A quantum extended Kalman filter

In quantum physics, a stochastic master equation (SME) estimates the state (density operator) of a quantum system in the Schrodinger picture based on a record of measurements made on the system. In the Heisenberg picture, the SME is a quantum filter. For a linear quantum system subject to linear measurements and Gaussian noise, the dynamics may be described by quantum stochastic differential equations (QSDEs), also known as quantum Langevin equations, and the quantum filter reduces to a so-called quantum Kalman filter. In this article, we introduce a quantum extended Kalman filter (quantum EKF), which applies a commutative approximation and a time-varying linearization to systems of nonlinear QSDEs. We will show that there are conditions under which a filter similar to a classical EKF can be implemented for quantum systems. The boundedness of estimation errors and the filtering problem with 'state-dependent' covariances for process and measurement noises are also discussed. We demonstrate the effectiveness of the quantum EKF by applying it to systems that involve multiple modes, nonlinear Hamiltonians, and simultaneous jump-diffusive measurements.

[1]  Moshe Zakai,et al.  On the Ultimate Boundedness of Moments Associated with Solutions of Stochastic Differential Equations , 1967 .

[2]  E. Davies,et al.  Quantum stochastic processes , 1969 .

[3]  E. Davies,et al.  Quantum stochastic processes II , 1970 .

[4]  S. Sakai C*-Algebras and W*-Algebras , 1971 .

[5]  Donald L. Snyder,et al.  Filtering and detection for doubly stochastic Poisson processes , 1972, IEEE Trans. Inf. Theory.

[6]  Thomas Kailath,et al.  The modeling of randomly modulated jump processes , 1975, IEEE Trans. Inf. Theory.

[7]  Example of a Lipschitz function of self-adjoint operators that gives a nonnuclear increment under a nuclear perturbation , 1975 .

[8]  G. Lindblad On the generators of quantum dynamical semigroups , 1976 .

[9]  E. B. Davies Quantum theory of open systems , 1976 .

[10]  Lennart Ljung,et al.  The Extended Kalman Filter as a Parameter Estimator for Linear Systems , 1979 .

[11]  V. C. L. Hutson,et al.  Applications of Functional Analysis and Operator Theory , 1980 .

[12]  Measuring processes in quantum mechanics I. Continuous observation and the watchdog effect , 1981 .

[13]  P. Krishnaprasad,et al.  Dynamic observers as asymptotic limits of recursive filters , 1982, 1982 21st IEEE Conference on Decision and Control.

[14]  A. Shiryayev,et al.  Statistics of Random Processes I: General Theory , 1984 .

[15]  Measuring processes in quantum mechanics. II. The classical behavior of measuring instruments , 1985 .

[16]  Viacheslav P. Belavkin,et al.  Nondemolition measurements, nonlinear filtering and dynamic programming of quantum stochastic processes , 1989 .

[17]  Yaakov Friedman,et al.  Operator differentiable functions , 1990 .

[18]  V. P. Belavkin,et al.  Measurements continuous in time and a posteriori states in quantum mechanics , 1991 .

[19]  V. P. Belavkin,et al.  Quantum continual measurements and a posteriori collapse on CCR , 1992 .

[20]  J. Grizzle,et al.  The Extended Kalman Filter as a Local Asymptotic Observer for Nonlinear Discrete-Time Systems , 1992, 1992 American Control Conference.

[21]  K. Parthasarathy An Introduction to Quantum Stochastic Calculus , 1992 .

[22]  Milburn,et al.  All-optical versus electro-optical quantum-limited feedback. , 1994, Physical review. A, Atomic, molecular, and optical physics.

[23]  Operator differentiable functions and derivations of operator algebras , 1996 .

[24]  M. Boutayeb,et al.  Convergence analysis of the extended Kalman filter used as an observer for nonlinear deterministic discrete-time systems , 1997, IEEE Trans. Autom. Control..

[25]  Konrad Reif,et al.  An EKF-Based Nonlinear Observer with a Prescribed Degree of Stability , 1998, Autom..

