Barycentric decomposition for quantum instruments

TL;DR

This paper introduces a barycentric decomposition framework for quantum instruments, channels, and measurements in finite-dimensional and separable Hilbert spaces, extending existing results and enabling finite-outcome representations.

Contribution

It generalizes the barycentric decomposition to a broader class of quantum instruments and formalizes finite-outcome representations for instruments between finite-dimensional Hilbert spaces.

Findings

Extended barycentric decomposition to quantum instruments, channels, and measurements.

Proved that instruments can be represented using finite-outcome instruments.

Generalized known decompositions for quantum measurements.

Abstract

We present a barycentric decomposition for quantum instruments whose output space is finite-dimensional and input space is separable. As a special case, we obtain a barycentric decomposition for channels between such spaces and for normalized positive-operator-valued measures in separable Hilbert spaces. This extends the known results by Ali and Chiribella et al. on decompositions of quantum measurements, and formalises the fact that every instrument between finite-dimensional Hilbert spaces can be represented using only finite-outcome instruments.