Limitations on quantum dimensionality reduction

Authors:
Aram W. Harrow;Ashley Montanaro;Anthony J. Short
Affiliations:
Department of Computer Science & Engineering, University of Washington, Seattle and Department of Mathematics, University of Bristol, Bristol, UK;Centre for Quantum Information and Foundations, DAMTP, University of Cambridge, Cambridge, UK;Centre for Quantum Information and Foundations, DAMTP, University of Cambridge, Cambridge, UK
Venue:
ICALP'11 Proceedings of the 38th international colloquim conference on Automata, languages and programming - Volume Part I
Year:
2011

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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.