Abstract
A solution to the quantum Zermelo problem for control Hamiltonians with general energy resource bounds is provided. Interestingly, the energy resource of the control Hamiltonian and the control time define a pair of conjugate variables that minimize the energy-time uncertainty relation. The resulting control protocol is applied to a single qubit as well as to a two-interacting qubit system represented by a Heisenberg spin dimer. For these low-dimensional systems, it is found that physically realizable control Hamiltonians exist only for certain, quantized, energy resources.
Highlights
Nature requires a quantum description [1]
Moved by the possibility to apply the above ideas to more general quantum problems, here we present a further development of the quantum Zermelo navigation problem, which provides a protocol that can be adapted to different physical scenarios
Given the actual position of a classical particle under the action of a given time-independent force field, there exists an optimal control velocity that, acting constantly on the particle, allows it to reach another position of interest in the least possible time
Summary
Nature requires a quantum description [1]. quantum-classical correspondence arguments are still in fashion because of their usefulness to understand and explain the behavior of quantum systems [2], and to devise new strategies to tackle quantum problems, as in the case of optimal control strategies [3,4,5,6,7]. Moved by the possibility to apply the above ideas to more general quantum problems, here we present a further development of the quantum Zermelo navigation problem, which provides a protocol that can be adapted to different physical scenarios In this regard, we have focused on a series of guidelines which stress the physics behind this approach with a limited abstract conceptualization of the problem. If so, we wanted to know how it looks like and whether it works optimally By proceeding this way we have been able to reach a general form for the condition specified by (2), where the left-hand side equals a general constant k, which, in turn, is related to the minimum time necessary to take the system from the initial to the final quantum state that we wish, circumventing the unwanted effects of the bare Hamiltonian.
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