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arXiv:1904.01183 · Physical Review A

Strict entanglement monotonicity under local operations and classical communication

Yu Jing Guo

Published Feb 28, 2019• 11 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physreva.99.022338

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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.
Abstract Quantum entanglement is known to be monogamous, i.e. it obeys strong constraints on how the entanglement can be distributed among multipartite systems. Almost all the entanglement monotones so far are shown to be monogamous. We explore here a family of entanglement monotones with the reduced functions are concave but not strictly concave and show that they are not monogamous. They are defined by four kinds of the ‘partial-norm’ of the reduced state, which we call them partial-norm of entanglement, minimal partial-norm of entanglement, reinforced minimal partial-norm of entanglement, and partial negativity, respectively. This indicates that, the previous axiomatic definition of the entanglement monotone needs supplemental agreement that the reduced function should be strictly concave since such a strict concavity can make sure that the corresponding convex-roof extended entanglement monotone is monogamous. Here, the reduced function of an entanglement monotone refers to the corresponding function on the reduced state for the measure on bipartite pure states.
It is proven that the logarithmic negativity does not increase on average under positive partial transpose preserving (PPT) operation including subselection (a set of operations that incorporate local operations and classical communication (LOCC) as a subset) and, in the process, a further proof is provided that the negativity does not increase on average under the same set of operations. Given that the logarithmic negativity is obtained from the negativity applying a concave function and is itself not a convex quantity this result is surprising as convexity is generally considered as describing the local physical process of losing information. The role of convexity and in particular its relation (or lack thereof) to physical processes is discussed in this context.
Much of the theory of entanglement concerns the transformations that are possible to a state under local operations with classical communication (LOCC); however, this set of operations is complicated and difficult to describe mathematically. An idea which has proven very useful is that of the entanglement monotone: a function of the state which is invariant under local unitary transformations and always decreases (or increases) on average after any local operation. In this paper we look on LOCC as the set of operations generated by infinitesimal local operations, operations which can be performed locally and which leave the state little changed. We show that a necessary and sufficient condition for a function of the state to be an entanglement monotone under local operations that do not involve information loss is that the function be a monotone under infinitesimal local operations. We then derive necessary and sufficient differential conditions for a function of the state to be an entanglement monotone. We first derive two conditions for local operations without information loss, and then show that they can be extended to more general operations by adding the requirement of convexity. We then demonstrate that a number of known entanglement monotones satisfy these differential criteria. Finally, as an application, we use the differential conditions to construct a new polynomial entanglement monotone for three-qubit pure states. It is our hope that this approach will avoid some of the difficulties in the theory of multipartite and mixed-state entanglement.
Gudder, in a recent paper, defined a candidate entanglement measure which is called the entanglement number. The entanglement number is first defined on pure states and then it extends to mixed states by the convex roof construction. In Gudder's article it was left as an open problem to show that Optimal Pure State Ensembles (OPSE) exist for the convex roof extension of the entanglement number from pure to mixed states. We answer Gudder's question in the affirmative, and therefore we obtain that the entanglement number vanishes only on the separable states. More generally we show that OPSE exist for the convex roof extension of any function that is norm continuous on the pure states of a finite dimensional Hilbert space. Further we prove that the entanglement number is an LOCC monotone, (and thus an entanglement measure), by using a criterion that was developed by Vidal in 2000. We present a simplified proof of Vidal's result where moreover we use an interesting point of view of tree representations for LOCC communications. Lastly, we generalize Gudder's entanglement number by producing a monotonic family of entanglement measures which converge in a natural way to the entropy of entanglement. Quanta 2020; 9: 22–36.
Abstract To quantify the entanglement is one of the most important topics in quantum entanglement theory. An entanglement measure is built from measures for pure states. Conditions when the entanglement measure is entanglement monotone and convex are presented, as well as the interpretation of smoothed one‐shot entanglement cost. Next, a difference between the measure under the local operation and classical communication and the separability‐preserving operations is presented. Then, the relation between the convex roof extended method and the way here for the entanglement measures built from the geometric entanglement measure for pure states, as well as the concurrence for pure states in two‐qubit systems are considered. It is also shown that the measure is monogamous for system.
We establish entanglement monotones in terms of an operational approach, which is closely connected with the state conversion from pure states to the objective state by the local operations and classical communications. It is shown that any good entanglement quantifier defined on pure states can induce an entanglement monotone for all density matrices. Particularly, we show that our entanglement monotone is the maximal one among all those having the same form for pure states. In some special cases, our proposed entanglement monotones turn to be equivalent to the convex roof construction, which hence gain an operational meaning. Some examples are given to demonstrate different cases.