Decomposition of any quantum measurement into extremals

TL;DR

This paper presents an efficient algorithm to decompose any quantum measurement into extremal components, reducing complexity and enabling tailored decompositions with potential applications in quantum information processing.

Contribution

The authors introduce a constructive algorithm that decomposes any quantum measurement into extremals with improved bounds, especially for rank-1 POVMs, surpassing previous methods.

Findings

Decomposition into at most (N-1)d+1 extremals for N-element measurements on d-dimensional space

Algorithm is efficient and allows for tailored decompositions with specific properties

Improves upon previous upper bounds scaling as d^2

Abstract

We design an efficient and constructive algorithm to decompose any generalized quantum measurement into a convex combination of extremal measurements. We show that if one allows for a classical post-processing step only extremal rank-1 POVMs are needed. For a measurement with $N$ elements on a $d$-dimensional space, our algorithm will decompose it into at most $(N-1)d+1$ extremals, whereas the best previously known upper bound scaled as $d^2$. Since the decomposition is not unique, we show how to tailor our algorithm to provide particular types of decompositions that exhibit some desired property.