Limitations on Quantum Dimensionality Reduction

  • Aram W. Harrow
  • Ashley Montanaro
  • Anthony J. Short
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6755)

Abstract

The Johnson-Lindenstrauss Lemma is a classic result which implies that any set of n real vectors can be compressed to O(logn) dimensions while only distorting pairwise Euclidean distances by a constant factor. Here we consider potential extensions of this result to the compression of quantum states. We show that, by contrast with the classical case, there does not exist any distribution over quantum channels that significantly reduces the dimension of quantum states while preserving the 2-norm distance with high probability. We discuss two tasks for which the 2-norm distance is indeed the correct figure of merit. In the case of the trace norm, we show that the dimension of low-rank mixed states can be reduced by up to a square root, but that essentially no dimensionality reduction is possible for highly mixed states.

Keywords

Dimensionality Reduction Failure Probability Quantum Channel Trace Norm State Discrimination 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Aram W. Harrow
    • 1
    • 2
  • Ashley Montanaro
    • 3
  • Anthony J. Short
    • 3
  1. 1.Department of Computer Science & EngineeringUniversity of WashingtonSeattleUSA
  2. 2.Department of MathematicsUniversity of BristolBristolUK
  3. 3.Centre for Quantum Information and Foundations, DAMTPUniversity of CambridgeCambridgeUK

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