[Tim May, in another thread]
Any finite system, which of course all systems are, can have all of its
quantum mechanics calculations done with finite-dimensional vector
spaces. The "full-blown machinery" of an infinite-dimensional Hilbert
space is nice to have, in the same way that Fourier analysis is more
elegantly done with all possible frequencies even though no actual
system (including the universe!) needs all frequencies.

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[J. Leao]
Another point worth making is that it seems unlikely that the recourse
to the infinite superposability of quantum states is going to be of any
help in this circunstance. It may be more profitable to look to
entanglement (which incidentaly is the trully novelty that QC brings
to the realm of computation) as the road to a trans-Turing class of
computations.
[SPK]
Entanglement is somewhat involved. See this paper:
http://www.arxiv.org/abs/quant-ph/0201143
And what about these "infinitely entangled states"?
s.
M. Keyl, D. Schlingemann, R. F. Werner
http://arxiv.org/abs/quant-ph/0212014
For states in infinite dimensional Hilbert spaces entanglement quantities
like the entanglement of distillation can become infinite. This leads
naturally to the question, whether one system in such an infinitely
entangled state can serve as a resource for tasks like the teleportation of
arbitrarily many qubits. We show that appropriate states cannot be obtained
by density operators in an infinite dimensional Hilbert space. However,
using techniques for the description of infinitely many degrees of freedom
from field theory and statistical mechanics, such states can nevertheless be
constructed rigorously. We explore two related possibilities, namely an
extended notion of algebras of observables, and the use of singular states
on the algebra of bounded operators. As applications we construct the
essentially unique infinite analogue of maximally entangled states, and the
singular state used heuristically in the fundamental paper of Einstein,
Rosen and Podolsky.