Limitations on quantum dimensionality reduction

Details Limitations on quantum dimensionality reduction Limitations on quantum dimensionality reduction

2010-12-10

arxiv.org/abs/1012.2262v2

Proc. of ICALP 2011, Volume 6755/2011, pp. 86-97

Physics

Quantum Physics

17 pages; v2: typo fixes and clarifications

Scientific paper

10.1007/978-3-642-22006-7_8

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.

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