I don't agree with Tim's suggestion that infinite-dimensional Hilbert spaces
are somewhat "ancilliary" in QM and that all systems are calculable in
finite dimensional modes. In fact infinite sets of spaces are the rule in
QM and
the finite dimensional subspaces only serve as toy systems.
Having said that, yes: entanglement is the "maximal peculiarity" of quantum
systems and it is more or less established that entanglement is the "resource"
that is reponsible for the quantum computational speed up. That does not
necessarily mean that it would lead to the "computational of the (Turing)
uncomputable" but it is the natural place to look. The KSW states described
in the paper you mention below are certainly worth investigating but finitely
entangled states already display most of the capabilities (teleportation, dense
coding, speed up, non-locality) that we came to associate with quantum
information processing. What we lack is a genuinely quantum model of
computation that could be mathematically tractable as the Turing or Post
models and can account for entanglement in all its glory.
-Joao Leao
scerir wrote:
> [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.
--
Joao Pedro Leao ::: jleao.domain.name.hidden
Harvard-Smithsonian Center for Astrophysics
1815 Massachussetts Av. , Cambridge MA 02140
Work Phone: (617)-496-7990 extension 124
VoIP Phone: (617)=384-6679
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"All generalizations are abusive (specially this one!)"
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Received on Tue Dec 31 2002 - 09:52:29 PST