V. P. Belavkin

Quantum stochastic calculus and quantum nonlinear filtering

A *-algebraic indefinite structure of quantum stochastic (QS) calculus is introduced and a continuity property of generalized nonadapted QS integrals is proved under the natural integrability conditions in an infinitely dimensional nuclear space. The class of nondemolition output QS processes in quantum open systems is characterized in terms of the QS calculus, and the problem of QS nonlinear filtering with respect to nondemolition, continuous measurements is investigated. The stochastic calculus of a posteriori conditional expectations in quantum observed systems is developed and a general quantum filtering stochastic equation for 2 QS process is derived. An application to the description of the spontaneous collapse of the quantum spin under continuous observation is given.

Quantum continual measurements and a posteriori collapse on CCR

A quantum stochastic model for the Markovian dynamics of an open system under the nondemolition unsharp observation which is continuous in time, is given. A stochastic equation for the posterior evolution of a quantum continuously observed system is derived and the spontaneous collapse (stochastically continuous reduction of the wave packet) is described. The quantum Langevin evolution equation is solved for the case of a quasi-free Hamiltonian in the initial CCR algebra with a linear output channel, and the posterior dynamics corresponding to an initial Gaussian state is found. It is shown for an example of the posterior dynamics of a quantum oscillator that any mixed state under a complete nondemolition measurement collapses exponentially to a pure Gaussian one.

A new wave equation for a continuous nondemolition measurement

Abstract A stochastic model for nondemolition continuous measurement in a quantum system is given. It is shown that the posterior dynamics, including a continuous collapse of the wave function, is described by a nonlinear stochastic wave equation. For a particle in an electromagnetic field it reduces to the Schrodinger equation with extra imaginary stochastic potentials.

Measurement, filtering and control in quantum open dynamical systems

Abstract A Markovian model for a quantum automata, i.e. an open quantum dynamical system with input and output channels and a feedback is described. A multi-stage version of the theory of quantum measurement and statistical decisions applied to the optimal control problem for quantum dynamical discrete-time objects is developed. Quantum analogies of Stratonovich nonstationary filtering and Bellman quantum dynamical programming for the discrete time are obtained. The Gaussian case of quantum one-dimensional linear Markovian dynamical system with a quantum linear transmission line is studied. The optimal quantum multi-stage decision rule consisting of the classical linear optimal control strategy and quantum optimal filtering procedure is found. The latter contains the optimal quantum coherent measurement on the output of the line and the recursive processing by Kalman-Busy filter. All the results are illustrated by an example of the optimal control problem for a quantum open oscillator at the input of a quantum wave transmission line.

Quantum communications and measurement

Quantum States and Input-Output Processes: The World of Quantum Noise and the Fundamental Output Process V.P. Belavkin, et al. Geometry of Quantum States S.L. Braunstein, C.M. Caves. Quantum Measurement Problem of Collapse: Mathematical Characterizations of Measurement Statistics M. Ozawa. Stochastic Dynamics of Continuously Observed Quantum Systems P. Staszewski. Quantum Jumps, Diffusion, and Localization: A Hamiltonian Solution to Quantum Collapse, State Diffusion, and Spontaneous Localization V.P. Belavkin, O. Melsheimer. Dissipation and Reduction of Superconducting States Due to Spontaneous Localization M. Buffa, et al. Quantum Channels, Entropy, and Information: State Change, Complexity and Fractals in Quantum Systems M. Ohya. Information in Mary Quantum Optical Communications: An Inequality Providing an Upper Limit A. Vourdas. Quantum Detection, Estimation, and Filtering: Quantum Filtering of Markov Signals with White Quantum Noise V.P. Belavkin. Covariant POVmeasures for W*Dynamical Systems T. Breuer. Quantum Optics, Experiments and Simulation: Quantum Noise Eaters C. Fabre, et al. Stochastic Simulations of Dissipation in Quantum Optics: Quantum Superpositions B.M. Garraway, P.L. Knight. 40 additional articles. Index.

A quantum nonadapted Ito formula and stochastic analysis in Fock scale

Abstract A generalized definition of quantum stochastic (QS) integrals and differentials is given in the free of adaptiveness and dimensionality form in terms of Malliavin derivative on a projective Fock space, and their uniform continuity with respect to the inductive limite convergence is proved. A new form of QS calculus based on an inductive ★-algebraic structure in an indefinite space is developed and a nonadaptive generalization of the QS Ito formula for its representation in Fock space is derived. 1 The problem of solution of general QS evolution equations in a Hilbert space is solved in terms of the constructed operator representation of chronological products, defined in the indefinite space, and the isometry and ∗-homomorphism property respectively for operators and maps of these solutions, corresponding to the peseudounitary and ★-homomorphism property of the QS integrable generators, is proved.

A posterior Schrödinger equation for continuous nondemolition measurement

A continuous model for a nondemolition observation of an atom is given. An equation for the corresponding instrument is found and a stochastic dissipative Schrodinger equation for the unnormalized posterior wave function of the atom is derived. It is shown that the continuously observed isolated atom relaxes to the ground state without mixing.

Bellman equations for optimal feedback control of qubit states

Using results from quantum filtering theory and methods from classical control theory, we derive an optimal control strategy for an open two-level system (a qubit in interaction with the electromagnetic field) controlled by a laser. The aim is to optimally choose the lasers amplitude and phase in order to drive the system into a desired state. The Bellman equations are obtained for the case of diffusive and counting measurements for vacuum field states. A full exact solution of the optimal control problem is given for a system with simpler, linear, dynamics. These linear dynamics can be obtained physically by considering a two-level atom in a strongly driven, heavily damped, optical cavity.

A quantum particle undergoing continuous observation

Abstract A stochastic model for the continuous nondemolition observation of the position of a quantum particle in a potential field and a boson reservoir is given. It is shown that the Gaussian wave packet evolving according to the posterior wave equation with a quadratic potential tends to the Gaussian stationary solution.

A continuous counting observation and posterior quantum dynamics

A stochastic model for a continuous photon counting measurement is proposed. An equation for the generating functional of the statistics of counts is found. A stochastic linear equation for the unnormalised posterior state vector of atoms, continuously collapsing without mixing, is derived by using quantum stochastic calculus methods.