Spectral theorem for the Lindblad equation for quadratic open fermionic systems

TL;DR

This paper proves a spectral theorem for the Lindblad equation governing quadratic open fermionic systems, explicitly constructing eigenspaces and steady states, advancing understanding of their quantum dynamics.

Contribution

It provides a rigorous spectral theorem for the Lindblad equation in quadratic fermionic systems, including explicit eigenspace and steady state constructions.

Findings

Explicit construction of invariant eigenspaces for all eigenvalues

Description of Jordan blocks for the Liouvillean dynamics

Algebraic method for steady state space construction

Abstract

The spectral theorem is proven for the quantum dynamics of quadratic open systems of n fermions described by the Lindblad equation. Invariant eigenspaces of the many-body Liouvillean dynamics and their largest Jordan blocks are explicitly constructed for all eigenvalues. For eigenvalue zero we describe an algebraic procedure for constructing (possibly higher dimensional) spaces of (degenerate) non-equilibrium steady states.