So far what I have learned is that von-Neumann entropy is a tool to measure or quantify information and therefore entanglement for a given pure state system. However, similar concepts emerge from the task of measuring entanglement in general, which would be entanglement cost and distillable entanglement. My question is what are the similarities and divergences of these concepts with von-Neumann entropy?
The reference from which I spotted this question is from this article.
The entanglement entropy (what you call "von Neumann entropy") is a good measure for entanglement of pure states in the asymptotic setting, i.e. when one is dealing with many copies. However, it is not a good measure for mixed states.
Distillable entanglement and entanglement cost are entanglement measures which apply to both pure and mixed states. Moreover, for pure states, all three measures are equal, which is desirable due to the special status of the entanglement entropy. Finally, entanglement cost and distillable entanglement are the largest and smallest entanglement measure which coincides with entanglement entropy for pure states, and which is operationally meaningful in an asymptotic scenario.
