Very Strong Irreversibility of Quantum Entanglement

In quantum information theory, the manipulation of entanglement is known to be irreversible: the rate at which one can extract pure entanglement (distillable entanglement) from a mixed state is generally lower than the rate required to create that state from pure entanglement (entanglement cost). While this is well-established for vanishing-error protocols, it remained unclear whether this irreversibility persists if one allows for a fixed, non-zero error. This paper addresses whether such a gap exists in the 'strong-converse' regime, where errors are allowed to be non-zero but must not approach the maximal possible value.

The authors investigate the strong-converse thresholds for both distillation and formation of entanglement. They resolve a conjecture by Lami and Regula by proving that for specific mixed states, the distillable entanglement is strictly smaller than the entanglement cost even when a fixed error is permitted. To achieve this, they derive a new semidefinite-programming (SDP) lower bound on the exponential strong-converse entanglement cost. They further extend their analysis to the class of completely PPT-preserving (C-PPT-P) operations, providing analytical criteria to identify states that exhibit this exponential irreversibility.

The study establishes that entanglement irreversibility is a robust, fundamental feature that does not disappear by relaxing the error requirements. Specifically, for the Lami-Regula state, the authors prove that the exponential strong-converse distillable entanglement is strictly less than the exponential strong-converse entanglement cost. They demonstrate that any protocol operating beyond these thresholds results in an error that decays exponentially with the number of copies. This confirms that the gap between distillation and formation is not an artifact of requiring perfect transformations, but an intrinsic property of entanglement theory.