Analyzing the Positivity Preservation of Numerical Methods for the Liouville-von Neumann Equation

TL;DR

This paper evaluates numerical methods for solving the Liouville-von Neumann equation in quantum mechanics, emphasizing the importance of positivity preservation and demonstrating that only the matrix exponential method maintains the necessary physical properties over time.

Contribution

It establishes a criterion for numerical methods to preserve quantum density matrix properties and assesses common methods, showing only the matrix exponential method is fully CPTP.

Findings

Only the matrix exponential method's update step is CPTP.

Runge-Kutta and predictor-corrector methods do not preserve positivity.

Ensures long-term physical validity of quantum simulations.

Abstract

The density matrix is a widely used tool in quantum mechanics. In order to determine its evolution with respect to time, the Liouville-von Neumann equation must be solved. However, analytic solutions of this differential equation exist only for simple cases. Additionally, if the equation is coupled to Maxwell's equations to model light-matter interaction, the resulting equation set -- the Maxwell-Bloch or Maxwell-Liouville-von Neumann (MLN) equations -- becomes nonlinear. In these advanced cases, numerical methods are required. Since the density matrix has certain mathematical properties, the numerical methods applied should be designed to preserve those properties. We establish the criterion that only methods that have a completely positive trace preserving (CPTP) update map can be used in long-term simulations. Subsequently, we assess the three most widely used methods -- the matrix exponential (ME) method, the Runge-Kutta (RK) method, and the predictor-corrector (PC) approach -- whether they provide this feature, and demonstrate that only the update step of the matrix exponential method is a CPTP map.