Communications in Mathematical Physics

, Volume 250, Issue 2, pp 371–391

Randomizing Quantum States: Constructions and Applications


    • Institute for Quantum InformationCaltech 107–81
    • Mathematical SciencesResearch Institute
  • Debbie Leung
    • Mathematical SciencesResearch Institute
  • Peter W. Shor
    • AT & T Labs ResearchFlorham Park
  • Andreas Winter
    • Department of Computer ScienceUniversity of Bristol
    • Mathematical SciencesResearch Institute

DOI: 10.1007/s00220-004-1087-6

Cite this article as:
Hayden, P., Leung, D., Shor, P. et al. Commun. Math. Phys. (2004) 250: 371. doi:10.1007/s00220-004-1087-6


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 d2 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.

Copyright information

© Springer-Verlag Berlin Heidelberg 2004