Qubit-qudit states with positive partial transpose

TL;DR

This paper investigates the properties of qubit-qudit states with positive partial transpose, establishing relationships between their rank and separability, and constructing explicit examples of states with various biranks.

Contribution

It introduces the concept of birank for qubit-qutrit states, characterizes the set of PPT entangled states, and provides explicit constructions of states with arbitrary feasible biranks.

Findings

The length of a qubit-qutrit separable state equals the maximum of its rank and the rank of its partial transpose.

The boundary of the set of two-qutrit PPT entangled states of rank four consists of separable states of length at most four.

Explicit examples of qubit-qudit PPT states of any feasible birank are constructed.

Abstract

We show that the length of a qubit-qutrit separable state is equal to the max(r,s), where r is the rank of the state and s is the rank of its partial transpose. We refer to the ordered pair (r,s) as the birank of this state. We also construct examples of qubit-qutrit separable states of any feasible birank (r,s). We determine the closure of the set of normalized two-qutrit entangled states of rank four having positive partial transpose (PPT). The boundary of this set consists of all separable states of length at most four. We prove that the length of any qubit-qudit separable state of birank (d+1,d+1) is d+1. We also show that all qubit-qudit PPT entangled states of birank (d+1,d+1) can be built in a simple way from edge states. If V is a subspace of dimension k<d in the tensor product of C^2 and C^d such that V contains no product vectors, we show that the set of all product vectors in the orthogonal complement of V is a vector bundle of rank d-k over the projective line. Finally, we explicitly construct examples of qubit-qudit PPT states (both separable and entangled) of any feasible birank.