Optimum Unambiguous Discrimination of Linearly Independent Pure States

TL;DR

This paper develops a comprehensive analytical framework for the optimal unambiguous discrimination of linearly independent pure quantum states, providing formulas, solutions, and geometric insights to maximize success probability.

Contribution

It introduces a general method and analytical formulas for optimal unambiguous state discrimination, including solutions for special cases and a simplified success probability calculation.

Findings

Derived a simple analytical formula for maximum success probability.

Provided detailed steps to obtain optimal measurement strategies.

Solved the discrimination problem for three states in special cases.

Abstract

Given $n$ linearly independent pure states and their prior probabilities, we study the problem of optimum unambiguous discrimination of these states. We derive the properties of the optimum solution and the equations that must be satisfied by the optimum measurement strategy which achieves the maximum average success probability, and also give the detailed steps to obtain the optimum solution and the optimum measurement strategy. The general method and results we obtain are also illustrated both numerically and geometrically. We derive a simple analytical formula of the maximum average success probability of unambiguous discrimination for a given set of pure states, and it can be used to simplify the calculation of the optimum solution in some situations. We also obtain the analytical solution of a generalized equal-probability measurement problem using the equations we introduce. Finally, as another application of our result, we study the optimum unambiguous discrimination problem of three linearly independent pure states in details and obtain analytical solutions for some special cases.