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HOW TO THINK OF QUANTUM MARKOV MODELS FROM AN ENGINEERING - PowerPoint PPT Presentation

Aug 27, 2022 •396 likes •729 views

HOW TO THINK OF QUANTUM MARKOV MODELS FROM AN ENGINEERING PERSPECTIVE John Gough (Aberystwyth) Mathematics of QIT May 6-10 Lorentz Center Input-Plant-Output Models Plant = System (state variable x ) Laplace domain

  1. HOW TO THINK OF QUANTUM MARKOV MODELS FROM AN ENGINEERING PERSPECTIVE John Gough (Aberystwyth) “ Mathematics of QIT ” May 6-10 Lorentz Center

  2. Input-Plant-Output Models • Plant = System (state variable x ) • Laplace domain • Transfer Function

  3. Block Diagrams • Series • Feedback

  4. Fractional Linear Transformations • “Open Loop” • “Closed Loop”

  5. Fractional Linear Transformation • The feedback reduction • Algebraic loops if

  6. Double Pass! • Special case of feedback reduction

  7. Networks and Feedback Control • Measurement Based Feedback Control • Coherent Feedback Control

  8. Quantum Inputs and Outputs Lamb Model / Caldeira-Leggett / Ford-Kac-Mazur / Thirring-Schwabl /Lewis-Maassen/ Yurke-Denker

  9. Non-Markov Models and Markov Limits Input-output relations Spectral Density Gardiner-Collett

  10. Quantum Markovian Dynamics • A semi-group of CP identity-preserving maps (Heisenberg picture!) • Generator (Lindblad) • Dilation auxiliary space , vector state , unitary on

  11. Quantum Input-Output Systems Hudson-Parthasarathy (1984) V.P. Belavkin (1979+) Gardiner-Collett (1985)

  12. Quantum Input Processes The “wires” are quantum fields! • Field quanta of type k annihilated at the system at time t : • Hilbert Space: • Default state is the (Fock) vacuum

  13. Quantum Stochastic Models Single input – Emission/Absorption Interaction • Wick-ordered form: • Heisenberg Picture • GKS-LindbladGenerator • Input-Output Relations

  14. Quantum Stochastic Models • Two inputs – pure scattering Wick-ordered form: Heisenberg Picture Input-Output Relations

  15. Quantum Ito Table • Fundamental Processes • Table • Product Rule

  16. SLH Formalism • Hamiltonian H • Coupling/Collapse Operators L • Scattering Operator S

  17. Quantum Stochastic Models • General ( S ,L , H ) case Wick-ordered form: Or better as a QSDE (quantum Ito stochastic calculus)

  18. Quantum Stochastic Models Heisenberg Picture Lindblad Generator Input-Output Relations

  19. This is Markovian! “Pyramidal” Multi -time Expectations In non- Markovian models there is no “state” in the usual sense!

  20. Quantum Networks • How to connect models? • Cascaded models • Algebraic loops • Feedback Control

  21. The Series Product

  22. Perturbations (Avron, Fraas, Graf) • Virtual displacement of the model Displacements Virtual work

  23. Local Asymptotic Normality • Suppose that the QMS has a unique faithful stationary state • (CCR) Algebra of fluctuations (Guta and Kiukas, Bouten) • Geometric structure closely related to the Series Product! • General perturbations (Bouten and JG) • PROBLEM: is this some form of de Bruijn identity?

  24. Network Rule # 1 Open loop systems in parallel

  25. Network Rule # 2 Feedback Reduction Formula

  27. The Network Rules are implemented in a workflow capture package QHDL QHDL (MabuchiLab) N. Tezak, et al., (2012) Phil. Trans. Roy. Soc. A, 370, 5270.

  28. Transfer Operator • Classical Transfer function • SLH version • Properties

  29. Adiabatic Elimination • An important model simplification split the systems into slow and fast subspaces • Mathematical this is also a fractional linear transformation • It commutes with feedback reduction! • JG, H. Nurdin, S. Wildfeuer, J. Math. Phys., 51, 123518 (2010); H. Nurdin, JG, Phil. Trans. R. Soc., A 370, 5422-36 (2012)

  30. Measurement • Homodyne • Compatibility • Filter Conditioned state Innovations

  31. Coherent Quantum Feedback Control

  32. Autonomous Quantum Error Correction

  33. Thank You For Your Attention,

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