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HOW TO THINK OF QUANTUM MARKOV MODELS FROM AN ENGINEERING PERSPECTIVE
John Gough
(Aberystwyth)
Input-Plant-Output Models
- Plant = System (state variable x)
- Laplace domain
- Transfer Function
HOW TO THINK OF QUANTUM MARKOV MODELS FROM AN ENGINEERING - - PowerPoint PPT Presentation
HOW TO THINK OF QUANTUM MARKOV MODELS FROM AN ENGINEERING - - PowerPoint PPT Presentation
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
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Block Diagrams
- Series
- Feedback
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Fractional Linear Transformations
- “Open Loop”
- “Closed Loop”
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Fractional Linear Transformation
- The feedback reduction
- Algebraic loops if
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Double Pass!
- Special case of feedback reduction
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Networks and Feedback Control
- Measurement Based
Feedback Control
- Coherent Feedback
Control
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Quantum Inputs and Outputs
Lamb Model / Caldeira-Leggett / Ford-Kac-Mazur / Thirring-Schwabl /Lewis-Maassen/ Yurke-Denker
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Non-Markov Models and Markov Limits
Gardiner-Collett Input-output relations Spectral Density
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Quantum Markovian Dynamics
- A semi-group of CP identity-preserving maps (Heisenberg picture!)
- Generator (Lindblad)
- Dilation
auxiliary space , vector state , unitary on
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Quantum Input-Output Systems
Gardiner-Collett (1985) Hudson-Parthasarathy (1984) V.P. Belavkin (1979+)
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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
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Quantum Stochastic Models
Single input – Emission/Absorption Interaction
- Wick-ordered form:
- Heisenberg Picture
- GKS-LindbladGenerator
- Input-Output Relations
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Quantum Stochastic Models
- Two inputs – pure scattering
Wick-ordered form: Heisenberg Picture Input-Output Relations
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Quantum Ito Table
- Fundamental Processes
- Table
- Product Rule
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SLH Formalism
- Hamiltonian H
- Coupling/Collapse Operators L
- Scattering Operator S
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Quantum Stochastic Models
- General (S ,L , H) case
Wick-ordered form: Or better as a QSDE (quantum Ito stochastic calculus)
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Quantum Stochastic Models
Heisenberg Picture Lindblad Generator Input-Output Relations
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This is Markovian!
“Pyramidal” Multi-time Expectations In non-Markovian models there is no “state” in the usual sense!
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Quantum Networks
- How to connect models?
- Cascaded models
- Algebraic loops
- Feedback Control
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The Series Product
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Perturbations (Avron, Fraas, Graf)
- Virtual displacement of the model
Displacements Virtual work
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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?
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Network Rule # 1 Open loop systems in parallel
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Network Rule # 2 Feedback Reduction Formula
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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.
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Transfer Operator
- Classical Transfer function
- SLH version
- Properties
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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)
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Measurement
- Homodyne
- Compatibility
- Filter
Conditioned state Innovations
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Coherent Quantum Feedback Control
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Autonomous Quantum Error Correction
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