Dear Joao and Friends,

    Since you mentioned the word "resource", let me add this paper by C.
Caves et al to the list of "must read" papers on QC.

http://arxiv.org/abs/quant-ph/0204157

from the abstract:
Climbing Mount Scalable: Physical-Resource Requirements for a Scalable
Quantum Computer
Authors: Robin Blume-Kohout (1), Carlton M. Caves (2), Ivan H. Deutsch (2)
((1) Los Alamos National Laboratory, (2) University of New Mexico)
Comments: LATEX, 24 pages, 1 color .eps figure. This new version has been
accepted for publication in Foundations of Physics

  The primary resource for quantum computation is Hilbert-space dimension.
Whereas Hilbert space itself is an abstract construction, the number of
dimensions available to a system is a physical quantity that requires
physical resources. Avoiding a demand for an exponential amount of these
resources places a fundamental constraint on the systems that are suitable
for scalable quantum computation. To be scalable, the effective number of
degrees of freedom in the computer must grow nearly linearly with the number
of qubits in an equivalent qubit-based quantum computer.

    In a related question, is anyone here familiar with the problem of
defining a Lebesgue measure for a infinitely dimensional Hilbert space?

http://www.lns.cornell.edu/spr/2000-04/msg0023750.html

Kindest regards,

Stephen

----- Original Message -----
From: "Joao Leao" <[EMAIL PROTECTED]>
To: "scerir" <[EMAIL PROTECTED]>
Cc: <[EMAIL PROTECTED]>
Sent: Tuesday, December 31, 2002 10:02 AM
Subject: Re: R: Quantum Probability and Decision Theory


> 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
snip


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