Joint measurability meets Birkhoff-von Neumann's theorem

TL;DR

This paper introduces doubly normalised tensors as a quantum generalisation of doubly stochastic matrices, establishing a Birkhoff-von Neumann type theorem and linking joint measurability to these mathematical structures.

Contribution

It defines doubly normalised tensors and proves a Birkhoff-von Neumann theorem analogue, connecting joint measurability with these tensors in quantum measurement theory.

Findings

DNTs generalise doubly stochastic matrices to quantum operators.

Joint measurability is characterized by DNTs in this framework.

DNTs naturally arise from specific joint measurability problems.

Abstract

Quantum measurements can be interpreted as a generalisation of probability vectors, in which non-negative real numbers are replaced by positive semi-definite operators. We extrapolate this analogy to define a generalisation of doubly stochastic matrices that we call doubly normalised tensors (DNTs), and formulate a corresponding version of Birkhoff-von Neumann's theorem, which states that permutations are the extremal points of the set of doubly stochastic matrices. We prove that joint measurability arises as a mathematical feature of DNTs in this context, needed to establish a characterisation similar to Birkhoff-von Neumann's. Conversely, we also show that DNTs emerge naturally from a particular instance of a joint measurability problem, remarking its relevance in general operator theory.