A note on the distributions in quantum mechanical systems

TL;DR

This paper explores the distributions in quantum systems, their controllability, and geometric structures like sub-Finsler manifolds, providing insights into steering quantum states and system decompositions.

Contribution

It introduces a geometric framework for quantum control using sub-Finsler manifolds and analyzes the controllability and distribution properties of quantum systems.

Findings

Decomposition of Lie group G into KAK form.

Description of a sub-Finsler manifold with geodesics.

Analysis of controllability and minimum time for quantum state steering.

Abstract

In this paper, we study the distributions and the affine distributions of the quantum mechanical system. Also, we discuss the controllability of the quantum mechanical system with the related question concerning the minimum time needed to steer a quantum system from a unitary evolution $U(0)=I$ of the unitary propagator to a desired unitary propagator $U_f$. Furthermore, the paper introduces a description of a $\mathfrak{k} \oplus \mathfrak{p}$ sub-Finsler manifold with its geodesics, which equivalents to the problem of driving the quantum mechanical system from an arbitrary initial state $U(0)=I$ to the target state $U_f$, some illustrative examples are included. We prove that the Lie group $G$ on a Finsler symmetric manifold $G/K$ can be decomposed into $KAK$.