# Entanglement entropy vs Entropy

• wam_mi
In summary: This is because in quantum mechanics, you can't always measure everything. So, in order to calculate the entanglement entropy, you need to average over all the possible outcomes of the unmeasured degrees of freedom. This allows you to focus on the entanglement aspect of the system rather than the inaccessible degrees of freedom.
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.

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.

Last edited:

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!

## 1. What is the difference between entanglement entropy and entropy?

Entanglement entropy is a measure of the amount of quantum entanglement between two or more particles, while entropy is a measure of the disorder or randomness in a system.

## 2. How are entanglement entropy and entropy related?

Entanglement entropy is a type of entropy that describes the correlations between particles in a quantum system. In contrast, entropy is a more general concept that can describe the disorder in both classical and quantum systems.

## 3. How is entanglement entropy calculated?

Entanglement entropy is calculated by tracing out the state of a subsystem from the total state of a quantum system. This involves measuring the entanglement between the subsystem and the rest of the system.

## 4. What are the applications of entanglement entropy and entropy?

Entanglement entropy is used in quantum information theory, quantum computation, and quantum field theory, while entropy has many applications in thermodynamics, information theory, and statistical mechanics.

## 5. Can entanglement entropy and entropy be used interchangeably?

No, entanglement entropy and entropy are distinct concepts that cannot be used interchangeably. They have different mathematical definitions and are used to describe different physical phenomena.

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