On Transfer Function Realizations for Linear Quantum Stochastic Systems
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.
Linear Quantum Stochastic Systems (LQSSs) are a class of models widely used in linear quantum optics and elsewhere [5, 6, 7]. In quantum optics, they describe a variety of devices, such as optical cavities, parametric amplifiers, etc., as well as networks of such devices. The mathematical framework for these models is provided by the theory of quantum Wiener processes and the associated Quantum Stochastic Differential Equations [8, 9, 10]. Potential applications of linear quantum optics include quantum information and photonic signal processing, see e.g. [11, 12, 13, 14, 15]. Another particularly important application of LQSSs is as coherent quantum feedback controllers for other quantum systems, i.e. controllers that do not perform any measurement on the controlled quantum system, and thus have the potential for increased performance compared to classical controllers, see e.g. [16, 17, 18, 19, 20, 21, 22, 23].
A problem of fundamental importance for applications of LQSSs, is the problem of realization/synthesis: Given a LQSS with specified parameters, how does one engineer that system using basic quantum optical devices, such as optical cavities, parametric amplifiers, phase shifters, beam splitters, squeezers etc.? The synthesis problem comes in two varieties. First, there is the strict realization problem which we just described. This type of realization is necessary in the case where the states of the quantum system are meaningful to the application at hand. Examples include quantum information processing algorithms [11, 12, 13] and state generation [24, 25]. In the case that only the input-output relation of the LQSS is important, we have the problem of transfer function realization. This is the case, for example, in controller synthesis [21, 22, 23].
In recent years, solutions have been proposed to both the strict and the transfer function realization problems. For the strict problem, [26, 27] propose a cascade of single-mode cavities realization. This allows for arbitrary couplings of the LQSS to its environment. However, not all possible interactions between cavity modes are possible, because the mode of a cavity can influence only modes of subsequent cavities. For this reason, direct Hamiltonian interactions  and feedback  between cavities have been used to “correct” the dynamics of the cascade to the desired form. In this article, we review two methods for the transfer function realization of LQSSs, proposed in [1, 2, 3, 4]. The first method uses a cascade of single-mode cavities. For the case of passive LQSSs,  has shown that such a realization is possible for any passive system, in which case all cavities needed to realize it are also passive. The result for the general case is established in , where it is shown that a cascade of cavities realization is possible for generic LQSSs. The second method  utilizes static linear quantum-optical networks that pre- and post-process the system inputs and outputs, thus leaving a simple “reduced” transfer function to be realized. This “reduced” transfer function can be realized, in turn, by a concatenation of single-mode cavities in a feedback interconnection through a static linear quantum-optical network. In the case of passive LQSSs, this realization is always possible, and all necessary devices needed for it are also passive.
In the case of passive LQSSs, the realization methods make crucial use of two classic theorems from Linear Algebra, namely Schur’s Unitary Triangularization theorem, and the Singular Value Decomposition , respectively. To extend them from passive LQSSs to general LQSSs that may contain active (quanta producing) quantum optical devices, we prove two analogous matrix factorizations for a class of even-dimensional structured matrices, the so-called doubled-up matrices [29, 30], in a class of complex spaces with indefinite scalar products, the so-called Krein spaces . Contrary to their classic counterparts, these factorizations do not hold for every doubled-up matrix.
2 Background Material
2.1 Notation and terminology
denotes the complex conjugate of a complex number or the adjoint of an operator , respectively. As usual, and denote the real and imaginary part of a complex number. The commutator of two operators and is defined as .
For a matrix with number or operator entries, , is the usual transpose, and . Also, for a vector with number or operator entries, we shall use the notation .
The identity matrix in dimensions will be denoted by , and a matrix of zeros will be denoted by . denotes the Kronecker delta symbol in dimensions, i.e. . is the block-diagonal matrix formed by the square matrices . is the horizontal concatenation of the matrices of equal row dimension.
We define , and . We have that and, . When the dimensions of , , or can be inferred from context, they will be denoted simply by , , and . Also, is the second Pauli matrix.
We define the Krein space (, ) as the vector space equipped with the indefinite inner product defined by , for any . The -norm of a vector is defined by , and if it is nonzero, a normalized multiple of is . For a matrix considered as a map from (, ) to (, ), its adjoint operator will be called -adjoint and denoted by , to distinguish it from its usual adjoint . One can show that . The -adjoint satisfies properties similar to the usual adjoint, namely , and .
Given two matrices , and , respectively, we can form the matrix . Such a matrix is said to be doubled-up . It is immediate to see that the set of doubled-up matrices is closed under addition, multiplication and taking (-) adjoints. Also, , if and only if is doubled-up When referring to a doubled-up matrix , and , will denote its upper-left and upper-right blocks.
