Commuting Kraus operators are normal

TL;DR

This paper proves that a set of commuting matrices satisfying a specific sum condition are necessarily normal and can be diagonalized simultaneously, advancing understanding of their structure.

Contribution

It establishes that commuting matrices with a sum of conjugate products equal to the identity are necessarily normal and simultaneously diagonalizable, a new structural insight.

Findings

Commuting matrices with sum of conjugate products equal to identity are normal.

Such matrices are simultaneously diagonalizable.

The result links commutativity, normality, and diagonalizability.

Abstract

Let $\{V_1, \dots, V_n \}$ be a set of mutually commuting matrices. We show that if $V_1^* V_1 + \cdots +V_n^* V_n = {\rm Id}$ then the matrices are normal and, in particular, simultaneously diagonalizable.