[26]  V. P. Belavkin,et al.  Measurement, filtering and control in quantum open dynamical systems , 1999 .

[27]  S. Sastry Nonlinear Systems: Analysis, Stability, and Control , 1999 .

[28]  J. Conway A course in operator theory , 1999 .

[29]  Konrad Reif,et al.  Nonlinear state observation using H∞-filtering Riccati design , 1999, IEEE Trans. Autom. Control..

[30]  R. Unbehauen,et al.  Stochastic stability of the continuous-time extended Kalman filter , 2000 .

[31]  G. Pedersen,et al.  Operator differentiable functions , 2000 .

[32]  V. J. Mathews,et al.  Polynomial Signal Processing , 2000 .

[33]  Hideo Mabuchi,et al.  Quantum feedback control and classical control theory , 1999, quant-ph/9912107.

[34]  Hideo Mabuchi,et al.  Feedback cooling of atomic motion in cavity QED (21 pages) , 2005, quant-ph/0509039.

[35]  A C Doherty,et al.  Optimal unravellings for feedback control in linear quantum systems. , 2005, Physical review letters.

[36]  Ramon van Handel,et al.  Quantum projection filter for a highly nonlinear model in cavity QED , 2005 .

[37]  J. Gough Quantum Stratonovich calculus and the quantum Wong-Zakai theorem , 2005, math-ph/0511046.

[38]  N. Yamamoto Robust observer for uncertain linear quantum systems , 2006, Proceedings of the 45th IEEE Conference on Decision and Control.

[39]  D. Wineland,et al.  High-fidelity adaptive qubit detection through repetitive quantum nondemolition measurements. , 2007, Physical review letters.

[40]  Matthew R. James,et al.  An Introduction to Quantum Filtering , 2006, SIAM Journal of Control and Optimization.

[41]  Matthew R. James,et al.  The Series Product and Its Application to Quantum Feedforward and Feedback Networks , 2007, IEEE Transactions on Automatic Control.

[42]  G. Milburn,et al.  Quantum Measurement and Control , 2009 .

[43]  Hendra Ishwara Nurdin,et al.  Network Synthesis of Linear Dynamical Quantum Stochastic Systems , 2008, SIAM J. Control. Optim..

[44]  D. Bernstein Matrix Mathematics: Theory, Facts, and Formulas , 2009 .

[45]  C. Pellegrini Markov chains approximation of jump-diffusion stochastic master equations , 2010 .

[46]  B. Rozovskii,et al.  The Oxford Handbook of Nonlinear Filtering , 2011 .

[47]  Mazyar Mirrahimi,et al.  Real-time quantum feedback prepares and stabilizes photon number states , 2011, Nature.

[48]  Hendra Ishwara Nurdin,et al.  Quantum filtering for systems driven by fields in single-photon states or superposition of coherent states , 2012 .

[49]  Masahito Ueda,et al.  Simultaneous continuous measurement of photon-counting and homodyne detection on a free photon field: dynamics of state reduction and the mutual influence of measurement backaction , 2012, 1212.0968.

[50]  M. Bloor,et al.  Applications of functional analysis , 2013 .

[51]  Neergaard-Nielsen Jonas Generation of single photons and Schrödinger kitten states of light , 2015 .

[52]  Mankei Tsang,et al.  Volterra filters for quantum estimation and detection , 2015, 1509.02133.

[53]  Markus Aspelmeyer,et al.  Optimal State Estimation for Cavity Optomechanical Systems. , 2015, Physical review letters.

[54]  Matthew J. Woolley,et al.  Quantum filtering for multiple diffusive and Poissonian measurements , 2015, 1503.04887.

[55]  Anja Walter,et al.  Introduction To Stochastic Calculus With Applications , 2016 .

[56]  A. Dąbrowska,et al.  Belavkin filtering with squeezed light sources , 2014, 1405.7795.

[57]  Rajeeva L. Karandikar,et al.  Introduction to stochastic calculus*** , 2018, Actuarial Finance.