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Aspects of Generic Entanglement

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. Abeyesinghe, A., Hayden, P., Smith, G., Winter, A.: Optimal superdense coding of entangled states. http://arxiv/org/list/quant-ph/0407061, 2004

  2. Ahlswede, R., Csiszár. I.: Common randomness in information theory and cryptography - Part II: CR-capacity. IEEE Trans. Inf. Theory 44, 225–240 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  ADS  MathSciNet  Google Scholar 

  4. Braunstein. S.L.: Geometry of quantum inference. Phys. Lett. A 247, 169 (1996)

    Google Scholar 

  5. Briegel, H.J., Raussendorf, R.: Persistent entanglement in arrays of interacting particles. Phys. Rev. Lett. 86(5), 910–913 (2001)

    Article  Google Scholar 

  6. Christandl, M., Winter, A.: Squashed entanglement – An additive entanglement measure. J. Math. Phys. 45(3), 829–840 (2004)

    Article  MathSciNet  Google Scholar 

  7. Dembo, A., Zeitouni, O.: Large Deviations Techniques and Applications. New York: Springer-Verlag, 1993

  8. Devetak, I.: The private classical capacity and quantum capacity of a quantum channel. IEEE Trans. Inf. Theory 51(1), 44–55 (2005)

    Article  MathSciNet  Google Scholar 

  9. Devetak, I., Winter, A.: Distilling common randomness from bipartite quantum states. IEEE Trans. Inf. Theory 50(12), 3183–3196 (2004)

    Article  MathSciNet  Google Scholar 

  10. Devetak, I., Winter, A.: Relating quantum privacy and quantum coherence: an operational approach. Phys. Rev. Lett. 93(8), 080501 (2004)

    Article  MathSciNet  Google Scholar 

  11. Devetak, I., Winter, A.: Distillation of secret key and entanglement from quantum states. Proc. Roy. Soc. Lond. A 461, 207–235 (2005)

    ADS  MathSciNet  MATH  Article  Google Scholar 

  12. Donald, M., Horodecki, M., Rudolph, O.: The uniqueness theorem for entanglement measures. J. Math. Phys. 43, 4252–4272 (2002)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  13. Duistermaat, J.J., Polk, J.A.C.: Lie Groups. Berlin: Springer-Verlag, 1999

  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)

    Article  ADS  MathSciNet  Google Scholar 

  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)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  16. Fuchs, C.A., van de Graaf, J.: Cryptographic distinguishability measures for quantum mechanical states. IEEE Trans. Inf. Theory 45, 1216–1227 (1999)

    Article  MATH  Google Scholar 

  17. Gurvits, L., Barnum, H., Separable balls around the maximally mixed multipartite quantum states. Phys. Rev. A 68, 042312 (2003)

    Google Scholar 

  18. Hall, M.J.W.: Random quantum correlations and density operator distributions. Phys. Lett. A 242, 123–129 (1998)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  19. Harris, J.: Algebraic Geometry: A First Course, Volume 133 of Graduate Texts in Mathematics. Berlin-Heidelberg-New York: Springer Verlag, 1992

  20. Harrow, A., Hayden, P., Leung, D.W.: Superdense coding of quantum states. Phys. Rev. Lett. 92, 187901 (2004)

    Article  ADS  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  22. Hayden, P., Leung, D.W., Shor, P.W., Winter, A.: Randomizing quantum states: Constructions and applications. Commun. Math. Phys. 250, 371–391 (2004)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  23. Hayden, P., Leung, D.W., Smith, G.: Multiparty data hiding of quantum information. Phys. Rev. A 71, 062339 (2005)

    Article  ADS  Google Scholar 

  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)

    MathSciNet  Google Scholar 

  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. Horodecki, M.: Entanglement measures. Quantum Inf. Comp. 1(1), 3–26 (2001)

    MathSciNet  Google Scholar 

  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)

    Article  ADS  Google Scholar 

  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)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  29. Jozsa, R.: Fidelity for mixed quantum states. J. Mod. Opt. 41, 2315–2323 (1994)

    ADS  MATH  MathSciNet  Google Scholar 

  30. Kendon, V., Życzkowski, K., Munro, W.: Bounds on entanglement in qudit subsystems. Phys. Rev. A. 66, 062310 (2002)

    Article  ADS  Google Scholar 

  31. Koashi, M.,Winter, A.: Monogamy of quantum entanglement and other correlations. Phys. Rev. A 69, 022309 (2004)

    Article  ADS  MathSciNet  Google Scholar 

  32. Ledoux, M.: The concentration of measure phenomenon, Volume 89 of Mathematical Surveys and Monographs. Providence, RI: American Math Soc, 2001

