Entanglement entropy vs Entropy

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Hi there,

I am currently reading some background materials about entanglement entropy relating to black holes. I got quite confused and can I just ask

(i) For example, if we have a bi-patite system, say A and B, separated by some shared boundary of the two sub-regions. Toni, an observer, who knows everything about region A but region B is completely unknown to him. Is it true to say that "Toni measures the entanglement entropy in region A by tracing over the degrees of freedom in region B, and it turns out the entanglement entropy in region A = the entanglement entropy in region B,"

(ii) I got the impression that entropy arises in a quantum state due to entanglement, and that happens because of the inaccessibility of the degrees of freedom that are hidden from the observer. So what is the difference between entanglement entropy and entropy then? Isn't it true that entropy is a measure of the uncertainty of a quantum state?

(iii) Why do we want to trace over the degrees of freedom? What does this physically mean?

That's all for now. Many thanks.
 

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(i) For example, if we have a bi-patite system, say A and B, separated by some shared boundary of the two sub-regions. Toni, an observer, who knows everything about region A but region B is completely unknown to him. Is it true to say that "Toni measures the entanglement entropy in region A by tracing over the degrees of freedom in region B, and it turns out the entanglement entropy in region A = the entanglement entropy in region B,"
That's correct.

(ii) I got the impression that entropy arises in a quantum state due to entanglement, and that happens because of the inaccessibility of the degrees of freedom that are hidden from the observer. So what is the difference between entanglement entropy and entropy then? Isn't it true that entropy is a measure of the uncertainty of a quantum state?
There are also other ways how entropy may arise in quantum mechanics. For example, if you know that the system is in a definite pure state, but you just don't know which one, then your knowledge can also be described by a mixed state and you can associate entropy with it. There is no entanglement in this case.

(iii) Why do we want to trace over the degrees of freedom? What does this physically mean?
It means statistical averaging over observables which you don't measure.
 
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