Low rank positive partial transpose states and their relation to product vectors

TL;DR

This paper investigates low rank entangled PPT states, especially rank five states near rank four states, exploring their geometry, construction, and relation to product vectors using perturbation theory.

Contribution

It introduces a method to construct rank five entangled PPT states close to known rank four states and analyzes their geometric properties and relation to product vectors.

Findings

Rank five entangled PPT states can be constructed near rank four states.

The geometry of low rank PPT states is characterized.

A method to reconstruct PPT states from product vectors in their kernel is presented.

Abstract

It is known that entangled mixed states that are positive under partial transposition (PPT states) must have rank at least four. In a previous paper we presented a classification of rank four entangled PPT states which we believe to be complete. In the present paper we continue our investigations of the low rank entangled PPT states. We use perturbation theory in order to construct rank five entangled PPT states close to the known rank four states, and in order to compute dimensions and study the geometry of surfaces of low rank PPT states. We exploit the close connection between low rank PPT states and product vectors. In particular, we show how to reconstruct a PPT state from a sufficient number of product vectors in its kernel. It may seem surprising that the number of product vectors needed may be smaller than the dimension of the kernel.