Quantum filtering: a reference probability approach

These notes are intended as an introduction to noncommutative (quantum) filtering theory. An introduction to quantum probability theory is given, focusing on the spectral theorem and the conditional expectation as the least squares estimate, and culminating in the construction of Wiener and Poisson processes on the Fock space. Next we describe the Hudson-Parthasarathy quantum Ito calculus and its use in the modelling of physical systems. Finally, we use a reference probability method to obtain quantum filtering equations, in the Belavkin-Zakai (unnormalized) form, for several system-observation models from quantum optics. The normalized (Belavkin-Kushner-Stratonovich) form is obtained through a noncommutative analogue of the Kallianpur-Striebel formula.

[1]  M. R. James.  A quantum Langevin formulation of risk-sensitive optimal control , 2004 .

[2]  Luigi Accardi,et al.  The weak coupling limit as a quantum functional central limit , 1990 .

[3]  Graham,et al.  Linear stochastic wave equations for continuously measured quantum systems. , 1994, Physical review. A, Atomic, molecular, and optical physics.

[4]  Robin L. Hudson,et al.  Quantum Ito's formula and stochastic evolutions , 1984 .

[5]  Luigi Accardi,et al.  A Representation Free Quantum Stochastic Calculus , 1992 .

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

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

[8]  Alberto Barchielli.  Direct and heterodyne detection and other applications of quantum stochastic calculus to quantum optics , 1990 .

[9]  R. Streater,et al.  Quasi-free quantum stochastic integrals for the CAR and CCR , 1983 .

[10]  Richard V. Kadison,et al.  Fundamentals of the Theory of Operator Algebras. Volume IV , 1998 .

[11]  A. S. Kholevo.  Quantum stochastic calculus , 1991 .

[12]  P. Meyer,et al.  Quantum Probability for Probabilists , 1993 .

[13]  John K. Stockton,et al.  Adaptive homodyne measurement of optical phase. , 2002, Physical review letters.

[14]  V. Belavkin.  THEORY OF THE CONTROL OF OBSERVABLE QUANTUM SYSTEMS , 2005 .

[15]  O. G. Smolyanov,et al.  Hamilton-Jacobi-Bellman equations for quantum optimal feedback control , 2005, quant-ph/0502155.

[16]  B. Kümmerer.  Markov dilations on W∗-algebras , 1985 .

[17]  M. Sentís.  Quantum theory of open systems , 2002 .

[19]  R. Kadison,et al.  Fundamentals of the Theory of Operator Algebras , 1983 .

[20]  Claus Köstler,et al.  Stationary Quantum Markov processes as solutions of stochastic differential equations , 1998 .

[21]  Luc Bouten,et al.  Stochastic Schrödinger equations , 2003 .

[22]  Luc Bouten,et al.  Stochastic Schr¨ odinger equations , 2004 .

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

[24]  Hans Maassen.  Quantum probability applied to the damped harmonic oscillator , 2004 .

[25]  Hideo Mabuchi,et al.  Quantum Kalman filtering and the Heisenberg limit in atomic magnetometry. , 2003, Physical review letters.

[26]  Viacheslav P. Belavkin,et al.  Non-Demolition Measurement and Control in Quantum Dynamical Systems , 1987 .

[27]  John K. Stockton,et al.  REVIEW ARTICLE: Modelling and feedback control design for quantum state preparation , 2005 .

[28]  V. P. Belavkin,et al.  Quantum Filtering of Markov Signals with White Quantum Noise , 2005, quant-ph/0512091.

[29]  Dénes Petz,et al.  An invitation to the algebra of canonical commutation relations , 1990 .

[30]  H. Carmichael.  An open systems approach to quantum optics , 1993 .

[31]  V. P. Belavkin,et al.  Optimal Quantum Filtering and Quantum Feedback Control , 2005 .

[32]  V. P. Belavkin,et al.  Quantum stochastic calculus and quantum nonlinear filtering , 1992 .

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

[34]  Harold J. Kushner,et al.  Jump-Diffusion Approximations for Ordinary Differential Equations with Wide-Band Random Right Hand Sides, , 1979 .

[35]  Hideo Mabuchi,et al.  Real-Time Quantum Feedback Control of Atomic Spin-Squeezing , 2004, Science.

[36]  Collett,et al.  Input and output in damped quantum systems: Quantum stochastic differential equations and the master equation. , 1985, Physical review. A, General physics.

[37]  Alberto Barchielli,et al.  Continual measurements in quantum mechanics , 2003 .

[38]  Anna Maria Paganoni,et al.  Detection theory in quantum optics: stochastic representation , 1996 .

[39]  Masamichi Takesaki,et al.  Conditional Expectations in von Neumann Algebras , 1972 .

[40]  John Gough.  Quantum Flows as Markovian Limit of Emission, Absorption and Scattering Interactions , 2003 .

[41]  David Williams,et al.  Probability with Martingales , 1991, Cambridge mathematical textbooks.

[42]  V. Belavkin.  Quantum Stochastic Positive Evolutions:¶Characterization, Construction, Dilation , 1997, math-ph/0512042.

[43]  V. P. Belavkin,et al.  Optimal Quantum Feedback Control via Quantum Dynamic Programming , 2005 .

[44]  Luc Bouten,et al.  Filtering and Control in Quantum Optics , 2004 .

[45]  Steven I. Marcus,et al.  An Introduction to Nonlinear Filtering , 1981 .

[46]  I. Segal.  Tensor algebras over Hilbert spaces. I , 1956 .

[48]  Hans Maassen,et al.  The essentially commutative dilations of dynamical semigroups onMn , 1987 .

[49]  C. cohen-tannoudji,et al.  Photons and Atoms: Introduction to Quantum Electrodynamics , 1989 .

[50]  R. Kadison,et al.  Fundamentals of the Theory of Operator Algebras, Volume II: Advanced Theory , 1997 .

[51]  Harold J. Kushner.  Diffusion approximations to output processes of nonlinear systems with wide-band inputs and applications , 1980, IEEE Trans. Inf. Theory.

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

[53]  S. Wills,et al.  A stochastic Stinespring Theorem , 2001 .

[54]  M. Zakai.  On the optimal filtering of diffusion processes , 1969 .

[55]  M. R. James,et al.  Risk-sensitive optimal control of quantum systems , 2004 .

[56]  Ramon van Handel,et al.  Feedback control of quantum state reduction , 2005, IEEE Transactions on Automatic Control.

[57]  V. Belavkin,et al.  Bellman equations for optimal feedback control of qubit states , 2004, quant-ph/0407192.

[58]  Milburn,et al.  Quantum theory of field-quadrature measurements. , 1993, Physical review. A, Atomic, molecular, and optical physics.

[59]  LU Y.G..  ON THE STOCHASTIC LIMIT FOR QUANTUM THEORY , 2003 .