Limitations on quantum dimensionality reduction
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Harrow_Limitations on.pdf
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199.2 KB
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Author(s)
Harrow, Aram W.
Montanaro, Ashley
Short, Anthony J.
Date Issued
June 2014
Journal
International Journal of Quantum Information
Publisher
World Scientific
Citation
Harrow, Aram W., Ashley Montanaro, and Anthony J. Short. “Limitations on Quantum Dimensionality Reduction.” Int. J. Quantum Inform. 13, no. 04 (June 2015): 1440001.
Version
Final published version
Abstract
The Johnson–Lindenstrauss Lemma is a classic result which implies that any set of n real vectors can be compressed to O(log n) 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.
MIT Department
Massachusetts Institute of Technology. Center for Theoretical Physics
Massachusetts Institute of Technology. Department of Physics
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Creative Commons Attribution
Persistent DSpace Link
DOI of Published Version
https://doi.org/10.1142/S0219749914400012
