Communications in Mathematical Physics

, Volume 265, Issue 1, pp 95–117 | Cite as

Aspects of Generic Entanglement

Article

Abstract

We study entanglement and other correlation properties of random states in high-dimensional bipartite systems. These correlations are quantified by parameters that are subject to the ``concentration of measure'' phenomenon, meaning that on a large-probability set these parameters are close to their expectation. For the entropy of entanglement, this has the counterintuitive consequence that there exist large subspaces in which all pure states are close to maximally entangled. This, in turn, implies the existence of mixed states with entanglement of formation near that of a maximally entangled state, but with negligible quantum mutual information and, therefore, negligible distillable entanglement, secret key, and common randomness. It also implies a very strong locking effect for the entanglement of formation: its value can jump from maximal to near zero by tracing over a number of qubits negligible compared to the size of the total system. Furthermore, such properties are generic. Similar phenomena are observed for random multiparty states, leading us to speculate on the possibility that the theory of entanglement is much simplified when restricted to asymptotically generic states. Further consequences of our results include a complete derandomization of the protocol for universal superdense coding of quantum states.

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References

  1. 1.
    Abeyesinghe, A., Hayden, P., Smith, G., Winter, A.: Optimal superdense coding of entangled states. http://arxiv/org/list/quant-ph/0407061, 2004
  2. 2.
    Ahlswede, R., Csiszár. I.: Common randomness in information theory and cryptography - Part II: CR-capacity. IEEE Trans. Inf. Theory 44, 225–240 (1998)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Bennett, G.H., DiVincenzo, D.P., Smolin, J.A., Wootters, W.K.: Mixed state entanglement and quantum error correction. Phys. Rev. A 54, 3824–3851 (1996)CrossRefADSMathSciNetGoogle Scholar
  4. 4.
    Braunstein. S.L.: Geometry of quantum inference. Phys. Lett. A 247, 169 (1996)Google Scholar
  5. 5.
    Briegel, H.J., Raussendorf, R.: Persistent entanglement in arrays of interacting particles. Phys. Rev. Lett. 86(5), 910–913 (2001)CrossRefGoogle Scholar
  6. 6.
    Christandl, M., Winter, A.: Squashed entanglement – An additive entanglement measure. J. Math. Phys. 45(3), 829–840 (2004)CrossRefMathSciNetGoogle Scholar
  7. 7.
    Dembo, A., Zeitouni, O.: Large Deviations Techniques and Applications. New York: Springer-Verlag, 1993Google Scholar
  8. 8.
    Devetak, I.: The private classical capacity and quantum capacity of a quantum channel. IEEE Trans. Inf. Theory 51(1), 44–55 (2005)CrossRefMathSciNetGoogle Scholar
  9. 9.
    Devetak, I., Winter, A.: Distilling common randomness from bipartite quantum states. IEEE Trans. Inf. Theory 50(12), 3183–3196 (2004)CrossRefMathSciNetGoogle Scholar
  10. 10.
    Devetak, I., Winter, A.: Relating quantum privacy and quantum coherence: an operational approach. Phys. Rev. Lett. 93(8), 080501 (2004)CrossRefMathSciNetGoogle Scholar
  11. 11.
    Devetak, I., Winter, A.: Distillation of secret key and entanglement from quantum states. Proc. Roy. Soc. Lond. A 461, 207–235 (2005)ADSMathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Donald, M., Horodecki, M., Rudolph, O.: The uniqueness theorem for entanglement measures. J. Math. Phys. 43, 4252–4272 (2002)CrossRefADSMATHMathSciNetGoogle Scholar
  13. 13.
    Duistermaat, J.J., Polk, J.A.C.: Lie Groups. Berlin: Springer-Verlag, 1999Google Scholar
  14. 14.
    Emerson, J., Weinstein, Y.S., Saraceno, M., Lloyd, S., Cory, D.G.: Pseudo-random unitary operators for quantum information processing. Science 302, 2098 (2003)CrossRefADSMathSciNetGoogle Scholar
