Optimal discrimination of quantum states with a fixed rate of inconclusive outcomes

TL;DR

This paper develops a general method for optimally discriminating quantum states with a fixed rate of inconclusive outcomes, unifying unambiguous and minimum error discrimination approaches.

Contribution

The authors introduce a versatile framework that interpolates between standard quantum state discrimination methods, providing a complete characterization of the minimum error as a function of inconclusive rate.

Findings

Identified a critical inconclusive rate matching minimum failure probability in unambiguous discrimination.

Derived a general solution applicable to various quantum state discrimination scenarios.

Illustrated the method with examples involving pure states and trine states.

Abstract

In this paper we present the solution to the problem of optimally discriminating among quantum states, i.e., identifying the states with maximum probability of success when a certain fixed rate of inconclusive answers is allowed. By varying the inconclusive rate, the scheme optimally interpolates between Unambiguous and Minimum Error discrimination, the two standard approaches to quantum state discrimination. We introduce a very general method that enables us to obtain the solution in a wide range of cases and give a complete characterization of the minimum discrimination error as a function of the rate of inconclusive answers. A critical value of this rate is identified that coincides with the minimum failure probability in the cases where unambiguous discrimination is possible and provides a natural generalization of it when states cannot be unambiguously discriminated. The method is illustrated on two explicit examples: discrimination of two pure states with arbitrary prior probabilities and discrimination of trine states.