An obstruction to small-time controllability of the bilinear Schr{\"o}dinger equation

TL;DR

This paper investigates how small-time controllability properties of classical Hamiltonian systems relate to their quantum counterparts, using the WKB method to analyze whether non-controllability persists after quantization.

Contribution

It provides conditions under which classical non-controllability implies quantum non-controllability, exploring the preservation of controllability properties through quantization.

Findings

Classical non-controllability often implies quantum non-controllability.

The WKB method is used to analyze controllability in quantum systems.

Conditions are identified for when controllability properties are preserved after quantization.

Abstract

In this article we discuss which controllability properties of classical Hamiltonian systems are preserved after quantization. We discuss some necessary and some sufficient conditions for small-time controllability of classical systems and quantum systems using the WKB method. In particular, we investigate the conjecture that if the classical system is not small-time controllable, then the corresponding quantum system is not small-time controllable either.