Discrimination with an error margin among three symmetric states of a qubit

TL;DR

This paper investigates optimal discrimination strategies for three symmetric qubit states, introducing an error margin to unify minimum-error and unambiguous discrimination, and derives analytical solutions for success probabilities.

Contribution

It provides a systematic analysis and analytical solutions for state discrimination with an error margin for both pure and mixed symmetric qubit states.

Findings

Optimal success probabilities are derived analytically for different error margin regimes.

Three measurement types are classified, with one being optimal depending on the error margin.

The framework unifies minimum-error and unambiguous discrimination approaches.

Abstract

We consider a state discrimination problem which deals with settings of minimum-error and unambiguous discrimination systematically by introducing a margin for the probability of an incorrect guess. We analyze discrimination of three symmetric pure states of a qubit. The measurements are classified into three types, and one of the three types is optimal depending on the value of the error margin. The problem is formulated as one of semidefinite programming. Starting with the dual problem derived from the primal one, we analytically obtain the optimal success probability and the optimal measurement that attains it in each domain of the error margin. Moreover, we analyze the case of three symmetric mixed states of a qubit.