As showed by Solovay here, in a non-separable Hilbert space $H$ there may be probability measures that cannot be written, for any $M$ closed subspace of $H$, as $\mu (M)=\mathrm{Tr}[\rho \mathbb{1}_M]$, for some positive self-adjoint trace class $\rho$ with trace 1 (density matrix). Here $\mathbb{1}_M$ denotes the orthogonal projection on $M$. [The proof of the existence of such "exotic measures" is undecidable in ZFC, however it is equivalent to the existence of a (real valued) measurable cardinal]

In some sense it means that in non-separable Hilbert spaces there may exist analogues to "normal quantum states" that are not density matrices$^\dagger$.

*Remark*: For normal quantum state I mean a ultraweakly positive continuous functional on the $C^*$-algebra of bounded operators that is interpreted to give their expectation value.

$^\dagger$: I mean that even if these states are probability measures with the countable additivity of orthogonal closed subspaces property, are not expressed as the trace of density matrices (while on separable Hilbert spaces this is always the case, and these measures are in one to one correspondence with normal states).

This post imported from StackExchange Physics at 2015-05-13 19:02 (UTC), posted by SE-user yuggib