期刊
SIAM JOURNAL ON OPTIMIZATION
卷 25, 期 4, 页码 2461-2493出版社
SIAM PUBLICATIONS
DOI: 10.1137/14097865X
关键词
quantum graph parameters; trace positive polynomials; copositive cone; chromatic number; quantum entanglement; nonlocal games
资金
- SIQS European project
We investigate the completely positive semidefinite cone CS+n, a new matrix cone consisting of all n x n matrices that admit a Gram representation by positive semidefinite matrices ( of any size). In particular, we study relationships between this cone and the completely positive and the doubly nonnegative cone, and between its dual cone and trace positive noncommutative polynomials. We use this new cone to model quantum analogues of the classical independence and chromatic graph parameters alpha(G) and chi(G), which are roughly obtained by allowing variables to be positive semidefinite matrices instead of 0/1 scalars in the programs defining the classical parameters. We can formulate these quantum parameters as conic linear programs over the cone CS+n. Using this conic approach we can recover the bounds in terms of the theta number and define further approximations by exploiting the link to trace positive polynomials.
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