On Fixed Points of L\"{u}ders Operation

TL;DR

This paper characterizes the fixed points of Lüders operations for finite commutative quantum measurements, showing they form the commutant, and provides a counterexample to a related conjecture about commutation.

Contribution

It proves the fixed points set of Lüders operations equals the commutant for finite commutative measurements and presents a counterexample to a conjecture on quantum effects.

Findings

Fixed points of Lüders operation equal the commutant for finite commutative measurements.

Constructed a Lüders operation with a fixed point that does not commute with all measurement effects.

Partially answers open problems in quantum measurement theory.

Abstract

In this paper, we prove that if $\mathcal{A}=\{E_i\}_{i=1}^{n}$ is a finite commutative quantum measurement, then the fixed points set of L\"{u}ders operation $L_{{\cal A}}$ is the commutant ${\cal A}'$ of ${\cal A}$, the result answers an open problem partially. We also give a concrete example of a L\"{u}ders operation $L_{{\cal A}}$ with $n=3$ such that $L_{{\cal A}}(B)=B$ does not imply that the quantum effect $B$ commutes with all $E_1, E_2$ and $E_3$, this example answers another open problem.