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

TL;DR

This paper proves that entanglement manipulation under PPT-preserving quantum operations is fundamentally irreversible, providing new bounds and insights into entanglement measures and their limitations.

Contribution

It introduces a new additive lower bound for the asymptotic relative entropy of entanglement with respect to PPT states, resolving a key open problem.

Findings

PPT operations cannot reversibly manipulate certain entangled states.

The entanglement cost under PPT operations equals one ebit for specific states.

Rains' bound and its regularization are strictly less than the asymptotic relative entropy of entanglement.

Abstract

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 $3\otimes3$ 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 byproduct, we find that for this class of states, both the Rains' 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.