(The questions How does the Schmidt decomposition generalize to tensor products of several finite-dimensional systems? and Is there a useful generalization of the Schmidt decomposition to the tensoring together of 3 or more vector spaces? seem to be asking something different from what follows, but perhaps I have not read them closely enough.) $\newcommand{\tp}{\otimes}$ $\newcommand{\ptp}{\hat{\otimes}}$ $\newcommand{\itp}{\check{\otimes}}$
In what follows: $\tp$ denotes the algebraic tensor product of complex vector spaces; $\ptp$ denotes the projective tensor product of Banach spaces; $\itp$ denotes the injective tensor product of Banach spaces; $\tp_2$ denotes the Hilbertian tensor product of two Hilbert spaces.
Let $H_1$, $H_2$ and $H_3$ be Hilbert spaces (you can assume finite--dimensional of arbitrary dimension, although ultimately I am after the separable infinite-dimensional case).
A consequence of the singular value decomposition (or Schmidt decomposition) is that each $w\in H_1\tp H_2$ can be written as $$ w = \sum_{i=1}^N \lambda_i e_i \tp f_i $$ where $$ \Vert w \Vert_{H_1\ptp H_2} = \sum_{i=1}^N |\lambda_i| \tag{1}$$ $$ \Vert w \Vert_{H_1\tp_2 H_2} = \left(\sum_{i=1}^N |\lambda_i|^2\right)^{1/2} \tag{2}$$ $$ \Vert w \Vert_{H_1\itp H_2} = \max_{1\leq i\leq N} |\lambda_i| \tag{$\infty$}$$
Question. Can we do the same for $H_1\tp H_2\tp H_3$?
That is, for given $w\in H_1\tp H_2\tp H_3$ we want vectors $(e_i)$, $(f_i)$, $(g_i)$ and scalars $(\lambda_i)$ such that $w=\sum_i \lambda_i e_i\tp f_i \tp g_i$ and the 3-variable analogues of $(1)$, $(2)$ and $(\infty)$ hold. I must admit this seems overly optimistic to me, so I wondered if there were standard counterexamples known, perhaps recorded in the quantum computing literature, or perhaps just folklore for specialists in Banach space theory.
(Remark: the SVD decomposition actually tells us that $(e_1,\dots, e_N)$ and $(f_1,\dots,f_N)$ are orthonormal. It isn't immediately clear to me if this is already forced by requiring $(1)$, $(2)$ and $(\infty)$, although I haven't given it any thought. In any case it doesn't seem to be directly needed in my intended application.)
