- Conic approach to quantum graph parameters using linear optimization over the completely positive semidefinite cone Monique Laurent (moniquecwi.nl) Teresa Piovesan (piovesancwi.nl) Abstract: We investigate the completely positive semidefinite cone $\mathcal{CS}_+^n$, a new matrix cone consisting of all $n\times n$ 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 doubly nonnegative cones, and between its dual cone and trace positive non-commutative 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 $\mathcal{CS}_+^n$. 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. Keywords: Quantum graph parameters, Trace positive polynomials, Copositive cone, Chromatic number, Quantum entanglement, Nonlocal games Category 1: Linear, Cone and Semidefinite Programming (Other ) Citation: Download: [PDF]Entry Submitted: 07/10/2014Entry Accepted: 07/10/2014Entry Last Modified: 10/05/2015Modify/Update this entry Visitors Authors More about us Links Subscribe, Unsubscribe Digest Archive Search, Browse the Repository Submit Update Policies Coordinator's Board Classification Scheme Credits Give us feedback Optimization Journals, Sites, Societies Optimization Online is supported by the Mathematical Optmization Society.