Evidence for Bound Entangled States with Negative Partial Transpose

Maximally entangled quantum states, when their two halves are shared between two parties, are a uniquely valuable resource for various information-processing tasks. Used in conjunction with a quantum communications channel, they can increase the classical data carrying capacity of that channel, in some cases by an arbitrarily large factor [1]. Possession of maximally entangled states can ensure perfect privacy of communication between the two parties by the use of quantum cryptography [2]. These states can facilitate the rapid performance of certain forms of distributed computations [3]. Of course, maximally entangled states are the key resource in quantum teleportation [4]. On the other hand, the surreptitious establishment of entanglement between two parties can thwart the establishment of trust between parties via bit commitment [5]. How can two parties come into the possession of a shared maximally entangled state? If the storage and transportation of quantum particles were perfect, then the state could have been synthesized in some laboratory long in the past and given to Alice and Bob (our personified parties) for storage until needed. In practice no such perfect infrastructure exists. Since the most interesting scenarios for the use of quantum entanglement are in cases where Alice and Bob are remote from one another, we will consider the long-distance transportation of quantum states needed to establish the shared entanglement to be difficult and imperfect, while the local processing of quantum information (unitary transformations, measurement) we will assume, for the sake of analysis, to be essentially perfect. Under these assumptions, when we wish to assess whether a given physical setup is or is not useful for entanglement assisted information processing, our analysis focuses on the mixed quantum state, ρ, in the hands of Alice and Bob after the difficult transportation step. We enquire whether ρ⊗n can be transformed, by LQ+CC operations, to a supply of maximally entangled states. Here the ⊗n notation indicates that n copies of the state ρ are available, and we will be concerned with asymptotic results as n is taken to infinity. LQ+CC operations (sometimes called LOCC in the literature) are obtained by any arbitrary sequence of local quantum operations (appending ancillae, performing unitary operations, discarding ancillae) supplemented by classical communication between Alice and Bob. An interesting fact about this possibility for the distillation of entanglement is that it is neither rare nor ubiquitous; a finite fraction of the set of all possible bipartite mixed states ρ can be successfully distilled [6], and a finite fraction cannot [7]. Much work has been focussed on whether ρ falls into the distillable or into the undistillable class, and this paper is primarily a contribution to this classification task. Before describing our new contributions, we will give a brief review of previous results on classifying states according to their distillability. Multipartite density matrices ρ are considered unentangled if there exists a decomposition of ρ into an ensemble of pure product states; for the bipartite case this means that we can write