Equivalence classes and canonical forms for two-qutrit entangled states of rank four having positive partial transpose

TL;DR

This paper classifies two-qutrit entangled states of rank four with positive partial transpose using equivalence classes and canonical forms, revealing new insights into their structure and parametrization.

Contribution

It introduces a topological classification of these states, constructs explicit canonical forms, and provides the first examples of extreme PPT states with differing ranks.

Findings

SLOCC equivalence classes are homeomorphic to a quotient of R^4 by A_5.

Explicit canonical forms can be parametrized by at most two real parameters.

First examples of extreme PPT states with rank different from their partial transpose.

Abstract

Let E' denote the set of non-normalized two-qutrit entangled states of rank four having positive partial transpose (PPT). We show that the set of SLOCC equivalence classes of states in E', equipped with the quotient topology, is homeomorphic to the quotient R/A_5 of the open rectangular box R in the Euclidean space R^4 by an action of the alternating group A_5. We construct an explicit map omega: Omega -> E', where Omega is the open positive orthant in R^4, whose image meets every SLOCC equivalence class E containeed in E'. Although the intersection of the image of omega and E is not necessarily a singleton set, it is always a finite set of cardinality at most 60. By abuse of language, we say that any state in this intersection is a canonical form of states rho in E. In particular, we show that all checkerboard PPT entangled states can be parametrized up to SLOCC equivalence by only two real parameters. We also summarize the known results on two-qutrit extreme PPT states and edge states, and examine which other interesting properties they may have. Thus we find the first examples of extreme PPT states whose rank is different from the rank of its partial transpose.