Relation between Von Neumann entropy (and other entanglement measures) and thermodynamical entropy

Question

Suppose I have a bipartite system (with Hilbert space $H = H_a \times H_b$) and the following state:

$$\sigma = \sum_{n} \frac{e^{-\beta E_n}}{Z} \rho_n$$

where $Z = \sum_n e^{- \beta E_n}$ and $\rho_n$ are arbitrary density matrices in $H$.

I'm interested in understanding if there are entanglement measures that can separate the 'thermodynamical' entropy (i.e., not knowing in which state the system is in) and the 'quantum' entropy (i.e., measuring the correlations and entanglement between sub-systems).

So, to $\sigma$ I'd associate a 'thermodynamical' entropy $$S = - \beta \frac{\partial \log Z}{\partial \beta} + \log Z,$$ but I don't know if there's an easy way to calculate the Von Neumann entropy, for instance.

If each $\rho_n$ was $| E_n \rangle \langle E_n|$ then the Von Neumann entropy and the 'thermodynamical' entropy would coincide, but I think this is the only case they would.

So, my question is the following: is there some entanglement measure that would divide into something like classical entropy + quantum entropy?

Answer 1

Since state $\sigma$ is not in thermal equilibrium I don't think one can use your definition of "thermodynamical" entropy. In fact, one should instead use Von Neumann entropy, which is a correct measure of statistical (so not quantum!) uncertainty. There is no other "classical" or "thermodynamical" entropy in quantum systems. As you mentioned, for a thermal state Von Neumann entropy is "thermodynamical" entropy.

As you probably know there are many entalgement measures, one of them being relative entropy of entalgement, based on quantum relative entropy. Note that Von Neumann entropy of the composite system is not itself a measure of entanglement between subsystems.

Additional comment: In fact, there is much recent research about possible generalizations of thermodynamics to small quantum systems out of equilibrium. For example, in the article "The second laws of quantum thermodynamics" (http://arxiv.org/abs/1305.5278) authors define a whole family of free energies based on Renyi divergences. One of them looks like classical free energy and is defined as: \begin{equation} F_1(\rho,\rho_{\beta})=\frac{1}{\beta} \left(S(\rho||\rho_{\beta})-\log{Z}\right) \end{equation} where $\rho_{\beta}$ stands for thermal state of Hamiltonian on which $\rho$ is defined, and $S(\rho||\rho_{\beta})$ is quantum relative entropy.