Dear Collegues!

During summer holidays we were involved in interesting debates on
Quantum and Classical Information Theories (which were tarnsformed
finally into a deep discussion on meaning of informartion and its reality).

I would like to present some concluding remarks. After this I propose
that everybody who is interested can send his own concluding remarks and
after that (may be in one--two weeks) session will be closed. I recall
my viewpoint: 

We all work with models of reality. We cannot <<understand reality>>,
but only describe it by using this or that model. The most advanced way
of modeling of reality is mathematical modeling. Sometimes reality is
even identified with a mathematical model. This happened with the modern
picture of space-time reality which is based on using of real continuum.
I pointed out that there were attempts to propose alternative models of
reality even for space-time, e.g., p-adic models. 

There are two main mathematical models of information:

a). Classical thermodynamical model in that information is defined from
entropy and the latter is based on the Kolmogorov measure-theoretic
definition of probability.

b). Quantum information model in that information is defined by using
linear algebra (operator theory).

Mathematical structures of models are different. In particular, QI is
noncommutative theory. The natural question arises: 

<<Is it just difference in mathematical models, in mathematical
descriptions, so just different mathematical representations of the same
kind of reality or one should consider two extremely different types of
physical phenomena, classical and quantum?>>

The conventional point of view is that there are two extremely different 
domains of physics, quantum and classical. The first one is about
microworld and the second is about macroworld. This is the Copenhagen
viewpoint: there are microscopic systems and macroscopic observers.
It induces many problems and paradoxes, but nevertheless it is
convenient in applications and it dominates in physics. One of the main
problems is the boundary between the quantum and classical domains. 

In the quantum domain a system can be in a superposition of a few
different states. This is precisely why quantum computers should work
quicker than classical ones. In classical it could not. For example, as
was pointed by Roger Penrose, a single neuron could not be at the same
time in the superposition of two states: firing and nonfiring. 

The famous Schrodinger cat was created by Schrrodinger to show
absurdness of Copenhagen interpretation. This example was proposed in
his letter to Einstein and it was a modification of an example from one
of Einsteins letters about pistolet and bomb. The main idea was that if
one assumes superposition of states for microscopic systems one would be
always able to lift this superposition to macroscopic systems.

My point was that two information theories are based on two probability
theories: classical Kolmogorov measure-theoretic probability and quantum 
von Neumann Hilbert space probability. In the second case we operate not
directly with probabilities but with complex probability amplitudes.

Some people think that quantum probability is more complicated than the
classical one. I do not think so. Theory of Lebesgues integral is
essentially deeper and more complicated from the mathematical viewpoint
than linear algebra, especially in finite dimensional spaces which are
used in quantum information theory.

I am trying to sell the idea that the whole quantum enterprise is about
simplification of description of extremely complex physical phenomena. 
I developed models in that the quantum probabilistic model appears as a
projection of more complex classical statistical model. 

Then I proceed: Wau! In such a case it seems that quantum probability
theory and quantum information could be used everywhere where we could
not provide the complete description of phenomena and we just try to
create a simplified representation in complex Hilbert space. 

So one can apply quantum information theory everywhere, from financial
mathematics to genetics.

Finally, about the last part of discussion about reality of information.
I understood that my rather restricted philosophic basis was not
sufficient to debate this problem on the same level as opponents of
non--reality of information. But I stay on my position: information is
not less real than mass or charge.

I agree with Søren Brier that the main problem is that in modern science
information is always reduced to probability:
<<Thus information as a basic quality in the world is something entirely
different from the present information theory that is based on
thermodynamics ensemble theory. That information concept is then more
basic than quantum theory.>>

There should be done something cardinally new...

I would like to thank all participants of out discussion.


Khrennikov A.Yu. ,p-adic valued distributions and their applications to
the mathematical physics, Kluwer, Dordreht, 1994.

Khrennikov A.Yu., Information dynamics in cognitive,
psychological, social,  and anomalous phenomena.Kluwer, Dordreht,2004.

Proceedings of Conference Quantum Theory: Reconsideration of
Foundations-3, American Institute of Physics, Ser. Conference
Proceedings, Melville, NY, 2006.

A. Yu. Khrennikov,  The principle of supplementarity: A
contextual probabilistic viewpoint to complementarity, the
interference of probabilities, and the incompatibility of variables
in quantum mechanics.Foundations of Physics,  35, N.
10, 1655 - 1693 (2005).

A. Yu. Khrennikov,  Interference in the classical probabilistic
model and its representation in complex Hilbert space.  Physica, E 29,
226-236 (2005).

A. A. Ezhov, A. Yu.  Khrennikov, Agents with left and right
dominant hemispheres and quantum statistics. Phys. Rev. E (3)
71 , N. 1, 016138-1 -8 (2005).

A. Yu. Khrennikov, Quantum-like brain: Intereference of
minds. BioSystems, 84, 225-241 (2005).

With Best Regards,

Andrei Khrennikov

Director of International Center for Mathematical Modeling in Physics,
Engineering, Economy and Cognitive Sc.,
University of Vaxjo, Sweden

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