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In Quantum mechanics entanglement is a concept that informs us about nature of states. It is a statement about non-product states, thus correlations. This is my rather foolish view of entanglement(correlations?). There is something named "entanglement entropy". I somewhat vaguely recall what the standard statistical mechanics definition for entropy is. I have heard there are many types of entropy, but I am not sure this is relevant. What is entanglement entropy?; and what is all the chatter about counting (of states) that always happens in papers about this? As I have a very limited physics background, I would prefer to explore answers that don't mention black holes, or quantum field theory if this is possible. I am hoping these can be precipitated in very basic quantum mechanics, or in some classical analogue dealing with say classical correlations and statistical mechanics.

Qmechanic
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  • Entanglement entropy is the Gibbs entropy for quantum systems. For a short account of the Gibbs entropy see http://physics.stackexchange.com/a/141324/28512. – alanf Jan 02 '16 at 11:54

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I assume you are puzzled by the incongruence between the concept of entropy as measure of "disorder" vs. that of entanglement entropy as measure of "correlations", given that many articles define the two by the same formula. The reason for the confusion is that entanglement entropy is often presented in a simplified version that "looks" like the regular entropy. Something is left out without being properly explained.

For any two quantum systems $A$ and $B$ the mutual entropy or mutual information is defined as the difference between the entropy of $A$ and $B$ in the absence of entanglement and their entropy in the presence of entanglement. If we denote the regular entropy as $S$ and mutual entropy as $I$, then we generally have $$ I(A+B) = S(A) + S(B) - S(A+B) $$ Like the regular entropies, the mutual entropy is always positive, $I(A+B) \ge 0$, and accounts for both entanglement and classical correlations. But for the particular case when the entangled state of $A$ and $B$ is a pure state, there are no classical correlations and the total entropy vanishes, $S(A+B) = 0$. It also happens that in this case we have necessarily $S(A) = S(B)$, so the mutual entropy reduces to $$ I(A+B) = 2S(A) $$ and becomes a measure of entanglement. The $2$ factor is eventually dropped for economy of language and notation. Keep in mind though that this is no longer true when the state of $A+B$ is a not a pure state, but a mixed state, and $S(A+B) > 0$.

udrv
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    If the total state AB is not pure, this is the mutual information and no more a measure of entanglement but of total correlations (quantum and classical). Calling it entanglement entropy is clearly a misnomer, and I'm not aware that people do so systematically. – Norbert Schuch Jan 02 '16 at 11:31
  • @NorbertSchuch You are right of course. Sorry about it, didn't use it in a while and got carried away. Is it correct now? – udrv Jan 02 '16 at 12:24
  • Sounds good. -- One noteworthy thing to point out about entanglement entropy is that the reason it is so popular in quantum information is that it uniquely quantifies the amount of entanglement in an asymptotic scenario (i.e., when we want to measure the entanglement in many copies of some pure state), which is of course great to have. Whether this asymptotic scenario is of course the relevant one e.g. in condensed matter applications is a different question. – Norbert Schuch Jan 02 '16 at 13:21
  • So basically mixed state measures by way of various purifications (distillation, cost, etc.)? – udrv Jan 02 '16 at 13:45
  • Entanglement of distillation, entanglement cost, etc. don't work through purifications (a purification is a way of thinking of a mixed state as part of a larger pure state), but by finding asymptotic protocols which relate it to pure state entanglement -- how many Bell pairs per copy can we extract, how many Bell pairs per copy are needed to build a state, etc.. But there is a whole load of mixed state entanglement measures -- there is no single nice entanglement measure. – Norbert Schuch Jan 02 '16 at 13:49
  • Got it. Bad wording about purification, had in mind what you said. – udrv Jan 02 '16 at 16:06
  • For me, it's always troublesome to talk about the entanglement of a mixed state by its density matrix since a density matrix does not really completely specify the 'state'. So the entanglement of a mixed state seems to be defined on a set of 'equivalent' states and we are always talking about the bounding property of the set. – XXDD Jan 03 '16 at 04:27
  • @X.Dong The local density matrix of an entangled system does not specify whether the system is entangled or not, but always characterizes completely the system's local state. That is, all the local observables of the system can be calculated from it. Entanglement, otoh, is visible only in the total density matrix of all systems involved - it is non-local. – udrv Jan 03 '16 at 14:27
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    @udrv Oh, I thought that Norbert was talking about the entanglement of mixed states, where the density matrix is not local, but on the mixed state composite system. Then as you said, the true state should be purification dependent, otherwise we can only talk about boundary properties of a set of 'equivalent' states described by the same density matrix. That does not sound a good definition of the entanglement of mixed state for me. – XXDD Jan 03 '16 at 15:48
  • @X.Dong Well, he was talking about non-local mixed states, but I think we had in mind different things. Did you mean mixed states are ambiguous wrt to their particular realization as statistical superpositions and so there is a question of whether entanglement may depend on the particular realization? By boundary properties you mean local properties? Not familiar with the term. – udrv Jan 03 '16 at 16:41
  • @udrv Yes. I think he is talking the non-local mixed state. Then the entanglement of a given density matrix is defined as the lower limit of the entanglement of all the possible realization of the density matrix. That's what I mean by 'boundary properties' of the density matrix, which in fact is correspondent to a set but not a specific 'state'. – XXDD Jan 04 '16 at 01:14
  • @X.Dong Correct, that makes sense now. Another word for "boundary" properties is "extremal" properties. – udrv Jan 04 '16 at 09:41