[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.

[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.