Qubit-qudit states with positive partial transpose

Abstract

We show that the length of a qubit-qutrit separable state is equal to $\mathrm{max}(r,s)$, where $r$ is the rank of the state and $s$ 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 having positive partial transpose (PPT) of rank 4. The boundary of this set consists of all separable states of length at most 4. We prove that the length of any qubit-qudit separable state of birank $(d+1,d+1)$ is equal to $d+1$. 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 a $2\ensuremath{\bigotimes}d$ space such that $V$ contains no product vectors, we show that the set of all product vectors in ${V}^{\ensuremath{\perp}}$ is a vector bundle of rank $d\ensuremath{-}k$ over the projective line. Finally, we explicitly construct examples of qubit-qudit PPT states (both separable and entangled) of any feasible birank.