Distillable entanglement under dually non-entangling operations

Abstract

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 [CMP 326, 63 (2014)]. 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.

10+24 pages; v2 is close to the published version

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• Distillation of secret key and entanglement from quantum states
• Quantum Communication With Zero-Capacity Channels
• Entanglement Theory and the Second Law of Thermodynamics
• Counterexamples to additivity of minimum output p-Renyi entropy for p close to 0
• Some Open Problems in Quantum Information Theory
• Reversibility of quantum resources through probabilistic protocols
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Cited by in corpus (6)

• Reversibility of quantum resources through probabilistic protocols
• A solution of the generalised quantum Stein's lemma
• Locally-Measured Rényi Divergences
• Asymptotic quantification of entanglement with a single copy
• Computable entanglement cost under positive partial transpose operations
• Nonexistence of maximally entangled mixed states for a fixed spectrum