Joint measurability meets Birkhoff-von Neumann's theorem
- URL: http://arxiv.org/abs/1809.07366v5
- Date: Tue, 9 May 2023 19:20:13 GMT
- Title: Joint measurability meets Birkhoff-von Neumann's theorem
- Authors: Leonardo Guerini and Alexandre Baraviera
- Abstract summary: 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.
We also show that DNTs emerge naturally from a particular instance of a joint measurability problem, remarking its relevance in general operator theory.
- Score: 77.34726150561087
- License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
- 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.
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