Communications in Mathematical Physics

, Volume 361, Issue 2, pp 661–708 | Cite as

Ultimate Data Hiding in Quantum Mechanics and Beyond

  • Ludovico LamiEmail author
  • Carlos Palazuelos
  • Andreas Winter


The phenomenon of data hiding, i.e. the existence of pairs of states of a bipartite system that are perfectly distinguishable via general entangled measurements yet almost indistinguishable under LOCC, is a distinctive signature of nonclassicality. The relevant figure of merit is the maximal ratio (called data hiding ratio) between the distinguishability norms associated with the two sets of measurements we are comparing, typically all measurements vs LOCC protocols. For a bipartite \({n\times n}\) quantum system, it is known that the data hiding ratio scales as n, i.e. the square root of the real dimension of the local state space of density matrices. We show that for bipartite \({n_A\times n_B}\) systems the maximum data hiding ratio against LOCC protocols is \({\Theta\left(\min\{n_A,n_B\}\right)}\). This scaling is better than the previously obtained upper bounds \({O\left(\sqrt{n_A n_B}\right)}\) and \({\min\{n_A^2, n_B^2\}}\), and moreover our intuitive argument yields constants close to optimal. In this paper, we investigate data hiding in the more general context of general probabilistic theories (GPTs), an axiomatic framework for physical theories encompassing only the most basic requirements about the predictive power of the theory. The main result of the paper is the determination of the maximal data hiding ratio obtainable in an arbitrary GPT, which is shown to scale linearly in the minimum of the local dimensions. We exhibit an explicit model achieving this bound up to additive constants, finding that quantum mechanical data hiding ratio is only of the order of the square root of the maximal one. Our proof rests crucially on an unexpected link between data hiding and the theory of projective and injective tensor products of Banach spaces. Finally, we develop a body of techniques to compute data hiding ratios for a variety of restricted classes of GPTs that support further symmetries.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Lami, L.: Non-classical correlations in quantum mechanics and beyond. Ph.D thesis (2017). arXiv:1803.02092
  2. 2.
    Terhal B.M., DiVincenzo D.P., Leung D.W.: Hiding bits in Bell states. Phys. Rev. Lett. 86, 5807 (2001) arXiv:quant-ph/0011042 CrossRefADSGoogle Scholar
  3. 3.
    DiVincenzo D.P., Leung D.W., Terhal B.M.: Quantum data hiding. IEEE Trans. Inf. Theory 48(3), 580–599 (2002) arXiv:quant-ph/0103098 MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Matthews W., Wehner S., Winter A.: Distinguishability of quantum states under restricted families of measurements with an application to quantum data hiding. Commun. Math. Phys. 291(3), 813–843 (2009)MathSciNetCrossRefzbMATHADSGoogle Scholar
  5. 5.
    Mackey G.: Mathematical Foundations of Quantum Mechanics. Benjamin, Amsterdam (1963)zbMATHGoogle Scholar
  6. 6.
    Davies E.B., Lewis J.T.: An operational approach to quantum probability. Commun. Math. Phys. 17(3), 239–260 (1970)MathSciNetCrossRefzbMATHADSGoogle Scholar
  7. 7.
    Ellis A.J.: The duality of partially ordered normed linear spaces. J. Lond. Math. Soc. 39(1), 730–744 (1964)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Edwards C.M.: Classes of operations in quantum theory. Commun. Math. Phys. 20(1), 26–56 (1971)MathSciNetCrossRefzbMATHADSGoogle Scholar
  9. 9.
    Edwards C.M.: The theory of pure operations. Commun. Math. Phys. 24(4), 260–288 (1972)MathSciNetCrossRefzbMATHADSGoogle Scholar
  10. 10.
    Ludwig, G.: An Axiomatic Basis of Quantum Mechanics, vols. 1, 2. Springer, Berlin (1985, 1987)Google Scholar
  11. 11.
    Kläy M., Randall C.H., Foulis D.J.: Tensor products and probability weights. Int. J. Theor. Phys. 26, 199 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Wilce A.: Tensor products in generalized measure theory. Int. J. Theor. Phys. 31, 1915 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Barrett J.: Information processing in generalized probabilistic theories. Phys. Rev. A 75, 032304 (2007) arXiv:quant-ph/0508211 CrossRefADSGoogle Scholar
