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This paper introduces a Lyapunov-based control strategy alongside two filtering methods for controlling and estimating the evolution of coherence vector elements from sequential homodyne measurements. The methods include traditional quantum filtering and a novel extended Kalman filter, which explicitly addresses the dynamics of a stochastic master equation with correlated noise, thereby ensuring the quantum properties of the estimated state variable by design. We also explore scenarios where the system’s Hamiltonian is unknown, demonstrating that both filters exhibit reduced performance with increased estimation errors. To address this, we propose a multiple model estimation scheme applicable to either filter. The estimated density operator is then controlled using the proposed switching-based Lyapunov scheme, which guarantees noise-to-state practical stability in probability. We demonstrate the effectiveness of our approach in stabilizing a qubit coupled to a leaky cavity under homodyne detection, even with uncertainty in the resonance frequency.

From the key composite quantum system made of a two-level system (qubit) and a harmonic oscillator (photon) with resonant or dispersive interactions, one derives the corresponding quantum Stochastic Master Equations (SME) when either the qubits or the photons are measured. Starting with an elementary discrete-time formulation based on explicit formulae for the interaction propagators, one shows how to include measurement imperfections and decoherence. This qubit/photon quantum system illustrates the Kraus-map structure of general discrete-time SME governing the dynamics of an open quantum system subject to measurement back-action and decoherence induced by the environment. Then, on the qubit/photon system, one explains the passage to a continuous-time mathematical model where the measurement signal is either a continuous real-value signal (typically homodyne or heterodyne signal) or a discontinuous and integer-value signal obtained from a counter. During this derivation, the Kraus map formulation is preserved in an infinitesimal way. Such a derivation provides also an equivalent Kraus-map formulation to the continuous-time SME usually expressed as stochastic differential equations driven either by Wiener or Poisson processes. From such Kraus-map formulation, simple linear numerical integration schemes are derived that preserve the positivity and the trace of the density operator, i.e. of the quantum state. [ABSTRACT FROM AUTHOR]

This paper gives an overview of parameter estimation and system identification for quantum input–output systems by continuous observation of the output field. We present recent results on the quantum Fisher information of the output with respect to unknown dynamical parameters. We discuss the structure of continuous-time measurements as solutions of the quantum Zakai equation, and their relationship to parameter estimation methods. Proceeding beyond parameter estimation, the paper also gives an overview of the emerging topic of quantum system identification for black-box modelling of quantum systems by continuous observation of a travelling wave probe, for the case of ergodic quantum input–output systems and linear quantum systems. Empirical methods for such black-box modelling are also discussed. [ABSTRACT FROM AUTHOR]

We give a tutorial exposition of the analogue of the filtering equation for quantum systems focusing on the quantum probabilistic framework and developing the ideas from the classical theory. Quantum covariances and conditional expectations on von Neumann algebras play an essential part in the presentation. [ABSTRACT FROM AUTHOR]

Quantum games represent the really twenty-first century branch of game theory, tightly linked to the modern development of quantum computing and quantum technologies. The main accent in these developments so far was made on stationary or repeated games. In this paper, we aim at initiating the truly dynamic theory with strategies chosen by players in real time. Since direct continuous observations are known to destroy quantum evolutions (so-called quantum Zeno paradox), the necessary new ingredient for quantum dynamic games must be the theory of non-direct observations and the corresponding quantum filtering. Apart from the technical problems in organizing feedback quantum control in real time, the difficulty in applying this theory for obtaining mathematically amenable control systems is due partially to the fact that it leads usually to rather non-trivial jump-type Markov processes and/or degenerate diffusions on manifolds, for which the corresponding control is very difficult to handle. The starting point for the present research is the remarkable discovery (quite unexpected, at least to the author) that there exists a very natural class of homodyne detections such that the diffusion processes on projective spaces resulting by filtering under such arrangements coincide exactly with the standard Brownian motions (BM) on these spaces. In some cases, one can even reduce the process to the plain BM on Euclidean spaces or tori. The theory of such motions is well studied making it possible to develop a tractable theory of related control and games, which can be at the same time practically implemented on quantum optical devices. [ABSTRACT FROM AUTHOR]

