Dunkel and Trigger [Phys. Rev. A {71}, 052102 (2005)] show that the Leipnik's joint entropy monotonously increases for the initially maximally classical Gaussian wave packet for a free particle. After expressing the joint entropy of the general Gaussian wave packets for quadratic Hamiltonians as
S(t)=ln(e/2)+ln(2Deltax(t)Deltap(t)/hbar), we show that a class of general Gaussian wave packets does not warrant the monotonous increase of the joint entropy. We propose that the random-phase entropy with respect to the squeeze angle always monotonously increases even for non-maximally classical states.
In the above-mentioned paper by Dunkel and Trigger [Phys. Rev. A 71, 052102 (2005)] a hypothesis has been pursued that the loss of information associated with the quantum evolution of pure states, quantified in terms of an increase in the time of so-called Leipnik's joint entropy, could be a rather general property shared by many quantum systems. This behavior has been confirmed for the unconfined model systems and properly tuned initial data (maximally classical states). We provide two particular examples which indicate a complexity of the quantum evolution. In the presence of a confining (harmonic) potential Leipnik's entropy there may be nonincreasing for maximally classical initial data. Another choice of initial data implies periodicity in the time of the Leipnik entropy.
We study an information-theoretic measure of uncertainty for quantum systems. It is the Shannon information I of the phase-space probability distribution 〈z\ensuremath{\Vert}\ensuremath{\rho}\ensuremath{\Vert}z〉, where \ensuremath{\Vert}z〉 are coherent states and \ensuremath{\rho} is the density matrix. As shown by Lieb I\ensuremath{\ge}1, and this bound represents a strengthened version of the uncertainty principle. For a harmonic oscillator in a thermal state, I coincides with von Neumann entropy, -Tr(\ensuremath{\rho}ln\ensuremath{\rho}), in the high-temperature regime, but unlike entropy, it is nonzero (and equal to the Lieb bound) at zero temperature. It therefore supplies a nontrivial measure of uncertainty due to both quantum and thermal fluctuations. We study I as a function of time for a class of nonequilibrium quantum systems consisting of a distinguished system coupled to a heat bath. We derive an evolution equation for I. For the harmonic oscillator, in the Fokker-Planck regime, we show that I increases monotonically, if the width of the coherent states is chosen to be the same as the width of the harmonic oscillator ground state. For other choices of the width, and for more general Hamiltonians, I settles down to a monotonic increase in the long run, but may suffer an initial decrease for certain initial states that undergo ``reassembly'' (the opposite of quantum spreading). Our main result is to prove, for linear systems, that I at each moment of time has a lower bound
Itmin, over all possible initial states.This bound is a generalization of the uncertainty principle to include thermal fluctuations in nonequilibrium systems, and represents the least amount of uncertainty the system must suffer after evolution in the presence of an environment for time t.
Itmin is an envelope, equal for each time t, to the time evolution of I for a certain initial state, which we calculate to be a nonminimal Gaussian.
Itmin coincides with the Lieb bound in the absence of an environment, and is related to von Neumann entropy in the long-time limit. The form of
Itmin indicates that the thermal fluctuations become comparable with the quantum fluctuations on a time scale equal to the decoherence time scale, in agreement with earlier work of Hu and Zhang. Our results are also related to those of Zurek, Habib, and Paz, who looked for the set of initial states generating the least amount of von Neumann entropy after a fixed period of nonunitary evolution.
Abstract A probabilistic description is essential for understanding the dynamics in many systems due to uncertainty or fluctuations. We show how to utilise time-dependent probability density functions to compute the information length , as a Lagrangian measure that counts the number of different states that a quantum system evolves through in time. Using , we examine the information change associated with the evolution of initial Gaussian wave packets and elucidate consequences of quantum effects.
A known limitation of time-dependent mean-field approaches is a lack of quantum tunneling for collective motions such as in sub-barrier fusion reactions. As a first step toward a solution, a time-dependent model is considered using a superposition of Gaussian wave packets, to describe the relative motion between two colliding nuclei, which may be simplified to a problem for one particle in one dimension. In this article, how the model describes the potential-barrier transmission is investigated by paying attention to the time evolution of the phase space distribution, which in particular reveals that the behavior of the free propagation of the incoming state is not trivial, depending on the number of superposed wave packets. Passage over the barrier can occur due to the high-momentum components in the incoming state corresponding to energies above the barrier height, which is, however, of classical nature and needs to be distinguished from the true quantum tunneling. Although a transmitted wave packet in some case may end up with an energy lower than the barrier, a difficulty is noticed in guaranteeing the energy conservation when the energies of different exit channels, e.g. of transmission and reflection, are individually measured. To overcome these issues for a description of quantum tunneling is still a challenging problem. This article mainly treats the same system with the same model as in the paper Hasegawa, Hagino and Tanimura (2020) [1]. However, we reach different conclusions.
The article intertwines the study of wave packet dynamics with information-theoretic measurements in one dimensional (D =1) system. A localized wave packet at time t=0 has been considered here and its change at later instant of time t is calculated for the position space wave function. The momentum space wave function is obtained by taking the Fourier transform of the position space wave function at time t= 0 . These wave functions are then employed to construct the wave packet's respective probability densities in position and momentum space, and later have been used to compute the corresponding position and momentum space Shannon (S) and Fisher information (I) entropies. It has been observed that although the Shannon and Fisher information entropies explicitly depend on the standard deviation, neither the Shannon entropy sum nor the product of Fisher information entropies is. Moreover, for D dimensional systems, the computed values of the Shannon and Fisher entropies are found to satisfy the lower bounds of the Bialynicki-Birula and Myceilski (BBM) inequality relation and the Stam-Cramer-Rao inequalities better known as the Fisher based uncertainty relation. Thus, our theoretical study explores the validation of information-theoretic measurements for wave packet dynamics using the basic formulations of information theory.