  33. Lloyd, S., Pagels, H.: Complexity as thermodynamic depth. Ann. Phys. 188(1), 186–213 (1988)

    MathSciNet  Google Scholar 

  34. Lubkin, K., Entropy of an n-system from its correlation with a k-reservoir. J. Math. Phys. 19, 1028 (1978)

    Google Scholar 

  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, 1986

  36. Nielsen, M.A., Chuang, I.L.: Quantum computation and quantum information. Cambridge, UK: Cambridge University Press, 2000

  37. Ohya, M., Petz, D.: Quantum entropy and its use. Texts and monographs in physics. Berlin: Springer-Verlag, 1993

  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. Page, D.N.: Average entropy of a subsystem. Phys. Rev. Lett. 71, 1291 (1993)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  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)

    MathSciNet  Article  Google Scholar 

  41. Peres, A.: Separability criterion for density matrices. Phys. Rev. Lett. 77, 1413 (1996)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  42. Sanchez-Ruiz, J.: Simple proof of Page's conjecture on the average entropy of a subsystem. Phys. Rev. E 52, 5653 (1995)

    Article  ADS  MathSciNet  Google Scholar 

  43. Schumacher, B., Westmoreland, M.D.: Approximate quantum error correction. Quantum Inf. Proc. 1(1–2), 5–12 (2002)

    Google Scholar 

  44. Sen, S.: Average entropy of a quantum subsystem. Phys. Rev. Lett. 77(1), 1–3 (1996)

    Article  Google Scholar 

  45. Shor, P.W.; Equivalence of additivity questions in quantum information theory. Commun. Math. Phys. 246(3), 453–472 (2004)

    Google Scholar 

  46. Sommers, H.-J., Życzkowski, K.: Statistical properties of random density matrices. J. Phys. A: Math. Gen. 37(35), 8457–8466 (2004)

    Article  MathSciNet  Google Scholar 

  47. Szarek, S.: The volume of separable states is super-doubly-exponentially small. Phys. Rev. A 72, 032304 (2005)

    Article  ADS  MathSciNet  Google Scholar 

  48. Uhlmann, A.: The `transition probability' in the state space of a *-algebra. Rep. Math. Phys. 9, 273 (1976)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  49. Umegaki, H.: Conditional expectations in an operator algebra IV (entropy and information). Kodai Math. Sem. Rep. 14, 59–85 (1962)

    MATH  MathSciNet  Google Scholar 

  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. Vidal, G., Cirac, J.I.: Irreversibility in asymptotic manipulations of entanglement. Phys. Rev. Lett. 86, 5803–5806 (2001)

    Article  ADS  Google Scholar 

  52. Vidal, G., Dür, W., Cirac, J.I.: Entanglement cost of mixed states. Phys. Rev. Lett. 89, 027901 (2002)

    Article  ADS  Google Scholar 

  53. von Neumann, J.: Thermodynamik quantenmechanischer Gesamtheiten. Nachr. der Gesellschaft der Wiss, Gött. 273–291 (1927)

  54. Weinstein, Y.S., Hellberg, C.S.: Matrix element randomness, entanglement, and quantum chaos. http://arxiv.org/list/quant-ph/0405053, 2004

  55. Young, R.M.: Euler's constant. Math. Gaz. 75, 187–190 (1991)

    MATH  Google Scholar 

  56. Zanardi, P., Zalka, C., Faoro, L.: On the entangling power of quantum evolutions. Phys. Rev. A 62, 030301 (2000)

    Article  ADS  MathSciNet  Google Scholar 

  57. Życzkowski, K: On the volume of the set of mixed entangled states II. Phys. Rev. A. 60, 3496 (1999)

    Article  ADS  MathSciNet  Google Scholar 

  58. Życzkowski, K., Horodecki, P., Sanpera, A., Lewenstein, M.: Volume of the set of separable states. Phys. Rev. A. 58, 883–892 (1998)

    Article  ADS  MathSciNet  Google Scholar 

  59. Życzkowski, K., Sommers, H.-J.: Induced measures in the space of mixed quantum states. J. Phys. A: Math. Gen. 34, 7111–7125 (2001)

    Article  ADS  MATH  Google Scholar 

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Correspondence to Patrick Hayden.

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Communicated by M. B. Ruskai

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Hayden, P., Leung, D. & Winter, A. Aspects of Generic Entanglement. Commun. Math. Phys. 265, 95–117 (2006). https://doi.org/10.1007/s00220-006-1535-6

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Keywords

  • Entropy
  • Quantum State
  • Mutual Information
  • Entangle State
  • Mixed State