Fixed points of commutative L\"uders operations

TL;DR

This paper proves that for a class of quantum operations with commuting effects forming a resolution of the identity, the fixed points are precisely those operators commuting with all operation elements.

Contribution

It confirms a conjecture by showing the fixed points set equals the commutant of the effects in a specific class of quantum operations.

Findings

Fixed points set equals the commutant of the effects.

Validates the conjecture for commuting effect-based quantum operations.

Enhances understanding of the structure of quantum fixed points.

Abstract

This paper verifies a conjecture posed in a pair of papers on the fixed point sets for a class of quantum operations. Specifically, it is proved that if a quantum operation has mutually commuting operation elements that are effects forming a resolution of the identity, then the fixed points set of the quantum operation is exactly the commutant of the operation elements.