement is not a projection-valued measure.

Abstract

We show that the canonical phase measurement is pure in the sense that the corresponding positive operator valued measure (POVM) is extremal in the convex set of all POVMs. This means that the canonical phase measurement cannot be interpreted as a noisy measurement, even if it is not a projection valued measure.