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 multipoint 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. [ABSTRACT FROM AUTHOR]

A cavity with nonlinear Kerr-medium is a fundamental component of quantum optics. This paper concerns the effect of nonlinear medium on the behavior of a Kerr-cavity. In order to reveal this effect, the cavity is driven by fields in either vacuum or single-photon states, and the scattered field is monitored by a homodyne or photon-counting detector. Then, the quantum filtering equation for the number operator is derived in both the Heisenberg and Schrödinger pictures considering measurements. The number of photons in the Kerr-cavity is estimated through the derived stochastic master equations in different situations of initial condition and input fields by considering different dimensions for the related Hilbert space. Afterward, the nonlinearity impact on both the number of photons in the cavity and the annihilation operator dynamic is investigated. In addition, the effect of truncating the corresponding Hilbert space dimension is investigated. In each case, the results are compared with a linear cavity. [ABSTRACT FROM AUTHOR]

There is an extensive literature on the dynamic law of large numbers for systems of quantum particles, that is, on the derivation of an equation describing the limiting individual behavior of particles in a large ensemble of identical interacting particles. The resulting equations are generally referred to as nonlinear Schrödinger equations or Hartree equations, or Gross–Pitaevskii equations. In this paper, we extend some of these convergence results to a stochastic framework. Specifically, we work with the Belavkin stochastic filtering of many-particle quantum systems. The resulting limiting equation is an equation of a new type, which can be regarded as a complex-valued infinite-dimensional nonlinear diffusion of McKean–Vlasov type. This result is the key ingredient for the theory of quantum mean-field games developed by the author in a previous paper. [ABSTRACT FROM AUTHOR]

We investigate the discrimination of pure-mixed (quantum filtering) and mixed-mixed states and compare their optimal success probability with the one for discriminating other pairs of pure states superposed by the vectors included in the mixed states. We prove that under the equal-fidelity condition, the pure-pure state discrimination scheme is superior to the pure-mixed (mixed-mixed) one. With respect to quantum filtering, the coherence exists only in one pure state and is detrimental to the state discrimination for lower dimensional systems; while it is the opposite for the mixed-mixed case with symmetrically distributed coherence. Making an extension to infinite-dimensional systems, we find that the coherence which is detrimental to state discrimination may become helpful and vice versa.

We present filtering equations for a class of hybrid quantum-classical systems where a quantum optical cavity system is continuously disturbed by a classical linear stochastic process. Another quantum cavity system is employed to describe the classical stochastic process as an analog. The initial hybrid quantum-classical system consisting of both a classical signal and a quantum cavity system is then modeled by a combined quantum system with two quantum cavity systems. The stochastic master equations, or filtering equations, are then derived using quantum filtering theory. A quantum extended Kalman filter method is adopted as a convenient solution to this filtering problem. The effectiveness of both methods is illustrated by numerical examples. [ABSTRACT FROM AUTHOR]

Quantum games and mean-field games (MFG) represent two important new branches of game theory. In a recent paper the author developed quantum MFGs merging these two branches. These quantum MFGs were based on the theory of continuous quantum observations and filtering of diffusive type. In the present paper we develop the analogous quantum MFG theory based on continuous quantum observations and filtering of counting type. However, proving existence and uniqueness of the solutions for resulting limiting forward-backward system based on jump-type processes on manifolds seems to be more complicated than for diffusions. In this paper we only prove that if a solution exists, then it gives an ϵ-Nash equilibrium for the corresponding N-player quantum game. The existence of solutions is suggested as an interesting open problem.

