- Contemporary messages sorted: [ by date ] [ by thread ] [ by subject ] [ by author ] [ by messages with attachments ]

From: scerir <scerir.domain.name.hidden>

Date: Tue, 31 Dec 2002 10:47:53 +0100

[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.

Received on Tue Dec 31 2002 - 04:40:45 PST

Date: Tue, 31 Dec 2002 10:47:53 +0100

[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.

Received on Tue Dec 31 2002 - 04:40:45 PST

*
This archive was generated by hypermail 2.3.0
: Fri Feb 16 2018 - 13:20:08 PST
*