copositive matrix – Optimization Online

Semidefinite hierarchies for diagonal unitary invariant bipartite quantum states

Published: 2025/12/18

Semi-definite Programming Completely positive matrix, copositive matrix, DPS hierarchy, entanglement, Moment approach, Quantum bipartite states, semidefinite programming, Separable states, sparsity, sum-of-squares polynomial

We investigate questions about the cone $\mathrm{SEP}_n$ of separable bipartite states, consisting of the Hermitian matrices acting on $\mathbb{C}^n\otimes\mathbb{C}^n$ that can be written as conic combinations of rank one matrices of the form $xx^*\otimes yy^*$ with $x,y\in\mathbb{C}^n$. Bipartite states that are not separable are said to be entangled. Detecting quantum entanglement is a fundamental task … Read more

Exactness of Parrilo’s conic approximations for copositive matrices and associated low order bounds for the stability number of a graph

Published: 2021/10/12

Monique Laurent

Luis Felipe Vargas

Approximation Algorithms, Semi-definite Programming alpha-critical graph, copositive matrix, semidefinite programming, Shor relaxation, stable set problem, sum-of-squares polynomial

De Klerk and Pasechnik (2002) introduced the bounds $\vartheta^{(r)}(G)$ ($r\in \mathbb{N}$) for the stability number $\alpha(G)$ of a graph $G$ and conjectured exactness at order $\alpha(G)-1$: $\vartheta^{(\alpha(G)-1)}(G)=\alpha(G)$. These bounds rely on the conic approximations $\mathcal{K}_n^{(r)}$ by Parrilo (2000) for the copositive cone $\text{COP}_n$. A difficulty in the convergence analysis of $\vartheta^{(r)}$ is the bad behaviour … Read more