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
Abeyesinghe, A., Hayden, P., Smith, G., Winter, A.: Optimal superdense coding of entangled states. http://arxiv/org/list/quant-ph/0407061, 2004
Ahlswede, R., Csiszár. I.: Common randomness in information theory and cryptography - Part II: CR-capacity. IEEE Trans. Inf. Theory 44, 225–240 (1998)
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)
Braunstein. S.L.: Geometry of quantum inference. Phys. Lett. A 247, 169 (1996)
Briegel, H.J., Raussendorf, R.: Persistent entanglement in arrays of interacting particles. Phys. Rev. Lett. 86(5), 910–913 (2001)
Christandl, M., Winter, A.: Squashed entanglement – An additive entanglement measure. J. Math. Phys. 45(3), 829–840 (2004)
Dembo, A., Zeitouni, O.: Large Deviations Techniques and Applications. New York: Springer-Verlag, 1993
Devetak, I.: The private classical capacity and quantum capacity of a quantum channel. IEEE Trans. Inf. Theory 51(1), 44–55 (2005)
Devetak, I., Winter, A.: Distilling common randomness from bipartite quantum states. IEEE Trans. Inf. Theory 50(12), 3183–3196 (2004)
Devetak, I., Winter, A.: Relating quantum privacy and quantum coherence: an operational approach. Phys. Rev. Lett. 93(8), 080501 (2004)
Devetak, I., Winter, A.: Distillation of secret key and entanglement from quantum states. Proc. Roy. Soc. Lond. A 461, 207–235 (2005)
Donald, M., Horodecki, M., Rudolph, O.: The uniqueness theorem for entanglement measures. J. Math. Phys. 43, 4252–4272 (2002)
Duistermaat, J.J., Polk, J.A.C.: Lie Groups. Berlin: Springer-Verlag, 1999
Emerson, J., Weinstein, Y.S., Saraceno, M., Lloyd, S., Cory, D.G.: Pseudo-random unitary operators for quantum information processing. Science 302, 2098 (2003)
Foong, S.K., Kanno, S.: Proof of Page's conjecture on the average entropy of a subsystem. Phys. Rev. Lett. 72, 1148–1151 (1994)
Fuchs, C.A., van de Graaf, J.: Cryptographic distinguishability measures for quantum mechanical states. IEEE Trans. Inf. Theory 45, 1216–1227 (1999)
Gurvits, L., Barnum, H., Separable balls around the maximally mixed multipartite quantum states. Phys. Rev. A 68, 042312 (2003)
Hall, M.J.W.: Random quantum correlations and density operator distributions. Phys. Lett. A 242, 123–129 (1998)
Harris, J.: Algebraic Geometry: A First Course, Volume 133 of Graduate Texts in Mathematics. Berlin-Heidelberg-New York: Springer Verlag, 1992
Harrow, A., Hayden, P., Leung, D.W.: Superdense coding of quantum states. Phys. Rev. Lett. 92, 187901 (2004)
Hayden, P., Horodecki, M., Terhal, B.M.: The asymptotic entanglement cost of preparing a quantum state. J. Phys. A 34(35), 6891–6898 (2001)
Hayden, P., Leung, D.W., Shor, P.W., Winter, A.: Randomizing quantum states: Constructions and applications. Commun. Math. Phys. 250, 371–391 (2004)
Hayden, P., Leung, D.W., Smith, G.: Multiparty data hiding of quantum information. Phys. Rev. A 71, 062339 (2005)
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)
Horodecki, K., Horodecki, M., Horodecki, P., Oppenheim, J.: Locking entanglement measures with a single qubit. http://arxiv.org/list/quant-ph/0404096, 2004
Horodecki, M.: Entanglement measures. Quantum Inf. Comp. 1(1), 3–26 (2001)
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)
Horodecki, M., Horodecki, P., Horodecki, R.: Mixed-state entanglement and distillation: Is there a ``bound'' entanglement in nature? Phys. Rev. Lett. 80, 5239 (1998)