  15. 15.
    Foong, S.K., Kanno, S.: Proof of Page's conjecture on the average entropy of a subsystem. Phys. Rev. Lett. 72, 1148–1151 (1994)CrossRefADSMATHMathSciNetGoogle Scholar
  16. 16.
    Fuchs, C.A., van de Graaf, J.: Cryptographic distinguishability measures for quantum mechanical states. IEEE Trans. Inf. Theory 45, 1216–1227 (1999)CrossRefMATHGoogle Scholar
  17. 17.
    Gurvits, L., Barnum, H., Separable balls around the maximally mixed multipartite quantum states. Phys. Rev. A 68, 042312 (2003)Google Scholar
  18. 18.
    Hall, M.J.W.: Random quantum correlations and density operator distributions. Phys. Lett. A 242, 123–129 (1998)CrossRefADSMATHMathSciNetGoogle Scholar
  19. 19.
    Harris, J.: Algebraic Geometry: A First Course, Volume 133 of Graduate Texts in Mathematics. Berlin-Heidelberg-New York: Springer Verlag, 1992Google Scholar
  20. 20.
    Harrow, A., Hayden, P., Leung, D.W.: Superdense coding of quantum states. Phys. Rev. Lett. 92, 187901 (2004)CrossRefADSMathSciNetGoogle Scholar
  21. 21.
    Hayden, P., Horodecki, M., Terhal, B.M.: The asymptotic entanglement cost of preparing a quantum state. J. Phys. A 34(35), 6891–6898 (2001)CrossRefMathSciNetGoogle Scholar
  22. 22.
    Hayden, P., Leung, D.W., Shor, P.W., Winter, A.: Randomizing quantum states: Constructions and applications. Commun. Math. Phys. 250, 371–391 (2004)CrossRefADSMATHMathSciNetGoogle Scholar
  23. 23.
    Hayden, P., Leung, D.W., Smith, G.: Multiparty data hiding of quantum information. Phys. Rev. A 71, 062339 (2005)CrossRefADSGoogle Scholar
  24. 24.
    Holevo, A.S.: Bounds for the quantity of information transmittable by a quantum communications channel. Problemy peredači Informacii 9(3), 3–11 (1973); English translation: Holevo, A.S.: probl. Inf. Transm. 9, 177–183 (1973)MathSciNetGoogle Scholar
  25. 25.
    Horodecki, K., Horodecki, M., Horodecki, P., Oppenheim, J.: Locking entanglement measures with a single qubit. http://arxiv.org/list/quant-ph/0404096, 2004
  26. 26.
    Horodecki, M.: Entanglement measures. Quantum Inf. Comp. 1(1), 3–26 (2001)MathSciNetGoogle Scholar
  27. 27.
    Horodecki, M., Horodecki, P., Horodecki, R.: Inseparable two spin-1/2 density matrices can be distilled to a singlet form. Phys. Rev. Lett. 78, 574 (1997)CrossRefADSGoogle Scholar
  28. 28.
    Horodecki, M., Horodecki, P., Horodecki, R.: Mixed-state entanglement and distillation: Is there a ``bound'' entanglement in nature? Phys. Rev. Lett. 80, 5239 (1998)CrossRefADSMATHMathSciNetGoogle Scholar
  29. 29.
    Jozsa, R.: Fidelity for mixed quantum states. J. Mod. Opt. 41, 2315–2323 (1994)ADSMATHMathSciNetGoogle Scholar
  30. 30.
    Kendon, V., Życzkowski, K., Munro, W.: Bounds on entanglement in qudit subsystems. Phys. Rev. A. 66, 062310 (2002)CrossRefADSGoogle Scholar
  31. 31.
    Koashi, M.,Winter, A.: Monogamy of quantum entanglement and other correlations. Phys. Rev. A 69, 022309 (2004)CrossRefADSMathSciNetGoogle Scholar
  32. 32.
    Ledoux, M.: The concentration of measure phenomenon, Volume 89 of Mathematical Surveys and Monographs. Providence, RI: American Math Soc, 2001Google Scholar
  33. 33.
    Lloyd, S., Pagels, H.: Complexity as thermodynamic depth. Ann. Phys. 188(1), 186–213 (1988)MathSciNetGoogle Scholar
  34. 34.
    Lubkin, K., Entropy of an n-system from its correlation with a k-reservoir. J. Math. Phys. 19, 1028 (1978)Google Scholar
  35. 35.
    Milman, V.D., Schechtman, G.: Asymptotic theory of finite dimensional normed spaces, Volume 1200 of Lecture Notes in Mathematics. Berlin-Heidelberg-New york: Springer-Verlag, 1986Google Scholar
  36. 36.
    Nielsen, M.A., Chuang, I.L.: Quantum computation and quantum information. Cambridge, UK: Cambridge University Press, 2000Google Scholar
  37. 37.
    Ohya, M., Petz, D.: Quantum entropy and its use. Texts and monographs in physics. Berlin: Springer-Verlag, 1993Google Scholar
