# A classification of Markovian fermionic Gaussian master equations

## TL;DR

This paper classifies the possible types of open fermionic Gaussian dynamics based on four criteria, revealing only nine feasible types and analyzing the implications of complete positivity.

## Contribution

It introduces a comprehensive classification scheme for fermionic Gaussian master equations, detailing the constraints and characteristics of each type.

## Key findings

- Only nine of sixteen theoretically possible dynamics are physically realizable.
- Complete positivity restricts dynamics to either unitary or active types.
- Active dynamics necessarily involve particle exchange with an environment.

## Abstract

We introduce a classification scheme for the generators of open fermionic Gaussian dynamics. We simultaneously partition the dynamics along the following four lines: (1) unitary versus non-unitary, (2) active versus passive, (3) state-dependent versus state-independent, and (4) single-mode versus multi-mode. We find that only nine of these 16 types of dynamics are possible. Using this partition we discuss the consequences of imposing complete positivity on fermionic Gaussian dynamics. In particular, we show that completely positive dynamics must be either unitary (and so can be implemented without a quantized environment) or active (and so must involve particle exchange with an environment).