A complex matrix R is called Bogoliubov if it is doubled-up and -unitary, i.e . The set of these matrices forms a non-compact Lie group known as the Bogoliubov group. Bogoliubov matrices are isometries of Krein spaces.
2.2 Linear Quantum Stochastic Systems
The material in this subsection is fairly standard, and our presentation aims mostly at establishing notation and terminology. To this end, we follow the review paper . For the mathematical background necessary for a precise discussion of LQSSs, some standard references are [8, 9, 10], while for a Physics perspective, see [5, 32]. The references [26, 33, 34, 35, 29] contain a lot of relevant material, as well.
The systems we consider in this work are collections of quantum harmonic oscillators interacting among themselves, as well as with their environment. The -th harmonic oscillator () is described by its position and momentum variables, and , respectively. These are self-adjoint operators satisfying the Canonical Commutation Relations (CCRs) , , and , for . We find it more convenient to work with the so-called annihilation and creation operators , and . They satisfy the CCRs , , and , . In the following, .
The environment is modelled as a collection of bosonic heat reservoirs. The -th heat reservoir () is described by the bosonic field annihilation and creation operators and , respectively. The field operators are adapted quantum stochastic processes with forward differentials , and . They satisfy the quantum Itô products , , , and . In the following, .
To describe the dynamics of the harmonic oscillators and the quantum fields (noises), we need to introduce certain operators. We begin with the class of annihilator only LQSSs. We also refer to such systems as passive LQSSs, because systems in this class describe optical devices such as damped optical cavities, that do not require an external source of quanta for their operation. First, we have the Hamiltonian operator , which specifies the dynamics of the harmonic oscillators in the absence of any environmental influence. is a Hermitian matrix referred to as the Hamiltonian matrix. Next, we have the coupling operator (vector of operators) that specifies the interaction of the harmonic oscillators with the quantum fields. depends linearly on the annihilation operators, and can be expressed as . is called the coupling matrix. Finally, we have the unitary scattering matrix , that describes the interactions between the quantum fields themselves. In practice, it represents the unitary transformation effected on the heat reservoir modes by a static passive linear optical network that precedes the LQSS, see Subsection 2.3.
In the Heisenberg picture of Quantum Mechanics, the joint evolution of the harmonic oscillators and the quantum fields is described by the following system of Quantum Stochastic Differential Equations (QSDEs):
The field operators , describe the outputs of the system. We can generalize (1) by allowing the system inputs to be not just quantum noises, but to contain a “signal part”, as well. Such is the case when the output of a passive LQSS is fed into another passive LQSS. So we substitute the more general input and output notations and , for and , respectively. The forward differentials and of -dimensional inputs and outputs, respectively, contain quantum noises, as well as linear combinations of variables of other systems. The resulting QSDEs are the following:
One can show that the structure of (2) is preserved under linear transformations of the state , if and only if is unitary. Under such a state transformation, the system parameters transform according to . From the point of view of Quantum Mechanics, must be unitary so that the new annihilation and creation operators satisfy the correct CCRs.
General LQSSs may contain active devices that require an external source of quanta for their operation, such as degenerate parametric amplifiers. In this case, system and field creation operators appear in the QSDEs for system and field annihilation operators, and vice versa. Since these are adjoint operators which have to be treated as separate variables, this leads to the appearance of doubled-up matrices in the corresponding QSDEs. To describe the most general linear dynamics of harmonic oscillators and quantum noises, we introduce generalized versions of the Hamiltonian operator, the coupling operator, and the scattering matrix defined above. We begin with the Hamiltonian operator
which specifies the dynamics of the harmonic oscillators in the absence of any environmental influence. The Hamiltonian matrix is Hermitian and doubled-up. Next, we have the coupling operator (vector of operators) that specifies the interaction of the harmonic oscillators with the quantum fields. depends linearly on the creation and annihilation operators, . We construct the doubled-up coupling matrix from and . Finally, we have the Bogoliubov generalized scattering matrix , that describes the interactions between the quantum fields themselves. In practice, it represents the Bogoliubov transformation effected on the heat reservoir modes by a general static linear quantum optical network that precedes the LQSS, see Subsection 2.3, and .
In the Heisenberg picture of Quantum Mechanics, the joint evolution of the harmonic oscillators and the quantum fields is described by the following system of Quantum Stochastic Differential Equations (QSDEs):
The forward differentials and of -dimensional inputs and outputs, respectively, contain quantum noises, as well as a signal part (linear combinations of variables of other systems). One can show that the structure of (3) is preserved under linear transformations of the state , if and only if is Bogoliubov. In that case the system parameters transform according to . From the point of view of Quantum Mechanics, must be Bogoliubov so that the new annihilation and creation operators satisfy the correct CCRs.
