We exhibit a two-parameter family of bipartite mixed states ${\rho }_{\mathrm{bc}},$ in a $d\otimes d$ Hilbert space, which are negative under partial transposition (NPT), but for which we conjecture that no maximally entangled pure states in $2\otimes 2$ can be distilled by local quantum operations and classical communication (LQ+CC). Evidence for this undistillability is provided by the result that, for certain states in this family, we cannot extract entanglement from any arbitrarily large number of copies of ${\rho }_{\mathrm{bc}}$ using a projection on $2\otimes 2.$ These states are canonical NPT states in the sense that any bipartite mixed state in any dimension with NPT can be reduced by LQ+CC operations to a NPT state of the ${\rho }_{\mathrm{bc}}$ form. We show that the main question about the distillability of mixed states can be formulated as an open mathematical question about the properties of composed positive linear maps.

• Received 7 October 1999

DOI:https://doi.org/10.1103/PhysRevA.61.062312