Compressibility of positive semidefinite factorizations and quantum models

Abstract

We investigate compressibility of the dimension of positive semidefinite matrices while approximately preserving their pairwise inner products. This can either be regarded as compression of positive semidefinite factorizations of nonnegative matrices or (if the matrices are subject to additional normalization constraints) as compression of quantum models. We derive both lower and upper bounds on compressibility. Applications are broad and range from the statistical analysis of experimental data to bounding the one-way quantum communication complexity of Boolean functions.

13 pages

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• Device-independent dimension tests in the prepare-and-measure scenario
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• Learning optimal quantum models is NP-hard
• Recommender systems inspired by the structure of quantum theory
• Quantum compression relative to a set of measurements
• Nonlocal games with noisy maximally entangled states are decidable
• Approximate tensor decompositions: disappearance of many separations