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Reservoir computing is a form of machine learning that utilizes nonlinear dynamical systems to perform complex tasks in a cost-effective manner when compared to typical neural networks. Many recent advancements in reservoir computing, in particular quantum reservoir computing, make use of reservoirs that are inherently stochastic. However, the theoretical justification for using these systems has not yet been well established. In this paper, we investigate the universality of stochastic reservoir computers, in which we use a stochastic system for reservoir computing using the probabilities of each reservoir state as the readout instead of the states themselves. In stochastic reservoir computing, the number of distinct states of the entire reservoir computer can potentially scale exponentially with the size of the reservoir hardware, offering the advantage of compact device size. We prove that classes of stochastic echo state networks, and therefore the class of all stochastic reservoir computers, are universal approximating classes. We also investigate the performance of two practical examples of stochastic reservoir computers in classification and chaotic time series prediction. While shot noise is a limiting factor in the performance of stochastic reservoir computing, we show significantly improved performance compared to a deterministic reservoir computer with similar hardware in cases where the effects of noise are small., Comment: 30 pages, 6 figures

Quantum Reservoir Computing leverages quantum systems to solve complex computational tasks with unprecedented efficiency and reduced energy consumption. This paper presents a novel QRC framework utilizing a quantum optical reservoir composed of two-level atoms within a single-mode optical cavity. Employing the Jaynes-Cummings and Tavis-Cummings models, we introduce a scalable and practically measurable reservoir that outperforms traditional classical reservoir computing in both memory retention and nonlinear data processing. We evaluate the reservoir's performance through two primary tasks: the prediction of time-series data via the Mackey-Glass task and the classification of sine-square waveforms. Our results demonstrate significant enhancements in performance with increased numbers of atoms, supported by non-destructive, continuous quantum measurements and polynomial regression techniques. This study confirms the potential of QRC to offer a scalable and efficient solution for advanced computational challenges, marking a significant step forward in the integration of quantum physics with machine learning technology., Comment: 9 pages, 8 figures

Quantum parameter estimation theory is an important component of quantum information theory and provides the statistical foundation that underpins important topics such as quantum system identification and quantum waveform estimation. When there is more than one parameter the ultimate precision in the mean square error given by the quantum Cram\'er-Rao bound is not necessarily achievable. For non-full rank quantum states, it was not known when this bound can be saturated (achieved) when only a single copy of the quantum state encoding the unknown parameters is available. This single-copy scenario is important because of its experimental/practical tractability. Recently, necessary and sufficient conditions for saturability of the quantum Cram\'er-Rao bound in the multiparameter single-copy scenario have been established in terms of i) the commutativity of a set of projected symmetric logarithmic derivatives and ii) the existence of a unitary solution to a system of coupled nonlinear partial differential equations. New sufficient conditions were also obtained that only depend on properties of the symmetric logarithmic derivatives. In this paper, key structural properties of optimal measurements that saturate the quantum Cram\'er-Rao bound are illuminated. These properties are exploited to i) show that the sufficient conditions are in fact necessary and sufficient for an optimal measurement to be projective, ii) give an alternative proof of previously established necessary conditions, and iii) describe general POVMs, not necessarily projective, that saturate the multiparameter QCRB. Examples are given where a unitary solution to the system of nonlinear partial differential equations can be explicitly calculated when the required conditions are fulfilled., Comment: 13 pages, no figures. Follow up to [H. I. Nurdin, IEEE Control Systems Lett., vol. 8, pp. 376-381, 2024] (arXiv:2402.11567). Comments are welcome

