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Matrices with high completely positive semidefinite rank

Sander Gribling(gribling***at***cwi.nl)
David de Laat(laat***at***cwi.nl)
Monique Laurent(laurent***at***cwi.nl)

Abstract: A real symmetric matrix M is completely positive semidefinite if it admits a Gram representation by positive semidefinite matrices (of any size d). The smallest such d is called the completely positive semidefinite rank of M, and it is an open question whether there exists an upper bound on this number as a function of the matrix size. We show that if such an upper bound exists, it has to be at least exponential in the matrix size. For this we exploit connections to quantum information theory and we construct extremal bipartite correlation matrices of large rank. We also exhibit a class of completely positive matrices with quadratic (in terms of the matrix size) completely positive rank, but with linear completely positive semidefinite rank, and we make a connection to the existence of Hadamard matrices.

Keywords:

Category 1: Linear, Cone and Semidefinite Programming

Citation: arXiv:1605.00988, CWI, 05/2016

Download: [PDF]

Entry Submitted: 05/13/2016
Entry Accepted: 05/18/2016
Entry Last Modified: 05/13/2016

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