2017/02/21 by Anupam Prakash, Prakash, Anupam, Antonios Varvitsiotis +1 · 2 citations
Computer Science · Mathematics · Physics and Astronomy · #FOS: Mathematics #Optimization and Control (math.OC) #Quantum Computing Algorithms and Architecture #Quantum Information and Cryptography #Quantum Mechanics and Applications #math.OC
paper · pdf · doi:10.48550/arxiv.1702.06305
New title. To appear in Linear & Multilinear Algebra. 15 pages. Comments welcome!
openalex publication_date 2017/02/21 · arxiv created 2018/09/28 · arxiv updated 2018/10/01 · openalex created_date 2022/10/02 · openalex updated_date 2026/07/28
We introduce a notion of matrix valued Gram decompositions for correlation matrices whose study is motivated by quantum information theory. We show that for extremal correlations, the matrices in such a factorization generate a Clifford algebra and thus, their size is exponential in terms of the rank of the correlation matrix. Using this we give a self-contained and succinct proof of the existence of completely positive semidefinite matrices with sub-exponential cpsd-rank, recently derived in the literature. This fact also underlies and generalizes Tsirelson's seminal lower bound on the local dimension of a quantum system necessary to generate an extreme quantum correlation.