The quantum Cram\'{e}r-Rao bound (QCRB) as the ultimate lower bound for precision in quantum parameter estimation is only known to be saturable in the multiparameter setting in special cases and under conditions such as full or average commutavity of the symmetric logarithmic derivatives (SLDs) associated with the parameters. Moreover, for general mixed states, collective measurements over infinitely many identical copies of the quantum state are generally required to attain the QCRB. In the important and experimentally relevant single-copy scenario, a necessary condition for saturating the QCRB in the multiparameter setting for general mixed states is the so-called partial commutativity condition on the SLDs. However, it is not known if this condition is also sufficient. This paper establishes necessary and sufficient conditions for saturability of the multiparameter QCRB in the single-copy setting in terms of the commutativity of a set of projected SLDs and the existence of a unitary solution to a system of nonlinear partial differential equations. New necessary conditions that imply partial commutativity are also obtained, which together with another condition become sufficient. Moreover, when the sufficient conditions are satisfied an optimal measurement saturating the QCRB can be chosen to be projective and explicitly characterized. An example is developed to illustrate the case of a multiparameter quantum state where the conditions derived herein are satisfied and can be explicitly verified., Comment: 10 + 3 pages, no figures. Version 5 appends a corrigendum (submitted for publication) that fixes a gap in the sufficiency of Condition 2 of Theorem 2 and restores the necessary and sufficient conditions for saturating the multiparameter QCRB. The corrigendum supplies additional examples. Comments are welcome

Reservoir computing, using nonlinear dynamical systems, offers a cost-effective alternative to neural networks for complex tasks involving processing of sequential data, time series modeling, and system identification. Echo state networks (ESNs), a type of reservoir computer, mirror neural networks but simplify training. They apply fixed, random linear transformations to the internal state, followed by nonlinear changes. This process, guided by input signals and linear regression, adapts the system to match target characteristics, reducing computational demands. A potential drawback of ESNs is that the fixed reservoir may not offer the complexity needed for specific problems. While directly altering (training) the internal ESN would reintroduce the computational burden, an indirect modification can be achieved by redirecting some output as input. This feedback can influence the internal reservoir state, yielding ESNs with enhanced complexity suitable for broader challenges. In this paper, we demonstrate that by feeding some component of the reservoir state back into the network through the input, we can drastically improve upon the performance of a given ESN. We rigorously prove that, for any given ESN, feedback will almost always improve the accuracy of the output. For a set of three tasks, each representing different problem classes, we find that with feedback the average error measures are reduced by $30\%-60\%$. Remarkably, feedback provides at least an equivalent performance boost to doubling the initial number of computational nodes, a computationally expensive and technologically challenging alternative. These results demonstrate the broad applicability and substantial usefulness of this feedback scheme., Comment: 32 pages, 6 figures

Quantum Markov models are employed ubiquitously 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 approximations for a wide range of quantum optical and mesoscopic 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., Comment: 6 pages, 3 figures. Minor typo corrected on p. 2 column 1 para 3: need should be need not. Published in Proceedings of the 62nd IEEE Conference on Decision and Control (Singapore, Dec. 12-15) pp. 5939-5944 (2023)

Reservoir computing is a machine learning framework that uses artificial or physical dissipative dynamics to predict time-series data using nonlinearity and memory properties of dynamical systems. Quantum systems are considered as promising reservoirs, but the conventional quantum reservoir computing (QRC) models have problems in the execution time. In this paper, we develop a quantum reservoir (QR) system that exploits repeated measurement to generate a time-series, which can effectively reduce the execution time. We experimentally implement the proposed QRC on the IBM's quantum superconducting device and show that it achieves higher accuracy as well as shorter execution time than the conventional QRC method. Furthermore, we study the temporal information processing capacity to quantify the computational capability of the proposed QRC; in particular, we use this quantity to identify the measurement strength that best tradeoffs the amount of available information and the strength of dissipation. An experimental demonstration with soft robot is also provided, where the repeated measurement over 1000 timesteps was effectively applied. Finally, a preliminary result with 120 qubits device is discussed.

In this paper, we consider the problem of open-loop control of a qubit that is coupled to an unknown fully quantum non-Markovian noise (either bosonic or fermionic). A graybox model that is empirically obtained from measurement data is employed to approximately represent the unknown quantum noise. The estimated model is then used to calculate the open-loop control pulses under constraints on the pulse amplitude and timing. For the control pulse optimization, we explore the use of gradient descent and genetic optimization methods. We consider the effect of finite sampling on estimating expectation values of observables and show results for single- and multi-axis control of a qubit.

