# Effects of parametric uncertainties in cascaded open quantum harmonic oscillators and robust generation of Gaussian invariant states

## TL;DR

This paper analyzes how parametric uncertainties affect the generation of Gaussian invariant states in cascaded open quantum harmonic oscillators and proposes a method to optimize robustness against such uncertainties.

## Contribution

It introduces an infinitesimal perturbation analysis for covariance matrices and develops a quadratic optimization approach for balancing oscillator realizations to enhance robustness.

## Key findings

- Sensitivity of the purity functional can be minimized through symplectic transformations.
- Effective solution for cascaded one-mode oscillators demonstrated numerically.
- Connection established between sensitivity index and classical statistical distances.

## Abstract

This paper is concerned with the generation of Gaussian invariant states in cascades of open quantum harmonic oscillators governed by linear quantum stochastic differential equations. We carry out infinitesimal perturbation analysis of the covariance matrix for the invariant Gaussian state of such a system and the related purity functional subject to inaccuracies in the energy and coupling matrices of the subsystems. This leads to the problem of balancing the state-space realizations of the component oscillators through symplectic similarity transformations in order to minimize the mean square sensitivity of the purity functional to small random perturbations of the parameters. This results in a quadratic optimization problem with an effective solution in the case of cascaded one-mode oscillators, which is demonstrated by a numerical example. We also discuss a connection of the sensitivity index with classical statistical distances and outline infinitesimal perturbation analysis for translation invariant cascades of identical oscillators. The findings of the paper are applicable to robust state generation in quantum stochastic networks.