Matrices with high completely positive semidefinite rank

详细信息    查看全文

• 作者：Sander Gribling^a ; ¹ ; ; David de Laat^a ; ¹ ; ^{mail@daviddelaat.nl" class="auth_mail" title="E-mail the corresponding author} ; ; Monique Laurent^a ; ^b ;
• 关键词：15B48 ; 15A23 ; 90C22
• 刊名：Linear Algebra and its Applications
• 出版年：2017
• 出版时间：15 January 2017
• 年：2017
• 卷：513
• 期：Complete
• 页码：122-148
• 全文大小：537 K

文摘

A real symmetric matrix M is completely positive semidefinite if it admits a Gram representation by (Hermitian) positive semidefinite matrices of any size d. The smallest such d is called the (complex) 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 construct completely positive semidefinite matrices of size 4k²+2k+2$4k^2+2k+2$ with complex completely positive semidefinite rank 2^k$2^k$ for any positive integer k. This shows 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.