  14. 14.
    Popescu S., Rohrlich D.: Quantum nonlocality as an axiom. Found. Phys. 24(3), 379–385 (1994)MathSciNetCrossRefADSGoogle Scholar
  15. 15.
    Barnum H., Barrett J., Leifer M., Wilce A.: Teleportation in general probabilistic theories. Proc. Symp. Appl. Math. 71, 25–48 (2012) arXiv:0805.3553 MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Janotta P., Gogolin C., Barrett J., Brunner N.: Limits on non-local correlations from the structure of the local state space. New J. Phys. 13, 063024 (2011)CrossRefADSGoogle Scholar
  17. 17.
    Barnum H., Barrett J., Leifer M., Wilce A.: Generalized no-broadcasting theorem. Phys. Rev. Lett. 99, 240501 (2007) arXiv:quant-ph/0611295 CrossRefADSGoogle Scholar
  18. 18.
    Barnum H., Gaebler C.P., Wilce A.: Ensemble steering, weak self-duality, and the structure of probabilistic theories. Found. Phys. 43(12), 1411–1427 (2013)MathSciNetCrossRefzbMATHADSGoogle Scholar
  19. 19.
    Pfister C., Wehner S.: An information-theoretic principle implies that any discrete physical theory is classical. Nat. Commun. 4, 1851 (2013)CrossRefADSGoogle Scholar
  20. 20.
    Chiribella G., Scandolo C.M.: Entanglement and thermodynamics in general probabilistic theories. New J. Phys. 17, 103027 (2015)CrossRefADSGoogle Scholar
  21. 21.
    Pfister, C.: One simple postulate implies that every polytopic state space is classical, Master thesis (2013). arXiv:1203.5622
  22. 22.
    Barnum H., Wilce A.: Information processing in convex operational theories. Electron. Notes Theor. Comput. Sci. 270(1), 3–15 (2011) arXiv:0908.2352 CrossRefzbMATHGoogle Scholar
  23. 23.
    Krein M.: Sur la decomposition minimale d’une fonctionnelle lineaire en composantes positives. Dokl. Akad. Nauk SSSR (NS) 28, 18–22 (1940)zbMATHGoogle Scholar
  24. 24.
    Boyd S., Vandenberghe L.: Convex Optimization. Cambridge University Press, Cambridge (2004)CrossRefzbMATHGoogle Scholar
  25. 25.
    Rudin W.: Functional Analysis. McGraw-Hill, New York (1991)zbMATHGoogle Scholar
  26. 26.
    Aubrun G., Szarek S.: Alice and Bob Meet Banach: The Interface of Asymptotic Geometric Analysis and Quantum Information Theory. American Mathematical Society, Providence (2017)CrossRefzbMATHGoogle Scholar
  27. 27.
    Namioka I., Phelphs R.R.: Tensor product of compact convex sets. Pac. J. Math. 31(2), 469–480 (1969)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Mulansky, B.: Tensor products of convex cones. In: Nürnberger, G., Schmidt, J.W., Walz, G. (eds.) Multivariate Approximation and Splines, vol. 125. ISNM, Lübeck (1997)Google Scholar
  29. 29.
    Kimura G., Miyadera T., Imai H.: Optimal state discrimination in general probabilistic theories. Phys. Rev. A 79, 062306 (2009)CrossRefADSGoogle Scholar
  30. 30.
    Helstrom C.W.: Quantum Detection and Estimation Theory. Academic Press, Cambridge (1976)zbMATHGoogle Scholar
  31. 31.
    Chitambar E., Leung D., Mančinska L., Ozols M., Winter A.: Everything you always wanted to know about LOCC (but were afraid to ask). Commun. Math. Phys. 328(1), 303–326 (2014)MathSciNetCrossRefzbMATHADSGoogle Scholar
  32. 32.
    Walgate J., Short A.J., Hardy L., Vedral V.: Local distinguishability of multipartite orthogonal quantum states. Phys. Rev. Lett. 85, 4972 (2000)CrossRefADSGoogle Scholar
  33. 33.
    Virmani S., Sacchi M.F., Plenio M.B., Markham D.: Optimal local discrimination of two multipartite pure states. Phys. Lett. A 288(2), 62–68 (2001)MathSciNetCrossRefzbMATHADSGoogle Scholar
  34. 34.
    Matthews W., Winter A.: On the Chernoff distance for asymptotic LOCC discrimination of bipartite quantum states. Commun. Math. Phys. 285(1), 161–174 (2008) arXiv:0710.4113 MathSciNetCrossRefzbMATHADSGoogle Scholar
  35. 35.
    Brandão F.G.S.L., Horodecki M.: Exponential decay of correlations implies area law. Commun. Math. Phys. 333(2), 761–798 (2015)MathSciNetCrossRefzbMATHADSGoogle Scholar
  36. 36.
    Vidal G., Tarrach R.: Robustness of entanglement. Phys. Rev. A 59, 141–155 (1999) arXiv:quant-ph/9806094 MathSciNetCrossRefADSGoogle Scholar
