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arXiv:quant-ph/9708052 · Physical Review A

Nonlocal-looking equations can make nonlinear quantum dynamics local

Marek Czachor

Published Jun 1, 1998• 36 citations
arXivHTMLPDF

Abstract

A general method for extending a nondissipative nonlinear Schr\"odinger and Liouville--von Neumann one-particle dynamics to an arbitrary number of particles is described. It is shown at a general level that the dynamics so obtained is completely separable, which is the strongest condition one can impose on dynamics of composite systems. It requires that for all initial states (entangled or not) not only can a subsystem not be influenced by any action undertaken by an observer in a separated system (strong separability), but additionally the self-consistency condition ${\mathrm{Tr}\mathrm{}}_{2}\ensuremath{\circ}{\ensuremath{\varphi}}_{1+2}^{t}={\ensuremath{\varphi}}_{1}^{t}\ensuremath{\circ}{\mathrm{Tr}\mathrm{}}_{2}$ is fulfilled. It is shown that a correct extension to $N$ particles involves integro-differential equations, which, in spite of their nonlocal appearance, make the theory fully local. As a consequence, a much larger class of nonlinearities satisfying the complete separability condition is allowed than has been assumed so far. In particular all nonlinearities of the form $F(|\ensuremath{\psi}(x)|)$ are acceptable. This shows that the locality condition does not single out logarithmic or one-homogeneous nonlinearities.