We end this subsection with the model of the single-mode optical cavity, which is the basic device for the proposed realization methods in this paper. It is described by its optical mode , with Hamiltonian matrix , where is the so-called cavity detuning. For a cavity with inputs/outputs, we let , and . and will be called the passive and the active coupling coefficient of the -th quantum noise to the cavity, respectively. When , the interaction of the cavity mode with the -th quantum noise will be referred to as (purely) passive, and when , it will be referred to as (purely) active. The model of a cavity with inputs/outputs, is the following:
, where . If a quantum noise couples passively to the cavity, the corresponding interaction may be realized with a partially transmitting mirror. For an interaction that has an active component, a more complicated implementation is needed, which makes use of an auxiliary cavity, see e.g.  for the details. From now on, we shall use the system-theoretic term port for any part of the experimental set-up that realizes an interaction of the cavity mode with a quantum noise (where an input enters and an output exits the cavity). Figure 1 is a graphical representation of a multi-port cavity modelled by equations (4).
2.3 Static Linear Optical Devices and Networks
Besides the single-mode cavities discussed above, the proposed realization methods make use of static linear quantum optical devices and networks, as well. Useful references for this material are [36, 26, 37, 38]. The most basic such devices are the following:
The phase shifter: This device produces a phase shift in its input optical field. That is, if and are its input and output fields, respectively, then . Notice that . This means that the energy of the output field is equal to that of the input field, and hence the device is passive.
The beam splitter: This device produces linear combinations of its two input fields. If we denote its inputs by and , and its outputs by and , then
is called the mixing angle of the beam splitter. and are phase differences in the two input and the two output fields, respectively, produced by phase shifters. is a common phase shift in both output fields. This form of corresponds to a general unitary matrix. Because , we can see that
and hence the total energy of the output fields is equal to that of the input fields.
The squeezer: This device reduces the variance in the real quadrature , or the imaginary quadrature of an input field , while increasing the variance in the other. Its operation is described by
is the squeezing parameter, and are phase shifts in the input and the output field, respectively, produced by phase shifters. This form of represents a general Bogoliubov matrix. It is easy to show that , for , and hence energy is not conserved. So, the squeezer is an active device.
By connecting various static linear optical devices, we may form static linear optical networks (multi-port devices). When a network is composed solely of passive devices, it is called passive. The input-output relation of a passive static network with inputs and outputs, and , respectively, is , with . Such a network is a multi-dimensional generalization of the beam splitter and is sometimes called a multi-beam splitter. It turns out that any passive static network can be constructed exclusively from phase shifters and beam splitters . This is due to the fact that an unitary matrix can be factorized in terms of matrices representing either phase shifting of an optical field in the network or beam splitting between two optical fields in the network, see Figure 2.
In the case of general static networks that may include active devices, the input-output relation takes the form , where is a Bogoliubov matrix. For every Bogoliubov matrix, the following factorization holds:
where and , with . This factorization is known as Bloch-Messiah reduction [26, 37, 38]. The physical interpretation of this equation is that a general static network may be implemented as a sequence of three static networks: First comes a passive static network (multi-beam splitter) implementing the unitary transformation . Then follows an active static network made of squeezers, each acting on an output of the first network, and finally, the outputs of the squeezers are fed into a second multi-beam splitter implementing the unitary transformation . This is depicted in Figure 3. Because of this structure, a general static network is sometimes called a multi-squeezer.
3 Realizations of Passive Linear Quantum Stochastic Systems
In this section, we present two transfer function realization methods for the case of passive LQSSs. Besides the importance of passive LQSSs in applications, they offer the simplest context in which to present the methods.
3.1 Cascade Realization
We begin with the cavity cascade realization previously obtained in  using the real quadrature form of a LQSS (position-momentum operators). Here, we present this cascade realization using a complex formalism (creation-annihilation operators) that simplifies the proof considerably, see also [40, Subsection 4.1]. We present this method in the following theorem:
Given a passive linear quantum stochastic system with Hamiltonian matrix , coupling operator , and scattering matrix , its transfer function can be realized by the following cascade of a multi-beam splitter and -port passive cavities:
The cavity parameters , and , , are determined as follows: Define , and let be a unitary matrix such that is lower-triangular. Then, , and .
Proof: It is a straightforward calculation to show that the cascade system (5) can be put in the following form:
where , , and
is lower-triangular. Now, given a passive LQSS with parameters , let . From Schur’s Unitary Triangularization theorem , there exists a unitary such that is lower-triangular. Using as a state transformation, we obtain a realization of the system dynamics in cascade form. The parameters of the cascade realization are given in terms of the original parameters by , and , where . Since the transfer function of a linear system is independent of its particular realization, it follows that the transfer function of a passive LQSS can always be realized by the cascade form given in the theorem.
We should point out that this realization is not unique, but depends on the order of appearance of the eigenvalues of on the diagonal of its lower-triangular form , which leads to different ’s and ’s. We demonstrate this method with an illustrative example.
Consider the 3-mode, 3-input passive linear quantum stochastic system with the following parameters:
and . We compute to be equal to
We compute a lower-triangular and a unitary , such that :
The parameters ,,, and ,, are given by , and
3.2 Realization Using Static Networks for Input/Output Processing and Feedback
Next, we present the realization method of  for the case of passive LQSSs, in the following theorem:
Given a passive linear quantum stochastic system with Hamiltonian matrix , coupling operator , and scattering matrix , let
be its transfer function. Let be the singular value decomposition of the coupling matrix , with
is the rank of , and , . Then, can be factorized as , where has the form
with . The first and last factors in this factorization of are unitary transformations of the output and the input, respectively, of the transfer function in the middle factor, and can be realized by multi-beam splitters. The transfer function is that of a passive LQSS with scattering matrix , coupling matrix , and Hamiltonian matrix . Moreover, can be realized by the following feedback network of 1-port and 2-port cavities:
Here, , and , where , and , are the cavity detuning and the coupling coefficient of the interconnection port, respectively, of the -th cavity, which can be chosen arbitrarily. The -dimensional vectors , and , contain the inputs/outputs of the system ports, and the -dimensional vectors , and , the inputs/outputs of the interconnection ports. Finally, the unitary interconnection matrix (feedback gain) is determined through the relations
From the fact that , , and are diagonal, all diagonal elements of are non-zero, and only diagonal elements of are non-zero, we see that (7) describes a collection of cavities, all of which have one interconnection port, but only have system ports. Hence, the feedback network consists of 1-port and 2-port cavities. Figure 5 provides a graphical representation of the realization method of Theorem 1.
Proof: It suffices to prove that is the transfer function of the system described by (7). To this end, we combine the last two equations in (7) to obtain the relation . At this point we introduce a variant of the Cayley transform for unitary matrices without unit eigenvalues , namely
The unitarity of implies that is skew-Hermitian. We can also solve uniquely for in terms of with the following result:
where is defined for all skew-Hermitian matrices , and can be seen to be unitary due to the skew-Hermitian nature of . It is easy to see that . Using the relation between and , and the definition of , the equations for the network take the following form:
These equations describe a passive linear quantum stochastic system with Hamiltonian matrix given by the expression
Given any values for the cavity parameters and , and any desired Hamiltonian matrix , we may determine the unique (and hence the unique ) that achieves this by the expression
Similarly to the cascade realization, there is non-uniqueness associated with the ordering of the singular values of on the diagonal of . However, there is additional non-uniqueness due to a continuum of choices for the values of and , . We demonstrate this method with an illustrative example.
For the system of Example 1, we have that the SVD of is given by , with
The Hamiltonian of the reduced system is given by
Letting and , equation (8) produces the following :
from which we calculate the feedback gain matrix using equation (9),
Figure 6 provides a graphical representation of the proposed implementation of the transfer function for this example.
4 Realizations of General Linear Quantum Stochastic Systems
In this section, we extend the transfer function realization methods for passive LQSSs presented in Section 3, to general LQSSs. These methods employed Schur’s Unitary Triangularization theorem, and the Singular Value Decomposition , respectively. To extend the methods to the general case, we prove versions of these two classic matrix decompositions for doubled-up matrices in (, ).
4.1 Cascade Realization
We begin with the analog of Schur’s Unitary Triangularization theorem for doubled-up matrices in Krein spaces. A version of this result for symplectic spaces has been derived in . Here, we prove the Krein space version in a way that closely follows the proof of the classic result in .
Let be a doubled-up matrix. Then, under Assumption I
Proof: First, we prove certain facts about the eigenstructure of . Let be an eigenvalue of with corresponding eigenvector , i.e. . We compute:
For nonreal , this implies that is also an eigenvalue of , with eigenvector . For a real , there are two possibilities: is either linearly independent from , or not. We show that, under the assumption that has non-zero -norm, the second possibility cannot occur. In fact, we shall prove that for any , is linearly independent from , if . Indeed, if , for some , then , and the two equations are compatible only if . At this point, we introduce two identities which shall be useful in the following:
for complex vectors . Both identities can be proven using the simple relation . Immediate consequences of these are that, and are -orthogonal and have opposite -norms, for any complex vector , i.e.
Then, we have that , which is excluded by the assumption that has non-zero -norm.
Let be an eigenvalue/eigenvector pair of . We shall assume that , and in particular that . This guarantees that and are linearly independent. If , we replace with , and with . Let be the -normalized version of . Then, , , and . We can always extend the set to a -orthonormal basis