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Register a new userMoment dynamics and observer design for a class of quasilinear quantum stochastic systems
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This paper is concerned with a class of open quantum systems whose dynamic variables have an algebraic structure, similar to that of the Pauli matrices pertaining to finite-level systems. The system interacts with external bosonic fields, and its Hamiltonian and coupling operators depend linearly on the system variables. This results in a Hudson-Parthasarathy quantum stochastic differential equation (QSDE) whose drift and dispersion terms are affine and linear functions of the system variables. The quasilinearity of the QSDE leads to tractable dynamics of mean values and higher-order multi-point moments of the system variables driven by vacuum input fields. This allows for the closed-form computation of the quasi-characteristic function of the invariant quantum state of the system and infinite-horizon asymptotic growth rates for a class of cost functionals. The tractability of the moment dynamics is also used for mean square optimal Luenberger observer design in a measurement-based filtering problem for a quasilinear quantum plant, which leads to a Kalman-like quantum filter.
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This paper is concerned with multimode open quantum harmonic oscillators and quadratic-exponential functionals (QEFs) as quantum risk-sensitive performance criteria. Such systems are described by linear quantum stochastic differential equations driven by multichannel bosonic fields. We develop a finite-horizon expansion for the system variables using the eigenbasis of their two-point commutator kernel with noncommuting position-momentum pairs as coefficients. This quantum Karhunen-Loeve expansion is used in order to obtain a Girsanov type representation for the quadratic-exponential functions of the system variables. This representation is valid regardless of a particular system-field state and employs the averaging over an auxiliary classical Gaussian random process whose covariance operator is defined in terms of the quantum commutator kernel. We use this representation in order to relate the QEF to the moment-generating functional of the system variables. This result is also specified for the invariant multipoint Gaussian quantum state when the oscillator is driven by vacuum fields.
This paper is concerned with quadratic-exponential functionals (QEFs) as risk-sensitive performance criteria for linear quantum stochastic systems driven by multichannel bosonic fields. Such costs impose an exponential penalty on quadratic functions of the quantum system variables over a bounded time interval, and their minimization secures a number of robustness properties for the system. We use an integral operator representation of the QEF, obtained recently, in order to compute its asymptotic infinite-horizon growth rate in the invariant Gaussian state when the stable system is driven by vacuum input fields. The resulting frequency-domain formulas express the QEF growth rate in terms of two spectral functions associated with the real and imaginary parts of the quantum covariance kernel of the system variables. We also discuss the computation of the QEF growth rate using homotopy and contour integration techniques and provide two illustrations including a numerical example with a two-mode oscillator.
This brief article gives an overview of quantum mechanics as a {em quantum probability theory}. It begins with a review of the basic operator-algebraic elements that connect probability theory with quantum probability theory. Then quantum stochastic processes is formulated as a generalization of stochastic processes within the framework of quantum probability theory. Quantum Markov models from quantum optics are used to explicitly illustrate the underlying abstract concepts and their connections to the quantum regression theorem from quantum optics.
This paper is concerned with a risk-sensitive optimal control problem for a feedback connection of a quantum plant with a measurement-based classical controller. The plant is a multimode open quantum harmonic oscillator driven by a multichannel quantum Wiener process, and the controller is a linear time invariant system governed by a stochastic differential equation. The control objective is to stabilize the closed-loop system and minimize the infinite-horizon asymptotic growth rate of a quadratic-exponential functional (QEF) which penalizes the plant variables and the controller output. We combine a frequency-domain representation of the QEF growth rate, obtained recently, with variational techniques and establish first-order necessary conditions of optimality for the state-space matrices of the controller.
Given a stochastic dynamical system modelled via stochastic differential equations (SDEs), we evaluate the safety of the system through characterisations of its exit time moments. We lift the (possibly nonlinear) dynamics into the space of the occupation and exit measures to obtain a set of linear evolution equations which depend on the infinitesimal generator of the SDE. Coupled with appropriate semidefinite positive matrix constraints, this yields a moment-based approach for the computation of exit time moments of SDEs with polynomial drift and diffusion dynamics. To extend the capability of the moment approach, we propose a state augmentation method which allows us to generate the evolution equations for a broader class of nonlinear stochastic systems and apply the moment method to previously unsupported dynamics. In particular, we show a general augmentation strategy for sinusoidal dynamics which can be found in most physical systems. We employ the methodology on an Ornstein-Uhlenbeck process and stochastic spring-mass-damper model to characterise their safety via their expected exit times and show the additional exit distribution insights that are afforded through higher order moments.
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