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Randomizing Quantum States: Constructions and Applications

Hayden, Patrick and Leung, Debbie and Shor, Peter W. and Winter, Andreas (2004) Randomizing Quantum States: Constructions and Applications. Communications in Mathematical Physics, 250 (2). pp. 371-391. ISSN 0010-3616. doi:10.1007/s00220-004-1087-6.

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The construction of a perfectly secure private quantum channel in dimension d is known to require 2 log d shared random key bits between the sender and receiver. We show that if only near-perfect security is required, the size of the key can be reduced by a factor of two. More specifically, we show that there exists a set of roughly d log d unitary operators whose average effect on every input pure state is almost perfectly randomizing, as compared to the d^2 operators required to randomize perfectly. Aside from the private quantum channel, variations of this construction can be applied to many other tasks in quantum information processing. We show, for instance, that it can be used to construct LOCC data hiding schemes for bits and qubits that are much more efficient than any others known, allowing roughly log d qubits to be hidden in 2 log d qubits. The method can also be used to exhibit the existence of quantum states with locked classical correlations, an arbitrarily large amplification of the correlation being accomplished by sending a negligibly small classical key. Our construction also provides the basic building block for a method of remotely preparing arbitrary d-dimensional pure quantum states using approximately log d bits of communication and log d ebits of entanglement.

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Additional Information:© 2004 Springer-Verlag. Received: 29 August 2003; Accepted: 14 November 2003; Published online: 8 July 2004. Communicated by M.B. Ruskai. We thank Daniel Gottesman, Leonid Gurvits, Karol Zyczkowski and, in particular, Charles Bennett for their helpful suggestions. PH and DL acknowledge the support of the Sherman Fairchild Foundation, the Richard C. Tolman Foundation, the Croucher Foundation and the US National Science Foundation under grant no. EIA-0086038.AWis supported by the U.K. Engineering and Physical Sciences Research Council.
Funding AgencyGrant Number
Sherman Fairchild FoundationUNSPECIFIED
Richard C. Tolman FoundationUNSPECIFIED
Croucher FoundationUNSPECIFIED
Engineering and Physical Sciences Research Council (EPSRC)UNSPECIFIED
Subject Keywords:randomization, quantum cryptography, private quantum channel, data hiding, locking correlations, entropic uncertainty relations
Issue or Number:2
Classification Code:PACS: 03.65.Ta, 03.67.Hk
Record Number:CaltechAUTHORS:20110817-091058387
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:24904
Deposited By: Tony Diaz
Deposited On:15 Sep 2011 16:18
Last Modified:09 Nov 2021 16:27

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