  In quantum optimal control theory, kinematic bounds are the minimum and
maximum values of the control objective achievable for any physically
realizable system dynamics. For a given initial state of the system, these
bounds depend on the nature and state of the controller. We consider a general
situation where the controlled quantum system is coupled to both an external
classical field (referred to as a classical controller) and an auxiliary
quantum system (referred to as a quantum controller). In this general
situation, the kinematic bound is between the classical kinematic bound (CKB),
corresponding to the case when only the classical controller is available, and
the quantum kinematic bound (QKB), corresponding to the ultimate physical limit
of the objective's value. Specifically, when the control objective is the
expectation value of a quantum observable (a Hermitian operator on the system's
Hilbert space), the QKBs are the minimum and maximum eigenvalues of this
operator. We present, both qualitatively and quantitatively, the necessary and
sufficient conditions for surpassing the CKB and reaching the QKB, through the
use of a quantum controller. The general conditions are illustrated by examples
in which the system and controller are initially in thermal states. The
obtained results provide a basis for the design of quantum controllers capable
of maximizing the control yield and reaching the ultimate physical limit.