DOI
10.1103/physreva.57.4122

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A general method for extending a nondissipative nonlinear Schr"odinger and Liouville--von Neumann one-particle dynamics to an arbitrary number of particles is described. It is shown at a general level that the dynamics so obtained is completely separable, which is the strongest condition one can impose on dynamics of composite systems. It requires that for all initial states (entangled or not) not only can a subsystem not be influenced by any action undertaken by an observer in a separated system (strong separability), but additionally the self-consistency condition Tr2\ensuremath∘\ensuremathφ1+2t=\ensuremathφ1t\ensuremath∘Tr2{\mathrm{Tr}\mathrm{}}_{2}\ensuremath{\circ}{\ensuremath{\varphi}}_{1+2}^{t}={\ensuremath{\varphi}}_{1}^{t}\ensuremath{\circ}{\mathrm{Tr}\mathrm{}}_{2}Tr2​\ensuremath∘\ensuremathφ1+2t​=\ensuremathφ1t​\ensuremath∘Tr2​ is fulfilled. It is shown that a correct extension to NNN particles involves integro-differential equations, which, in spite of their nonlocal appearance, make the theory fully local. As a consequence, a much larger class of nonlinearities satisfying the complete separability condition is allowed than has been assumed so far. In particular all nonlinearities of the form F(∣\ensuremathψ(x)∣)F(|\ensuremath{\psi}(x)|)F(∣\ensuremathψ(x)∣) are acceptable. This shows that the locality condition does not single out logarithmic or one-homogeneous nonlinearities.
We show an example of benign non-separability in an apparently separable system consisting of nnn free non-correlated quantum particles, solitonic solutions to the nonlinear phase modification of the Schrödinger equation proposed recently. The non-separability manifests itself in the wave function of a single particle being influenced by the very presence of other particles. In the simplest case of identical particles, it is the number of particles that affects the wave function of each particle and, in particular, the width of its Gaussian probability density. As a result, this width, a local property, is directly linked to the mass of the entire Universe in a very Machian manner. In the realistic limit of large nnn if the width in question is to be microscopic, the coupling constant must be very small resulting in an ``almost linear'' theory. This provides a model explanation of why the linearity of quantum mechanics can be accepted with such a high degree of certainty even if the more fundamental underlying theory could be nonlinear. We also demonstrate that when such non-correlated solitons are coupled to harmonic oscilators they lead to a faster-than-light nonlocal telegraph since changing the frequency of one oscillator affects instantaneously the probability density of particles associated with other oscillators. This effect can be alleviated by fine-tuning the parameters of the solution. Exclusion rules of a novel kind that we term supersuperselection rules also emerge from these solutions. They are similar to the mass and the univalence superselection rules in linear quantum mechanics. The effects in question and the exclusion rules do not appear if a weakly separable extension to nnn-particles is employed.
We develop a dynamical theory, based on a system of ordinary differential equations describing the motion of particles which reproduces the results of quantum mechanics. The system generalizes the Hamilton equations of classical mechanics to the quantum domain, and turns into them in the classical limit ℏ→0\hbar\rightarrow 0ℏ→0. The particles' motions are completely determined by the initial conditions. In this theory, the wave function ψψψ of quantum mechanics is equal to the exponent of an action function, obtained by integrating some Lagrangian function along particle trajectories, described by equations of motion. Consequently, the equation for the logarithm of a wave function is related to the equations of motion in the same way as the Hamilton-Jacobi equation is related to the Hamilton equations in classical mechanics. We demonstrate that the probability density of particles, moving according to these equations, should be given by a standard quantum-mechanical relation, ρ=∣ψ∣2ρ=|ψ|^2ρ=∣ψ∣2. The theory of quantum measurements is presented, and the mechanism of nonlocal correlations between results of distant measurements with entangled particles is revealed. In the last section, we extend the theory to particles with nonzero spin.
A simple example of completely positive dynamics is considered for which both the generator of a nonlocal integro-differential equation leading to such dynamics and the time-local generator can be calculated explicitly. It is demonstrated that introducing a small parameter in this simple example allows one to reproduce some nonperturbative phenomena that occur in more realistic models. In addition, a special case of fermionic dynamics is considered, and it is shown that one can actually find families of moments whose dynamics is linear but satisfies non-Markovian equations.
A nonlinear modification of the Schrödinger equation is proposed in which the Lagrangian density for the Schrödinger equation is extended by terms polynomial in Δmln⁡(Ψ∗/Ψ)Δ^{m}\ln (Ψ^{*}/Ψ)Δmln(Ψ∗/Ψ) multiplied by Ψ∗ΨΨ^{*}ΨΨ∗Ψ. This introduces a homogeneous nonlinearity in a Galilean invariant manner through the phase SSS rather than the amplitude RRR of the wave function Ψ=Rexp⁡(iS)Ψ=R\exp (iS)Ψ=Rexp(iS). From this general scheme we choose the simplest minimal model defined in some reasonable way. The model in question offers the simplest way to modify the Bohm formulation of quantum mechanics so as to allow a leading phase contribution to the quantum potential and a leading quantum contribution to the probability current removing asymmetries present in Bohm's original formulation. It preserves most of physically relevant properties of the Schrödinger equation including stationary states of quantum-mechanical systems. It can be thought of as the simplest model of nonlinear quantum mechanics of extended objects among other such models that also emerge within the general scheme proposed. The extensions of this model to nnn particles and the question of separability of compound systems are studied. It is noted that there exists a weakly separable extension in addition to a strongly separable one. The place of the general modification scheme in a broader spectrum of nonlinear modifications of the Schrödinger equation is discussed. It is pointed out that the models it gives rise to have a unique definition of energy in that the field-theoretical energy functional coincides with the quantum-mechanical one. It is found that the Lagrangian for its simplest variant represents the Lagrangian for a restricted version of the Doebner-Goldin modification of this equation.
It is demonstrated here that local dynamics have the ability to strongly modify the entangling power of unitary quantum gates acting on a composite system. The scenario is common to numerous physical systems, in which the time evolution involves local operators and nonlocal interactions. To distinguish between distinct classes of gates with zero entangling power we introduce a complementary quantity called gate typicality and study its properties. Analyzing multiple, say nnn, applications of any entangling operator, UUU, interlaced with random local gates we prove that both investigated quantities approach their asymptotic values in a simple exponential form. These values coincide with the averages for random nonlocal operators on the full composite space and could be significantly larger than that of Un{U}^{n}Un. This rapid convergence to equilibrium, valid for subsystems of arbitrary size, is illustrated by studying multiple actions of diagonal unitary gates and controlled unitary gates.