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 modeling of quantum systems by continuous observation of a traveling wave probe, for the case of ergodic quantum input-output systems and linear quantum systems. Empirical methods for such black-box modeling are also discussed., Comment: 25 pages, to appear in Annual Reviews in Control

A general theory of quantum stochastic processes was formulated by Accardi, Frigerio and Lewis in 1982 within the operator-algebraic framework of quantum probability theory, as a non-commutative extension of the Kolmogorovian classical stochastic processes. More recently, studies on non-Markovian quantum processes have led to the discrete-time process tensor formalism in the Schr\"{o}dinger picture to describe the outcomes of sequential interventions on open quantum systems. However, there has been no treatment of the relationship of the process tensor formalism to the quantum probabilistic theory of quantum stochastic processes. This paper gives an exposition of quantum stochastic processes and the process tensor and the relationship between them. In particular, it is shown how the latter emerges from the former via extended correlation kernels incorporating ancillas., Comment: 14 pages standard LaTeX article, no figures. To appear in Proceedings of the 60th IEEE Conference on Decision and Control (CDC), Dec. 13-15, 2021 (https://2021.ieeecdc.org/). Some corrections in Example 4 and section 6. Comments are welcome

Convergent, contractive or incremental stability properties of nonlinear systems have attracted interest for control tasks such as observer design, output regulation and synchronization. The convergence property plays a central role in the neuromorphic (brain-inspired) computing of reservoir computing, which seeks to harness the information processing capability of nonlinear systems. This paper presents a small-gain theorem for discrete-time output-feedback interconnected systems to be uniformly input-to-output convergent (UIOC) with outputs converging to a bounded reference output uniquely determined by the input. A small-gain theorem for interconnected time-varying discrete-time uniform input-to-output stable systems that could be of separate interest is also presented as an intermediate result. Applications of the UIOC small-gain theorem are illustrated in the design of observer-based controllers and interconnected nonlinear classical and quantum dynamical systems (as reservoir computers) for black-box system identification., Comment: 12 pages, 6 figures. Submitted for publication. Comments are welcome

In commercial inverters, an LCL filter is considered an integral part to filter out the switching harmonics and generate a sinusoidal output voltage. The existing literature on the averaged virtual oscillator controller (VOC) dynamics is for current feedback before the output LCL filter that contains the switching harmonics or for inductive filters ignoring the effect of filter capacitance. In this work, a new version of averaged VOC dynamics is presented for islanded inverters with current feedback after the LCL filter thus avoiding the switching harmonics going into the VOC. The embedded droop-characteristics within the averaged VOC dynamics are identified and a parameter design procedure is presented to regulate the output voltage magnitude and frequency according to the desired ac-performance specifications. Further, a power dispatch technique based on this newer version of averaged VOC dynamics is presented to simultaneously regulate both the active and reactive output power of two parallel-connected islanded inverters. The control laws are derived and a power security constraint is presented to determine the achievable power set-point. Simulation results for load transients and power dispatch validate the proposed version of averaged VOC dynamics., Comment: Accepted in IECON 2020 - 46th Annual Conference of the IEEE Industrial Electronics Society, Singapore

The combination of machine learning and quantum computing has emerged as a promising approach for addressing previously untenable problems. Reservoir computing is an efficient learning paradigm that utilizes nonlinear dynamical systems for temporal information processing, i.e., processing of input sequences to produce output sequences. Here we propose quantum reservoir computing that harnesses complex dissipative quantum dynamics. Our class of quantum reservoirs is universal, in that any nonlinear fading memory map can be approximated arbitrarily closely and uniformly over all inputs by a quantum reservoir from this class. We describe a subclass of the universal class that is readily implementable using quantum gates native to current noisy gate-model quantum computers. Proof-of-principle experiments on remotely accessed cloud-based superconducting quantum computers demonstrate that small and noisy quantum reservoirs can tackle high-order nonlinear temporal tasks. Our theoretical and experimental results pave the path for attractive temporal processing applications of near-term gate-model quantum computers of increasing fidelity but without quantum error correction, signifying the potential of these devices for wider applications including neural modeling, speech recognition and natural language processing, going beyond static classification and regression tasks., Comment: 9 pages main text, 14 pages appendices, 13 figures. Added implementation scheme using QND measurements and proposal of more efficient implementation schemes without and with QND measurements. To appear in Physical Review Applied

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., Comment: 14 pages, invited article for the second edition of Springer's Encyclopedia of Systems and Control (to appear). Comments welcome

