Zonoids and sparsification of quantum measurements

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. We also show that similar (but weaker) approximation results actually hold for any POVM on $\mathbf{C}^d$. By defining an appropriate notion of tensor product for zonoids, we are then able to extend our results to the multipartite setting: we show, roughly speaking, that local POVMs may be sparsified locally. In particular, the local uniform POVM on $\mathbf{C}^{d_1}\otimes\cdots\otimes\mathbf{C}^{d_k}$ can be approximated by a discrete POVM which is local and has $O(d_1^2\times\cdots\times d_k^2)$ outcomes.