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 SÃ¸ren 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. References: 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 _______________________________________________ fis mailing list fis@listas.unizar.es http://webmail.unizar.es/mailman/listinfo/fis