Commutator Relations Reveal Solvable Structures in Unambiguous State Discrimination

TL;DR

This paper introduces a criterion based on commutator relations to determine when two matrices can be simultaneously simplified into a block-diagonal form, aiding in solving unambiguous state discrimination problems.

Contribution

The paper develops a new commutator-based criterion to identify solvable structures in unambiguous state discrimination, facilitating problem reduction.

Findings

Criterion effectively detects reducibility to block-diagonal form.

Application to unambiguous state comparison demonstrates practical utility.

Provides a systematic test for problem simplification in quantum state discrimination.

Abstract

We present a criterion, based on three commutator relations, that allows to decide whether two self-adjoint matrices with non-overlapping support are simultaneously unitarily similar to quasidiagonal matrices, i.e., whether they can be simultaneously brought into a diagonal structure with 2x2-dimensional blocks. Application of this criterion to unambiguous state discrimination provides a systematic test whether the given problem is reducible to a solvable structure. As an example, we discuss unambiguous state comparison.