Matrices With High Completely Positive Semidefinite Rank

David de Laat, Sander Gribling , Monique Laurent

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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.
Original languageEnglish
Place of PublicationIthaca
PublisherCornell University Library
Number of pages17
Publication statusPublished - 3 May 2016

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