, Volume 20, Issue 1, pp 1–23 | Cite as

Zonoids and sparsification of quantum measurements

  • Guillaume AubrunEmail author
  • Cécilia Lancien


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 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 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 \(\mathbf {C}^{d_1}\otimes \cdots \otimes \mathbf {C}^{d_k}\) can be approximated by a discrete POVM which is local and has \(O(d_1^2 \times \cdots \times d_k^2)\) outcomes.


Positive operator-valued measure Zonoid Sparsification 

Mathematics Subject Classification

52A21 81P15 81P45 



We thank Andreas Winter for having first raised the general question of finding POVMs with few outcomes but good discriminating power. We also thank Marius Junge for suggesting the possible connection between POVMs and zonoids, and for pointing out to us relevant literature.


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© Springer Basel 2015

Authors and Affiliations

  1. 1.Institut Camille JordanUniversité Claude Bernard Lyon 1Villeurbanne cedexFrance
  2. 2.Física Teòrica: Informació i Fenomens QuànticsUniversitat Autònoma de BarcelonaBellaterra (Barcelona)Spain

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