  37. 37.
    Horodecki M., Horodecki P.: Reduction criterion of separability and limits for a class of protocols of entanglement distillation. Phys. Rev. A 59, 4206 (1999)CrossRefADSGoogle Scholar
  38. 38.
    Bennett C.H., Brassard G., Crépeau C., Jozsa R., Peres A., Wootters W. K.: Teleporting an unknown quantum state via dual classical and Einstein–Podolsky–Rosen channels. Phys. Rev. Lett. 70, 1895–1899 (1993)MathSciNetCrossRefzbMATHADSGoogle Scholar
  39. 39.
    Werner R.F.: Quantum states with Einstein–Podolsky–Rosen correlations admitting a hidden-variable model. Phys. Rev. A. 40(8), 4277–4281 (1989)CrossRefzbMATHADSGoogle Scholar
  40. 40.
    Vollbrecht K.G.H., Werner R.F.: Entanglement measures under symmetry. Phys. Rev. A 64, 062307 (2001)CrossRefADSGoogle Scholar
  41. 41.
    Wilce, A.: Four and a half axioms for finite dimensional quantum mechanics (2009). arXiv:0912.5530
  42. 42.
    Hardy, L.: Quantum theory from five reasonable axioms (2001). arXiv:quant-ph/0101012
  43. 43.
    Masanes L., Müller M.P., Augusiak R., Pérez-García D.: Existence of an information unit as a postulate of quantum theory. Proc. Natl Acad. Sci. USA 110, 16373–16377 (2013)CrossRefADSGoogle Scholar
  44. 44.
    Koecher M.: Die geoodätischen von Positivitaätsbereichen. Math. Ann. 135, 192–202 (1958)MathSciNetCrossRefGoogle Scholar
  45. 45.
    Vinberg E.B.: Homogeneous cones. Dokl. Acad. Nauk. SSSR 141, 270–273 (1960) (English trans. Sov. Math. Dokl. 2, 1416–1619 (1961))zbMATHGoogle Scholar
  46. 46.
    Defant A., Floret K.: Tensor Norms and Operator Ideals. North-Holland, Amsterdam (1993)zbMATHGoogle Scholar
  47. 47.
    Ryan R.A.: Introduction to Tensor Products of Banach Spaces. Springer, Berlin (2002)CrossRefzbMATHGoogle Scholar
  48. 48.
    Pisier G.: Counterexamples to a conjecture of Grothendieck. Acta Math. 151(1), 181–208 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    Pisier, G.: Finite rank projections on Banach spaces and a conjecture of Grothendieck. In: Proceedings of the International Congress of Mathematicians, vol. 2, Warszawa, pp. 1027–1039 (1983)Google Scholar
  50. 50.
    Lindenstrauss J., Tzafriri L.: Classical Banach Spaces I and II. Springer, Berlin (1977)CrossRefzbMATHGoogle Scholar
  51. 51.
    Erdös P., Spencer J.: Probabilistic Methods in Combinatorics. Academic Press, Cambridge (1974)zbMATHGoogle Scholar
  52. 52.
    RödlP. Frankl V., Wilson R. M.: The number of submatrices of a given type in a Hadamard matrix and related results. J. Combin. Theory Ser. B 44(3), 317–328 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  53. 53.
    Alon N.: Approximating sparse binary matrices in the cut-norm. Linear Algebra Appl. 486, 409–418 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  54. 54.
    Alon N., Spencer J.H.: The Probabilistic Method, 3rd edn. Wiley, New York (2015)zbMATHGoogle Scholar
  55. 55.
    de Launey W., Gordon D.M.: A comment on the Hadamard conjecture. J. Combin. Theory Ser. A 95(1), 180–184 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  56. 56.
    Szarek S.: On the best constants in the Khinchin Inequality. Stud. Math. 58(2), 197–208 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  57. 57.
    Haagerup U.: The best constants in the Khintchine inequality. Stud. Math. 70(3), 231–283 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  58. 58.
    Latala R., Oleszkiewicz K.: On the best constant in the Khintchine-Kahane inequality. Stud. Math. 109(1), 101–104 (1994)zbMATHGoogle Scholar
  59. 59.
    Hayden P., Leung D.W., Winter A.: Aspects of generic entanglement. Commun. Math. Phys. 265(1), 95–117 (2006)MathSciNetCrossRefzbMATHADSGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Física Teòrica: Informació i Fenòmens Quàntics, Departament de FísicaUniversitat Autònoma de BarcelonaBellaterraSpain
  2. 2.Departamento de Análisis y Matemática AplicadaUniversidad Complutense de MadridMadridSpain
  3. 3.Instituto de Ciencias MatemáticasUniversidad Complutense de MadridMadridSpain
  4. 4.ICREA – Institució Catalana de Recerca i Estudis AvançatsBarcelonaSpain

Personalised recommendations