Global geometric difference between separable and Positive partial transpose states

TL;DR

This paper investigates the geometric differences between PPT and separable states in quantum systems, revealing that convex combinations of extreme points can lie in the interior, highlighting fundamental distinctions.

Contribution

It demonstrates that a small number of extreme points can generate interior points in the convex set of PPT states, unlike separable states, clarifying their geometric relationship.

Findings

Convex combinations of two extreme PPT states can be interior points.

At least mn extreme points are needed for interior points in separable states.

Distinct geometric properties differentiate PPT and separable states.

Abstract

In the convex set of all $3\ot 3$ states with positive partial transposes, we show that one can take two extreme points whose convex combinations belong to the interior of the convex set. Their convex combinations may be even in the interior of the convex set of all separable states. In general, we need at least $mn$ extreme points to get an interior point by their convex combination, for the case of the convex set of all $m\ot n$ separable states. This shows a sharp distinction between PPT states and separable states. We also consider the same questions for positive maps and decomposable maps.