Automata Simulating Quantum Logics

Automata Simulating Quantum Logics - International Journal...

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International Journal of Theoretical Physics, Vol. 29, No. 2, 1990 Automata Simulating Quantum Logics A. A. Grib ~ and R. R. Zapatrin 2 Received August 8, 1988 The idea of computational complementarity is developed further. A special class of macroscopic automata to imitate quantum and classical systems is described. The simplest automaton imitating a spin-l/2 particle is completely considered. 1. INTRODUCTION Niels Bohr was the first to think that the discovery of quantum mechanics was something more than that of new laws of microphysics; it was a new point of view which could be useful in different areas, for example, biology or economics. The aim of this paper is to find some examples which have nothing to do with microphysics but use the same mathematical formalism as quantum mechanics. These examples can be found in economics, sociology, and theory of automata. The possibility of constructing macroscopic automata imitating quantum systems is very important for the general problem of imitating the modeling of microphysical processes and constructing special computers for this aim. The main idea is to find "different representations" of quantum logics. By quantum logic we mean, following Birkhoff and von Neumann (1936), some nondistributive lattice which corresponds on one hand to some quan- tum microsystem and on the other to some classical system. D. Finkelstein was the first to see the correspondence between quantum lattices and graphs (Finkelstein and Finkelstein, 1983) which makes it possible to find macro- scopic realizations of quantum logics. To illustrate the idea, consider a very simple quantum system: a particle with spin one-half which is described by two projections of spin Sz and S~. The lattice of properties of this particle ~Leningrad Financial and Economical Institute, Leningrad, USSR. 2Leningrad Polytechnical Institute, Leningrad, USSR. 113 0020-7748/90/0200-0113506,00/0 ~) 1990 Plenum Publishing Corporation
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114 Grib and Zapatrin is an orthomodular nondistributive lattice: To this lattice there corresponds the graph (Finkelstein and Finkelstein, 1983). or, briefly, ( ) ) ) Now consider the opposite question: in what sense does the nondis- tributive lattice correspond to the graph? Let 1, 2, 3, 4 be states of some system (e.g., an economic one) and suppose that there is an observer who tries to check the state of the system by putting questions to it. The system has the following property: it can answer "yes" to the question "are you in 2" not only if it is in 2, but also if it is in 1 or 3. It can change its state by one step responding to the question if and only if corresponding states are connected by an arc. But let the observer be clever enough to know this property of the system: then he must use some "negative logics." He concludes that the system is in 2 if to the question "are you in 4?" a negative answer is obtained. So by a negative answer to a complementary question he can know the real state of the system. But then it is easy to see that one
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Automata Simulating Quantum Logics - International Journal...

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