In this paper, we develop a theory of learning nonlinear input-output maps with fading memory by dissipative quantum systems, as a quantum counterpart of the theory of approximating such maps using classical dynamical systems. The theory identifies the properties required for a class of dissipative quantum systems to be {\em universal}, in that any input-output map with fading memory can be approximated arbitrarily closely by an element of this class. We then introduce an example class of dissipative quantum systems that is provably universal. Numerical experiments illustrate that with a small number of qubits, this class can achieve comparable performance to classical learning schemes with a large number of tunable parameters. Further numerical analysis suggests that the exponentially increasing Hilbert space presents a potential resource for dissipative quantum systems to surpass classical learning schemes for input-output maps., Comment: 33 pages, 12 figures, 2 tables. Close to published version in Quantum Information Processing (https://rdcu.be/bCBUf)

We develop a notion of dephasing under the action of a quantum Markov semigroup in terms of convergence of operators to a block-diagonal form determined by irreducible invariant subspaces. If the latter are all one-dimensional, we say the dephasing is maximal. With this definition, we show that a key necessary requirement on the Lindblad generator is bistochasticity, and focus on characterizing whether a maximally dephasing evolution may be described in terms of a unitary dilation with only classical noise, as opposed to a genuine non-commutative Hudson-Parthasarathy dilation. To this end, we make use of a seminal result of K\"{u}mmerer and Maassen on the class of commutative dilations of quantum Markov semigroups. In particular, we introduce an intrinsic quantity constructed from the generator, which vanishes if and only if the latter admits a self-adjoint representation and which quantifies the degree of obstruction to having a classical diffusive noise model., Comment: submitted 12 November to AHP

In this paper we investigate system identification for general quantum linear systems. We consider the situation where the input field is prepared as stationary (squeezed) quantum noise. In this regime the output field is characterised by the power spectrum, which encodes covariance of the output state. We address the following two questions: (1) Which parameters can be identified from the power spectrum? (2) How to construct a system realisation from the power spectrum? The power spectrum depends on the system parameters via the transfer function. We show that the transfer function can be uniquely recovered from the power spectrum, so that equivalent systems are related by a symplectic transformation., Comment: 10 pages, 2 figures, submitted to Automatica. arXiv admin note: text overlap with arXiv:1608.01227

Kerr nonlinear cavities displaying optical thresholding have been proposed for the realization of ultra-low power photonic logic gates. In the ultra-low photon number regime, corresponding to energy levels in the attojoule scale, quantum input-output models become important to study the effect of unavoidable quantum fluctuations on the performance of such logic gates. However, being a quantum anharmonic oscillator, a Kerr-cavity has an infinite dimensional Hilbert space spanned by the Fock states of the oscillator. This poses a challenge to simulate and analyze photonic logic gates and circuits composed of multiple Kerr nonlinearities. For simulation, the Hilbert of the oscillator is typically truncated to the span of only a finite number of Fock states. This paper develops a quasi-principal components approach to identify important subspaces of a Kerr-cavity Hilbert space and exploits it to construct an approximate reduced model of the Kerr-cavity on a smaller Hilbert space. Using this approach, we find a reduced dimension model with a Hilbert space dimension of 15 that can closely match the magnitudes of the mean transmitted and reflected output fields of a conventional truncated Fock state model of dimension 75, when driven by an input coherent field that switches between two levels. For the same input, the reduced model also closely matches the magnitudes of the mean output fields of Kerr-cavity-based AND and NOT gates and a NAND latch obtained from simulation of the full 75 dimension model., Comment: 33 pages (with 17 pages of appendices), 5 figures. Closer to published version. Published in Journal of Physics D

We derive the explicit analytical form of the time-dependent coupling parameter to an external field for perfect absorption of traveling single photon fields with arbitrary temporal profiles by a tunable single input-output open quantum system, which can be realized as either a single qubit or single resonator system. However, the time-dependent coupling parameter for perfect absorption has a singularity at $t=0$ and constraints on real systems prohibit a faithful physical realization of the perfect absorber. A numerical example is included to illustrate the absorber's performance under practical limitations on the coupling strength., Comment: 15 pages, 1 figure. A preliminary and condensed version of this paper will appear in Proceedings of the 55th IEEE Conference on Decision and Control (Las Vegas, NV, Dec. 12-14, 2016), http://cdc2016.ieeecss.org/

