I've rewritten the question in math notation, and I've left the old version in physics bra-ket notation here.

## Background

A simple consequence of the singular value decomposition is that any vector $v$ in a vector space $V$ formed by the tensor product of two smaller spaces ("subsystems") $U$ and $W$ of dimension $d_U$ and $d_W$,

$v \in V = U \otimes W$,

has a special decomposition in terms of rank-one tensors (aka product states)

$v = \sum_{i=1}^d \lambda_i u_i \otimes w_i $, $\qquad u_i \in U$, $\qquad w_i \in W$, $\qquad d = \mathrm{min}(d_U,d_W)$

built from the fixed orthonormal bases $\{ u_i \}$ and $\{ w_i \}$.

This decomposition has many nice properties, and it's simple to see that there's no way to generalize it to the case of 3 or more subsystems while keeping all of them. For instance, a generic state in $V = \bigotimes_{n=1}^N V_n$ cannot be expressed as a sum of fewer than $\tilde{d}$ simple tensors, with $\tilde{d} \gg d_n = \mathrm{dim}V_n$ for all $n=1,\ldots,N$.

## Vague Question

Is there a useful/natural/canonical decomposition of a vector into a "small" number of orthogonal simple tensors for the case of 3 or more subsystems? At the very least, the GHZ state should take it's canonical form in such a decompostion:

$v_{GHZ} = a_1 \otimes \cdots \otimes a_N + b_1 \otimes \cdots \otimes b_N$, $\qquad a_n, b_n \in V_n$, $\qquad \langle a_n ; b_n \rangle = 0$ for all $n$.

## Specific Question

If we guess that the entropy function

$H[\{ p_i \} ] = -\sum_i p_i \ln p_i$

is appropriate, we can define the "minimum-entropy product-state decomposition" to be the decomposition

$v = \sum_{i=1}^{\tilde{d}} \lambda_i v_i$, $\qquad v_i = \bigotimes_{n=1}^N \psi_i^n$, $\qquad \psi_i^n \in V_n$

with the minimum value for the entropy $H[\{ p_i = \lambda_i^2 \} ]$, under the condition that the $\{ v_i \}$ are orthonormal. Note that we allow $\langle \psi_i^n ; \psi_j^n \rangle \neq 0$ for $i \neq j$.

The natural questions to ask are: Is this decomposition generically unique (i.e. unique except for some set of measure zero in the global vector space)? Is it continuous? (What other properties should this decomposition have to satisfy to be useful?)

I am 99% sure that in the case of $N=2$, this reduces to the Schmidt decomposition and that the answers to both questions is "yes".

Is any of this sensitive to our choice of the entropy function $H$, as opposed to some other permutation-invariant and majoritization-preserving function of the spectrum?

product stateisrank-one tensor,simple tensororpure tensor. You may want to avoid the last one since it overlaps with the notion ofpure statesand generates some confusion. $\endgroup$