Entanglement entropy vs Entropy

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SUMMARY

This discussion focuses on the concepts of entanglement entropy and its relationship to entropy in quantum mechanics, particularly in bipartite systems. It is established that Toni, an observer with complete knowledge of region A but none of region B, measures entanglement entropy in region A by tracing over the degrees of freedom in region B, leading to the conclusion that the entanglement entropy in both regions is equal. The conversation clarifies that while entanglement entropy arises from inaccessible degrees of freedom, entropy can also arise from uncertainty in a quantum state without entanglement, as seen in mixed states. Tracing over degrees of freedom is described as a method of statistical averaging over unmeasured observables.

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wam_mi
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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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wam_mi said:
(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.

wam_mi said:
(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.

wam_mi said:
(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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