What is entanglement entropy? and all those stories about counting

Question

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.

Answer 1

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 $S(A+B) > 0$.