Abstract An approximate exponential quantum projection filtering scheme is developed for a class of open quantum systems described by Hudson–Parthasarathy quantum stochastic differential equations, aiming to reduce the computational burden associated with online calculation of the quantum filter. By using a differential geometric approach, the quantum trajectory is constrained in a finite-dimensional differentiable manifold consisting of an unnormalized exponential family of quantum density operators, and an exponential quantum projection filter is then formulated as a number of stochastic differential equations satisfied by the finite-dimensional coordinate system of this manifold. A convenient design of the differentiable manifold is also presented through reduction of the local approximation errors, which yields a simplification of the quantum projection filter equations. It is shown that the computational cost can be significantly reduced by using the quantum projection filter instead of the quantum filter. It is also shown that when the quantum projection filtering approach is applied to a class of open quantum systems that asymptotically converge to a pure state, the input-to-state stability of the corresponding exponential quantum projection filter can be established. Simulation results from an atomic ensemble system example are provided to illustrate the performance of the projection filtering scheme. It is expected that the proposed approach can be used in developing more efficient quantum control methods. [ABSTRACT FROM AUTHOR]

The possibility of engineering the coherent propagation of electrons through a confined medium with potential energy barriers of various spatial shapes is investigated. Using basic quantum mechanical principles, a semi-analytical method is developed to compute the current density of the particle's localization probability in a nanoscale strip, across regions with different values of potential energy and with various geometries. The results show uneven spatial distribution of the probability current density, with possible extreme contrast between electron flows through neighbouring areas. Thus, interactions of the confined electronic wave with customized shapes of potential energy barriers are shown to lead to either focusing or filtering effects of the electron propagation. It is further demonstrated that the method can be adapted for studying the probability flow scattering by insular potential energy obstacles. Though the present method is illustrated on a narrow two-dimensional strip, it can be straightforwardly generalized for three-dimensional cases. [ABSTRACT FROM AUTHOR]

Summary: The single‐photon quantum filtering problems have been investigated recently with applications in quantum computing. In practice, the detector responds with a quantum efficiency of less than unity since there exists some mode mismatch between the detector and the system and the single‐photon signal may be corrupted by quantum white noise. Consequently, quantum filters based on multiple measurements are designed in this paper to improve estimation performance. More specifically, the filtering equations for a 2‐level quantum system driven by a single‐photon input state and under multiple measurements are presented in this paper. Four scenarios, ie, (1) 2 diffusive measurements with Q‐P quadrature form, (2) 2 diffusive measurements with Q‐Q quadrature form, (3) diffusive plus Poissonian measurements, and (4) 2 Poissonian measurements, are considered. It is natural to compare the filtering results, ie, measuring a single channel or both channels, which one is better? By the simulation where we use a single photon to excite an atom, it seems that multiple measurements enable us to excite the atom with higher probability than only measuring a single channel. In addition, a measurement back‐action phenomenon is revealed by the simulation results. [ABSTRACT FROM AUTHOR]

Interaction of a two level atom with a single photon packet is investigated by considering full interaction model of Rabi model. A homodyne measurement providence information about the photon-atom system. Quantum filtering equations for this system are found and the results are discussed. Effects of adding an external magnetic field that is explained. [ABSTRACT FROM AUTHOR]

The Proton magnetic resonance (1H-MRS) spectrum contains information about the concentration of tissue metabolites within a predefined region of interest (a voxel). The conventional spectrum in some cases obscures information about less abundant metabolites due to limited separation and complex splitting of the metabolite peaks. One method to detect these metabolites is to reduce the complexity of the spectrum using editing. This review provides an overview of the one-dimensional editing methods available to interrogate these obscured metabolite peaks. These methods include sequence optimizations, echo-time averaging, J-difference editing methods (single BASING, dual BASING, and MEGA-PRESS), constant-time PRESS, and multiple quantum filtering. It then provides an overview of the brain metabolites whose detection can benefit from one or more of these editing approaches, including ascorbic acid, γ-aminobutyric acid, lactate, aspartate, N-acetyl aspartyl glutamate, 2-hydroxyglutarate, glutathione, glutamate, glycine, and serine. Magn Reson Med 77:1377-1389, 2017. © 2017 International Society for Magnetic Resonance in Medicine. [ABSTRACT FROM AUTHOR]

