Limits of optimal control yields achievable with quantum controllers

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 where 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.

Figure 1

Block diagrams of quantum control systems: (a) a quantum plant manipulated by a classical control; (b) a quantum plant manipulated by a classical control and a quantum controller.

Figure 2

Schematics of two typical trap-free control landscapes. For both landscapes, the search is in a two-dimensional control space. (a) The maximum submanifold is one-dimensional (with codimension 1) and there are no saddles; (b) the maximum submanifold is two-dimensional (with codimension 0) but there is a one-dimensional saddle submanifold.

Figure 3

Schematic diagram of typical spectra of $P$, illustrating condition (11) for surpassing the upper CKB on the control yield. The spectrum of $P$ is decomposed into bands ${\mathcal{B}}_i$ $(i=1,...,r)$ such that all elements of ${\mathcal{B}}_i$ multiply the same distinct eigenvalue ${\overline{\theta }}_i$ in $J_{{\sigma }_0}$ [cf. Eq. (9)]. When none of the bands overlap (left), the upper kinematic bound $J_{max}$ remains the same as the upper CKB $J_{{\sigma }_0}=J_{max}^{\mathrm{cl}}$. Only when at least two of the bands overlap (right) does $J_{max}$ exceed the upper CKB.

Figure 4

The kinematic bounds on the control yield $J$ for a two-level quantum system coupled to a collection of $M$ identical spins. Both the system and the controller are initially in thermal equilibrium states. The system's target observable is ${\sigma }_z$. The bounds $J_{max}$ and $J_{min}$ are shown as functions of the controller's parameter ${\lambda }_{\mathrm{c}}$ for a fixed ${\lambda }_{\mathrm{s}}=1$ and two values of $M$. The solid (blue) curve corresponds to $M=10$, and the dashed (red) curve to $M=2$. The CKB $J_{max}^{\mathrm{cl}}=tanh({\lambda }_{\mathrm{s}}/2)\approx 0.4621$ is surpassed for ${\lambda }_{\mathrm{c}}>{\lambda }_{\mathrm{s}}/M$.

Figure 5

The difference $J_{max}-J_{max}^{\mathrm{cl}}$ as a function of the controller's parameter ${\lambda }_{\mathrm{c}}$, for a two-spin quantum system (with a fixed ${\lambda }_{\mathrm{s}}=1)$ coupled to a collection of $M=10$ identical spins. Both the system and the controller are initially in thermal equilibrium states. The solid (blue) curve corresponds to the system's target observable $\theta$ chosen as the projector ${\Pi }_0$ on the subspace $\left\{\right|\uparrow \downarrow \rangle ,|\downarrow \uparrow \rangle \}$, and the dashed (green) curve corresponds to the system's target observable $\theta$ chosen as the projector ${\Pi }_1$ on the state $|\uparrow \uparrow \rangle$. For $\theta ={\Pi }_0$, the kinematic bound $J_{max}$ is always above the CKB, while for $\theta ={\Pi }_1,J_{max}$ only surpasses the CKB when ${\lambda }_{\mathrm{c}}$ exceeds the threshold value ${\lambda }_{\mathrm{s}}/M=0.1$.