# Zonoids and sparsification of quantum measurements

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 $$\mathbf {C}^d$$ C d , the uniform POVM (the most symmetric POVM) can be sparsified, i.e. approximated by a discrete POVM having only $$O(d^2)$$ O ( d 2 ) outcomes. We also show that similar (but weaker) approximation results actually hold for any POVM on $$\mathbf {C}^d$$ C d . By considering an appropriate notion of tensor product for zonoids, we 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 $$\mathbf {C}^{d_1}\otimes \cdots \otimes \mathbf {C}^{d_k}$$ C d 1 ⊗ ⋯ ⊗ C d k can be approximated by a discrete POVM which is local and has $$O(d_1^2 \times \cdots \times d_k^2)$$ O ( d 1 2 × ⋯ × d k 2 ) outcomes. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

# Zonoids and sparsification of quantum measurements

, Volume 20 (1) – Apr 30, 2015
23 pages

/lp/springer_journal/zonoids-and-sparsification-of-quantum-measurements-qxWr47Y0py
Publisher
Springer Journals
Subject
Mathematics; Fourier Analysis; Operator Theory; Potential Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1007/s11117-015-0337-5
Publisher site
See Article on Publisher Site

### Abstract

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 $$\mathbf {C}^d$$ C d , the uniform POVM (the most symmetric POVM) can be sparsified, i.e. approximated by a discrete POVM having only $$O(d^2)$$ O ( d 2 ) outcomes. We also show that similar (but weaker) approximation results actually hold for any POVM on $$\mathbf {C}^d$$ C d . By considering an appropriate notion of tensor product for zonoids, we 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 $$\mathbf {C}^{d_1}\otimes \cdots \otimes \mathbf {C}^{d_k}$$ C d 1 ⊗ ⋯ ⊗ C d k can be approximated by a discrete POVM which is local and has $$O(d_1^2 \times \cdots \times d_k^2)$$ O ( d 1 2 × ⋯ × d k 2 ) outcomes.

### Journal

PositivitySpringer Journals

Published: Apr 30, 2015

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