Why are there so many different measures of entanglement? I have seen various articles (e.g. Introduction to Entanglement Measures)  discuss the various entanglement measures that exist. These include: concurrence, entanglement cost, distillable entanglement, entanglement of formation, etc.
My questions are: why are there so many different measures of entanglement? Do they have any physical significance? Can they be differentiated experimentally? And from a practical point of view, how do we choose which one we should focus on (for theoretical or experimental purposes)?
 A: Since entanglement just means non-classical correlation (basically) and correlation is a complex phenomenon, it is natural that there will not be any one number that you can pin on it to explain all the phenomena.
But how do you quantify correlations at all? You can, for instance, try to describe the set of correlations with purely mathematical means:


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*The set of states is a convex set, so you can define notions of geometry. For instance, you can ask the "distance" between an entangled state and the set of separable states. One such "distance" (it's not really a distance) can be computed using the relative entropy (which pops up in many different contexts as well).

*For more than two parties, the geometry can become quite complex. A lot of measures have been proposed, I don't want to go into whether they are useful or not. For a small list see for instance:
https://en.wikipedia.org/wiki/Multipartite_entanglement#Multipartite_entanglement_measures
One of the most important ideas is to find operational measures. An operational measure measures the ability to perform a certain task. What can you do with entanglement? Well, all sorts of stuff:


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*Given an entangled state, you can ask which other states can be produced from it by using only local quantum operations (i.e. acting on each subsystem of the entangled system only) and classical communication. This is a classical problem of state interconvertibility. For pure bipartite entangled states, the answer is majorisation of the spectrum, which is therefore something like an entanglement measure.


This naturally leads to the question of whether there are "maximally entangled states", i.e. states from which you can produce all other states with LOCC (local op. and classical communication). While such states exist among bipartite (pure) states, they do not necessarily exist among multipartite states. I refer to the classic papers
https://arxiv.org/abs/quant-ph/0005115
https://arxiv.org/abs/quant-ph/0109033
and the more recent work by Barbara Kraus' group, for instance:
https://arxiv.org/abs/1510.09164
https://arxiv.org/abs/1503.00615
For instance, if there are many "maximally entangled" states, you can define measures measuring what percentage of all states can be reached from a single state or something similar.


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*You can ask the same question when given a huge number of the same state. Does it help to have several states? For bipartite pure qubit states for instance, it turns out that having enough states means you can even "distill" maximally entangled states. The ration of states needed to produce a maximally entangled state in the limit of infinitely many states is called the distillable entanglement.

*You can ask the converse question: Given maximally entangled states (if they exist) and separable states, how much "entanglement" is there to be distributed to different states? Given another state, how many of these states can I produce with an entangled state. In the limit of infinitely many copies, this is known as the entanglement of formation

*In cryptography, you are not really interested in the number of maximally entangled states you can produce but how much of that entanglement can be used for cryptographic protocols using private keys. There are operational measures measuring exactly that such as the distillable key

*So far, I have mostly talked about state interconvertibility given either one copy of a state or infinitely many copies of a state. Obviously, you can ask the question in between and you will get a whole host of measures similar to the entanglement of formation and the distillable entanglement (see for instance here: https://arxiv.org/abs/1701.08679)
And this is not all. There are many other scenarios that people consider that are interesting from a physical perspective and ultimately resolve to entanglement measures.
So, what about your questions?


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*Do the measures have physical significance? Yes, at least the operational ones do. 

*Can they be differentiated experimentally? Yes and no. Most measures refer to a physical state, for instance turning a state into another using LOCC, hence they only make sense in that context a priori. But you can show that not all measures are the same. For instance, at least for mixed states, the distillable entanglement and the entanglement of formation are definitely not the same. 

*What measures to focus on? For this question, I cannot give an answer. While you should certainly know the von Neumann entropy, the relative entropy and the distillable entanglement and entanglement of formation, everything beyond must be specified in context. Depending on the sort of problem you want to study, you must choose the (mostly operational) measures related to your problem.
