Positivity and entanglement of polynomial Gaussian integral operators

We investigate the positivity of self-adjoint polynomial Gaussian integral operators κPG, that is, the multivariable kernel κPG is a product of a polynomial P and a Gaussian kernel κG. We show that κPG can be only positive if the Gaussian part is positive. This has an important corollary for the bipartite entanglement of the density operators κPG: if the Gaussian density operator κG fails the Peres--Horodecki criterion, then the corresponding polynomial Gaussian density operators κPG also fail the criterion for all P, hence they are all entangled. We introduce a new preorder on Gaussian kernels such that if κG₀ κG₁ then κPG₀≥ 0 implies κPG₁≥ 0 for all polynomials P. Therefore, deciding the positivity of a polynomial Gaussian operator determines the positivity of a lot of another polynomial Gaussian operators which have the same polynomial factor. This preorder has implication for the entanglement problem, too. We also prove that polynomial Gaussian operators with polynomials of odd degree cannot be positive semidefinite.