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

Abstract

Quantum measurements can be interpreted as a generalization of probability vectors, in which non-negative real numbers are replaced by positive semi-definite operators. We extrapolate this analogy to define a generalization of doubly stochastic matrices that we call doubly normalized tensors (DNTs), and investigate 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 appears naturally as a mathematical feature of DNTs in this context and that this feature is necessary and sufficient for a characterization similar to Birkhoff–von Neumann's. Conversely, we also show that DNTs arise from a particular instance of a joint measurability problem, remarking the relevance of this quantum-theoretical property in general operator theory.