Irreversibility of entanglement distillation for a class of symmetric states

We investigate the irreversibility of entanglement distillation for a symmetric $\left(d+1\right)$-parameter family of mixed bipartite quantum states acting on Hilbert spaces of arbitrary dimension $d\times d$. We prove that in this family the entanglement cost is generically strictly larger than the distillable entanglement, so that the set of states for which the distillation process is asymptotically reversible is of measure zero. This remains true even if the distillation process is catalytically assisted by pure-state entanglement and every operation is allowed, which preserves the positivity of the partial transpose. It is shown that reversibility occurs only in cases where the state is a tagged mixture. The reversible cases are shown to be completely characterized by minimal uncertainty vectors for entropic uncertainty relations.

Figure 1

The relative amount of undistillable entanglement contained in ${\rho }_{\lambda \left(\nu \right)}$ for $d=2,10,100$ (solid, dashed, and dotted curve, respectively). Whereas near the separable boundary $\left(\nu =0\right)$ almost all the entanglement is undistillable, close to the maximally entangled state $\left(\nu =1\right)$ most of the entanglement can be revealed by entanglement distillation.