States of charged quantum fields and their statistical properties in the presence of critical potential steps

Evolution of charged quantum fields under the action of constant nonuniform electric fields is studied. To this end we construct a special generating functional for density operators of the quantum fields with different initial conditions. Then we study some reductions of the density operators. For example, reductions to electron or positron subsystems, reduction induced by measurements, and spatial reduction to the left or to the right subsystems of final particles. We calculate von Neumann entropy for the corresponding reduced density operators, estimating in such a way an information loss. Then we illustrate the obtained results by calculations in a specific background of a strong constant electric field between two infinite capacitor plates separated by a finite distance $L$.