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

Abstract

Let denote the set of non-normalized two-qutrit entangled states of rank four having positive partial transpose (PPT). We show that the set of stochastic local operations and classical communications (SLOCC) equivalence classes of states in, equipped with the quotient topology, is homeomorphic to the quotient R/A5 of the open rectangular box R⊂ R4 by an action of the alternating group A5. We construct an explicit map, where Ω is the open positive orthant in R4, whose image ω(Ω) meets every SLOCC equivalence class. Although the intersection ω(Ω) ∩ 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 ω(Ω) ∩ E is a canonical form of any ρ ∈ 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. © 2012 American Institute of Physics.