# Is there a known generalization of the Schmidt decomposition based on a maximal set of “locally orthogonal” vectors?

I came across the following unusual generalization of the Schmidt decomposition in my work, which I describe below. I would like to know if this structure has been studied before so I can read more about it. I try to use mostly math terminology, but there are some physics (quantum info) implications that are important, which I discuss.

Suppose we are given a normalized vector $v$ from a vector space composed of $N$ smaller spaces ('subsystems') that are tensored together:

$v \in V = V^{(1)} \otimes \cdots \otimes V^{(N)}$.

Now I am looking for a preferred decomposition

$v = \sum_i \sqrt{p_i} v_i$

of $v$ expressed as a sum (weighted by the positive values $\sqrt{p_i}$) of orthonormal vectors $v_i$ such that the $v_i$ live on orthogonal subspaces of each subsystem. More precisely, we require

$v_i \in \bigotimes_n V^{(n)}_i$

where

$V^{(n)} = \bigoplus_i V^{(n)}_i$

for every $n$ (with $1 \le n \le N$).

This is interesting for quantum mechanics because it means that the $i$-conditional reduced density matrices of any subsystem $V^{(n)}$,

$\rho^{(n)}_i = \mathrm{Tr}_{\overline{V^{(n)}}} \left[ v_i v_i^\star \right]$

are restricted to the same orthogonal subspaces: $\rho^{(n)}_i \in \mathcal{M}[V^{(n)}_i]$. (Here, $\mathcal{M}[V^{(n)}_i]$ denotes the set of operators on the subspace $V^{(n)}_i$, and the trace is over all spaces except $V^{(n)}$.) Therefore, observers can make local measurements on any of the systems $V^{(n)}$ and determine which "branch" $v_i$ they are on.

This structure is special for the following reason. (a) There exists a unique decomposition which maximizes the number of vectors in the sum while still satisfying the requirement of orthogonal records in each of the subsystems. (b) This maximal decomposition can be obtained through a mechanical (i.e. algorithmic) fine-graining procedure; the same decomposition is obtained regardless of the order in which the fine-graining is done. (c) The quantity $E = - \sum_i p_i \ln p_i$ is maximized for the maximal decomposition and defines a measure of the global entanglement (with respect to the particular choice of subsystems); if $v$ is such that any of the subsystems are unentangled with the rest, this quantity vanishes. (d) This structure reduces to the Schmidt decomposition (and $E$ reduces to the traditional entropy of entanglement) when $N=2$. (e) This structure, and the corresponding value of $E$, is insensitive to pairwise entangling interactions between subsystems (so long as they do not destroy the orthogonality of the conditional local states $\rho_i^{(n)}$) even when the strength/type of these entangling interactions are conditional on the branch $i$.

So: has this been explored before?