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arXiv:quant-ph/0702136 · Open Systems & Information Dynamics

Remarks on the Equivalence of Full Additivity and Monotonicity for the Entanglement Cost

Fernando G. S. L. Brandão, M. Horodecki, Martin Bodo Plenio, S. Virmani

Published Sep 1, 2007• 10 citations
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Abstract

We analyse the relationship between the full additivity of the entanglement cost and its full monotonicity under local operations and classical communication. We show that the two properties are equivalent for the entanglement cost. The proof works for the regularization of any convex, subadditive, and asymptotically continuous entanglement monotone, and hence also applies to the asymptotic relative entropy of entanglement.

DOI
10.1007/s11080-007-9056-0

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We analyse the relationship between the full additivity of the entanglement cost and its full monotonicity under local operations and classical communication. We show that the two properties are equivalent for the entanglement cost. The proof works for the regularization of any convex, subadditive, and asymptotically continuous entanglement monotone, and hence also applies to the asymptotic relative entropy of entanglement.
Entanglement monotone is defined as a convex measure of entanglement that does not increase on average under local operations and classical communication (LOCC). Here we call an entanglement monotone a strict entanglement monotone (SEM) if it decreases strictly on average under LOCC. We show that, for any convex roof extended entanglement monotone that on pure states is given by a function of the reduced states, if the function is strictly concave, then it is a SEM. Moreover, we prove that the negativity and the relative entropy of entanglement, which are not defined by the convex roof structure, are also SEMs. In addition, if the squashed entanglement could be obtained by some optimal extension, then it is a SEM as well. Our results imply that entanglement is strictly decreasing on average under LOCC.
We investigate the properties of three entanglement measures that quantify the statistical distinguishability of a given state with the closest disentangled state that has the same reductions as the primary state. In particular, we concentrate on the relative entropy of entanglement with reversed entries. We show that this quantity is an entanglement monotone which is strongly additive, thereby demonstrating that monotonicity under local quantum operations and strong additivity are compatible in principle. In accordance with the presented statistical interpretation which is provided, this entanglement monotone, however, has the property that it diverges on pure states, with the consequence that it cannot distinguish the degree of entanglement of different pure states. We also prove that the relative entropy of entanglement with respect to the set of disentangled states that have identical reductions to the primary state is an entanglement monotone. We finally investigate the trace-norm measure and demonstrate that it is also a proper entanglement monotone.
We investigate the properties of three entanglement measures that quantify the statistical distinguishability of a given state with the closest disentangled state that has the same reductions as the primary state. In particular, we concentrate on the relative entropy of entanglement with reversed entries. We show that this quantity is an entanglement monotone which is strongly additive, thereby demonstrating that monotonicity under local quantum operations and strong additivity are compatible in principle. In accordance with the presented statistical interpretation which is provided, this entanglement monotone, however, has the property that it diverges on pure states, with the consequence that it cannot distinguish the degree of entanglement of different pure states. We also prove that the relative entropy of entanglement with respect to the set of disentangled states that have identical reductions to the primary state is an entanglement monotone. We finally investigate the trace-norm measure and demonstrate that it is also a proper entanglement monotone.
Properties of the max-relative entropy of entanglement, defined in Ref. 10, are investigated, and its significance as an upper bound to the one-shot rate for perfect entanglement dilution, under a particular class of quantum operations, is discussed. It is shown that it is in fact equal to another known entanglement monotone, namely the log robustness, defined in Ref. 7. It is known that the latter is not asymptotically continuous and it is not known whether it is weakly additive. However, by suitably modifying the max-relative entropy of entanglement we obtain a quantity which is seen to satisfy both these properties. In fact, the modified quantity is shown to be equal to the regularized relative entropy of entanglement.
We demonstrate the convexity of the difference between the regularized entanglement of purification and the entropy, as a function of the state. This is proved by means of a new asymptotic protocol to prepare a state from pre-shared entanglement and by local operations only. We go on to employ this convexity property in an investigation of the additivity of the (single-copy) entanglement of purification: using numerical results for two-qubit Werner states we find strong evidence that the entanglement of purification is different from its regularization, hence that entanglement of purification is not additive.