Lami Ludovico, Regula Bartosz
QuSoft, Amsterdam, the Netherlands.
Korteweg-de Vries Institute for Mathematics, University of Amsterdam, Amsterdam, the Netherlands.
Nat Commun. 2024 Nov 22;15(1):10120. doi: 10.1038/s41467-024-54201-5.
Computing the exact rate at which entanglement can be distilled from noisy quantum states is one of the longest-standing questions in quantum information. We give an exact solution for entanglement distillation under the set of dually non-entangling (DNE) operations-a relaxation of the typically considered local operations and classical communication, comprising all channels which preserve the sets of separable states and measurements. We show that the DNE distillable entanglement coincides with a modified version of the regularised relative entropy of entanglement in which the arguments are measured with a separable measurement. Ours is only the second known regularised formula for the distillable entanglement under any class of free operations in entanglement theory, after that given by Devetak and Winter for (one-way) local operations and classical communication. An immediate consequence of our finding is that, under DNE, entanglement can be distilled from any entangled state. As our second main result, we construct a general upper bound on the DNE distillable entanglement, using which we prove that the separably measured relative entropy of entanglement can be strictly smaller than the regularisation of the standard relative entropy of entanglement, solving an open problem posed by Li and Winter. Finally, we study also the reverse task of entanglement dilution and show that the restriction to DNE operations does not change the entanglement cost when compared with the larger class of non-entangling operations. This implies a strong form of irreversiblility of entanglement theory under DNE operations: even when asymptotically vanishing amounts of entanglement may be generated, entangled states cannot be converted reversibly.
计算从噪声量子态中可提取纠缠的精确速率，是量子信息领域长期存在的问题之一。我们给出了在对偶非纠缠（DNE）操作集下纠缠提取的精确解——这是对通常考虑的局部操作和经典通信的一种放宽，包括所有保持可分态集和测量的信道。我们表明，DNE可提取纠缠与纠缠正则化相对熵的一个修改版本一致，其中论据是用可分测量来度量的。我们的结果是纠缠理论中在任何一类自由操作下可提取纠缠的第二个已知正则化公式，第一个是由德维塔克和温特给出的关于（单向）局部操作和经典通信的公式。我们这一发现的一个直接结果是，在DNE操作下，纠缠可从任何纠缠态中提取。作为我们的第二个主要结果，我们构造了一个关于DNE可提取纠缠的一般上界，利用它我们证明了可分测量的纠缠相对熵可以严格小于标准纠缠相对熵的正则化，从而解决了李和温特提出的一个开放问题。最后，我们还研究了纠缠稀释的逆任务，并表明与更大的非纠缠操作类相比，对DNE操作的限制不会改变纠缠代价。这意味着在DNE操作下纠缠理论具有一种强形式的不可逆性：即使渐近消失量的纠缠可能被生成，纠缠态也不能被可逆地转换。
