Constructing Fermionic Dynamics with Closed Moment Hierarchies

TL;DR

This paper develops a framework for fermionic dynamics using completely positive maps with closed moment hierarchies, enabling efficient computation of low-order moments in fermionic systems.

Contribution

It introduces a class of fermionic maps with explicit formulas and invariance properties, connecting to second quantization and post-selection techniques.

Findings

Explicit formulas for fermionic maps in terms of minors of transformation matrices

Invariant span of monomials up to fixed order for even environment states

Closed equations for low-order moments enable efficient computation

Abstract

We construct a broad class of completely positive maps and Go\-rini--Kossakowski--Sudarshan-Lindblad generators for fermionic systems induced by linear transformations of system and environment modes. For these maps, we derive explicit Heisenberg-picture formulas for arbitrary normally ordered monomials in terms of minors of the underlying mode-transformation matrices and environment correlation tensors. We show that for even environment states the linear span of monomials up to any fixed order is invariant, which yields closed equations for low-order moments and makes their computation efficient. We also discuss the relation of this construction to second quantization of non-Hermitian one-particle contractions and extend the formalism to completely positive maps arising from post-selection.