Solving the $KP$ problem with the Global Cartan Decomposition

Geometric methods have useful application for solving problems in a range of quantum information disciplines, including the synthesis of time-optimal unitaries in quantum control. In particular, the use of Cartan decompositions to solve problems in optimal control, especially lambda systems, has given rise to a range of techniques for solving the so-called $KP$-problem, where target unitaries belong to a semi-simple Lie group manifold $G$ whose Lie algebra admits a $\mathfrak{g}=\mathfrak{k}\oplus \mathfrak{p}$ decomposition and time-optimal solutions are represented by subRiemannian geodesics synthesised via a distribution of generators in $\mathfrak{p}$. In this paper, we propose a new method utilising global Cartan decompositions $G=KAK$ of symmetric spaces $G\mathrm{/}K$ for generating time-optimal unitaries for targets $-iX\in [\mathfrak{p},\mathfrak{p}]\subset \mathfrak{k}$ with controls $-iH(t)\in \mathfrak{p}$. Target unitaries are parametrised as $U=kac$ where $k,c\in K$ and $a=e^{i\mathrm{\Theta }}$ with $\mathrm{\Theta }\in \mathfrak{a}$. We show that the assumption of $d\mathrm{\Theta }=0$ equates to the corresponding time-optimal unitary control problem being able to be solved analytically using variational techniques. We identify how such control problems correspond to the holonomies of a compact globally Riemannian symmetric space, where local translations are generated by $\mathfrak{p}$ and local rotations are generated by $[\mathfrak{p},\mathfrak{p}]$.