We investigate the discrimination of pure-mixed (quantum filtering) and mixed-mixed states and compare their optimal success probability with the one for discriminating other pairs of pure states superposed by the vectors included in the mixed states. We prove that under the equal-fidelity condition, the pure-pure state discrimination scheme is superior to the pure-mixed (mixed-mixed) one. With respect to quantum filtering, the coherence exists only in one pure state and is detrimental to the state discrimination for lower dimensional systems; while it is the opposite for the mixed-mixed case with symmetrically distributed coherence. Making an extension to infinite-dimensional systems, we find that the coherence which is detrimental to state discrimination may become helpful and vice versa. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to determine quantum master and filter equations for systems coupled to fields in certain non-classical continuous-mode states. Specifically, we consider two types of field states (i) single photon states, and (ii) superpositions of coherent states. The system and field are described using a quantum stochastic unitary model. Master equations are derived from this model and are given in terms of systems of coupled equations. The output field carries information about the system, and is continuously monitored. The quantum filters are determined with the aid of an embedding of the system into a larger non-Markovian system, and are given by a system of coupled stochastic differential equations. [ABSTRACT FROM AUTHOR]

Quantum mechanical systems exhibit an inherently probabilistic behavior upon measurement which excludes in principle the singular case of direct observability. The theory of quantum stochastic time continuous measurements and quantum filtering was earlier developed by the author on the basis of non-Markov conditionally-independent increment models for quantum noise and quantum nondemolition observability. Here this theory is generalized to the case of demolition indirect measurements of quantum unstable systems satisfying the microcausality principle. The exposition of the theory is given in the most general algebraic setting unifying quantum and classical theories as particular cases. The reduced quantum feedback-controlled dynamics is described equivalently by linear quasi-Markov and nonlinear conditionally-Markov stochastic master equations. Using this scheme for diffusive and counting measurements to describe the stochastic evolution of the open quantum system under the continuous indirect observation and working in parallel with classical indeterministic control theory, we derive the Bellman equations for optimal feedback control of the a posteriori stochastic quantum states conditioned upon these measurements. The resulting Bellman equation for the diffusive observation is then applied to the explicitly solvable quantum linear-quadratic-Gaussian problem which emphasizes many similarities with the corresponding classical control problem. [ABSTRACT FROM AUTHOR]

We review the recent developments in quantum control and its contribution to quantum information processing. [ABSTRACT FROM AUTHOR]

When are quantum filters asymptotically independent of the initial state? We show that this is the case for absolutely continuous initial states when the quantum stochastic model satisfies an observability condition. When the initial system is finite dimensional, this condition can be verified explicitly in terms of a rank condition on the coefficients of the associated quantum stochastic differential equation. [ABSTRACT FROM AUTHOR]

This paper provides an introduction to quantum filtering theory. An introduction to quantum probability theory is given, focusing on the spectral theorem and the conditional expectation as a least squares estimate, and culminating in the construction of Wiener and Poisson processes on the Fock space. We describe the quantum ItÔ calculus and its use in the modeling of physical systems. We use both reference probability and innovations methods to obtain quantum filtering equations for system-probe models from quantum optics. [ABSTRACT FROM AUTHOR]

We investigate the discrimination of pure-mixed (quantum filtering) and mixed-mixed states and compare their optimal success probability with the one for discriminating other pairs of pure states superposed by the vectors included in the mixed states. We prove that under the equal-fidelity condition, the pure-pure state discrimination scheme is superior to the pure-mixed (mixed-mixed) one. With respect to quantum filtering, the coherence exists only in one pure state and is detrimental to the state discrimination for lower dimensional systems; while it is the opposite for the mixed-mixed case with symmetrically distributed coherence. Making an extension to infinite-dimensional systems, we find that the coherence which is detrimental to state discrimination may become helpful and vice versa.