Is there a useful generalization of the Schmidt decomposition to the tensoring together of 3 or more vector spaces?
An answer to your "vague question" may be provided by http://en.wikipedia.org/wiki/Higher-order_singular_value_decomposition, but as far as I know "finding a good generalization of the SVD for 3-tensors" is kind of a research problem in linear algebra, too. Check out Kolda, Bader "Tensor decompositions and applications". Your idea of minimizing an entropy function is new, as far as I can tell, but I am not a tensor expert.
Vague question:
The hierarchical higher order SVD would lead you to the canonical form for the GHZ state. When you have $H_1 \otimes H_2 \otimes \ldots \otimes H_n$ you basically first Schmidt decompose your state $\psi$ relative to $H_1 \otimes (H_2 \otimes \ldots \otimes H_N)$ and then continue like this until you finish.
In more generality, the Tucker decomposition is the most general multilinear generalization of the SVD. It is not, however, unique. The hierarchical higher order SVD I mentioned above is a special case of the Tucker decomposition.
Specific question:
I don't know the answer to your question, but I did notice this: If you start with $\psi = \sum_{jk} \gamma_{jk} \alpha_j \otimes \beta_k$ then your entropy is essentially the entropy of the post measurement mixed state with measurement basis being $\alpha_j \otimes \beta_k$. So what you want to look at is which local measurement scheme gives you the minimum post-measurement entropy. So in the bipartite case all you need to do is take a partial trace and whatever the entropy of that matrix, that is the minimum post-measurement entropy. I.e. you get entanglement, which confirms your suspicion that the Schmidt decomposition is what you get in bipartite case. A similar strategy might work in the multipartite case.
There is some recent work on tensors (and also the special cases of symmetric and alternating tensors) that admit orthogonal or unitary decompositions (resp., symmetric or alternating decompositions). I will point out, particularly, the following three papers by Elina Robeva and coauthors:
- https://arxiv.org/abs/1409.6685
- https://arxiv.org/abs/1512.08031
- https://arxiv.org/abs/1603.09004