# A Structure of Minimum Error Discrimination for Linearly Independent States

## TL;DR

This paper investigates the minimum error discrimination problem for linearly independent quantum states, establishing a bijective map linking ensembles and their optimal measurements, and simplifying the conditions for optimality.

## Contribution

It introduces a bijective map for LI state ensembles, relates PGM to optimal measurements, and simplifies the optimality conditions for these ensembles.

## Key findings

- The bijective map relates ensembles to their optimal measurements.
- Fixed points of the map correspond to ensembles where PGM is optimal.
- Optimality conditions for LI states are simplified.

## Abstract

In this paper we study the Minimum Error Discrimination problem (MED) for ensembles of linearly independent (LI) states. We define a bijective map from the set of those ensembles to itself and we show that the Pretty Good Measurement (PGM) and the optimal measurement for the MED are related by the map. In particular, the fixed points of the map are those ensembles for which the PGM is the optimal measurement. Also, we simplify the optimality conditions for the measurement of an ensemble of LI states.