Positive partial transpose from spectra
Roland Hildebrand · Physical Review A · 2007
In this paper we solve the following problem. Let ${\mathcal{H}}_{nm}$ be a Hilbert space of dimension $nm$, and let $A$ be a positive semidefinite self-adjoint linear operator on ${\mathcal{H}}_{nm}$. Under which conditions on the spectrum has $A$ a positive partial transpose (is PPT) with respect to any partition ${\mathcal{H}}_{n}\ensuremath{\bigotimes}{\mathcal{H}}_{m}$ of the space ${\mathcal{H}}_{nm}$ as a tensor product of an $n$-dimensional and an $m$-dimensional Hilbert space? We show that the necessary and sufficient conditions can be expressed as a set of linear matrix inequalities (LMIs) on the eigenvalues of $A$.