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Time-minimum control of quantum purity for 2-level Lindblad equations

  • * Corresponding author: Anthony Bloch

    * Corresponding author: Anthony Bloch 

A. Bloch and W. Clark were supported by NSF grant DMS-1613819 and A. Bloch was also supported by AFSOR grant FA9550-18-1-0028. L. Colombo was partially supported by MINECO (Spain) grant MTM2016-76072-P and Juan de la Cierva Incorporación Fellowship

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  • We study time-minimum optimal control for a class of quantum two-dimensional dissipative systems whose dynamics are governed by the Lindblad equation and where control inputs acts only in the Hamiltonian. The dynamics of the control system are analyzed as a bi-linear control system on the Bloch ball after a decoupling of such dynamics into intra- and inter-unitary orbits. The (singular) control problem consists of finding a trajectory of the state variables solving a radial equation in the minimum amount of time, starting at the completely mixed state and ending at the state with the maximum achievable purity.

    The boundary value problem determined by the time-minimum singular optimal control problem is studied numerically. If controls are unbounded, simulations show that multiple local minimal solutions might exist. To find the unique globally minimal solution, we must repeat the algorithm for various initial conditions and find the best solution out of all of the candidates. If controls are bounded, optimal controls are given by bang-bang controls using the Pontryagin minimum principle. Using a switching map we construct optimal solutions consisting of singular arcs. If controls are bounded, the analysis of our model also implies classical analysis done previously for this problem.

    Mathematics Subject Classification: Primary: 49K15; Secondary: 81Q05, 81Q93.

    Citation:

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  • Figure 1.  Trajectory and controls of the $7^{th}$ order curve. The black ellipse is the escape chimney

    Figure 2.  Left: Example where $ b = [0;0] $, $ \alpha = -3 $, and $ \beta = -0.6 $. Right: Example where $ b = [0;0] $, $ \alpha = -0.8 $, and $ \beta = -0.6 $

    Figure 3.  Example where $ b = [-2;-1] $, $ \alpha = -4 $, and $ \beta = -3 $. The red curves show the trajectory when $ u = -1 $ and blue when $ u = 1 $

    Table 1.  Numerical results from time-minimal controls with solutions of various orders

    MTimeEnergy
    11.93727.5830
    31.93668.6873
    51.93611.6368
    71.93591.3765
     | Show Table
    DownLoad: CSV

    Table 2.  Switching times for the example in Figure 3

    Initial controlTime until first switchTime between first and second switches
    $u = 1$2.19430.4685
    $u = -1$1.15320.4905
     | Show Table
    DownLoad: CSV
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