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2013

Zonoids and sparsification of quantum measurements

Aubrun, Guillaume, Lancien, Cécilia

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In this paper, we establish a connection between zonoids (a concept from classical convex geometry) and the distinguishability norms associated to quantum measurements, or POVMs (Positive Operator-Valued Measures), recently introduced in quantum information theory.

This correspondence allows us to state and prove the POVM version of classical results from the local theory of Banach spaces about the approximation of zonoids by zonotopes.
We show that on $\mathbf{C}^d$, the uniform POVM (the most symmetric POVM) can be sparsified, i.e.
approximated by a discrete POVM, the latter having only $O(d^2)$ outcomes.

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