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 \(\mathbf {C}^d\), 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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Ambainis, A., Emerson, J.: Quantum t-designs: t-wise independence in the quantum world. In: Proceedings of 22nd IEEE Conference on Computational Complexity, pp. 129–140. Piscataway, NJ (2007). arXiv:quant-ph/0701126
Aubrun, G.: On almost randomizing channels with a short Kraus decomposition. Commun. Math. Phys. 288(3), 1103–1116 (2009). arXiv:0805.2900
Barvinok, A.: A course in convexity, vol. 54. American Mathematical Soc. (2002)
Batson, J., Spielman, D.A., Srivatsava, N.: Twice-Ramanujan sparsifiers. arXiv:0808.0163
Bennett, G.: Schur multipliers. Duke Math. J. 44(3), 603–639 (1977)
Bolker, E.D.: A class of convex bodies. Trans. AMS 145, 323–345 (1969)
Bourgain, J., Lindenstrauss, J., Milman, V.: Approximation of zonoids by zonotopes. Acta Mathematica 162(1), 73–141 (1989)
Brandão, F.G.S.L., Christandl, M., Yard, J.T.: Faithful squashed entanglement. Commun. Math. Phys. 306, 805–830 (2011). arXiv:1010.1750 [quant-ph]
Chafaï, D., Guédon, O., Lecué, G., Pajor, A.: Interactions between compressed sensing, random matrices and high dimensional geometry
Figiel, T., Johnson, W.B.: Large subspaces of \(\ell _{\infty }^n\) and estimates of the Gordon-Lewis constant. Israel J. Math. 37(1-2), 92–112 (1980)
Figiel, T., Lindenstrauss, J., Milman, V.D.: The dimension of almost spherical sections of convex bodies. Acta Mathematica 139(1-2), 53–94 (1977)
Goodey, P., Weil,W.: Zonoids and Generalizations. Handbook of Convex Geometry, vol. B, pp. 1296–1326. North-Holland, Amsterdam (1993)
Gordon, Y.: Some inequalities for Gaussian processes and applications. Israel J. Math. 50(4), 265–289 (1985)
Harrow, A.W.: The Church of the Symmetric Subspace. arXiv:1308.6595 [quant-ph]
Harrow, A.W., Montanaro, A., Short, A.J.: Limitations on quantum dimensionality reduction. In: Proceedings of ICALP’11 LNCS 6755, pp. 86–97. Springer, Berlin Heidelberg (2011). arXiv:1012.2262 [quant-ph]
Helstrom, C.W.: Quantum detection and estimation theory. Academic Press, New York (1976)
Holevo, A.S.: Statistical decision theory for quantum systems. J. Mult. Anal. 3, 337–394 (1973)
Indyk, P.: Uncertainty principles, extractors, and explicit embeddings of \(L_2\) into \(L_1\). In: 39th ACM Symposium on Theory of Computing (2007)
Indyk, P., Szarek, S.: Almost-Euclidean subspaces of \(l_1^N\) via tensor products: a simple approach to randomness reduction. In: RANDOM 2010, LNCS 6302, pp. 632–641. Springer, Berlin Heidelberg (2010). arXiv:1001.0041 [math.MG]
Lancien, C., Winter, A.: Distinguishing multi-partite states by local measurements. Commun. Math. Phys. 323, 555–573 (2013). arXiv:1206.2884 [quant-ph]
Lovett, S., Sodin, S.: Almost Euclidean sections of the \(N\)-dimensional cross-polytope using \(O(N)\) random bits. Commun. Contemp. Math. 10(4), 477–489 (2008). arXiv:math/0701102
Matthews, W., Wehner, S., Winter, A.: Distinguishability of quantum states under restricted families of measurements with an application to data hiding. Comm. Math. Phys. 291(3) (2009). arXiv:0810.2327[quant-ph]
Pisier, G.: The volume of convex bodies and banach spaces geometry, Cambridge tracts in mathematics, 94th edn. Cambridge University Press, Cambridge (1989)
Rosental, H., Szarek, S.: On tensor products of operators from \(L^p\) to \(L^q\). Functional Analysis, pp. 108–132. Springer, Berlin Heidelberg (1991)
Rudin, W.: Trigonometric series with gaps. J. Math. Mech 9(2), 203–227
Rudin, W.: Functional analysis. McGraw-Hill International Series in pure and applied Mathematics, Singapore (1973)
Rudin, W.: Real and complex analysis. McGraw-Hill International Editions, Mathematics Series, Singapore (1987)
Schechtman, G.: More on embedding subspaces of \(L_p\) in \(l^n_r\). Compositio Math. 61(2), 159–169 (1987)
Schenider, R., Weil, W.: Zonoids and related topics. Convexity Appl. 296–317 (1983)
Sen, P.: Random measurement bases, quantum state distinction and applications to the hidden subgroup problem. In: Proceedings of 21st IEEE Conference on Computational Complexity, Piscataway, NJ (2006). arXiv:quant-ph/0512085
Talagrand, M.: Embedding subspaces of \(L_1\) into \(\ell _1^N\). Proc. Am. Math. Soc. 108(2), 363–369 (1990)
Acknowledgments
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.
Author information
Authors and Affiliations
Corresponding author
Additional information
This research was supported by the ANR project OSQPI ANR-11-BS01-0008.
Rights and permissions
About this article
Cite this article
Aubrun, G., Lancien, C. Zonoids and sparsification of quantum measurements. Positivity 20, 1–23 (2016). https://doi.org/10.1007/s11117-015-0337-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11117-015-0337-5