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arXiv:quant-ph/0502170 · Physical Review A

Positive partial transpose from spectra

Roland Hildebrand

Published Nov 28, 2007• 59 citations
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Abstract

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 $A$.

DOI
10.1103/physreva.76.052325

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