The realization of transfer functions of Linear Quantum Stochastic Systems (LQSSs) is an issue of fundamental importance for the practical applications of such systems, especially as coherent controllers for other quantum systems. In this paper, we review two realization methods proposed by the authors in [1], [2], [3], [4]. The first one uses a cascade of a static linear quantum-optical network and single-mode optical cavities, while the second uses a feedback network of such cavities, along with static linear quantum-optical networks that pre- and post-process the cavity network inputs and outputs., Comment: Extended version of an article appearing in the Proceedings of the 55nd IEEE Conference on Decision and Control, Las Vegas, December 2016

Effective state transfer is one of the most important problems in quantum information processing. Typically, a quantum information device is composed of many subsystems with multi-input ports. In this paper, we develop a general theory describing the condition for perfect state transfer from the multi-input ports to the internal system components, for general passive linear quantum systems. The key notion used is the zero of the transfer function matrix. Application to entanglement generation and distribution in a quantum network is also discussed., Comment: 6 pages, 3 figures. A preliminary condensed version of this work will appear in Proceedings of the 55th IEEE Conference on Decision and Control

This paper presents formulae for Einstein-Podolsky-Rosen (EPR) entanglement generated from $N$ nondegenerate optical parametric amplifiers (NOPAs) interconnected in a linear coherent feedback (CFB) chain in the idealized lossless scenario and infinite bandwidth limit. The lossless scenario sets the ultimate EPR entanglement (two-mode squeezing) that can be achieved by this linear chain of NOPAs while the infinite bandwidth limit simplifies the analysis but gives an accurate approximation to the EPR entanglement at low frequencies of interest. Two adjustable phase shifts are placed at the outputs of the system to achieve the best EPR entanglement by selecting appropriate quadratures of the output fields., Comment: 23 pages, 4 figures. Condensed version of this paper to appear in Proceedings of the 2016 American Control Conference (ACC), Boston, MA, USA, July 6-8, 2016. Corrections to references. arXiv admin note: text overlap with arXiv:1508.04584

The purpose of this paper is to prove a local optimality property of a recently proposed coherent feedback configuration for distributed generation of EPR entanglement using two nondegenerate optical parametric amplifiers (NOPAs) in the idealized infinite bandwidth limit. This local optimality is with respect to a class of similar coherent feedback configurations but employing different unitary scattering matrices, representing different scattering of propagating signals within the network. The infinite bandwidth limit is considered as it significantly simplifies the analysis, allowing local optimality criteria to be explicitly verified. Nonetheless, this limit is relevant for the finite bandwidth scenario as it provides an accurate approximation to the EPR entanglement in the low frequency region where EPR entanglement exists., Comment: 17 pages, 5 figures. Condensed version of this paper to appear in Proceedings of the 54th IEEE Conference on Decision and Control (CDC), Osaka, Japan, Dec. 15-18, 2015

This paper is concerned with linear quantum networks of $N$ nondegenerate optical parametric amplifiers (NOPAs), with $N$ up to 6, which are interconnected in a coherent feedback chain. Each network connects two communicating parties (Alice and Bob) over two transmission channels. In previous work we have shown that a dual-NOPA coherent feedback network generates better Einstein-Podolsky-Rosen (EPR) entanglement (i.e., more two-mode squeezing) between its two outgoing Gaussian fields than a single NOPA, when the same total pump power is consumed and the systems undergo the same transmission losses over the same distance. This paper aims to analyze stability, EPR entanglement between two outgoing fields of interest, and bipartite entanglement of two-mode Gaussian states of cavity modes of the $N$-NOPA networks under the effect of transmission and amplification losses, as well as time delays. It is numerically shown that, in the absence of losses and delays, the network with more NOPAs in the chain requires less total pump power to generate the same degree of EPR entanglement. Moreover, we report on the internal entanglement synchronization that occurs in the steady state between certain pairs of Gaussian oscillator modes inside the NOPA cavities of the networks., Comment: 29 pages, 11 figures, to appear in Quantum Information and Computation (QIC)

