Time-optimal quantum transformations with bounded bandwidth

Abstract

In this paper, we derive sharp lower bounds, also known as quantum speed limits, for the time it takes to transform a quantum system into a state such that an observable assumes its lowest average value. We assume that the system is initially in an incoherent state relative to the observable and that the state evolves according to a von Neumann equation with a Hamiltonian whose bandwidth is uniformly bounded. The transformation time depends intricately on the observable's and the initial state's eigenvalue spectrum and the relative constellation of the associated eigenspaces. The problem of finding quantum speed limits consequently divides into different cases requiring different strategies. We derive quantum speed limits in a large number of cases, and we simultaneously develop a method to break down complex cases into manageable ones. The derivations involve both combinatorial and differential geometric techniques. We also study multipartite systems and show that allowing correlations between the parts can speed up the transformation time. In a final section, we use the quantum speed limits to obtain upper bounds on the power with which energy can be extracted from quantum batteries.

Minimum-Time Quantum Control and the Quantum Brachistochrone Equation

“…In general, the boundary condition dictates how | ψ( T ) changes and therefore determines | δψ(T ) . In the current context, the final boundary condition (26) dictates that…”

“…This is tantamount to imposing the boundary conditions in Eqs. ( 25)- (26). Therefore, it is sufficient to consider the case where the initial state is fixed while the final boundary condition is movable, according to the U(1) gauge transformation.…”

“…To reach this goal, we first leave the final state undetermined; it will be eventually specified by imposing Eq. (26). In leaving the final state unfixed, one can focus on the highly nontrivial part of solving the QB problem, i.e., determining the dynamics of the Lagrange multipliers:…”

“…The QB program by CHKO results in a differential equation with boundary conditions, which we shall refer to as the CHKO equation in what follows. It has inspired a number of theoretical works [20][21][22][23][24][25][26] and experiments [27]. The QB problem has close connections to other fundamental notions in nonequilibrium quantum physics, such as the quantum speed limit for driven systems [28,29] and counterdiabatic driving [14,15,17].…”

From quantum speed limits to energy-efficient quantum gates

“…§ Note that we are searching for the optimal protocol that drives the quantum state along a geodesic of the Riemannian manifold corresponding to the cost functional(14) [62,[66][67][68].…”