On critical points of the objective functional for maximization of qubit observables

TL;DR

This paper proves that for controlling a two-level quantum system, all critical points of the objective functional are global when the final time T is at least π/2, halving the previously known minimal time for trap absence.

Contribution

The authors demonstrate that the minimal time to ensure all critical points are global is reduced from π to π/2 for a two-level quantum control system.

Findings

All maxima and minima are global if T ≥ π/2.

Minimal time for trap-free control is halved from π to π/2.

Reduces the known control time threshold for global optimality.

Abstract

We study unconstrained control of a two-level quantum system and analyse critical points of the objective functional which represents quantum average of system observable at some final time $T$. In Proc. Steklov Inst. Math. 285, 233-240 (2014) it was shown that all maxima and minima of the objective functional are global if $T\ge \pi$ (in suitable units). In the present work we show that all maxima and minima are global as soon as $T\ge \pi/2$. Hence we reduce by the factor of two the minimal time for which traps, i.e., local but not global maxima or minima of the objective functional, do not exist.