Irreversibility of Asymptotic Entanglement Manipulation Under Quantum Operations Completely Preserving Positivity of Partial Transpose

We demonstrate the irreversibility of asymptotic entanglement manipulation under quantum operations that completely preserve the positivity of partial transpose (PPT), resolving a major open problem in quantum information theory. Our key tool is a new efficiently computable additive lower bound for the asymptotic relative entropy of entanglement with respect to PPT states, which can be used to evaluate the entanglement cost under local operations and classical communication (LOCC). We find that for any rank-two mixed state supporting on the $3\otimes 3$ antisymmetric subspace, the amount of distillable entanglement by PPT operations is strictly smaller than one entanglement bit (ebit) while its entanglement cost under PPT operations is exactly one ebit. As a by-product, we find that for this class of states, both the Rains’s bound and its regularization are strictly less than the asymptotic relative entropy of entanglement. So, in general, there is no unique entanglement measure for the manipulation of entanglement by PPT operations. We further show a computable sufficient condition for the irreversibility of entanglement distillation by LOCC (or PPT) operations.

Figure 1

The amount of Bell pairs distilled from the state is insufficient to reconstruct the given state under PPT operations in the asymptotic regime.

Figure 2

The PPT-relative entropy of entanglement is defined as the smallest quantum relative entropy from $\rho$ to the state $\sigma$ taken from the set of PPT states $\mathrm{\Gamma }$. Assume that ${\rho }_0$ and ${\sigma }_0$ give the smallest quantum relative entropy from $D(\rho )$ to $\mathrm{\Gamma }$. It is clear that $E_R(\rho )=S(\rho ||\sigma )\ge S({\rho }_0||{\sigma }_0)$ and we show $S({\rho }_0||{\sigma }_0)\ge -\log \mathrm{Tr}P{\sigma }_0\ge E_{\eta }(\rho )$ in Proposition 1 where $P$ is the projection onto the support of $\rho$. This lower bound $E_{\eta }(\rho )$ is powerful since it still works in the asymptotic setting due to its additivity under tensor product.