Extend the geometric probe construction beyond commuting generators

Extend the geometric construction of optimal probe states beyond commuting quantum generators, where the joint spectrum is no longer a fixed finite set, by developing a suitable general operator-geometric framework.

Background

The paper derives optimal probe states for distributed multiparameter quantum sensing when all parameter generators commute. In that setting, the generators possess a common eigenbasis, and their joint eigenvalue vectors form a fixed finite point set whose enclosing balls and ellipsoids determine optimal probe supports, populations, and precision bounds.

For noncommuting generators, a common eigenbasis and fixed joint spectrum generally do not exist. The authors explicitly identify extending the construction to this setting as an unresolved direction and indicate that the ellipsoid description would need to be replaced by a more general operator-geometric framework.