An Exponential Quantum Projection Filter for Open Quantum Systems \thanksreffootnoteinfo
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.
PolyU]Qing Gao, PolyU]Guofeng Zhang, ANU]Ian R. Petersen
Department of Applied Mathematics, Hong Kong Polytechnic University, Hong Kong SAR, China.
Research School of Engineering, Australian National University, Canberra, ACT, 2601, Australia.
Key words: Open quantum systems; quantum filtering; quantum information geometry; exponential quantum projection filter.
The past decades have witnessed tremendous advances in quantum technologies which allow us to effectively probe and manipulate matter at the level of atoms (e.g., [Akimoto & Hayashi (2011)], [Viola & Knill (2003)], [Hamerly & Mabuchi (2012)], [Liu et al. (2016)]). A basic requirement in realizing these technologies is to infer the unknown quantum system states from measurements. Nevertheless, two fundamental nonclassical features manifested by quantum systems are that i) any quantum measurement scheme can extract in principle only partial information from the observed quantum system; and ii) any quantum measurement inevitably changes the quantum system states in a probabilistic way ([Breuer & Petruccione (2002)], [Gardiner & Zoller (2000)], [Wiseman & Milburn (2010)]). As a result, any measurement based quantum feedback control problem is essentially a problem of stochastic control theory with partial observations and can generally be converted into a control problem for a quantum filter with fully accessible states, as in classical stochastic control theory ([van Handel et al. (2005a)], [van Handel et al. (2005c)], [Mirrahimi & van Handel (2007)], [Ticozzi & Viola (2008)], [Ticozzi & Viola (2009)]). In this context, the quantum system and observations are modelled as a pair of quantum stochastic differential equations, while the quantum filter, also known as the quantum trajectory, is a dynamic equation driven by the classical output signal of a laboratory measuring device ([Belavkin (1992)], [Bouten et al. (2007)], [Gao et al. (2016)]). A quantum filter recursively updates the information state of a quantum system undergoing continual measurements and provides real-time information that can be fed into the quantum controller. Therefore, real time solution of the quantum filter equations is essential in implementing a quantum feedback control setup, which, however, tends to be computationally expensive, especially when the quantum system has a high dimension ([Song et al. (2016)]).
In order to make the implementation more efficient, several approaches have been proposed in the literature concerning the approximation or model reduction of quantum filter, to mention a few, see ([Emzir et al. (2016)], [Rouchon & Ralph (2015)]). In [Emzir et al. (2016)], an extended Kalman filtering approach was developed for a class of open quantum systems subject to continuous measurement, where time-varying linearization was applied to the system dynamics and a Kalman filter was designed for the linearized system. The proposed approach performs well for nearly linear quantum systems. A numerical approach to reducing the computational burden associated with calculating quantum trajectories was discussed in [Rouchon & Ralph (2015)] and was used to demonstrate a two-qubit feedback control scheme. It was shown in simulation studies that a high approximation accuracy can be achieved even when a small number of integration steps is involved.
The main goal of this paper is to approximate the optimal quantum filter using a lower-dimensional quantum projection filter, motivated by the pioneering work on projection filtering for classical stochastic systems by Brigo, Hanzon and LeGland ([Brigo et al. (1998)], [Brigo et al. (1999)]). The basic idea of projection filtering is to constrain the optimal filter to remain in a finite-dimensional submanifold embedded in the state space of the filter. Then the projection filter can be expressed as a set of dynamic equations satisfied by the local coordinates of this submanifold. The problem of quantum projection filtering has been addressed in [van Handel & Mabuchi (2005b)] where the information state of a highly nonlinear quantum model of a strongly coupled two-level atom in an optical cavity was approximately determined by a tractable set of stochastic differential equations. However, the approach in [van Handel & Mabuchi (2005b)] requires exact prior knowledge of an invariant set of the solutions to the quantum filter equations. In other words, a finite-dimensional family of densities is already known to be a good approximation of the information state. This restrictive assumption was removed in [Nielsen et al. (2009)] where an unsupervised learning identification algorithm was developed to determine the structure of the submanifold. However, the identification algorithm itself could be time consuming when a more general and complex open quantum system is considered instead of the simple two-level quantum system in [Nielsen et al. (2009)]. In this paper, we design an exponential quantum projection filter for a general atom-laser interaction system subject to continuous homodyne detection, using a differential-geometric method in quantum information geometry theory. We propose a finite-dimensional differentiable submanifold consisting of an unnormalized exponential family of quantum density operators, on which a quantum Fisher metric structure is rigorously defined. Then through a projection operation, the solutions to the unnormalized quantum filter equations are maintained in this submanifold. In other words, the resulting quantum trajectory becomes a curve on the finite-dimensional manifold and the unnormalized quantum filter equation reduces to a set of recursive equations satisfied on the corresponding finite-dimensional coordinate system. We also present a convenient design of the differentiable manifold, by which the local approximation errors are significantly reduced and the quantum projection filter equations are simplified. In addition, it is shown that when the projection filtering strategy 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.
