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Positivity

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

Zonoids and sparsification of quantum measurements

  • Guillaume AubrunEmail author
  • Cécilia Lancien
Article

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.

Keywords

Positive operator-valued measure Zonoid Sparsification 

Mathematics Subject Classification

52A21 81P15 81P45 

Notes

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.

References

  1. 1.
    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
  2. 2.
    Aubrun, G.: On almost randomizing channels with a short Kraus decomposition. Commun. Math. Phys. 288(3), 1103–1116 (2009). arXiv:0805.2900
  3. 3.
    Barvinok, A.: A course in convexity, vol. 54. American Mathematical Soc. (2002)Google Scholar
  4. 4.
    Batson, J., Spielman, D.A., Srivatsava, N.: Twice-Ramanujan sparsifiers. arXiv:0808.0163
  5. 5.
    Bennett, G.: Schur multipliers. Duke Math. J. 44(3), 603–639 (1977)CrossRefMathSciNetzbMATHGoogle Scholar
  6. 6.
    Bolker, E.D.: A class of convex bodies. Trans. AMS 145, 323–345 (1969)CrossRefMathSciNetzbMATHGoogle Scholar
  7. 7.
    Bourgain, J., Lindenstrauss, J., Milman, V.: Approximation of zonoids by zonotopes. Acta Mathematica 162(1), 73–141 (1989)CrossRefMathSciNetzbMATHGoogle Scholar
  8. 8.
    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]
  9. 9.
    Chafaï, D., Guédon, O., Lecué, G., Pajor, A.: Interactions between compressed sensing, random matrices and high dimensional geometryGoogle Scholar
  10. 10.
    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)CrossRefMathSciNetzbMATHGoogle Scholar
  11. 11.
    Figiel, T., Lindenstrauss, J., Milman, V.D.: The dimension of almost spherical sections of convex bodies. Acta Mathematica 139(1-2), 53–94 (1977)CrossRefMathSciNetzbMATHGoogle Scholar
  12. 12.
    Goodey, P., Weil,W.: Zonoids and Generalizations. Handbook of Convex Geometry, vol. B, pp. 1296–1326. North-Holland, Amsterdam (1993)Google Scholar
  13. 13.
    Gordon, Y.: Some inequalities for Gaussian processes and applications. Israel J. Math. 50(4), 265–289 (1985)CrossRefMathSciNetzbMATHGoogle Scholar
  14. 14.
    Harrow, A.W.: The Church of the Symmetric Subspace. arXiv:1308.6595 [quant-ph]
  15. 15.
    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]
  16. 16.
    Helstrom, C.W.: Quantum detection and estimation theory. Academic Press, New York (1976)zbMATHGoogle Scholar
  17. 17.
    Holevo, A.S.: Statistical decision theory for quantum systems. J. Mult. Anal. 3, 337–394 (1973)CrossRefMathSciNetzbMATHGoogle Scholar
  18. 18.
    Indyk, P.: Uncertainty principles, extractors, and explicit embeddings of \(L_2\) into \(L_1\). In: 39th ACM Symposium on Theory of Computing (2007)Google Scholar
  19. 19.
    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]
  20. 20.
    Lancien, C., Winter, A.: Distinguishing multi-partite states by local measurements. Commun. Math. Phys. 323, 555–573 (2013). arXiv:1206.2884 [quant-ph]
  21. 21.
    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
  22. 22.
    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]
  23. 23.
    Pisier, G.: The volume of convex bodies and banach spaces geometry, Cambridge tracts in mathematics, 94th edn. Cambridge University Press, Cambridge (1989)CrossRefGoogle Scholar
  24. 24.
    Rosental, H., Szarek, S.: On tensor products of operators from \(L^p\) to \(L^q\). Functional Analysis, pp. 108–132. Springer, Berlin Heidelberg (1991)Google Scholar
  25. 25.
    Rudin, W.: Trigonometric series with gaps. J. Math. Mech 9(2), 203–227Google Scholar
  26. 26.
    Rudin, W.: Functional analysis. McGraw-Hill International Series in pure and applied Mathematics, Singapore (1973)Google Scholar
  27. 27.
    Rudin, W.: Real and complex analysis. McGraw-Hill International Editions, Mathematics Series, Singapore (1987)Google Scholar
  28. 28.
    Schechtman, G.: More on embedding subspaces of \(L_p\) in \(l^n_r\). Compositio Math. 61(2), 159–169 (1987)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Schenider, R., Weil, W.: Zonoids and related topics. Convexity Appl. 296–317 (1983)Google Scholar
  30. 30.
    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
  31. 31.
    Talagrand, M.: Embedding subspaces of \(L_1\) into \(\ell _1^N\). Proc. Am. Math. Soc. 108(2), 363–369 (1990)MathSciNetzbMATHGoogle Scholar

Copyright information

© 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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