The purpose of this paper is to set out the problems of modeling quantum communication and signal processing where the communication between systems is via a non-Markovian channel. This is a general feature of quantum transmission lines. Our ultimate objective is to extend the network rules that have been developed for Markovian models. To this end we recall the Hamiltonian description of such non-Markov models of transmission lines and their quantization. These have occurred in the context of non-equilibrium thermodynamics, but our interest is in the transmission lines as carriers of information rather than heat baths. We show that there is an analytic scattering matrix associated with these models and that stability may be formulated in terms of the lossless bounded real property. Noting that the input and output fields do not separately satisfy a non-self-demolition principle, we discuss the rigorous limit in which such models appear Markov and so amenable to standard approaches of quantum filtering and control.
The purpose of this paper is to set out the problems of modeling quantum communication and signal processing where the communication between systems is via a non-Markovian channel. This is a general feature of quantum transmission lines. Our ultimate objective is to extend the network rules that have been developed for Markovian models. To this end we recall the Hamiltonian description of such non-Markov models of transmission lines and their quantization. These have occurred in the context of non-equilibrium thermodynamics, but our interest is in the transmission lines as carriers of information rather than heat baths. We show that there is an analytic scattering matrix associated with these models and that stability may be formulated in terms of the lossless bounded real property. Noting that the input and output fields do not separately satisfy a non-self-demolition principle, we discuss the rigorous limit in which such models appear Markov and so amenable to standard approaches of quantum filtering and control.
The concept of a controlled flow of a dynamical system, especially when the controlling process feeds information back about the system, is of central importance in control engineering. In this paper, we build on the ideas presented by Bouten and van Handel [Quantum Stochastics and Information: Statistics, Filtering and Control (World Scientific, 2008)] and develop a general theory of quantum feedback. We elucidate the relationship between the controlling processes, Z, and the measured processes, Y, and to this end we make a distinction between what we call the input picture and the output picture. We should note that the input-output relations for the noise fields have additional terms not present in the standard theory but that the relationship between the control processes and measured processes themselves is internally consistent—we do this for the two main cases of quadrature measurement and photon-counting measurement. The theory is general enough to include a modulating filter which post-processes the measurement readout Y before returning to the system. This opens up the prospect of applying very general engineering feedback control techniques to open quantum systems in a systematic manner, and we consider a number of specific modulating filter problems. Finally, we give a brief argument as to why most of the rules for making instantaneous feedback connections [J. Gough and M. R. James, Commun. Math. Phys. 287, 1109 (2009)] ought to apply for controlled dynamical networks as well.
Enabled by rapidly developing quantum technologies, it is possible to network quantum systems at a much larger scale in the near future. To deal with non-Markovian dynamics that is prevalent in solid-state devices, we propose a general transfer function based framework for modeling linear quantum networks, in which signal flow graphs are applied to characterize the network topology by flow of quantum signals. We define a noncommutative ring D and use its elements to construct Hamiltonians, transformations and transfer functions for both active and passive systems. The signal flow graph obtained for direct and indirect coherent quantum feedback systems clearly show the feedback loop via bidirectional signal flows. Importantly, the transfer function from input to output field is derived for non-Markovian quantum systems with colored inputs, from which the Markovian input-output relation can be easily obtained as a limiting case. Moreover, the transfer function possesses a symmetry structure that is analogous to the well-known scattering transformation in Schrödinger picture. Finally, we show that these transfer functions can be integrated to build complex feedback networks via interconnections, serial products and feedback, which may include either direct or indirect coherent feedback loops, and transfer functions between quantum signal nodes can be calculated by the Riegle’s matrix gain rule. The theory paves the way for modeling, analyzing and synthesizing non-Markovian linear quantum feedback networks in the frequency-domain.
Enabled by rapidly developing quantum technologies, it is possible to network quantum systems at a much larger scale in the near future. To deal with non-Markovian dynamics that is prevalent in solid-state devices, we propose a general transfer function based framework for modeling linear quantum networks, in which signal flow graphs are applied to characterize the network topology by flow of quantum signals. We define a noncommutative ring D and use its elements to construct Hamiltonians, transformations and transfer functions for both active and passive systems. The signal flow graph obtained for direct and indirect coherent quantum feedback systems clearly show the feedback loop via bidirectional signal flows. Importantly, the transfer function from input to output field is derived for non-Markovian quantum systems with colored inputs, from which the Markovian input-output relation can be easily obtained as a limiting case. Moreover, the transfer function possesses a symmetry structure that is analogous to the well-know scattering transformation in \sd picture. Finally, we show that these transfer functions can be integrated to build complex feedback networks via interconnections, serial products and feedback, which may include either direct or indirect coherent feedback loops, and transfer functions between quantum signal nodes can be calculated by the Riegle's matrix gain rule. The theory paves the way for modeling, analyzing and synthesizing non-Markovian linear quantum feedback networks in the frequency-domain.
