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arXiv:quant-ph/0401168

Random Solitons Realization of Quantum Mechanics and Stochastic Qubits

Yu. P. Rybakov, T. F. Kamalov

Published Jan 27, 2004• 0 citations
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Abstract

Stochastic realization of the wave function in quantum mechanics, with the inclusion of soliton representation of extended particles, is discussed. The concept of Stochastic Qubits is used for quantum computing modeling.
DOI
10.48550/arxiv.quant-ph/0401168

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Stochastic realization of the wave function in quantum mechanics with the inclusion of soliton representation of extended particles is discussed. Two-soliton configurations are used for constructing entangled states in generalized quantum mechanics dealing with extended particles, endowed with nontrivial spin S. Entangled solitons construction being introduced in the nonlinear spinor field model, the Einstein-Podolsky-Rosen (EPR) correlation is calculated and shown to coincide with the quantum mechanical one for the 1/2-spin particles. The concept of stochastic qubits is used for quantum computing modeling.
Special stochastic representation of the wave function in Quantum Mechanics (QM), based on soliton realization of extended particles, is suggested with the aim to model quantum states via classical computer. Entangled solitons construction being introduced in the nonlinear spinor field model, the Einstein, Podolsky, Rosen (EPR) spin correlation is calculated and shown to coincide with the quantum mechanical one for the spin-1/2 particles in the singlet state. The concept of stochastic qubits is used for quantum computing modelling.
Starting with the projective interpretation of the Hilbert space special stochastic representation of the wave function in Quantum Mechanics (QM), based on soliton realization of extended particles, is considered with the aim to model quantum states via classical computer. Entangled solitons construction having been earlier introduced in the nonlinear spinor field model for the calculation of the Einstein-Podolsky-Rosen (EPR) spin correlation for the spin-1/2 particles in the singlet state, the other example is now studied. The latter concerns the entangled envelope solitons in Kerr dielectric with cubic nonlinearity, where we use two-solitons configurations for modeling the entangled states of photons. Finally, the concept of stochastic qubits is used for quantum computing modeling.
Background Using the Einstein's idea on particles-solitons, the stochastic representation of the wave function as a large sum of solitons with random phases is suggested.Purpose To illustrate the main idea of the stochastic representation, let us recall the lectures by N. Wiener on nonlinear problems in random theory.Method In these lectures Wiener used the so-called 𝛼-representation of the wave function for a nonrelativistic particle in three-dimensional space, the wave function being considered as an element of the random Hilbert space with the Gaussian dispersion.Results This representation appears to be equivalent, in accordance with the central limiting theorem, to the sum of many complex solitons with random phases.Conclusions On the basis of this representation it is possible to substantiate the Born's rule for measuring physical observables and also to explain the "spin-statistics" correlation in quantum mechanics.
By comparing Schrödinger's cat with its classical counterpart, I show that a quantum superposition should be understood as an expectation over possible eigenstates weighted by wave-like probabilities. Upon the occurrence of a certain event, the quantum system is randomly realized into one of the possible eigenstates due to its intrinsic stochasticity. While the randomness of a single realization cannot be controlled or predicted, the overall distribution can be regulated via experimental setup and converges as the number of events increases. A measurement is indeed an activity employing a certain event to convert a quantum effect into a macroscopic outcome. Consequently, the puzzling concepts of wavefunction collapse, many worlds, and decoherence become unnecessary for understanding quantum superposition. This expectation-realization interpretation, which integrates probability theory with wave mechanics, can also be extended to quantum pathways. Moreover, it reframes tests of Bell's inequalities as validating the wave-like probability nature of quantum mechanics, with no need to invoke the mysterious notions of quantum non-locality and "spooky action at a distance".