Non-negative symmetric polynomials and entangled Bosons

The fundamental relation between quantum entanglement and convex algebraic geometry has unveiled a set of powerful tools, imported from the former to the study the latter. The space of separable mixed states is convex and so is the space of the corresponding observable values. Therefore, the problem of determining whether a set of observable values come from an entangled state is tantamount to checking for membership in a convex set, of a point with coordinates given by the set of observable values. Here, we use techniques from convex algebraic geometry to develop powerful criteria for entanglement in a many-body system of Bosonic atoms with a non-zero spin. The experimentally accessible observables are the spin expectation values and which, upon truncating at rank two, are 9 independent quantities. Recently, entanglement criteria in terms of 3 of these 9 quantities have been derived. We develop entanglement criteria using all of these 9 quantities. If we consider these numbers as coordinates of a point in a 9 dimensional space, those with a separable parent state lie within a convex region, also known as the moment cone. Therefore, the problem is to characterize this moment cone. Owing to the Bosonic exchange symmetry, this moment cone is the dual of the cone of non-negative symmetric polynomials. Together with a characterization of this cone, we also develop an entanglement criterion that is asymptotically tight, for large atom numbers.M.S

Symmetry reduction to optimize a graph-based polynomial from queueing theory

For given integers n and d, both at least 2, we consider a homogeneous multivariate polynomial fd of degree d in variables indexed by the edges of the complete graph on n vertices and coefficients depending on cardinalities of certain unions of edges. Cardinaels, Borst and Van Leeuwaarden (arXiv:2111.05777, 2021) asked whether fd, which arises in a model of job-occupancy in redundancy scheduling, attains its minimum over the standard simplex at the uniform probability vector. Brosch, Laurent and Steenkamp [SIAM J. Optim. 31 (2021), 2227--2254] proved that fd is convex over the standard simplex if d=2 and d=3, implying the desired result for these d. We give a symmetry reduction to show that for fixed d, the polynomial is convex over the standard simplex (for all n≥2) if a constant number of constant matrices (with size and coefficients independent of n) are positive semidefinite. This result is then used in combination with a computer-assisted verification to show that the polynomial fd is convex for d≤9