Extreme points of absolutely PPT states with exactly three distinct eigenvalues

TL;DR

This paper characterizes the boundary and extreme points of full-rank two-qutrit states with three distinct eigenvalues, revealing their structure and relation to known extreme points, advancing understanding in entanglement theory.

Contribution

It provides an explicit characterization of boundary and extreme points of two-qutrit AP states with three eigenvalues, a problem previously open in entanglement theory.

Findings

All boundary points are extreme except one.

Extreme points are described by at most one parameter.

Most points approach known extreme points as parameters reach interval ends.

Abstract

Whether the sets of absolutely separable (AS) and absolutely two-qutrit positive-partial-transpose (AP) states are the same has been an open problem in entanglement theory for decades. Since they are both convex sets, we investigate the boundary and extreme points of full-rank two-qutrit AP states with exactly three distinct eigenvalues. We show that every boundary point is an extreme point, with exactly one exception. We explicitly characterize the expressions of such points, each of which turns out to contain at most one parameter in some intervals. When the parameter approaches the ends of intervals, most points become the known extreme points of exactly two distinct eigenvalues. We present our results by tables and figures.