Surprisal of a quantum state: Dynamics, compact representation, and coherence effects

Abstract

Progress towards quantum technologies continues to provide essential new insights on the microscopic dynamics of systems in phase space. This highlights coherence effects whether these are due to ultrafast lasers whose energy width spans several states all the way to the output of quantum computing. Surprisal analysis has provided seminal insights on the probability distributions of quantum systems from elementary particle and also nuclear physics, through molecular reaction dynamics to system biology. It is therefore necessary to extend surprisal analysis to the full quantum regime where it characterizes not only the probabilities of states but also their coherence. In principle this can be done by the maximal entropy formalism but in the full quantum regime its application is far from trivial [E.g., S. Dagan and Y. Dothan, Phys Rev D 26, 248 1982] because an exponential function of not commuting operators is not easily accommodated. Starting from an exact dynamical approach we develop a description of the dynamics where the quantum mechanical surprisal, a linear combination of operators, plays a central role. We provide an explicit route to the Lagrange multipliers of the system and identify those operators that act as the dominant constraints.

The density matrix via few dominant observables: The quantum interference in the isotope effect for atto-pumped N2

“…Here we propagate the surprisal in a finite basis representation that achieves numerical convergence. 11 The choice of the basis states is dictated on the one hand by the fact that the initial wave packets pumped by an ultrafast excitation are localized in the Franck Condon region. On the other hand the highest excited state has a potential with a shifted equilibrium distance.…”

“…(11) becomes exact. 9,11 If the evolution of the operators in Eq. ( 10) is not closed, the set of constraints in Eq.…”

“…Yet it is often the case that using a finite and small set of operators already offer a good approximation for the surprisal. 11 In the absence of coupling between the diabatic states closure is enabled by using a time-dependent basis of coherent vibrational states on the shifted electronic state, Eq. (4).…”

“…To describe the dynamics of coupled electronic states we implement an algebraic formulation of quantum dynamics that is motivated by an information-theoretic approach. [9][10][11] It is a predictive, dynamical, version of the procedure known as surprisal analysis. 12,13 In essence the dynamical approach seeks to identify such operators, that we call 'constraints', whose mean values are sufficient to provide an exact or realistically approximate description of the dynamics.…”

“…The mean values of these operators serve as constraints in lowering the information entropy of the density matrix. The final result isρ = exp − k λ k (t)Â k 9,11 with the time dependence carried by the Lagrange multipliers λ k (t). Below we discuss computing the time dependence of the Lagrange multipliers for the quantum multi electronic state problem.…”