[Paper Review] On Exponential Stabilization of Spin-1/2 Systems

Weichao Liang, Nina H. Amini+1 more|arXiv (Cornell University)|Mar 28, 2018

Quantum Information and Cryptography16 references17 citations

TL;DR

This paper proposes a parametrized continuous-time feedback controller for exponential stabilization of spin-1/2 quantum systems toward the ground or excited state under continuous-time measurements. Using stochastic Lyapunov techniques, it proves almost sure global exponential stabilization and provides a lower bound on the convergence rate, while analyzing the impact of controller parameters on performance.

ABSTRACT

In this paper, we study the stabilization problem of quantum spin-1/2 systems under continuous-time measurements. In the case without feedback, we show exponential stabilization around the excited and ground state by providing a lower bound of the convergence rate. Based on stochastic Lyapunov techniques, we propose a parametrized measurement-based feedback which ensures exponential convergence toward the excited state. Moreover, we give a lower bound of the convergence rate for this case. Then, we discuss the effect of each parameter appeared in the control law in the convergence rate. Finally, we illustrate the efficiency of such feedback law through simulations.

Motivation & Objective

• To establish exponential stabilization of spin-1/2 systems toward the ground or excited state without feedback under continuous measurement.
• To design a continuous feedback controller ensuring almost sure global exponential stabilization toward a target eigenstate.
• To derive a lower bound on the convergence rate for the feedback-controlled system.
• To analyze the influence of controller parameters on the convergence rate.
• To validate the controller's effectiveness through numerical simulations.

Proposed method

• Uses the stochastic master equation to model the continuous-time dynamics of a spin-1/2 system under imperfect measurements.
• Applies stochastic Lyapunov techniques with a candidate Lyapunov function based on the standard deviation of σz to analyze stability.
• Introduces a parametrized feedback law involving control input ut that ensures local convergence to the target state and avoids antipodal states.
• Employs Itô’s formula and the Doob’s martingale convergence theorem to prove almost sure exponential stability.
• Derives a lower bound on the Lyapunov exponent as −ηM/2 for the open-loop case and extends it to the feedback case.
• Uses the infinitesimal generator of the Lyapunov function to analyze the asymptotic behavior of quantum trajectories.

Experimental results

Research questions

• RQ1 What is the convergence rate of quantum state reduction toward the ground or excited state in the absence of feedback?
• RQ2 Can a continuous feedback controller ensure almost sure global exponential stabilization of a spin-1/2 system toward a target eigenstate?
• RQ3 How do the parameters of the feedback controller affect the rate of convergence to the target state?
• RQ4 What is the lower bound on the convergence rate for the feedback-controlled system?
• RQ5 How does the system's trajectory behave asymptotically under the proposed feedback law?

Key findings

• Without feedback, the system exhibits almost sure exponential stabilization toward the set of eigenstates {ρg, ρe} with a Lyapunov exponent lower bounded by −ηM/2.
• The probability of convergence to a specific eigenstate ρe or ρg is equal to Tr(ρ0ρe) or Tr(ρ0ρg), respectively.
• With the proposed feedback controller, the system is almost surely globally exponentially stabilized toward the target eigenstate, with a convergence rate lower bounded by a positive constant depending on controller parameters.
• The controller structure ensures local attraction to the target state while repelling trajectories from the antipodal state, preventing convergence to the wrong eigenstate.
• The convergence rate is sensitive to the controller parameters, and their influence is analytically quantified in the paper.
• Simulations confirm the effectiveness of the feedback controller in achieving fast and robust stabilization toward the desired quantum state.