  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.