Recent theoretical investigations on quantum coherent feedback networks have found that with the same pump power, the Einstein-Podolski-Rosen (EPR)-like entanglement generated via a dual nondegenerate optical parametric amplifier (NOPA) system placed in a certain coherent feedback loop is stronger than the EPR-like entangled pairs produced by a single NOPA. In this paper, we present a linear quantum system consisting of two NOPAs and a static linear passive network of optical devices. The network has six inputs and six outputs, among which four outputs and four inputs are connected in a coherent feedback loop with the two NOPAs. This passive network is represented by a $6 \times 6$ complex unitary matrix. A modified steepest descent method is used to find a passive complex unitary matrix at which the entanglement of this dual-NOPA network is locally maximized. Here we choose the matrix corresponding to a dual-NOPA coherent feedback network from our previous work as a starting point for the modified steepest descent algorithm. By decomposing the unitary matrix obtained by the algorithm as the product of so-called two-level unitary matrices, we find an optimized configuration in which the complex matrix is realized by a static optical network made of beam splitters., Comment: 17 pages, 4 figures. To appear in Proceedings of the 2015 American Control Conference (ACC), http://acc2015.a2c2.org. A more recent review paper on continuous-variable quantum information added

Recent work has shown that deploying two nondegenerate optical parametric amplifiers (NOPAs) separately at two distant parties in a coherent feedback loop generates stronger Einstein-Podolski-Rosen (EPR) entanglement between two propagating continuous-mode output fields than a single NOPA under same pump power, decay rate and transmission losses. The purpose of this paper is to investigate the stability and EPR entanglement of a dual-NOPA coherent feedback system under the effect of phase shifts in the transmission channel between two distant parties. It is shown that, in the presence of phase shifts, EPR entanglement worsens or can vanish, but can be improved to some extent in certain scenarios by adding a phase shifter at each output with a certain value of phase shift. In ideal cases, in the absence of transmission and amplification losses, existence of EPR entanglement and whether the original EPR entanglement can be recovered by the additional phase shifters are decided by values of the phase shifts in the path., Comment: 19 pages, 6 figures. Condensed version of this paper to appear in Proceedings of the 53rd IEEE Conference on Decision and Control (CDC), Los Angeles, CA, Dec. 15-17, 2014. Continues upon the earlier work in http://dx.doi.org/10.1007/s11128-014-0845-4

In this paper, we show that quantum memory for qudit states encoded in a single photon pulsed optical field has a conceptually simple modular realization using only passive linear optics and coherent feedback. We exploit the idea that two decaying optical cavities can be coupled in a coherent feedback configuration to create an internal mode of the coupled system which is isolated and decoherence-free for the purpose of qubit storage. The qubit memory can then be switched between writing/read-out mode and storage mode simply by varying the routing of certain freely propagating optical fields in the network. It is then shown that the qubit memories can be interconnected with one another to form a qudit quantum memory. We explain each of the phase of writing, storage, and read-out for this modular quantum memory scheme. The results point a way towards modular architectures for complex compound quantum memories., Comment: 24 pages, 8 figures. Submitted. Comments welcome

In this paper, we treat the quantum filtering problem for multiple input multiple output (MIMO) Markovian open quantum systems coupled to multiple boson fields in an arbitrary zero-mean jointly Gaussian state, using the reference probability approach formulated by Bouten and van Handel as a quantum version of a well-known method of the same name from classical nonlinear filtering theory, and exploiting the generalized Araki-Woods representation of Gough. This includes Gaussian field states such as vacuum, squeezed vacuum, thermal, and squeezed thermal states as special cases. The contribution is a derivation of the general quantum filtering equation (or stochastic master equation as they are known in the quantum optics community) in the full MIMO setup for any zero-mean jointy Gaussian input field states, up to some mild rank assumptions on certain matrices relating to the measurement vector., Comment: 19 pages, no figures. Published in a special issue of the Russian Journal of Mathematical Physics dedicated to the memory of Slava Belavkin

