Can anyone help me out with von Neumann entropy please?

[lhj2010]

Can anyone help me out with von Neumann entropy please??

I've been wondering if I could be able to calculate the entropy of a system which is part of a bigger system.

For example, let's say that there's a cavity and two atoms(atoms are two-level system).
The hamiltonian of the system would roughly be like this;
[tex] \[
H = \sum_{j} \omega s_{jz} + \omega a^{\dag} a + \sum_{j} g ( s_{j-} a^{\dag} + s_{j+} a ),
\][/tex]
And you know the density matrix. I put the state in this form;
[tex] \[
|\Psi \ket = |\Psi_{atom1} \ket \otimes |\Psi_{atom2} \ket \otimes |\Psi_{cav} \ket = a|1_1, 0_2, 0_3 \ket + b|0_1, 1_2, 0_3 \ket + c|0_1, 0_2, 1_3 \ket
\][/tex]

where 1 is atom1, 2 is atom 2, and 3 is a cavity.

and got this density matrix;

[tex] \[
\rho^{ABC} =
\left( {\begin{array}{ccc}
|a|^2 & ab^\ast & ac^\ast \\
a^\ast b & |b|^2 & bc^\ast \\
a^\ast c & b^\ast c & |c|^2 \\
\end{array} } \right)
\]
[/tex]




Now, I'm trying to see the entropy between the two atoms, and I was wondering about this. I don't know if it's really simple thing or something that's not really been cleared.

First, I thought I had to reduce the matrix to kill the degree of freedom of the cavity. Which gives me this.

[tex] \[
\rho^{AB} =
\left( {\begin{array}{ccc}
|a|^2 & ab^\ast & 0 \\
a^\ast b & |b|^2 & 0 \\
0 & 0 & |c|^2 \\
\end{array} } \right)
\][/tex]

Then I thought now I had to reduce the matrix once more in order to evaluate the entropy. And this gives me a funny result. By choosing different atom to reduce, we get different entropy. The average entropy in a period is the same, but shouldn't it be identical at any time?? I mean, it's the same system. Is there some kind of restriction when you use reduced matrix for entropy like, 'you shouldn't use the reduced matrix to reduce it for the entropy' or something like that?

I really don't know if I'm beeing stupid or anything, but even if I am, let me know.
Thanks in advance! :D

 

[eljose79]

Hello! Calculating von Neumann entropy for a system that is part of a bigger system can be a bit tricky, but it is definitely possible. First, it's important to understand that entropy is a measure of the amount of uncertainty or randomness in a system. In quantum mechanics, this uncertainty is represented by the density matrix, which you have already calculated for your system.

To calculate the entropy between the two atoms, you can use the reduced density matrix as you have done. However, it's important to note that the reduced density matrix needs to be normalized before using it to calculate entropy. This means that the sum of the diagonal elements in the reduced density matrix should add up to 1. In your calculation, this would mean dividing the reduced density matrix by the sum of the diagonal elements.

Once you have a normalized reduced density matrix, you can use the formula for von Neumann entropy, which is given by S = -Tr(ρ ln ρ), where Tr is the trace operator. This will give you the entropy between the two atoms in your system.

As for your observation about getting different entropy values when reducing the matrix for different atoms, this is actually expected. The reduced density matrix depends on which subsystem you are considering, and therefore the entropy will also depend on which subsystem you are calculating it for. However, as you mentioned, the average entropy over a period of time should be the same, as the system is the same.

I hope this helps! Let me know if you have any further questions.