Evidence for bound entangled states with negative partial transpose

We exhibit a two-parameter family of bipartite mixed states ρ_bc, in a d⊗d Hilbert space, which are negative under partial transposition (NPT), but for which we conjecture that no maximally entangled pure states in 2⊗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 ρ_bc using a projection on 2⊗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 ρ_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.