Limitations on quantum dimensionality reduction

TL;DR

This paper demonstrates fundamental limitations in reducing the dimensionality of quantum states while preserving their distances, contrasting classical results, and highlights specific cases where some reduction is possible.

Contribution

It extends classical dimensionality reduction concepts to quantum states, showing that significant reduction while preserving distances is generally impossible, with some exceptions.

Findings

No distribution over quantum channels can significantly reduce quantum state dimensions while preserving 2-norm distances.

Low-rank quantum states can have their dimension reduced up to a square root in the trace norm.

No meaningful dimensionality reduction is possible for highly mixed quantum states.

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.