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Aspects of Generic Entanglement

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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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Authors and Affiliations

  1. Institute for Quantum Information, Caltech 107–81, Pasadena, CA, 91125, USA

    Patrick Hayden & Debbie W. Leung

  2. Department of Computer Science, McGill University, Montreal, Quebec, H3A 2A7, Canada

    Patrick Hayden

  3. Department of Mathematics, University of Bristol, Bristol, BS8 1TW, United Kingdom

    Andreas Winter

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  1. Patrick Hayden
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  2. Debbie W. Leung
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  3. Andreas Winter
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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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