I have the same questions for Eliezer as Jadagul and Toby Ord, namely:

- Why would the space of amplitude distributions have uncountable dimension? Unless I've misunderstood, it sounds like it would be something like L^2, which is separable (has countable orthogonal dimension). (Of course, maybe by "dimension" you just meant the cardinality of a Hamel basis, in which case you're right -- there's no Hilbert space with Hamel dimension aleph_0. However, "dimension" in the context of Hilbert spaces nearly always refers to orthogonal dimension.)

- Assuming you really did mean uncountable, how do you know aleph_1 is the right cardinality, rather than, say, 2^aleph_0? Are you assuming the continuum hypothesis?