The reachable set of single-mode unstable quadratic Hamiltonians

TL;DR

This paper investigates the reachability of Gaussian unitary transformations generated by unstable quadratic Hamiltonians in one degree of freedom, revealing limitations and providing partial analytical and numerical characterizations of the reachable set.

Contribution

It offers the first analysis of the reachable set for unstable quadratic Hamiltonians, showing certain symplectic operations are unreachable and combining analytical and numerical methods.

Findings

Orthogonal symplectic operations are unreachable with unstable controls.

The rank criterion is necessary but not sufficient for reachability in this context.

Numerical algorithms help fully characterize the reachable set in specific cases.

Abstract

The question of open-loop control in the Gaussian regime may be cast by asking which Gaussian unitary transformations are reachable by turning on and off a given set of quadratic Hamiltonians. For compact groups, including finite dimensional unitary groups, the well known Lie algebra rank criterion provides a sufficient and necessary condition for the reachable set to cover the whole group. Because of the non-compact nature of the symplectic group, which corresponds to Gaussian unitary transformations, this criterion turns out to be still necessary but not sufficient for Gaussian systems. If the control Hamiltonians are unstable, in a sense made rigorous in the main text, the peculiar situation may arise where the rank criterion is satisfied and yet not all symplectic transformations are reachable. Here, we address this situation for one degree of freedom and study the properties of the reachable set under unstable control Hamiltonians. First, we provide a partial analytical characterisation of the reachable set and prove that no orthogonal (`energy-preserving' or `passive' in the literature) symplectic operations may be reached with such controls. Then, we apply numerical optimal control algorithms to demonstrate a complete characterisation of the set in specific cases.