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arXiv:quant-ph/0005126 · Physical Review A

Additivity of the entanglement of formation

Fabio Benatti, Heide Narnhofer

Published Mar 16, 2001• 46 citations
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Abstract

We address the question of whether the entanglement of formation is additive over tensor products of full matrix algebras, and derive a necessary and sufficient condition for optimality of vector states. By discussing two particular cases, we suggest that such a condition may be used either to prove or to find a counterexample to additivity.

DOI
10.1103/physreva.63.042306

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We address the question of whether the entanglement of formation is additive over tensor products of full matrix algebras, and derive a necessary and sufficient condition for optimality of vector states. By discussing two particular cases, we suggest that such a condition may be used either to prove or to find a counterexample to additivity.
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An important measure of bipartite entanglement is the entanglement of formation, which is defined as the minimum average pure state entanglement of all decompositions realizing a given state. A decomposition which achieves this minimum is called an optimal decomposition. However, as for the entanglement of formation, there is not much known about the structure of such optimal decompositions, except for some special cases, like states of two qubits or isotropic states. Here we present a necessary and sufficient condition for a set of pure states of a finite dimensional bipartite system to form an optimal decomposition. This condition is well suited to treat the question, whether the entanglement of formation is additive or not.
In this paper, I discuss the additivity conjecture in quantum information theory.The additivity conjecture was originally a set of at least four conjectures.These conjectures said that certain functions of quantum states and channels were additive under tensor products.While some of these conjectures were previously known to be stronger than others, they have recently all been proved equivalent.This conjecture is a very intriguing mathematical question which the best efforts of a large number of quantum information theorists have not been able to resolve for nearly a decade.It is a mathematically elegant question that is one of the most important open questions in the field of quantum information and computation.This paper then is intended to be both an exposition of the conjecture, its background, and some of the methods that have been used to yield partial results for it, as well as a plea for help in resolving this conjecture.Very recently (summer 2007), substantial progress has been made on this conjecture, in that counterexamples to a set of stronger conjectures have been found.These will be described briefly.
Recently Shor proved equivalence of several open (sub)additivity problems related to the Holevo capacity and the entanglement of formation [15]. In our previous note [6] equivalence of these to the additivity of the Holevo capacity for channels with arbitrary linear constraints was shown. This note is the development of the previous one in the direction of channels with general constraints. Introducing input constraints provides greater flexibility in the treatment of the additivity conjecture. The Holevo capacity of arbitrarily constrained channel is considered and the characteristic property of an optimal ensemble for such channel is derived, generalizing the maximal distance property of Schumacher and Westmoreland (proposition 1). It is shown that the additivity conjecture for two channels with single linear constraints is equivalent to the similar conjecture for two arbitrarily constrained channels and, hence, to an interesting subadditivity property of the χχχ-function for the tensor product of these channels (theorem 1). We also propose an alternative way of proving that the additivity conjecture for any two unconstrained channels implies strong superadditivity of the entanglement of formation. The arguments from the convex analysis provide another characterization of channels for which subadditivity of the χχχ-function holds (theorem 3). This characterization and some modification of Shor's channel extension provide a simple way of proving that global additivity of the minimum output entropy for unconstrained channels implies global subadditivity of the χχχ-function and strong superadditivity of the entanglement of formation.
We present a new method of constructing sets of states with known entanglement of formation. The method realizes the optimal decomposition families of states. We also derive the additivity of the entanglement of formation and thus calculate the entanglement cost. These results are used to find out the undistillable entanglement of some states. We illustrate our method by investigating the two-qubit state, the separable state, the maximally correlated state, the isotropic state and the Werner state. We also discuss the realization of Werner state in experiment.