Simple Derivation of the Lindblad Equation

TL;DR

This paper provides a straightforward derivation of the Lindblad equation, a fundamental tool in quantum theory for describing the evolution of density matrices, using basic linear algebra concepts suitable for students.

Contribution

It offers a simple derivation of the Lindblad equation utilizing only eigenvector and eigenvalue decompositions, making it accessible for learners.

Findings

Includes elementary examples of the Lindblad equation

Derivation accessible to students with basic quantum algebra

Clarifies the properties ensuring complete positivity

Abstract

The Lindblad equation is an evolution equation for the density matrix in quantum theory. It is the general linear, Markovian, form which ensures that the density matrix is hermitian, trace 1, positive and completely positive. Some elementary examples of the Lindblad equation are given. The derivation of the Lindblad equation presented here is "simple" in that all it uses is the expression of a hermitian matrix in terms of its orthonormal eigenvectors and real eigenvalues. Thus, it is appropriate for students who have learned the algebra of quantum theory. Where helpful, arguments are first given in a two-dimensional hilbert space.