  38. 38.
    Oppenheim, J., Horodecki, K., Horodecki, M., Horodecki, P., Horodecki, R.; A new type of complementarity between quantum and classical information. Phys. Rev. A 68, 022307 (2003)Google Scholar
  39. 39.
    Page, D.N.: Average entropy of a subsystem. Phys. Rev. Lett. 71, 1291 (1993)CrossRefADSMATHMathSciNetGoogle Scholar
  40. 40.
    Parthasarathy, K.R.: On the maximal dimension of a completely entangled subspace for finite level quantum systems. Proc. Indian Acad. Sci. (Math. Sci.) 114(4), 365–374 (2004)MathSciNetCrossRefGoogle Scholar
  41. 41.
    Peres, A.: Separability criterion for density matrices. Phys. Rev. Lett. 77, 1413 (1996)CrossRefADSMATHMathSciNetGoogle Scholar
  42. 42.
    Sanchez-Ruiz, J.: Simple proof of Page's conjecture on the average entropy of a subsystem. Phys. Rev. E 52, 5653 (1995)CrossRefADSMathSciNetGoogle Scholar
  43. 43.
    Schumacher, B., Westmoreland, M.D.: Approximate quantum error correction. Quantum Inf. Proc. 1(1–2), 5–12 (2002)Google Scholar
  44. 44.
    Sen, S.: Average entropy of a quantum subsystem. Phys. Rev. Lett. 77(1), 1–3 (1996)CrossRefGoogle Scholar
  45. 45.
    Shor, P.W.; Equivalence of additivity questions in quantum information theory. Commun. Math. Phys. 246(3), 453–472 (2004)Google Scholar
  46. 46.
    Sommers, H.-J., Życzkowski, K.: Statistical properties of random density matrices. J. Phys. A: Math. Gen. 37(35), 8457–8466 (2004)CrossRefMathSciNetGoogle Scholar
  47. 47.
    Szarek, S.: The volume of separable states is super-doubly-exponentially small. Phys. Rev. A 72, 032304 (2005)CrossRefADSMathSciNetGoogle Scholar
  48. 48.
    Uhlmann, A.: The `transition probability' in the state space of a *-algebra. Rep. Math. Phys. 9, 273 (1976)CrossRefMATHMathSciNetADSGoogle Scholar
  49. 49.
    Umegaki, H.: Conditional expectations in an operator algebra IV (entropy and information). Kodai Math. Sem. Rep. 14, 59–85 (1962)MATHMathSciNetGoogle Scholar
  50. 50.
    Vedral, V., Plenio, M.B., Rippin, M.A., Knight, P.L.; Quantifying entanglement. Phys. Rev. Lett. 78(12), 2275–2279 (1997)Google Scholar
  51. 51.
    Vidal, G., Cirac, J.I.: Irreversibility in asymptotic manipulations of entanglement. Phys. Rev. Lett. 86, 5803–5806 (2001)CrossRefADSGoogle Scholar
  52. 52.
    Vidal, G., Dür, W., Cirac, J.I.: Entanglement cost of mixed states. Phys. Rev. Lett. 89, 027901 (2002)CrossRefADSGoogle Scholar
  53. 53.
    von Neumann, J.: Thermodynamik quantenmechanischer Gesamtheiten. Nachr. der Gesellschaft der Wiss, Gött. 273–291 (1927)Google Scholar
  54. 54.
    Weinstein, Y.S., Hellberg, C.S.: Matrix element randomness, entanglement, and quantum chaos. http://arxiv.org/list/quant-ph/0405053, 2004
  55. 55.
    Young, R.M.: Euler's constant. Math. Gaz. 75, 187–190 (1991)MATHGoogle Scholar
  56. 56.
    Zanardi, P., Zalka, C., Faoro, L.: On the entangling power of quantum evolutions. Phys. Rev. A 62, 030301 (2000)CrossRefADSMathSciNetGoogle Scholar
  57. 57.
    Życzkowski, K: On the volume of the set of mixed entangled states II. Phys. Rev. A. 60, 3496 (1999)CrossRefADSMathSciNetGoogle Scholar
  58. 58.
    Życzkowski, K., Horodecki, P., Sanpera, A., Lewenstein, M.: Volume of the set of separable states. Phys. Rev. A. 58, 883–892 (1998)CrossRefADSMathSciNetGoogle Scholar
  59. 59.
    Życzkowski, K., Sommers, H.-J.: Induced measures in the space of mixed quantum states. J. Phys. A: Math. Gen. 34, 7111–7125 (2001)CrossRefADSMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Patrick Hayden
    • 1
    • 2
  • Debbie W. Leung
    • 1
  • Andreas Winter
    • 3
  1. 1.Institute for Quantum InformationPasadenaUSA
  2. 2.Department of Computer ScienceMcGill UniversityMontrealCanada
  3. 3.Department of MathematicsUniversity of BristolBristolUnited Kingdom

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