This paper is organized as follows. Section 2 introduces some preliminaries on the quantum system model, quantum filter and quantum information geometry. Section 3 presents the main contributions of this paper. Here, we first derive the exponential quantum projection filter equations and provide a convenient design of the differentiable manifold in Subsection 3.1. Then we apply the projection filtering strategy to an asymptotically stable open quantum system and analyze the behaviour of the corresponding exponential quantum projection filter in Subsection 3.2. Section 4 applies the proposed approach to an atomic ensemble interacting with an electromagnetic field and demonstrates the approximation performance through simulation studies. Section 5 concludes this paper.
Notation. i=. Here we use the Roman type character i to distinguish the imaginary unit from the index . represents the complex conjugate transpose of matrix . are the singular values of matrix which are arranged in decreasing order, i.e., . is the trace of matrix . is the commutator of matrices and . is the identity matrix. is the max norm of matrix . and represent the -dimensional real vector space and complex vector space, respectively.
2.1 System Formulation and Quantum Filter
We sketch the open quantum system model under consideration in this section; a more detailed description can be found in ([Bouten et al. (2007)], [Breuer & Petruccione (2002)], [Song et al. (2016)]) and the references therein.
In this paper, we consider a typical physical scenario from quantum optics. An arbitrary quantum system , e.g., an atomic ensemble, is in weak interaction with an external single-channel laser field that is initially in the vacuum state. A cavity is used to increase the interaction strength between the light and the quantum system. One of the cavity mirrors, through which a forward mode of the electromagnetic field scatters off, is made slightly leaky such that information about the quantum system is extracted using a homodyne detector. The single-channel probe laser field has an annihilation operator and a creation operator , which are operators defined on a symmetric Fock space that can be decomposed into the past and future components in the form of a tensor product . Let and be integrated annihilation and creation field operators on , respectively. In this paper, the laser field is supposed to be canonical, that is,
Let us denote by the Hilbert space of the quantum system and suppose . The composite system composed of the atomic system and the field is assumed to be isolated. Then its temporal Heisenberg-picture evolution can be described by a unitary operator on the tensor product Hilbert space , which satisfies the following Hudson-Parthasarathy quantum stochastic differential equation\@xfootnoteWe have assumed =1 by using atomic units in this paper.:
with the initial condition , where is the initial Hamiltonian of the quantum system , and is a coupling operator, or measurement operator that describes how the system interact with the input field. The joint system state is given by some quantum state in and the vacuum state .
In the Heisenberg picture, an initial system operator evolves to at time . Using the quantum rules, satisfies the following quantum master equation:
where is the so-called Lindblad generator:
A homodyne detector measures the observable where is the real quadrature of the input laser field and generates a classical photocurrent signal. The so-called self-nondemolition property, i.e., for all enables monitoring continuously and interpreting as a classical signal (photocurrent). By the rules, satisfies
Equations (1) and (3) form the system-observation pair of our model. As in classical stochastic control theory, the goal of quantum filtering is to find the least-mean-square estimate of the system observable given the prior observations , that is, to derive an expression for the quantum conditional expectation where is the commutative von Neumann algebra generated by the observation process . From the so-called nondemolition condition, i.e., for all , can be isomorphically interpreted as a classical conditional expectation and is thus well defined ([Bouten et al. (2007)], [Gao et al. (2016)]). The dynamic equation satisfied by has been derived as ([Belavkin (1992)], [Bouten et al. (2007)]):
with . Here is the adjoint Lindblad generator:
Note that the quantum filter (5) is a classical stochastic differential equation that is driven by the Wiener type classical photocurrent signal and can thus be conveniently implemented on a classical signal processor. Equation (5) has been widely used in applications including quantum state estimation and quantum feedback control ([Hamerly & Mabuchi (2012)], [van Handel et al. (2005a)], [van Handel et al. (2005c)]), where in time calculation of (5) is essential. However, one has to solve a system of recursive stochastic differential equations in order to determine the conditional probability density defined on . A high computational cost will arise if the atomic system has a large number of energy levels. The main goal of this paper is to reduce the dimension of the filtering equations while guaranteeing acceptable approximation performance.
2.2 Quantum Information Geometry
This subsection will introduce some foundations of quantum information geometry theory. A more detailed formulation can be found in Chapter 7 of the book ([Amari & Nagaoka (2000)]). Let the set of all self-adjoint operators on the Hilbert space be denoted by
Subsequently, we focus on the geometry of the set of nonnegative self-adjoint operators which is denoted by
Hence is a closed subset of and is naturally regarded as a real manifold with dimension . Apparently, the tangent space at each point to , which is denoted by , is identified with . When a tangent vector is considered as an element of by this identification, we denote it by and call it the mixture representation (m-representation) of . When a coordinate system is given on so that each state is parameterised as , the representation of the natural basis vector of the tangent vector space is identified with
where . Naturally, are linearly independent and
A differentiable manifold is not naturally endowed with an inner product structure. We need to add a Riemannian structure to the manifold. To be specific, we define a Riemannian metric on . The symmetrized inner product is employed to define the inner product on ([Amari & Nagaoka (2000)]):
Based on this inner product, we define another useful representation called the of a tangent vector as the self-adjoint operator satisfying
Using the representation defined above, we define an inner product on by
Then forms a Riemmanian metric on which may be regarded as a quantum version of the Fisher metric. The components of this metric are given by
Example 2.1. Consider a qubit system, the Hilbert space is identified with . Denote
where are Pauli matrices described by
Then . Each can be represented by
In this case, one has
respectively. Then, given any , its representation is a linear combination of and its representation can be derived from (12).
3 An Exponential Quantum Projection Filter: Design and Analysis
In this section, we propose a projection filtering approach to approximating the quantum filter equation in (5), using differential geometric methods in quantum information geometry theory. The basic idea of the projection filtering strategy is illustrated in Fig. 1. We consider to apply a projection operation to a space of unnormalized quantum density operators and map the optimal quantum filter equation onto a fixed lower-dimensional submanifold. A natural basis will be derived for the tangent space at each point of this submanifold, and a local projection operation can be defined with respect to a quantum Fisher metric to map the infinitesimal increments generated by the quantum filter equation onto such tangent spaces. The resulting stochastic vector field on the submanifold then defines the dynamics of the approximation filter. In this paper, we consider to use a submanifold consisting of an unnormalized exponential family of quantum density operators. It is noted that quantum density operators in the exponential form is useful in practice, examples being Gaussian states and general thermal states ([Amari & Nagaoka (2000)], [Jiang (2014)]).
3.1 Design of the Quantum Projection Filter
The quantum projection filter equation will be derived in this subsection. We start from the unnormalized version of the quantum filter equation in (5):
where is the unnormalized information state corresponding to such that . is initially set to be . The unnormalized filter equation (20) is used since its linear form is easier to manipulate compared with the nonlinear filter equation in (5).
It is worth mentioning that in order to illustrate the unnormalized quantum filter using a differential manifold structure, one must interpret the stochastic differential equation in (20) using Stratonovich integral theory because ’s rule is incompatible with a manifold structure [Brigo et al. (1998)]. We have the following result.
Lemma 3.1. The quantum stochastic differential equation in (20) can be equivalently rewritten as the following Stratonovich quantum stochastic differential equation:
Proof. The proof of Lemma 3.1 is given in Appendix.
Now we design the quantum projection filter following the scheme illustrated in Fig. 1. On one hand, it follows from (20) that is nonnegative and self-adjoint. Thus the totality of the unnormalized quantum density matrix is identified with the set in (8). It can be verified that the two terms and on the right hand side of (21) are vectors in , or equivalently, operators belonging to the set in (7). On the other hand, the submanifold is designed to be a manifold consisting of an exponential family of unnormalized quantum density operators:
where the submanifold operators are mutually commutating and pre-designed. We suppose that the entire submanifold can be covered by a single coordinate chart , where is an open subset of containing the origin. Then we have .
According to the chain rule in Stratonovich stochastic calculus, we have
where . Assuming the set is linearly independent, then this set forms an representation of the natural basis of ; i.e., the tangent vector space at each point to . We have
A direct calculation using Stratonovich stochastic calculus yields
because the operator is self-adjoint. The quantum Fisher information matrix is an dimensional real matrix given by . Then an orthogonal projection operation can be defined for every as follows:
where the matrix is the inverse of the quantum information matrix .
Consider a curve in around the point to be of the form . This corresponds to a real curve in around the real vector , through the coordinate chart . Let us consider that the curve starts from the initial condition that , or equivalently, the curve starts from . The unnormalized exponential quantum projection filter is then defined as the following quantum stochastic differential equation on the -dimensional differentiable manifold :
From the definition of the manifold in (23), the projection quantum filter can be equivalently written using the equations satisfied by the real curve in . Denote . An explicit form of the curve equations is given in the following theorem.
Theorem 3.1. The real curve satisfies the following recursive stochastic differential equation:
with the initial conditions , where and are both dimensional column vectors of real functions on . The th elements of these quantities are given by
The stochastic differential equation (30) combined with the equation (23) determines the unnormalized projection quantum density operator. In this paper, (29) or (30) is called the quantum projection filter. The approximate quantum information state can be then simply obtained as
It can be observed that only a system of stochastic differential equations is needed to be calculated in order to determine . Recall that one has to calculate a collection of stochastic differential equations in determining the information state in the quantum filter equation (5). Thus the computational cost would be reduced significantly if the number is chosen to be small.
The above design procedure requires the predesign of the submanifold operators . The remainder of this subsection will be devoted to a convenient design method for these self-adjoint operators through reduction of the local approximation errors. In fact, the proposed approximation scheme in Theorem 3.1 is implemented through two steps. First, the right-hand side of (21) is evaluated at the current projection filter quantum density operator on , instead of the true density operator . However, the right-hand side vectors , and will generally make the solutions leave the manifold . Thus a second approximation is made by projecting these vector fields onto the linear tangent vector space . In the remainder of this subsection, we will present a design of the submanifold by considering the local errors for the quantum projection filter occurring in the second approximation step at time .
Following similar ideas as in [Brigo et al. (1998)], we define at each point the prediction residual as
and two correction residuals as
Although it is not required in Theorem 3.1, the following assumption will be essential in the subsequent analysis in this paper.
Assumption 3.1. The coupling operator is self-adjoint, i.e., .
This assumption is practically reasonable in many experimental settings; e.g., trapping a cold atomic ensemble in an optical cavity ([van Handel et al. (2005c)], [Thomsen et al. (2002)]). Since is self-adjoint, it admits a spectral decomposition , where is the number of nonzero eigenvalues of , the set contains all of the nonzero real eigenvalues of , and is a set of projection operators that satisfies . Then one has the following result:
Theorem 3.2. The correction residuals and are both identically zero for all , if the submanifold in (23) is designed according to
Moreover, the exponential quantum projection filter (30) becomes
where and .
Through the design method in Theorem 3.1, the components of the quantum Fisher metric in (27) are given by
and the quantum Fisher matrix becomes a diagonal matrix
The th elements of the vector functions and in (30) are given by
It has been shown in Theorem 3.1 that, by using the design scheme as in (36), the correction residuals and are both eliminated while the prediction residual still exists. In general, it is difficult to analyze which depends on the trajectory of the quantum projection filter. However, in a special case, an upper bound of can be derived and the exponential quantum projection filter (37) can be further simplified.
The unnormalized quantum filter (20) and the exponential quantum filter (29) are both driven by the classical photocurrent which is a Wiener process with bounded drift under some classical probability measure . Using Girsanov’s theorem, however, one can always find a measure that is equivalent to such that is a Wiener process with zero drift on the interval , where is a fixed time called the final time (Page 458, [Mirrahimi & van Handel (2007)]). Let denote the expectation operation with respect to the measure . One has the following result.
and the correction residuals and are both identically zero for all . Moreover, the prediction residual satisfies
It then follows from Lemma A2 in the Appendix that
By using the rules, one can calculate from (44) that
Under some conditions, the exponential quantum projection filter could be an exact expression for the quantum filter (20). The following model reduction result is a corollary of Theorem 3.2.
Corollary 3.1. When the system Hamiltonian , if the submanifold is designed according to (36).
3.2 Practical Stability of the Quantum Projection Filter
In this subsection, we will analyze the time behaviour of the exponential quantum projection filter on the interval . Before proceeding, the following notation is introduced.
Let be an orthonormal basis of . For the quantum filter equation in (5) and for any , let
The stable states , , are called pointer states of the quantum filter.
Let be a particular pointer state of the quantum filter (5). Then the Hilbert space can be decomposed in the direct sum where . This yields a convenient decomposition of all matrices on , that is, by choosing an appropriate basis, can be written as
where and are operators from to , to , to , and to , respectively.
Denote and the orthogonal projectors on and , respectively.
Definition 3.2. The quantum filter (5) is said to be strongly globally asymptotically stable (), if it fulfils a nondemolition condition for an orthonormal basis and there is a pointer state such that, ,