# Solutions of generic bilinear master equations for a quantum oscillator -- positive and factorized conditions on stationary states

## TL;DR

This paper derives solutions for a broad class of bilinear master equations governing quantum oscillators, characterizing stationary states' positivity and factorization, and identifying conditions for positive evolution even with non-completely positive generators.

## Contribution

It provides a unified framework for solving bilinear master equations, characterizes stationary states via Minkowski vectors, and explores positivity conditions beyond complete positivity.

## Key findings

- Solutions in Gaussian form for generic bilinear master equations.
- Characterization of stationary states using Minkowski vectors.
- Conditions for positive evolution with non-completely positive generators.

## Abstract

We obtain the solutions of the generic bilinear master equation for a quantum oscillator with constant coefficients in the Gaussian form. The well-behavedness and positive semidefiniteness of the stationary states could be characterized by a three-dimensional Minkowski vector. By requiring the stationary states to satisfy a factorized condition, we obtain a generic class of master equations that includes the well-known ones and their generalizations, some of which are completely positive. A further subset of the master equations with the Gibbs states as stationary states is also obtained. For master equations with not completely positive generators, an analysis on the stationary states suggests conditions on the coefficients of the master equations that generate positive evolution for a given initial state.