Evidence for bound entangled states with negative partial transpose

We exhibit a two-parameter family of bipartite mixed states ₁₂, in a d Hilbert space, which are negative under partial transposition (NPT), but for which we conjecture that no maximally entangled pure states in 22 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 ₁₂ using a projection on 22. 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 ₁₂ 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.