This paper concerns the recently proposed quasi-balanced truncation model reduction method for linear quantum stochastic systems. It has previously been shown that the quasi-balanceable class of systems (i.e. systems that can be truncated via the quasi-balanced method) includes the class of completely passive systems. In this work, we refine the previously established characterization of quasi-balanceable systems and show that the class of quasi-balanceable systems is strictly larger than the class of completely passive systems. In particular, we derive a novel characterization of completely passive linear quantum stochastic systems solely in terms of the controllability Gramian of such systems. Exploiting this result, we prove that all linear quantum stochastic systems with a pure Gaussian steady-state (active systems included) are all quasi-balanceable, and establish a new complete parameterization for this important class of systems. Examples are provided to illustrate our results., Comment: 12 pages. Submitted for publication

We show that in the regime of ground-state cooling, simple expressions can be derived for the performance of resolved-sideband cooling --- an example of coherent feedback control --- and optimal linear measurement-based feedback cooling for a harmonic oscillator. These results are valid to leading order in the small parameters that define this regime. They provide insight into the origins of the limitations of coherent and measurement-based feedback for linear systems, and the relationship between them. These limitations are not fundamental bounds imposed by quantum mechanics, but are due to the fact that both cooling methods are restricted to use only a linear interaction with the resonator. We compare the performance of the two methods on an equal footing --- that is, for the same interaction strength --- and confirm that coherent feedback is able to make much better use of the linear interaction than measurement-based feedback. We find that this performance gap is caused not by the back-action noise of the measurement but by the projection noise. We also obtain simple expressions for the maximal cooling that can be obtained by both methods in this regime, optimized over the interaction strength., Comment: 14 pages, 2 png figures; v2: revised for publication

We introduce a new class of continuous matrix product (CMP) states and establish the stochastic master equations (quantum filters) for an arbitrary quantum system probed by a bosonic input field in this class of states. We show that this class of CMP states arise naturally as outputs of a Markovian model, and that input fields in these states lead to master and filtering (quantum trajectory) equations which are matrix-valued. Furthermore, it is shown that this class of continuous matrix product states include the (continuous-mode) single photon and time-ordered multi-photon states., Comment: 17 pages, 2 figures

The purpose of this paper is to present a network realization theory for a class of mixed quantum-classical linear stochastic systems. Two forms, the standard form and the general form, of this class of linear mixed quantum-classical systems are proposed. Necessary and sufficient conditions for their physical realizability are derived. Based on these physical realizability conditions, a network synthesis theory for this class of linear mixed quantum-classical systems is developed, which clearly exhibits the quantum component, the classical component, and their interface. An example is used to illustrate the theory presented in this paper., Comment: Accepted version, to appear in the journal Automatica (https://www.journals.elsevier.com/automatica/), 20 pages, 5 figures. A preliminary version of the paper appeared in Proceedings of the 51st IEEE Conference on Decision and Control (CDC), pp. 1093-1098, Maui, Hawaii, USA, 2012

The purpose of this paper is to develop a model reduction theory for linear quantum stochastic systems that are commonly encountered in quantum optics and related fields, modeling devices such as optical cavities and optical parametric amplifiers, as well as quantum networks composed of such devices. Results are derived on subsystem truncation of such systems and it is shown that this truncation preserves the physical realizability property of linear quantum stochastic systems. It is also shown that the property of complete passivity of linear quantum stochastic systems is preserved under subsystem truncation. A necessary and sufficient condition for the existence of a balanced realization of a linear quantum stochastic system under sympletic transformations is derived. Such a condition turns out to be very restrictive and will not be satisfied by generic linear quantum stochastic systems, thus necessary and sufficient conditions for relaxed notions of simultaneous diagonalization of the controllability and observability Gramians of linear quantum stochastic systems under symplectic transformations are also obtained. The notion of a quasi-balanced realization is introduced and it is shown that all asymptotically stable completely passive linear quantum stochastic systems have a quasi-balanced realization. Moreover, an explicit bound for the subsystem truncation error on a quasi-balanceable linear quantum stochastic system is provided. The results are applied in an example of model reduction in the context of low-pass optical filtering of coherent light using a network of optical cavities., Comment: 28 pages, 3 figures. Preliminary parts of this paper have been presented at the 2013 American Control Conference (http://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=06580217) and 2012 Australian Control Conference (http://search.informit.com.au/fullText;dn=230324713737486;res=IELENG). Submitted for publication

