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Physical Review A

Nonlinear quantum evolution does not imply supraluminal communication

Miguel Ferrero, David Cienfuegos Salgado, J. L. Sánchez‐Gómez

Published Jul 7, 2004• 21 citations
Publisher

Abstract

We build up simple models of quantum evolution showing that nonlinearity does not necessarily imply either the possibility of faster-than-light communications or the violation of the second law of thermodynamics. We offer counterexamples proving that conditions of no supraluminal communication considered in the literature as necessary and sufficient are actually only necessary. We formulate conditions and show that this preclusion only entails the linearity of reduced dynamics, thus reopening the possibility to unexplored nonlinear joint evolutions. We make constant use of the Bloch representation of finite-dimensional quantum systems to express our results.

DOI
10.1103/physreva.70.014101

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We build up simple models of quantum evolution showing that nonlinearity does not necessarily imply either the possibility of faster-than-light communications or the violation of the second law of thermodynamics. We offer counterexamples proving that conditions of no supraluminal communication considered in the literature as necessary and sufficient are actually only necessary. We formulate conditions and show that this preclusion only entails the linearity of reduced dynamics, thus reopening the possibility to unexplored nonlinear joint evolutions. We make constant use of the Bloch representation of finite-dimensional quantum systems to express our results.
Hybrid classical-quantum approaches are instrumental in numerous fields, from condensed matter physics to quantum information science. We recently proposed to describe hybrid systems starting from a set of natural axioms for measurement probabilities without adding any underlying mathematical structure. The so defined probability measures fulfill a no-signaling condition that ensures that instantaneous communication is impossible. We formulate here a dynamical generalization of this condition. It means that, for two independent systems, the outcome probabilities of a measurement made on one of them are not affected by a measurement performed earlier on the other. Analogous requirements are satisfied for usual classical and quantum bipartite systems and violating them would make faster-than-light signaling possible. The dynamical no-signaling condition has important consequences for classical-quantum interactions that depend on the hybrid approach used. For no-signaling hybrid dynamics with classical trajectories, the classical degrees of freedom can influence the quantum ones but the latter cannot react on the former. If pure states of quantum systems remain pure then the dynamical no-signaling condition implies the absence of classical reaction. When all hybrid states are allowed, there are no genuine classical-quantum interactions for no-signaling hybrid dynamics that do not generate correlations between the classical and the quantum degrees of freedom. In all these cases, the proposed condition is equivalent to the convex-linearity of the probability measure transformations describing finite-time evolutions.
Starting with the same form of atomic nonlinearity, Weinberg [Ann. Phys. (NY) 194, 336 (1989)] and W\'odkiewicz and Scully [Phys. Rev. A 42, 5111 (1990)] obtained contradictory results concerning an evolution of the atomic inversion w in a two-level atom in Weinberg's nonlinear quantum mechanics: If the atom is initially in a ground state then either the evolution of w (1) can be linear if one uses a nonlinear generalization of the Jaynes-Cummings Hamiltonian, or (2) is always nonlinear if one uses the nonlinear Bloch equations derived from the nonlinear atomic Hamiltonian function. It is shown that the difference is rooted in inequivalent descriptions of the composite ``atom-plus-field'' system. The linear evolution of w results from a ``faster-than-light communication'' between the atom and the field. If one applies a description without the ``faster-than-light telegraph'' then the calculations based on a suitably modified Jaynes-Cummings Hamiltonian lead to the same dynamics of w as is found in semiclassical calculations based on Bloch equations. It is shown also that a nonlinear quantum mechanics based on a nonlinear Schr\"odinger equation does not possess a natural probability interpretation.
Abstract There is a growing interest, both from the theoretical as well as experimental side, to test the validity of the quantum superposition principle, and of theories which explicitly violate it by adding nonlinear terms to the Schrödinger equation. We review the original argument elaborated by Gisin (1989 Helv. Phys. Acta 62 363), which shows that the non-superluminal-signaling condition implies that the dynamics of the density matrix must be linear. This places very strong constraints on the permissible modifications of the Schrödinger equation, since they have to give rise, at the statistical level, to a linear evolution for the density matrix. The derivation is done in a heuristic way here and is appropriate for the students familiar with the textbook quantum mechanics and the language of density matrices.
The vast majority of the literature dealing with quantum dynamics is concerned with linear evolution of the wave function or the density matrix. A complete dynamical description requires a full understanding of the evolution of measured quantum systems, necessary to explain actual experimental results. The dynamics of such systems is intrinsically nonlinear even at the level of distribution functions, both classically as well as quantum mechanically. Aside from being physically more complete, this treatment reveals the existence of dynamical regimes, such as chaos, that have no counterpart in the linear case. Here, we present a short introductory review of some of these aspects, with a few illustrative results and examples.
Remarkable progress of quantum information theory (QIT) allowed to formulate mathematical theorems for conditions that data-transmitting or data-processing occurs with a non-negative entropy gain. However, relation of these results formulated in terms of entropy gain in quantum channels to temporal evolution of real physical systems is not thoroughly understood. Here we build on the mathematical formalism provided by QIT to formulate the quantum H-theorem in terms of physical observables. We discuss the manifestation of the second law of thermodynamics in quantum physics and uncover special situations where the second law can be violated. We further demonstrate that the typical evolution of energy-isolated quantum systems occurs with non-diminishing entropy.