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.

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Acknowledgments

The authors are thankful to T. Perche and M. T. Quintino for fruitful discussions and suggestions. AB is partially supported by CNPq.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 From this point on, we abuse the notation and write ${\mathrm{\Pi }}_l$ both for the permutation map ${\mathrm{\Pi }}_l:\{1,\dots ,n\}\to \{1,\dots ,n\}$ and for its matrix representation.

Additional information

Funding

LG was supported by the São Paulo Research Foundation (FAPESP) [grant number 2016/01343-7], [grant number 2018/04208-9].