This paper studies a scheme of two spatially distant oscillator systems that are connected by Gaussian fields and examines distributed entanglement generation between two continuous-mode output Gaussian fields that are radiated by the oscillators. It is demonstrated that using measurement-feedback control while a non-local effective entangling operation is on can help to enhance the Einstein-Podolski-Rosen (EPR)-like entanglement between the output fields. The effect of propagation delays and losses in the fields interconnecting the two oscillators, and the effect of other losses in the system, are also considered. In particular, for a range of time delays the measurement feedback controller is able to maintain stability of the closed-loop system and the entanglement enhancement, but the achievable enhancement is only over a smaller bandwidth that is commensurate with the length of the time delays., Comment: 11 pages, 10 figures. Almost identical to the published version (the latter has some final minor editorial changes)

We show that, for arbitrary quantum feedback networks consisting of several quantum mechanical components connected by quantum fields, the limit of adiabatic elimination of fast oscillator modes in the components and the limit of instantaneous transmission along internal quantum field connections commute. The underlying technique is to show that both limits involve a Schur complement procedure. The result shows that the frequently used approximations, for instance to eliminate strongly coupled optical cavities, are mathematically consistent., Comment: 38 pages, 10 figures, minor typos corrected and minor editorial changes. Published in Journal of Mathematical Physics

Recently, it has been demonstrated that an arbitrary linear quantum stochastic system can be realized as a cascade connection of simpler one degree of freedom quantum harmonic oscillators together with a direct interaction Hamiltonian which is bilinear in the canonical operators of the oscillators. However, from an experimental point of view, realizations by pure cascading, without a direct interaction Hamiltonian, would be much simpler to implement and this raises the natural question of what class of linear quantum stochastic systems are realizable by cascading alone. This paper gives a precise characterization of this class of linear quantum stochastic systems and then it is proved that, in the weaker sense of transfer function realizability, all passive linear quantum stochastic systems belong to this class. A constructive example is given to show the transfer function realization of a two degrees of freedom passive linear quantum stochastic system by pure cascading., Comment: Revised version, incorporating reviewers' comments. Accepted for publication in IEEE Transactions on Automatic Control (2010). 15 pages, 1 figure

Sideband cooling is a technique that potentially allows mechanical resonators to be prepared in their ground states, important for future applications in quantum technologies. Tian has recently shown that side-band cooling can be implemented by modulating the coupling between a nano-resonator and a superconducting oscillator, a process of frequency conversion [L. Tian, PRB 79, 193407 (2009)]. While side-band cooling is usually treated in the steady-state regime, the effective resonant coupling will also generate near perfect state-swapping from the superconductor to the mechanical resonator. We perform numerical simulations of this system, examining the ground-state cooling achieved by the state-swapping. Further, we show that the superconducting oscillator can be used to control the amplitude and phase of the resonator, while simultaneously cooling it, and thus act as a coherent "quantum feedback controller"., Comment: revtex4-1, 4 pages, 1 png figure. v3: further results on coherent feedback control

Quantum error correction (QEC) is fundamental for quantum information processing but entails a substantial overhead of classically-controlled quantum operations, which can be architecturally cumbersome to accommodate. Here we discuss a novel approach to designing elementary QEC memory cells, in which all control operations are performed autonomously by an embedded optical feedback loop. Our approach is natural for nanophotonic implementations in which each qubit can be coupled to its own optical resonator, and our design for a memory cell based on the quantum bit-flip or phase-flip code requires only five qubit-cavities (three for the register and two for the controller) connected by wave-guides. The photonic QEC circuit is entirely on-chip, requiring no external clocking or control, and during steady-state operation would only need to be powered by the injection of constant-amplitude coherent fields., Comment: 18 pages, 6 figures

In this paper, we consider a linear quantum network composed of two distantly separated cavities that are connected via a one-way optical field. When one of the cavity is damped and the other is undamped, the overall cavity state obtains a large amount of entanglement in its quadratures. This entanglement however immediately decays and vanishes in a finite time. That is, entanglement sudden-death occurs. We show that the direct measurement feedback method proposed by Wiseman can avoid this entanglement sudden-death, and further, enhance the entanglement. It is also shown that the entangled state under feedback control is robust against signal loss in a realistic detector, indicating the reliability of the proposed direct feedback method in practical situations., Comment: 11 pages