Year: 2018
Type: article
Abstract: A symmetric n×n matrix X is completely positive semidefinite (cpsd) if there exist d×d positive semidefinite matrices {Pi}i=1n (for some d∈N) such that Xij=Tr(PiPj), for all i,j∈{1,…,n}. The cpsd−rank of a cpsd matrix is the smallest d∈N 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.
Source: Linear and Multilinear Algebra
Authors: Anupam Prakash, Antonios Varvitsiotis
Language: English
Subfield: Numerical Analysis
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Awards: G1988020301, G870216602
OpenAlex ID: W2893897940