Tl;dr: The nice thing about the partition function is that all relevant properties of a thermodynamic system can easily be derived from it. In quantum information theory, multipartite entanglement is poorly understood and very difficult and we don't have a similar partition function yet - and it might not exist at all. Maybe you can get a "partition function" in quantum thermodynamics, but then it'll be the thermodynamic one.
In information theory, entropy is a measure of information - in particular, it is not a thermodynamic quantity and it is not at all clear that it can be a thermodynamic quantity at all.
Quantum Information Theory
In quantum information theory, which takes its basic notion of "entropy" from information theory, what you call "entropy of entanglement" is a measure of entanglement for bipartite systems. Given a bipartite (pure) quantum system $\rho_{AB}$, what is the entropy of the reduced system $\rho_A$ and $\rho_B$? If the state $\rho_{AB}$ is pure, then a high entropy of $\rho_A$ corresponds to a lot of entanglement between the two systems. Since entanglement is correlations, this means that the entanglement entropy measures the bipartite correlations (and therefore in a sense the "information" shared between the two systems). Let's first consider this idea of entropy, which is a priori not a thermodynamic quantity!
Already when $\rho_{AB}$ is mixed, you have to be careful, because it could be that although both reduced states have considerable entropy, the state $\rho_{AB}$ is a product state.
Now, as you correctly remark, for a multipartite system, correlations can be shared between more than just one party. This makes matters much more difficult and multipartite entanglement has only been poorly understood as of now.
And the problem is that your scheme won't work: Operationally speaking, there is not just one type of entanglement between three particles (or four, etc.). In QI, we are interested in doing something with entangled particles and that often means to transform a state (or a bunch of states) into some other state with only local quantum operations and maybe some classical communication (entanglement is useful, because it can be shared over distances). This is known as convertibility under LOCC transformations. Sadly, in order to fully describe entanglement between three pure qubits, this already requires two parameters just to characterise the entanglement where all three qubits are entangled. For four qubits, this set of parameters grows to nine. A complete operational classification of entanglement for three mixed qubits has only been achieved recently and requires more than just one entropy.
What does this mean for your question? In my view this means that there is no partition function as you want it for entanglement entropy. First of all, entanglement entropy is no thermodynamic quantity and therefore, it is not connected to the usual thermodynamic partition functions. Furthermore, entanglement entropy is not sufficient to describe multipartite entanglement alone. The basic appeal about the partition function is that you can derive about every important thermodynamic quantity from it by easy manipulations such as derivatives. Since entanglement between multi-partite systems is poorly understood, this implies that we still don't have a good idea how such a quantity would look like.
Some Thermodynamics
As mentioned above, although the formula for thermodynamic entropy is the same as information theoretic entropy, the two entropies are different things in information theory. Under certain circumstances you can think of entanglement entropy as something related to thermodynamics. There are several examples where this has been studied to some degree:
- condensed matter systems (spin chains)
- conformal field theories
- quantum thermodynamics
In an example of the first application, the authors consider bipartite entanglement in spin chains, showing that if we take as a bipartition some region of size $L$ and its complement in the system, the entropy scales with $L$ and the scaling depends on the quantum system. Correlations increase as the parameter varies. Note that everything is done at zero temperature - we don't really have a thermodynamic system, instead, the usual entropy will sit on top of the entanglement entropy. Also, I don't see a notion of "partition function" and I don't see how it can be useful.
I don't know much about the second example - but it seems connected to the Bekenstein-Hawking entropy (see here for one of the older papers). Maybe there is some thermodynamics here, I don't know.
For the third application, I have also only limited knowledge. While this is probably in the spirit of what you thought about, I guess your question can also easily be answered here: the partition function is the usual partition function as long as thermodynamic and entanglement entropy are the same. Part of the field is to actually about how thermodynamic entropy is connected to information theoretic entropy. They are definitely not the same - in fact, you might want to consider different measures altogether. If it was the same, you could of course use the usual partition function. However, others argue that you can should replace basic axioms of statistical mechanics in order to arrive at entanglement entropy (once again: the world is bipartite with a "system" part and an "environment" part). Part of the heuristic here if I remember correctly is that natural states of low thermodynamic entropy (complete magnetisation for example) are product state if one considers a part of the system as "system" and the other part as "environment". If you increase the system, the thermal fluctuations will increase the correlations between system and environment, thus increasing entropy. In other words, information will "leak" into the environment, further increasing entropy between the systems. This basically means that the fundamental laws of statistical mechanics can also hold for the entanglement entropy. Using this, they can replace the "equiprobability axiom" from thermodynamiscs.
