Correlation matrices, Clifford algebras, and completely positive semidefinite rank

Abstract

ABSTRACT A symmetric matrix X is completely positive semidefinite (cpsd) if there exist positive semidefinite matrices (for some ) such that for all . The of a cpsd matrix is the smallest for which such a representation is possible. It was shown independently in Prakash A, Sikora J, Varvitsiotis A, et al. [Completely positive semidefinite rank. 2016 Apr. arXiv:1604.07199] and Gribling S, de Laat D, Laurent M. [Matrices with high completely positive semidefinite rank. Linear Algebra Appl. 2017 May;513:122–148] that there exist completely positive semidefinite matrices with sub-exponential cpsd-rank. Both proofs were obtained using fundamental results from the quantum information literature as a black-box. In this work we give a self-contained and succinct proof of the existence of completely positive semidefinite matrices with sub-exponential cpsd-rank. For this, we introduce matrix valued Gram decompositions for correlation matrices and show that for extremal correlations, the matrices in such a factorization generate a Clifford algebra. Lastly, we show that this fact underlies and generalizes Tsirelson's results concerning the structure of quantum representations for extremal quantum correlation matrices.