Bounding the separable rank via polynomial optimization

Understand

We investigate questions related to the set $\mathcal{SEP}_d$ consisting of the linear maps $\rho$ acting on $\mathbb{C}^d\otimes \mathbb{C}^d$ that can be written as a convex combination of rank one matrices of the form $xx^*\otimes yy^*$.

• Such maps are known in quantum information theory as the separable bipartite states, while nonseparable states are called entangled.
• In particular we introduce bounds for the separable rank $\mathrm{rank_{sep}}(\rho)$, defined as the smallest number of rank one states $xx^*\otimes yy^*$ entering the decomposition of a separable state $\rho$.
• Our approach relies on the moment method and yields a hierarchy of semidefinite-based lower bounds, that converges to a parameter $\tau_{\mathrm{sep}}(\rho)$, a natural convexification of the combinatorial parameter $\mathrm{rank_{sep}}(\rho)$.