Optimal Unambiguous Discrimination of Quantum States

TL;DR

This paper provides an exact analytical solution for optimally distinguishing nonorthogonal quantum states unambiguously, using linear programming techniques and the Lewenstein-Sanpera decomposition, with practical examples illustrating its effectiveness.

Contribution

It introduces a novel method reducing the semi-definite programming problem to a linear programming problem solvable by the simplex method, and connects this with Lewenstein-Sanpera decomposition for optimal measurement design.

Findings

Exact analytic solutions for state discrimination problems.

Reduction of semi-definite programming to linear programming.

Explicit examples demonstrating method effectiveness.

Abstract

Unambiguously distinguishing between nonorthogonal but linearly independent quantum states is a challenging problem in quantum information processing. In this work, an exact analytic solution to an optimum measurement problem involving an arbitrary number of pure linearly independent quantum states is presented. To this end, the relevant semi-definite programming task is reduced to a linear programming one with a feasible region of polygon type which can be solved via simplex method. The strength of the method is illustrated through some explicit examples. Also using the close connection between the Lewenstein-Sanpera decomposition(LSD) and semi-definite programming approach, the optimal positive operator valued measure for some of the well-known examples is obtain via Lewenstein-Sanpera decomposition method. {\bf Keywords:} Optimal Unambiguous State Discrimination, Linear Programming, Lewenstein-Sanpera decomposition. {\bf PACs Index: 03.67.Hk, 03.65.Ta, 42.50.-p