Which notion of energy for bilinear quantum systems?

TL;DR

This paper compares different L^p norms to describe control in bilinear quantum systems, finding that the L^1 norm is most suitable for steering states with minimal control effort, and provides explicit cost calculations.

Contribution

It demonstrates that the L^1 norm is optimal for controlling bilinear quantum systems and computes explicit control costs for this norm.

Findings

L^1 norm effectively describes control energy in quantum systems

Arbitrary state steering is possible with small L^p control norms for p>1

Explicit optimal control costs are derived for the L^1 norm

Abstract

In this note we investigate what is the best L^p-norm in order to describe the relation between the evolution of the state of a bilinear quantum system with the L^p-norm of the external field. Although L^2 has a structure more easy to handle, the L^1 norm is more suitable for this purpose. Indeed for every p>1, it is possible to steer, with arbitrary precision, a generic bilinear quantum system from any eigenstate of the free Hamiltonian to any other with a control of arbitrary small L^p norm. Explicit optimal costs for the L^1 norm are computed on an example.