Minimum-error discrimination between two sets of similarity-transformed quantum states

TL;DR

This paper investigates the problem of minimum-error discrimination between two sets of quantum states transformed by similarity operations, providing explicit solutions for irreducible cases and methods for reducible cases, with applications to qubits and generalized Bloch states.

Contribution

It derives closed-form solutions for optimal measurements in irreducible cases and offers a systematic approach for reducible cases, advancing quantum state discrimination theory.

Findings

Closed-form optimal measurement for irreducible representations.

Procedural method for reducible representations.

Maximum success probability calculated for specific quantum states.

Abstract

Using the necessary and sufficient conditions, minimum error discrimination among two sets of similarity transformed equiprobable quantum qudit states is investigated. In the case that the unitary operators are generating sets of two irreducible representations, the optimal set of measurements and the corresponding maximum success probability of discrimination are determined in closed form. In the case of reducible representations, there exists no closed-form formula in general, but the procedure can be applied in each case accordingly. Finally, we give the maximum success probability of optimal discrimination for some important examples of mixed quantum states, such as qubit states together with three special cases and generalized Bloch sphere m-qubit states.