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arXiv:1812.01449 · Linear and Multilinear Algebra

On decomposable correlation matrices

Benjamin Lovitz

Published Aug 31, 2019• 3 citations
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Abstract

A correlation matrix is a positive semidefinite matrix with ones on the diagonal. In this work, we introduce and study r-decomposable correlation matrices: those that can be written as the Schur product of correlation matrices of rank at most r. We find that for all r≥ 2, every (r+1)×(r+1) correlation matrix is r-decomposable, and we construct (2r+1)×(2r+1) correlation matrices that are not r-decomposable. One question this leaves open is whether every 4×4 correlation matrix is 2-decomposable, which we make partial progress towards resolving. Motivations in quantum information are discussed.

DOI
10.1080/03081087.2019.1661347

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A correlation matrix is a positive semidefinite matrix with ones on the diagonal. In this work, we introduce and study r-decomposable correlation matrices: those that can be written as the Schur product of correlation matrices of rank at most r. We find that for all r≥ 2, every (r+1)×(r+1) correlation matrix is r-decomposable, and we construct (2r+1)×(2r+1) correlation matrices that are not r-decomposable. One question this leaves open is whether every 4×4 correlation matrix is 2-decomposable, which we make partial progress towards resolving. Motivations in quantum information are discussed.
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.
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.
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We propose a method to detect genuine quantum correlation for arbitrary quantum states in terms of the rank of coefficient matrices associated with the pure state. We then derive a necessary and sufficient condition for a quantum state to possess genuine correlation, namely that all corresponding coefficient matrices have rank larger than 1. We demonstrate an approach to decompose the genuine quantum correlated state with high rank coefficient matrix into the form of product states with no genuine quantum correlation for a pure state.
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