Minimum Error Discrimination for an Ensemble of Linearly Independent Pure States

TL;DR

This paper develops a new method using differential equations and analytic continuation to solve the minimum error discrimination problem for ensembles of linearly independent pure quantum states, improving computational efficiency and simplicity.

Contribution

It introduces a differential equation approach to find optimal measurements, extending previous work and providing a more efficient, simpler solution method.

Findings

The method effectively solves the discrimination problem for linearly independent pure states.

It is computationally comparable or superior to SDP-based methods.

The approach simplifies implementation and offers a new analytical perspective.

Abstract

Inspired by the work done by Belavkin [Belavkin V. P., Stochastics, 1, 315 (1975)], and independently by Mochon, [Phys. Rev. A 73, 032328, (2006)], we formulate the problem of minimum error discrimination of any ensemble of $n$ linearly independent pure states by stripping the problem of its rotational covariance and retaining only the rotationally invariant aspect of the problem. This is done by embedding the optimal conditions in a matrix equality as well as matrix inequality. Employing the implicit function theorem in these conditions we get a set of first-order coupled ordinary non-linear differential equations which can be used to drag the solution from an initial point (where solution is known) to another point (whose solution is sought). This way of obtaining the solution can be done through a simple Taylor series expansion and analytic continuation when required. Thus, we \emph{complete} the work done by Belavkin and Mochon by ultimately leading their theory to a solution for the minimum error discrimination problem of linearly independent pure state ensembles. We also compare the computational complexity of our technique with a barrier-type interior point method of SDP and show that our technique is computationally as efficient as (actually, a bit more than) the SDP algorithm, with the added advantage of being much simpler to implement.