Distillability and PPT entanglement of low-rank quantum states

Abstract

The bipartite quantum states ρ, with rank strictly smaller than the maximum of the ranks of the reduced states ρA and ρB, are distillable by local operations and classical communication (Horodecki P, Smolin J A, Terhal B M and Thapliyal A V 2003 Theor. Comput. Sci. 292 589–96; 1999 arXiv:quant-ph/9910122). Our first main result is that this is also true for NPT states with rank equal to this maximum. (A state is PPT if the partial transpose of its density matrix is positive semidefinite, and otherwise it is NPT.) This was conjectured first in 1999 in the special case when the ranks of ρA and ρB are equal (see (Horodecki P, Smolin J A, Terhal B M and Thapliyal A V 2003 Theor. Comput. Sci. 292 589–96; 1999 arXiv:quant-ph/9910122). Our second main result provides a complete solution of the separability problem for bipartite states of rank 4. Namely, we show that such a state is separable if and only if it is PPT and its range contains at least one product state. We also prove that the so-called checkerboard states are distillable if and only if they are NPT.