Complete solution for unambiguous discrimination of three pure states with real inner products

TL;DR

This paper provides a complete analytic solution for unambiguous discrimination of three pure states with real inner products, including general results and conditions, and explores extensions to complex inner products.

Contribution

It offers the first closed-form analytic solutions for three pure states with real inner inner products and clarifies conditions for reducing the problem to fewer states.

Findings

Analytic solutions for three pure states with real inner products

Conditions for problem reduction to fewer states

Partial solutions for certain complex inner product cases

Abstract

Complete solutions are given in a closed analytic form for unambiguous discrimination of three general pure states with real mutual inner products. For this purpose, we first establish some general results on unambiguous discrimination of n linearly independent pure states. The uniqueness of solution is proved. The condition under which the problem is reduced to an (n-1)-state problem is clarified. After giving the solution for three pure states with real mutual inner products, we examine some difficulties in extending our method to the case of complex inner products. There is a class of set of three pure states with complex inner products for which we obtain an analytical solution.