کد مقاله کد نشریه سال انتشار مقاله انگلیسی نسخه تمام متن
4598422 1631082 2017 27 صفحه PDF دانلود رایگان
عنوان انگلیسی مقاله ISI
Matrices with high completely positive semidefinite rank
ترجمه فارسی عنوان
ماتریس با درجه بالا نیمه قطعی کاملا مثبت
کلمات کلیدی
مخروط نیمه قطعی کاملا مثبت؛ فاکتور ماتریس؛ همبستگی کوانتومی؛ جبری Clifford؛ ماتریس هادامارد
موضوعات مرتبط
مهندسی و علوم پایه ریاضیات اعداد جبر و تئوری
چکیده انگلیسی

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+24k2+2k+2 with complex completely positive semidefinite rank 2k2k 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.

ناشر
Database: Elsevier - ScienceDirect (ساینس دایرکت)
Journal: Linear Algebra and its Applications - Volume 513, 15 January 2017, Pages 122–148
نویسندگان
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