Minimum Error Discrimination of Linearly Independent Pure States: Analytic Properties of POVM

TL;DR

This paper investigates the analytic properties of POVMs in minimum error discrimination of linearly independent pure states, revealing the complexity of solutions and proposing a method to smoothly transition solutions between different state ensembles.

Contribution

It introduces a novel approach using the implicit functions theorem and RK4 to find optimal POVMs by evolving from known solutions to general cases.

Findings

Polynomial equations explain the difficulty in closed-form solutions.

Optimal POVMs vary smoothly with state ensemble parameters.

RK4 method achieves low error in solving differential equations.

Abstract

The optimization conditions for minimum error discrimination of linearly independent pure states comprise of two kinds: stationary conditions over the space of rank one projective measurements and the global maximization conditions. A discrete number of projective measurments will solve th former of which a unique one will solve the latter. In the case of three real linearly independent pure states we show that the stationary conditions translate to a system of simultaneous polynomial (non linear) equations in three variabes thus explaining why it's so difficult to obtain a closed-form solution for the optimal POVM. Additionally, our method suggests that as an ensemble of LI pure states is varied as a smooth function of some independent parameters, the optimal POVM will also vary smoothly as a function of the same parameters. By employing the implicit functions theorem we exploit this fact to obtain a technique to find the solution of MED of LI pure states by dragging the solution from a known example (say, pure orthogonal states) to any general linearly indepenent ensemble of pure states in the same Hilbert space. By employing RK4 to solve the first order coupled non-linear differential equations find that the resulting error is within the RK4 error performance.