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I came across an unusual multi-partite generalization of the Schmidt decomposition in my work, which I describe below. Usually, when people say "a multi-partite Schmidt decomposition", they mean a sum of product states which is in some sense minimal, and there are different decomposition depending on what metric is used to measure minimality. In contrast, I'm looking at a decomposition of a state into a maximal sum of locally-orthogonal (but not necessarily product) states. I would like to know if this structure has been studied before so I can read more about it.

Suppose we are given a normalized quantum state $\vert \psi \rangle$ from a Hilbert space $\mathcal{H}$ composed of $N$ smaller subsystems $\mathcal{H}^{(n)}$ that are tensored together:

$\vert \psi \rangle \in \mathcal{H} = \mathcal{H}^{(1)} \otimes \cdots \otimes \mathcal{H}^{(N)}$.

Now I am looking for a preferred decomposition

$\vert \psi \rangle = \sum_i \sqrt{p_i} \vert \psi_i \rangle$

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

$\vert \psi_i \rangle \in \bigotimes_n \mathcal{H}^{(n)}_i$


$\mathcal{H}^{(n)} = \bigoplus_i \mathcal{H}^{(n)}_i$

is a decomposition of $\mathcal{H}^{(n)}$ into orthogonal subspaces.

This is interesting because it means that the $i$-conditional reduced density matrices of any subsystem $\mathcal{H}^{(n)}$,

$\rho^{(n)}_i = \mathrm{Tr}_{\overline{\mathcal{H}^{(n)}}} \left[ \vert \psi_i \rangle \langle \psi_i \vert \right]$

are restricted to the same orthogonal subspaces. (Here, the trace is over all spaces except $\mathcal{H}^{(n)}$.) Therefore, observers can make local measurements on any of the systems $\mathcal{H}^{(n)}$ and determine which "branch" $\vert \psi_i \rangle$ they are on.


So: has this been explored before?

(I also submitted essentially the same question in math notation here.)

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Dear Jess Riedel. In general, it is frown upon to cross-post simultaneously, because it may waste potential answerer's time. As a minimum OP should mention the cross-post (on both sites!). The preferred procedure is to not cross-post, and if the post hasn't received an acceptable answer after, say, a couple of days, then OP could flag for migration. –  Qmechanic Sep 28 '13 at 16:34
OK, I will follow that advice in the future. If I can get anyone (anyone!) to look at one of these, I'll close the other one and have it redirect. –  Jess Riedel Sep 29 '13 at 1:03
Spooky! I'd never heard this kind of thing called the Schmidt decomposition before, but right this minute I finished writing a post on Glauber's work in showing that a general quantum state (of the EM field) is a superposition of tensor products of coherent states - not quite the same thing, but eerily alike in flavour. Now I know the general abstraction of this idea, thanks to your question. –  WetSavannaAnimal aka Rod Vance Sep 29 '13 at 10:30
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