Algebra and Hilbert space structures induced by quantum probes

In the general setting of quantum controls, it is unrealistic to control all of the degrees of freedom of a quantum system. We consider a scenario where our direct access is restricted to a small subsystem $S$ that is constantly interacting with the rest of the system $E$. What we investigate here is the fundamental structure of the Hilbert space that is caused solely by the restrictedness of the direct control. We clarify the intrinsic space structure of the entire system and that of the operations which could be activated through $S$. The structures hereby revealed would help us make quantum control problems more transparent and provide a guide for understanding what we can implement. They can be deduced by considering an algebraic structure, which is the Jordan algebra formed from Hermitian operators, naturally induced by the setting of limited access. From a few very simple assumptions about direct operations, we elucidate rich structures of the operator algebras and Hilbert spaces that manifest themselves in quantum control scenarios.