Comparison of the relative entropy of entanglement and negativity

It is well known that for two qubits the upper bounds of the relative entropy of entanglement (REE) for a given concurrence as well as the negativity for a given concurrence are reached by pure states. We show that, by contrast, there are two-qubit mixed states for which the REE for some range of a fixed negativity is higher than that for pure states. Moreover, we demonstrate that a mixture of a pure entangled state and pure separable state orthogonal to it is likely to give the maximal REE. By noting that the negativity is a measure of entanglement cost under operations preserving positivity of partial transpose, our results provide an explicit example of operations such that, even though the entanglement cost for an exact preparation is the same, the entanglement of distillation of a mixed state can exceed that of pure states. This means that the entanglement manipulation via a pure state can result in a larger entanglement loss than that via a mixed state.