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Given two Hilbert spaces [itex]H_1[/itex] and [itex]H_2[/itex], we can construct their tensor product [itex]H=H_1\otimes H_2[/itex]. This is another Hilbert space.

What I'm wondering is if there are any theorems about what sort of decompositions we can make if we're given a Hilbert space [itex]H[/itex], and want to express it as a tensor product of two "smaller" Hilbert spaces. Can we pick an arbitrary subspace and call it [itex]H_1[/itex], and then construct [itex]H_2[/itex] from [itex]H[/itex] and [itex]H_1[/itex]?

I'm interested in subsystems in QM, and not just ensembles of identically prepared systems. I want to know e.g. if we can always decompose the Hilbert space of the universe (in the many-worlds interpretation) into "this guy" [itex]\otimes[/itex] "everything else".

Maybe I'm phrasing the question wrong. Maybe I should focus on the observables instead of the states. I don't know. If you do, let me know.