Bounding the separable rank via polynomial optimization

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 … Read more

Lower bounds on matrix factorization ranks via noncommutative polynomial optimization

We use techniques from (tracial noncommutative) polynomial optimization to formulate hierarchies of semidefinite programming lower bounds on matrix factorization ranks. In particular, we consider the nonnegative rank, the positive semidefinite rank, and their symmetric analogues: the completely positive rank and the completely positive semidefinite rank. We study the convergence properties of our hierarchies, compare them … Read more

Bounds on entanglement dimensions and quantum graph parameters via noncommutative polynomial optimization

In this paper we study bipartite quantum correlations using techniques from tracial polynomial optimization. We construct a hierarchy of semidefinite programming lower bounds on the minimal entanglement dimension of a bipartite correlation. This hierarchy converges to a new parameter: the minimal average entanglement dimension, which measures the amount of entanglement needed to reproduce a quantum … Read more

Matrices with high completely positive semidefinite rank

A real symmetric matrix M is completely positive semidefinite if it admits a Gram representation by positive semidefinite matrices (of any size d). The smallest such d is called the 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 … Read more