Optimal Discrimination of Qubit States - Methods, Solutions, and Properties

TL;DR

This paper introduces a geometric approach for optimally discriminating qubit states, providing closed-form solutions for equal priors and insights into how state properties influence guessing probabilities.

Contribution

It presents a systematic geometric method for minimum-error discrimination of qubit states, including closed-form solutions for equal priors and characterization of optimal measurements.

Findings

Guessing probability depends on intrinsic properties of the state set.

Closed-form solutions are provided for equal prior probabilities.

Optimal measurements are characterized and methods to find them are given.

Abstract

We show a geometric formulation for minimum-error discrimination of qubit states, that can be applied to arbitrary sets of qubit states given with arbitrary a priori probabilities. In particular, when qubit states are given with equal \emph{a priori} probabilities, we provide a systematic way of finding optimal discrimination and the complete solution in a closed form. This generally gives a bound to cases when prior probabilities are unequal. Then, it is shown that the guessing probability does not depend on detailed relations among given states, such as angles between them, but on a property that can be assigned by the set of given states itself. This also shows how a set of quantum states can be modified such that the guessing probability remains the same. Optimal measurements are also characterized accordingly, and a general method of finding them is provided.