Birkhoff–von Neumann's theorem, doubly normalized tensors, and joint measurability

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 doubly stochastic matrices that we call normalised tensors (DNTs), and formulate corresponding version Birkhoff-von Neumann's theorem, states permutations the extremal points set matrices. prove joint measurability arises mathematical feature DNTs context, needed establish characterisation similar Neumann's. Conversely, also show emerge naturally from particular instance problem, remarking its relevance general operator theory.