Quantum Markov models are employed ubiqui-tously in quantum physics and in quantum information theory due to their relative simplicity and analytical tractability. In particular, these models are known to give accurate approxi-mations for a wide range of quantum optical and meso scopic systems. However, in general, the validity of the Markov approximation entails assumptions regarding properties of the system of interest and its environment, which may not be satisfied or accurate in arbitrary physical systems. Therefore, developing useful modelling tools for general non-Markovian quantum systems for which the Markov approximation is inappropriate or deficient is an undertaking of significant importance. This work considers non-Markovian principal quantum systems that can be embedded in a larger Markovian quantum system with one or more compound baths consisting of an auxiliary quantum system and a quantum white noise field, and derives a set of coupled stochastic and quantum master equations for embedded non-Markovian quantum systems. The case of a purely Hamiltonian coupling between the principal and auxiliary systems as a closed system without coupling to white noises is included as a special case. The results are expected to be of interest for (open-loop and feedback) control of continuous-time non-Markovian systems and studying reduced models for numerical simulation of such systems. They may also shed more light on the general structure of continuous-time non-Markovian quantum systems.
Quantum input-output response analysis is a useful method for modeling the dynamics of complex quantum networks, such as those for communication or quantum control via cascade connections. Non-Markovian effects are expected to be important in networks realized using mesoscopic circuits, but such effects have not yet been studied. Here we extend the Markovian input-output network formalism to non-Markovian networks. The general formalism can be applied to various examples: (i) we show how non-Markovian coherent feedback can reduce the speed of decoherence for an atom in an optical cavity; (ii) we examine the effect of finite-cavity bandwidths in a superconducting circuit.
We propose a general theoretical framework that is suitable to study a wide class of stabilization problems for quantum Markovian dynamical systems. Building on systemtheoretic ideas, we propose definitions of invariant and attractive quantum subsystem, characterize Markovian invariance properties, and provide sufficient conditions for attraction. The general framework and results are illustrated by addressing the potential of output-feedback Markovian control strategies for quantum pure state-stabilization. In particular, constructive results for the synthesis of stabilizing semigroups in arbitrary finite-dimensional Markovian systems are established. I. BACKGROUND AND MOTIVATIONS Stabilization problems are of central relevance for many quantum control applications, ranging from state preparation of quantum-optical and nano-mechanical systems to generation of noise-protected realizations of quantum information in realistic devices [1]. Dynamical systems undergoing Markovian evolution [2], [3] are both widely relevant from a physical standpoint and present distinctive control challenges – preventing, in particular, open-loop quantum-engineering and stabilization methods based on dynamical decoupling to be viable [4], [5]. However, we show here how a wide class of stabilization problems can be effectively treated in a general framework, provided by attractive quantum subsystems. After introducing the main ideas and definitions along with some general results, we shall explore their application to pure-state preparation problems for Markovian output-feedback control. We refer to the forthcoming journal version of the present paper [6] for detailed proofs we shall omit or merely sketch in the following sections. Consider a separable Hilbert space H over the complex field C. Let B(H) represent the set of linear bounded operators on H, H(H) denoting the real subspace of Hermitian operators, with I, O being the identity and the zero operator, respectively. In the standard statistical formulation of quantum mechanics [7], [8], the dimension of the Hilbert space H associated with the quantum system of interest, Q, is determined by the physics of the problem. In what follows, we consider finite-dimensional systems, i.e. dim(H) 0 being described by a TracePreserving, Completely-Positive (TPCP) map Tt(·) [12], [1]. A differential equation for the density operator of I may be derived provided that a forward composition law holds: Definition 1 (QDS): A quantum dynamical semigroup is a one-parameter family of TPCP maps {Tt(·), t ≥ 0} that satisfies: (i) T0 = I, (ii) Tt ◦ Ts = Tt+s, ∀t, s > 0, (iii) trace(Tt(ρ)X) is a continuous function of t, ∀ρ ∈ D(HI), ∀X ∈ B(HI). Due to the trace and positivity preserving assumptions, a QDS is a semigroup of contractions. It has been proved [10], [13] that the Hille-Yoshida generator for a QDS exists and can be cast in the following canonical form: