Yu, QiDong, DaoyiPetersen, Ian R.Gao, Qing2026-07-022026-07-02ORCID:/0000-0002-7425-3559/work/219056959ORCID:/0000-0003-4856-9450/work/219057266https://hdl.handle.net/1885/733812282A filtering problem for a class of quantum systems disturbed by a classical stochastic process is investigated in this paper. The classical disturbance process, which is assumed to be described by a linear stochastic differential equation, is modeled by a quantum cavity model. Then the hybrid quantum-classical system is described by a combined quantum system consisting of two quantum cavity subsystems. Quantum filtering theory and a quantum extended Kalman filter method are employed to estimate the states of the combined quantum system. An estimate of the classical stochastic process is derived from the estimate of the combined quantum system.1. INTRODUCTION 1. INTRODUCTION 1. INTRODUCTION Characterizing unknown quantum states has been a fundamen- Characterizing unknown quantum states has been a fundamen- tal task in quantum computation, quantum metrology and quan- tal task in quantum computation, quantum metrology and quan- tum control. To estimate an unknown static quantum state, state tal task in quantum computation, quantum metrology and quan- tum control. To estimate an unknown static quantum state, state tum control. To estimate an unknown static quantum state, state tomography methods such as maximum likelihood estimation (Paris and Rehácˇek (2004)), Bayesian mean estimation (Paris (Paris and Rˇ ehácˇek (2004)), Bayesian mean estimation (Paris and Rehácˇeˇk (2004)) and linear regression estimation (Hou (Parisˇand Rehácˇek (2004)), Bayesian mean estimation (Paris and Rˇ ehácˇek (2004)) and linear regression estimation (Hou et al.ˇ(2016); Qi et al. (2013)) have been developed. For es- et al. (2016); Qi et al. (2013)) have been developed. For es- timating a dynamic quantum state, a quantum filtering theory et al. (2016); Qi et al. (2013)) have been developed. For es- timating a dynamic quantum state, a quantum filtering theory has been developed (Bouten et al. (2007, 2009)). Quantum has been developed (Bouten et al. (2007, 2009)). Quantum filtering theory was introduced by Belavkin in the 1980’s as filtering theory was introduced by Belavkin in the 1980’s as documented in a series of articles (Belavkin (1991)). The basic documented in a series of articles (Belavkin (1991)). The basic premise is to build a non-commutative counterpart for classical documented in a series of articles (Belavkin (1991)). The basic premise is to build a non-commutative counterpart for classical probability theory so that approaches to deriving the classi- premise is to build a non-commutative counterpart for classical probability theory so that approaches to deriving the classi- cal filtering equation can be adapted to quantum dynamical probability theory so that approaches to deriving the classi- cal filtering equation can be adapted to quantum dynamical systems. The main difference between this theory and clas- systems. The main difference between this theory and clas- sical filtering theory is that non-commutative observables in systems. The main difference between this theory and clas- sical filtering theory is that non-commutative observables in quantum systems cannot be jointly represented on a single quantum systems cannot be jointly represented on a single classical probability space. Quantum filtering theory enables quantum systems cannot be jointly represented on a single classical probability space. Quantum filtering theory enables us to optimally estimate the quantum system state using non- us to optimally estimate the quantum system state using non- demolition measurements. It plays a crucial role in many areas us to optimally estimate the quantum system state using non- demolition measurements. It plays a crucial role in many areas such as quantum control (van Handel et al. (2005), Armen such as quantum control (van Handel et al. (2005), Armen et al. (2002)). Recently, quantum filtering theory has been et al. (2002)). Recently, quantum filtering theory has been successfully applied in experimental designs such as trapped et al. (2002)). Recently, quantum filtering theory has been successfully applied in experimental designs such as trapped ions (Hume et al., 2007), cavity QED systems (Sayrin et al., ions (Hume et al., 2007), cavity QED systems (Sayrin et al., 2011), and optomechanical systems (Wieczorek et al., 2015). 2011), and optomechanical systems (Wieczorek et al., 2015). In practice, physical quantum systems are unavoidably affected 2011), and optomechanical systems (Wieczorek et al., 2015). In practice, physical quantum systems are unavoidably affected by classical signals (Ralph et al., 2011; Wang and Dong, 2017), In practice, physical quantum systems are unavoidably affected by classical signals (Ralph et al., 2011; Wang and Dong, 2017), and a number of researchers are becoming interested in the and a number of researchers are becoming interested in the filtering problem for ‘hybrid’ quantum-classical systems where and a number of researchers are becoming interested in the filtering problem for ‘hybrid’ quantum-classical systems where the quantum systems are subject to a classical process. Relevant filtering problem for ‘hybrid’ quantum-classical systems where the quantum systems are subject to a classical process. Relevant results can be found in e.g., Tsang’s work on quantum smooth- results can be found in e.g., Tsang’s work on quantum smooth- r★esults can be found in e.g., Tsang’s work on quantum smooth- ★ This work was supported by the Australian Research Council’s Discovery ★ This work was supported by the Australian Research Council’s Discovery Projects funding scheme under Project DP130101658 and Laureate Fellowship This work was supported by the Australian Research Council’s Discovery Pro1je1c0t1s0f0u0n2d0in, gansdchtheemAeiurnFdoerrcPerOojfeficcteDoPf 1S3c0ie1n0t1if6ic58ReasnedaLrcahuurenadteerFaeglrleoewmsheinpt Projects funding scheme under Project DP130101658 and Laureate