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arXiv:1705.00878 · International Journal of Quantum Chemistry

A study of entangled systems in the many‐body signed particle formulation of quantum mechanics

Jean Michel Sellier, Kristina G. Kapanova

Published Aug 21, 2017• 3 citations
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Abstract

Abstract Recently a new formulation of quantum mechanics has been introduced, based on signed classical field‐less particles interacting with an external field by means of only creation and annihilation events. In this article, we extend this novel theory to the case of many‐body problems. We show that, when restricted to spatial finite domains and discrete momentum space, the proposed extended theory is equivalent to the time‐dependent many‐body Wigner Monte Carlo method. In this new picture, the treatment of entangled systems comes naturally and, therefore, we apply it to the study of quantum entangled systems. The latter is represented in terms of two Gaussian wave packets moving in opposite directions. We introduce the presence of a dissipative background and show how the entanglement is affected by different (controlled) configurations.

DOI
10.1002/qua.25447

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Abstract Recently a new formulation of quantum mechanics has been introduced, based on signed classical field‐less particles interacting with an external field by means of only creation and annihilation events. In this article, we extend this novel theory to the case of many‐body problems. We show that, when restricted to spatial finite domains and discrete momentum space, the proposed extended theory is equivalent to the time‐dependent many‐body Wigner Monte Carlo method. In this new picture, the treatment of entangled systems comes naturally and, therefore, we apply it to the study of quantum entangled systems. The latter is represented in terms of two Gaussian wave packets moving in opposite directions. We introduce the presence of a dissipative background and show how the entanglement is affected by different (controlled) configurations.
The Wigner formulation of quantum mechanics based on the concept of Wigner function (WF), is particularly suitable for the study of quantum transport in mesoscopic systems since it allows to describe quantum mechanical effects in an (r,p) phase space, in analogy to what is done for classical systems. Recently we have introduced the concept of Wigner paths (WPs) in phase space which both provide a pictorial representation of the quantum evolution of the system of interest and constitute a useful tool for providing numerical solutions of the quantum equation describing the time evolution of the system. The theory has been applied to the following cases: (a) The study of the transport properties of the transient regime for homogeneous systems. (b) The comparison between quantum and classical treatment of the scattering mechanism. (c) The study of the reflection of the WF from an infinite potential barrier. Furthermore, the Wigner approach, originally formulated in terms of single-time WF has been extended by introducing the two-time WF. The Fourier transform of this function contains independent information about both momentum and energy of the electron. The Monte Carlo procedure based on the generation of WPs can be used to evaluate the two-time WF of an electron interacting with the phonon gas.
Exactly solvable many-body systems are few and far between, and the utility of approximate methods cannot be overestimated. Entanglement mean field theory is an approximate method to handle such systems. While mean field theories reduce the many-body system to an effective single-body one, entanglement mean field theory reduces it to a two-body system. And in contrast to mean field theories where the self-consistency equations are in terms of single-site physical parameters, those in entanglement mean field theory are in terms of both single- and two-site parameters. Hitherto, the theory has been applied to predict properties of the static states, like ground and thermal states, of many-body systems. Here we give a method to employ it to predict properties of time-evolved states. The predictions are then compared with known results of paradigmatic spin Hamiltonians.
Recently the use of neural networks has been introduced in the context of the signed particle formulation of quantum mechanics to rapidly and reliably compute the Wigner kernel of any provided potential. This new technique has introduced two important advantages over the more standard finite difference/element methods: 1) it reduces the amount of memory required for the simulation of a quantum system. As a matter of fact, it does not require storing the kernel in a (expensive) multi-dimensional array, and 2) a consistent speedup is obtained since now one can compute the kernel on the cells of interest only, i.e. the cells occupied by signed particles. Although this certainly represents a step forward into the direction of rapid simulations of quantum systems, it comes at a price: the number of hidden neurons is constrained by design to be equal to the number of cells of the discretized real space. It is easy to see how this limitation can quickly become an issue when very fine meshes are necessary. In this work, we continue to ameliorate this previous approach by reducing the complexity of the neural network and, consequently, by introducing an additional speedup. More specifically, we propose a new network architecture which requires less neurons than the previous approach in its hidden layer. For validation purposes, we apply this novel technique to a well known simple, but very indicative, one-dimensional quantum system consisting of a Gaussian wave packet interacting with a potential barrier. In order to clearly show the validity of our suggested approach, time-dependent comparisons with the previous technique are presented. In spite of its simpler architecture, a good agreement is observed, thus representing one step further towards fast and reliable simulations of time-dependent quantum systems.
Quantum mechanics for a four‐state‐system is derived from classical statistics. Entanglement, interference, the difference between identical fermions or bosons and the unitary time evolution find an interpretation within a classical statistical ensemble. Quantum systems are subsystems of larger classical statistical systems, which include the environment or the vacuum. They are characterized by incomplete statistics in the sense that the classical correlation function cannot be used for sequences of measurements in the subsystem.
Abstract We study properties of entangled systems in the (mainly non-relativistic) second quantization formalism. This is then applied to interacting and non-interacting bosons and fermions and the differences between the two are discussed. We present a general formalism to show how entanglement changes with the change of modes of the system. This is illustrated with examples such as the Bose condensation and the Unruh effect. It is then shown that a non-interacting collection of fermions at zero temperature can be entangled in spin, providing that their distances do not exceed the inverse Fermi wavenumber. Beyond this distance all bipartite entanglement vanishes, although classical correlations still persist. We compute the entanglement of formation as well as the mutual information for two spin-correlated electrons as a function of their distance. The analogous, non-interacting collection of bosons displays no entanglement in the internal degrees of freedom. We show how to generalize our analysis of the entanglement in the internal degrees of freedom to an arbitrary number of particles.
The expectation value of the total energy of a wave packet, associated with the speed (moments) space by means of the Lorentz transformation, allows us to obtain by means of quantum wave formulation an energetic expression equivalent, and at the same time less restricted (more versatile and informative), to the one obtained by means of a corpuscular treatment. On the one hand, the energetic formants of the matter appear, on the other the expression in undulatory terms of it, of that which we call mass, and finally an additional phase factor as the last vestige of the undulatory nature after the formation process, that is to say, as an accrediting element of that nature, and mechanism of restitution of it, responsible for the wave-corpuscle duality, of the related creation-annihilation processes, and of the relations among the different generations of elementary particles, as it was shown in https://vixra.org/abs/2003.0001.