Entanglement entropy vs Entropy

[wam_mi]

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.

 

[Demystifier]

(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.

 

[Entropia]

Hello,

(i) Yes, that is correct. Entanglement entropy is a measure of the amount of entanglement between two systems, in this case regions A and B. By tracing over the degrees of freedom in region B, Toni is essentially measuring the entanglement between A and B, which is equal to the entanglement between B and A. This is because entanglement is a symmetric property, meaning that it is the same regardless of which system is being observed.

(ii) Entropy is a measure of the disorder or randomness in a system. In quantum mechanics, this disorder can arise due to the entanglement between different parts of the system. Entanglement entropy is a specific type of entropy that measures the amount of entanglement present in a system. It is not the same as the overall entropy of the system, but it is related to it.

(iii) Tracing over the degrees of freedom means that we are ignoring or averaging out the information from certain parts of the system. In the case of entanglement entropy, this allows us to focus on the entanglement between different parts of the system, rather than the individual properties of each part. Physically, this can represent the idea of an observer having limited knowledge or access to certain parts of the system, and only being able to measure the overall entanglement between different parts.

I hope this helps clarify the differences between entanglement entropy and entropy. Keep in mind that both are important concepts in understanding the behavior of quantum systems, and they are often used together to gain a more complete understanding of a system. Let me know if you have any further questions. Happy studying!