Minimum distance of the boundary of the set of PPT states from the maximally mixed state using the geometry of the positive semidefinite cone

Abstract

Using a geometric measure of entanglement quantification based on Euclidean distance of the Hermitian matrices (Patel and Panigrahi in Geometric measure of entanglement based on local measurement, 2016. arXiv:1608.06145), we obtain the minimum distance between the set of bipartite n-qudit density matrices with a positive partial transpose and the maximally mixed state. This minimum distance is obtained as 1dn(dn-1), which is also the minimum distance within which all quantum states are separable. An idea of the interior of the set of all positive semidefinite matrices has also been provided. A particular class of Werner states has been identified for which the PPT criterion is necessary and sufficient for separability in dimensions greater than six. 2019, Springer Science+Business Media, LLC, part of Springer Nature.