In between trips,
I have been trying to keep track of our discussion
under the banner "Quantum Information."
We have covered a lot of issues,
not all closely related to the main theme.
I would like to comment on a couple of points:
1. Classical information is quantum information.
It is certain that classical information
is a special case of quantum information
(with commuting operators/diagonal matrices,
and/or in the limit as Planck's constant becomes negligble).
So far, I have not seen cogent arguments in the reverse direction,
that quantum information always reduces to classical information.
This remains the key question.
2. Pauli predated Shannon.
In connection with the previous point, remember
that Pauli introduced infomation-theoretic entropy
15 years before Shannon
(see Entropy 3 (2001), p. 2 )
3. Does the quantum need the continuum?
Despite several postings that discussed wave functions,
it is worth recalling that quantum information theory
does not depend on the continuum.
For example, the matrices I discussed in the initial posting
in connection with the qubit (e.g. spinning electron)
may be taken to have entries from more general fields
than the complex numbers.
Provided these fields have a quadratic structure over their prime field
(i.e. mirror the relationship of the complex field to the real field),
then much of the quantum formalism applies.
(In other cases, some order structure is required.)
4. Context is vital.
As in classical physics,
where consideration of context may resolve paradoxes such as Gibbs',
the FIS discussion has emphasized the crucial point
that context is also very important in quantum theory,
even if physicists conventionally underplay it.
5. Quantum Kolmogorov?
Mention of Kolmogorov in connection with classical probabilities
prompts repetition of the question as to whether
his reduction of classical probability to classical computational complexity
-- the negative of the logarithm of the probability of an object is essentially
the length of the shortest program generating it --
can be extended to give a reduction of quantum probabilty
to quantum computation
(see Entropy 3 (2001), p. 3,
where the names of Martin-Loef, Li, and Vitanyi also arise).
For this programme to get started, it would have to be shown that
quantum computation can do more than just resolve one bit.
6. Quantum theory outside physics.
Another question that has arisen is whether quantum formalism
only applies to the physics of microscopic
or aggregated microscopic systems.
Alain Connes' book on "Non-commutative geometry"
gives an example of quantum formalism in number theory.
Regards, Jonathan Smith.
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Received on Wed Jun 28 23:23:23 2006