Jozsa, R.: Fidelity for mixed quantum states. J. Mod. Opt. 41, 2315–2323 (1994)
Kendon, V., Życzkowski, K., Munro, W.: Bounds on entanglement in qudit subsystems. Phys. Rev. A. 66, 062310 (2002)
Koashi, M.,Winter, A.: Monogamy of quantum entanglement and other correlations. Phys. Rev. A 69, 022309 (2004)
Ledoux, M.: The concentration of measure phenomenon, Volume 89 of Mathematical Surveys and Monographs. Providence, RI: American Math Soc, 2001
Lloyd, S., Pagels, H.: Complexity as thermodynamic depth. Ann. Phys. 188(1), 186–213 (1988)
Lubkin, K., Entropy of an n-system from its correlation with a k-reservoir. J. Math. Phys. 19, 1028 (1978)
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
Nielsen, M.A., Chuang, I.L.: Quantum computation and quantum information. Cambridge, UK: Cambridge University Press, 2000
Ohya, M., Petz, D.: Quantum entropy and its use. Texts and monographs in physics. Berlin: Springer-Verlag, 1993
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)
Page, D.N.: Average entropy of a subsystem. Phys. Rev. Lett. 71, 1291 (1993)
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)
Peres, A.: Separability criterion for density matrices. Phys. Rev. Lett. 77, 1413 (1996)
Sanchez-Ruiz, J.: Simple proof of Page's conjecture on the average entropy of a subsystem. Phys. Rev. E 52, 5653 (1995)
Schumacher, B., Westmoreland, M.D.: Approximate quantum error correction. Quantum Inf. Proc. 1(1–2), 5–12 (2002)
Sen, S.: Average entropy of a quantum subsystem. Phys. Rev. Lett. 77(1), 1–3 (1996)
Shor, P.W.; Equivalence of additivity questions in quantum information theory. Commun. Math. Phys. 246(3), 453–472 (2004)
Sommers, H.-J., Życzkowski, K.: Statistical properties of random density matrices. J. Phys. A: Math. Gen. 37(35), 8457–8466 (2004)
Szarek, S.: The volume of separable states is super-doubly-exponentially small. Phys. Rev. A 72, 032304 (2005)
Uhlmann, A.: The `transition probability' in the state space of a *-algebra. Rep. Math. Phys. 9, 273 (1976)
Umegaki, H.: Conditional expectations in an operator algebra IV (entropy and information). Kodai Math. Sem. Rep. 14, 59–85 (1962)
Vedral, V., Plenio, M.B., Rippin, M.A., Knight, P.L.; Quantifying entanglement. Phys. Rev. Lett. 78(12), 2275–2279 (1997)
Vidal, G., Cirac, J.I.: Irreversibility in asymptotic manipulations of entanglement. Phys. Rev. Lett. 86, 5803–5806 (2001)
Vidal, G., Dür, W., Cirac, J.I.: Entanglement cost of mixed states. Phys. Rev. Lett. 89, 027901 (2002)
von Neumann, J.: Thermodynamik quantenmechanischer Gesamtheiten. Nachr. der Gesellschaft der Wiss, Gött. 273–291 (1927)
Weinstein, Y.S., Hellberg, C.S.: Matrix element randomness, entanglement, and quantum chaos. http://arxiv.org/list/quant-ph/0405053, 2004
Young, R.M.: Euler's constant. Math. Gaz. 75, 187–190 (1991)
Zanardi, P., Zalka, C., Faoro, L.: On the entangling power of quantum evolutions. Phys. Rev. A 62, 030301 (2000)
Życzkowski, K: On the volume of the set of mixed entangled states II. Phys. Rev. A. 60, 3496 (1999)
Życzkowski, K., Horodecki, P., Sanpera, A., Lewenstein, M.: Volume of the set of separable states. Phys. Rev. A. 58, 883–892 (1998)
Życzkowski, K., Sommers, H.-J.: Induced measures in the space of mixed quantum states. J. Phys. A: Math. Gen. 34, 7111–7125 (2001)
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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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DOI: https://doi.org/10.1007/s00220-006-1535-6