An Improved Bound on Distillable Entanglement

The best bound known on 2-locally distillable entanglement is that of Vedral and Plenio, involving a certain measure of entanglement based on relative entropy. It turns out that a related argument can be used to give an even stronger bound; we give this bound, and examine some of its properties. In particular, and in contrast to the earlier bounds, the new bound is not additive in general. We give an example of a state for which the bound fails to be additive, as well as a number of states for which the bound is additive. One of the central problems in quantum information theory is that of entanglement distillation [1], the production of maximally entangled states from partially entangled states. The purpose of the present note is to give a new upper bound on the rate at which entanglement can be distilled. In general, if C is a given class of physical operations, we define C-distillable entanglement as follows: Definition. The C-distillable entanglement of a state ρ on a state space VA ⊗ VB is the maximum number DC(ρ) such that there exists a sequence Ti : (VA ⊗ VB) → Vi ⊗ Vi of operations from C, with ni → ∞, 1 ni log2 dimVi → DC(ρ), and F (Ti(ρ)) → 1. See [2] for a discussion of why this is a valid definition of distillable entanglement. Here F (σ) is the fidelity of the state σ, defined by F (σ) = Φ(V )σΦ(V ), Φ(V ) = 1 √ dimV ∑