The Minimum Distance of PPT Bound Entangled States from the Maximally Mixed State

TL;DR

This paper quantifies the minimum Euclidean distance between PPT bound entangled states and the maximally mixed state, revealing bounds relevant for entanglement detection and providing an alternative proof of PPT bound entanglement existence.

Contribution

It introduces a geometric measure of entanglement based on Euclidean distance, deriving explicit bounds for PPT bound entangled states and separable states, and offers an alternative proof of PPT bound entanglement.

Findings

Derived the minimum distance for PPT bound entangled states as 1/√(√(d^n(d^n-1))+1)

Established the minimum distance for separable states as R/(1+d^{n-1})

Provided an alternative proof for the non-emptiness of PPT bound entangled states

Abstract

Using a geometric measure of entanglement quantification based on Euclidean distance of the Hermitian matrices, we obtain the minimum distance between a bipartite bound entangled $n$- qudit density matrix and the maximally mixed state.This minimum distance for which entangled density matrices necessarily have positive partial transpose (PPT) is obtained as $\frac{1}{\sqrt{\sqrt{d^n(d^n-1)}+1}}$, which is also a lower limit for the existence of 1-distillable entangled states. The separable states necessarily lie within a minimum distance of $\frac{R}{1+d^{n-1}}$ from the Identity,where R is the radius of the closed ball homeomorphic to the set of density matrices, which is lesser than the limit for the limit for PPT bound entangled states. Furthermore an alternate proof on the non-emptiness of the PPT bound entangled states has also been given.