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Physics of Particles and Nuclei

Maximization of the Uhlmann–Jozsa Fidelity for an Open Two-Level Quantum System with Coherent and Incoherent Controls

Oleg Vasilevich Morzhin, Alexander Nikolaevich Pechen

Published Jul 1, 2020• 13 citations
Publisher

Abstract

Abstract In the paper, the problem of maximization of the Uhlmann–Jozsa fidelity, which shows the degree of coincidence between the given target density matrix and the final density matrix of a two-level open quantum system exposed to a simultaneous action of coherent and incoherent controls, is considered. For some initial and target density matrices, it was shown analytically that the zero coherent and incoherent controls are singular controls that satisfy the second order necessary conditions for the optimality. In the general case, a numerical method with reduction to finite-dimensional optimization is considered. Numerical results were obtained and presented.

DOI
10.1134/s1063779620040516

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Abstract In the paper, the problem of maximization of the Uhlmann–Jozsa fidelity, which shows the degree of coincidence between the given target density matrix and the final density matrix of a two-level open quantum system exposed to a simultaneous action of coherent and incoherent controls, is considered. For some initial and target density matrices, it was shown analytically that the zero coherent and incoherent controls are singular controls that satisfy the second order necessary conditions for the optimality. In the general case, a numerical method with reduction to finite-dimensional optimization is considered. Numerical results were obtained and presented.
The article considers a two-level open quantum system evolving under the action of coherent and incoherent controls using the general control method proposed in Phys. Rev. A. 73, 062102 (2006). Coherent control determines the Hamiltonian aspects of the dynamics whereas incoherent control determines the dissipative aspects. The goal is to find controls which steer the initial density matrix into a state which maximizes overlap with a predefined target density matrix. The controlled dynamics is represented as evolution in the Bloch ball and is analyzed analytically using Pontryagin maximum principle and numerically using optimization either in the functional space of controls (with the conditional and projected gradient methods) or through reduction to a finite-dimensional control space.
This paper considers some control problems for closed and open two-level quantum systems. The closed system's dynamics is governed by the Schrödinger equation with coherent control. The open system dynamics is governed by the Gorini-Kossakowski-Sudarshan-Lindblad master equation whose Hamiltonian depends on coherent control and superoperator of dissipation depends on incoherent control. For the closed system, we consider the problem of generation of the phase shift gate for some values of phases and final times for which we numerically show that zero coherent control, which is a stationary point of the objective functional, is not optimal; it gives an example of subtle point for practical solving quantum control problems. The two-stage method, which was developed in [48] for generic NNN-level open quantum systems for approximate generation of a given target density matrix, is modified here for the case of two-level systems. We modify the first ("incoherent") stage by numerically optimizing piecewise constant incoherent control instead of using constant incoherent control analytically computed using eigenvalues of the target density matrix. Exact analytical formulas are derived for the system state evolution, the objective functions and their gradients for the modified first stage. These formulas are then applied in the two-step gradient projection method. The numerical simulations show that the modified first stage duration can be significantly less than the unmodified first stage duration, but at the cost of performing optimization in the class of piecewise constant controls.
Abstract In this work, we consider a pair of qubits controlled by coherent and incoherent controls. The dynamics of the two-qubit system is driven by a Gorini–Kossakowsky–Sudarchhan–Lindblad master equation where coherent control enters into the Hamiltonian and incoherent control inters into both the Hamiltonian (via Lamb shift) and the dissipative superoperator. Two classes of interaction between the system and the coherent field are considered. For this system, we analyze the control problem of generating a given target density matrix which is formulated as minimizing the Hilbert–Schmidt distance between the final density matrix and the target density matrix. Incoherent control is modeled as a sum of constant in time Gaussians with centers related with the transitions frequencies between the energy levels of the qubits. Coherent control in general formulation is considered as measurable function and in numerical experiments as piecewise constant function with constraints on magnitudes and variations. Finite-dimensional numerical optimization is performed using the dual annealing method; the corresponding results are described for some initial and target density matrices and for some set of the parameters of the control problem.
This article considers some control problems for closed and open two-level quantum systems. The closed system's dynamics is governed by the Schrödinger equation with coherent control. The open system's dynamics is governed by the Gorini-Kossakowski-Sudarshan-Lindblad master equation whose Hamiltonian depends on coherent control and superoperator of dissipation depends on incoherent control. For the closed system, we consider the problem for generation of the phase shift gate for some values of phases and final times for which we numerically show that zero coherent control, which is a stationary point of the objective functional, is not optimal; it gives an example of subtle point for practical solving quantum control problems. The two-stage method, which was developed in [A. Pechen, Phys. Rev. A., 84, 042106 (2011)] for generic NNN-level open quantum systems for approximate generation of a given target density matrix, is modified here for the case of two-level systems. We modify the first ("incoherent") stage by numerically optimizing piecewise constant incoherent control instead of using constant incoherent control analytically computed using eigenvalues of the target density matrix. Exact analytical formulas are derived for the system's state evolution, the objective functions and their gradients for the modified first stage. These formulas are then applied in the two-step gradient projection method. The numerical simulations show that the modified first stage's duration can be significantly less than the unmodified first stage's duration, but at the cost of performing optimization in the class of piecewise constant controls. Bibliography: 72 titles.
This work considers two-qubit open quantum systems driven by coherent and incoherent controls. Incoherent control induces time-dependent decoherence rates via time-dependent spectral density of the environment which is used as a resource for controlling the system. The system evolves according to the Gorini–Kossakowski– Sudarshan–Lindblad master equation with time-dependent coefficients. For two types of interaction with coherent control, three types of objectives are considered: 1) maximizing the Hilbert–Schmidt overlap between the final and target density matrices; 2) minimizing the Hilbert–Schmidt distance between these matrices; 3) steering the overlap to a given value. For the first problem, we develop the Krotov type methods directly in terms of density matrices with or without regularization for piecewise continuous controls with constaints and find the cases where the methods produce (either exactly or with some precision) zero controls which satisfy the Pontryagin maximum principle and produce the overlap’s values close to their upper bounds. For the problems 2) and 3), we find cases when the dual annealing method steers the objectives close to zero and produces a non-zero control.
The article considers a two-level open quantum system, whose evolution is governed by the Gorini–Kossakowski– Lindblad–Sudarshan master equation with Hamiltonian and dissipation superoperator depending, correspondingly, on piecewise constant coherent and incoherent controls with constrained magnitudes. Additional constraints on controls’ variations are also considered. The problem is analyzed using Bloch parametrization of system’s density matrix. We adapt the section method for obtaining outer parallelepipedal and pointwise estimations of reachable and controllability sets in the Bloch ball via solving a number of problems for optimizing coherent and incoherent controls with respect to some objective criteria. The differential evolution and dual annealing optimization methods are used. The numerical results show how the reachable sets’ estimations depend on distances between the system’s initial states and the Bloch ball’s center point, final times, constraints on controls’ magnitudes and variations.