Fellowship FL110100020, and the Air Force Office of Scientific Research under agreement FL110100020, and the Air Force Office of Scientific Research under agreement number FA2386-16-1-4065. number FA2386-16-1-4065. C24o0p5y-r8i9g6h3t ©© 22001177, IIFFAC (International Federation of Automatic Cont1r2o2l)3 H5 osting by Elsevier Ltd. All rights reserved. CPoeepry rreigvhietw © u 2n0d1e7r rIFesApConsibility of International Federation of Autom1a2ti2c3 C5ontrol. Copyright © 2017 IFAC 12235 10.1016/j.ifacol.2017.08.1948 ing (Tsang (2009a,b)) where a concept of hybrid quantum-ing (Tsang (2009a,b)) where a concept of hybrid quantum-classical density operator was used as the main technical tool. ing (Tsang (2009a,b)) where a concept of hybrid quantum-classical density operator was used as the main technical tool. Recently, Gao et al. (2016a,b) developed a quantum-classical classical density operator was used as the main technical tool. Recently, Gao et al. (2016a,b) developed a quantum-classical Bayesian inference approach to solve fault tolerant quantum Recently, Gao et al. (2016a,b) developed a quantum-classical Bayesian inference approach to solve fault tolerant quantum filtering and fault detection problems for a class of quantum Bayesian inference approach to solve fault tolerant quantum filtering and fault detection problems for a class of quantum optical systems subject to stochastic faults. filtering and fault detection problems for a class of quantum optical systems subject to stochastic faults. In this paper, we extend the previous work (Gao et al., 2016a) In this paper, we extend the previous work (Gao et al., 2016a) to the case that the disturbance process has a continuous value In this paper, we extend the previous work (Gao et al., 2016a) to the case that the disturbance process has a continuous value space and our main goal is to estimate both the quantum state space and our main goal is to estimate both the quantum state and the classical process using non-demolition quantum mea-space and our main goal is to estimate both the quantum state and the classical process using non-demolition quantum measurements. We consider a system-probe model with a time-surements. We consider a system-probe model with a time-varying Hamiltonian that depends on a classical stochastic pro-surements. We consider a system-probe model with a time-varying Hamiltonian that depends on a classical stochastic process. This hybrid quantum-classical stochastic system is ana-varying Hamiltonian that depends on a classical stochastic process. This hybrid quantum-classical stochastic system is analyzed by building a quantum analog of the classical stochastic cess. This hybrid quantum-classical stochastic system is analyzed by building a quantum analog of the classical stochastic process; see also (Wang et al., 2013). The idea of using an process; see also (Wang et al., 2013). The idea of using an aritificial quantum system to model noise has been considered process; see also (Wang et al., 2013). The idea of using an aritificial quantum system to model noise has been considered in (Xue et al., 2017, 2015a,b). However, the authors only con-aritificial quantum system to model noise has been considered in (Xue et al., 2017, 2015a,b). However, the authors only consider the disturbance to be quantum noise. Then, in our case, sider the disturbance to be quantum noise. Then, in our case, quantum filtering theory can be utilized to investigate the filter-sider the disturbance to be quantum noise. Then, in our case, quantum filtering theory can be utilized to investigate the filtering problem. The estimation tasks are accomplished by using a quantum filtering theory can be utilized to investigate the filtering problem. The estimation tasks are accomplished by using a quantum extended Kalman filter (QEKF) approach. ing problem. The estimation tasks are accomplished by using a quantum extended Kalman filter (QEKF) approach. The structure of this paper is as follows. In Section 2, we briefly The structure of this paper is as follows. In Section 2, we briefly introduce quantum probability theory and quantum filtering introduce quantum probability theory and quantum filtering theory. Section 3 is devoted to the modeling of the classical sig-introduce quantum probability theory and quantum filtering theory. Section 3 is devoted to the modeling of the classical signal using a quantum cavity model. A stochastic master equation theory. Section 3 is devoted to the modeling of the classical signal using a quantum cavity model. A stochastic master equation (SME) is then obtained to solve the filtering problem. A QEKF nal using a quantum cavity model. A stochastic master equation (SME) is then obtained to solve the filtering problem. A QEKF approach is also employed to estimate both the quantum state approach is also employed to estimate both the quantum state and the classical process in Section 4. Section 5 concludes this approach is also employed to estimate both the quantum state and the classical process in Section 4. Section 5 concludes this paper. and the classical process in Section 4. Section 5 concludes this ppaappeerr.. paper. † Notation:ANotation:Am×n denotesanmrowandn⊤ columnmatrix;A†denotesanmrowandncolumnmatrix;A Notation: Am×n denotes an m row and n column matrix; A† denotes conmju×gnate and transpose of A; A is the transpose of Notation: Am×n denotes an m row and n⊤column matrix; A deno∗tes conjugate and transpose of A; A is the transpose of deno∗∗tes conjugate and transpose of A; A is the transpose of A; A∗ is the conjugate of A; Tr(A) is the trace of A; X is used to A; A is the conjugate of A; Tr(A) is the trace of A; X is used to denote any operators and x is a vector of those operators; ρ is a denote any operators and x is a vector of those operators; ρ is a density operator representing a quantum state;√ â is the estimate of a; i means the imaginary unit, i.e., i = −1.6en©2017 The authorsquantum extended Kalman filterquantum filteringquantum system201710.1016/j.ifacol.2017.08.194885044871261