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 qua...

In 1983 L. G. Brown introduced a spectral distribution measure for non-normal elements in a nite von Neumann algebra M with respect to a xed normal faithful tracial state. In this paper we compute Brown's spectral distribution measure in case T has a polar decomposition T = UH where U is a Haar unitary and U and H are-free. (When Ker T = f0g this is equivalent to that (T; T) is an R-diagonal pa...

Motivated by recent progress in quantum information theory, this article aims at optimizing trace polynomials, i.e., polynomials noncommuting variables and traces of their products. A novel Positivstellensatz certifying positivity subject to constraints is presented, a hierarchy semidefinite relaxations converging monotonically the optimum polynomial tracial provided. This can be seen as analog...

Let k be a eld, S = k[xv : v 2 V ] be the polynomial ring over the nite set of variables (xv : v 2 V), and m = (x v : v 2 V) the ideal dening the origin of Spec S. It is theoretically known (see e.g. Alonso et al. 1991) that the algorithmic ideas for the computation of ideal (and module) intersections, quotients, deciding radical membership etc. in S may be adopted not only for computations in ...

We describe a subset of the level{1, level{2, and level{3 BLAS implemented for each node of the Connection Machine system CM{200. The routines, collectively called LBLAS, have interfaces consistent with languages with an array syntax such as Fortran 90. One novel feature, important for distributed memory architectures, is the capability of performing computations on multiple instances of object...

Let k be a field, S = k[xv : v ∈ V ] be the polynomial ring over the finite set of variables (xv : v ∈ V ), and m = (xv : v ∈ V ) the ideal defining the origin of Spec S. It is theoretically known (see e.g. Alonso et al. 1991) that the algorithmic ideas for the computation of ideal (and module) intersections, quotients, deciding radical membership etc. in S may be adopted not only for computati...

Abstract. We present some generalizations of quantum information inequalities involving tracial positive linear maps between C∗-algebras. Among several results, we establish a noncommutative Heisenberg uncertainty relation. More precisely, we show that if Φ : A → B is a tracial positive linear map between C∗-algebras , ρ ∈ A is a Φ-density element and A,B are self-adjoint operators of A such th...

We consider the reduction of problems on general noncommutative Lp-spaces to the corresponding ones on those associated with finite von Neumann algebras. The main tool is an unpublished result of the first-named author which approximates any noncommutative Lpspace by tracial ones. We show that under some natural conditions a map between two von Neumann algebras extends to their crossed products...