On synthesis of linear quantum stochastic systems by pure cascading111This research is supported by the Australian Research Council.
Recently, it has been demonstrated that an arbitrary linear quantum stochastic system can be realized as a cascade connection of simple 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.
Keywords: Linear quantum stochastic systems, quantum system realization, quantum networks, quantum control, linear quantum optics
1 Background and Motivation
Recently, there has been interest in the literature on control of a linear quantum stochastic system with a controller which is a quantum system of the same type [1, 2, 3, 4], often referred to as “coherent-feedback control”. The potential applications for linear quantum stochastic systems include quantum information processing and photonic signal processing. For instance, they can act as the coherent photonic circuitry subsystem in a cavity QED system, the latter system being realized by placing suitable atoms inside the optical cavities in a linear quantum stochastic system. Cavity QED networks are of interest for quantum information processing (see, e.g., ), such as in the quantum internet , whilst the controller realized in  is an early sample application of linear quantum stochastic systems to photonic signal processing.
The studies on coherent-feedback control naturally led to the consideration of the network synthesis problem for linear quantum stochastic systems , which may be viewed as a quantum analogue of the network synthesis problem for linear electrical systems . Nurdin, James and Doherty  have shown that any linear quantum stochastic system can, in principle, be synthesized by a cascade of simple one degree of freedom harmonic oscillators together with a direct interaction Hamiltonian between the canonical operators of these oscillators. Alternative schemes have subsequently been proposed in [9, 10], but we note that  considers a weaker type of realizability than in [7, 9], i.e., transfer function realizability (cf. section 3), and the results therein limited to a certain sub-class of linear quantum stochastic systems.
From an experimental perspective, direct bilinear interaction Hamiltonians between independent harmonic oscillators are challenging to implement for systems that have more than just a few degrees of freedom and therefore it becomes important to investigate what kind of systems can be realized by a pure cascade connection. A key result of this paper is a necessary and sufficient condition for a linear quantum stochastic system to be realizable by only a cascade connection of one degree of freedom oscillators, without any direct interaction Hamiltonian. Moreover, we also show that the associated transfer functions of all passive linear quantum stochastic systems can always be realized by a cascade connection, proving in general the partial results of  without the additional assumptions made therein.
The organization of this paper is as follows. Section 2 sets up the notations and gives a brief overview of linear quantum stochastic systems. Section 3 defines the synthesis problem and discusses the notions of strict realizability and transfer function realizability. Section 4 derives a necessary and sufficient condition for a linear quantum system to be realizable by a pure cascade connection of one degree of freedom quantum harmonic oscillators, in both the strict and transfer function sense of realizability. Section 5 then introduces the class of passive linear quantum systems and proves that all such systems are transfer functions realizable by a pure cascade connection. Finally, section 6 offers some conclusions of this paper.
We shall use the following notations: , denotes the adjoint of a linear operator as well as the conjugate of a complex number. If then , and , where denotes matrix transposition. and , and denote the identity matrix by whenever its size can be inferred from context and use to denote an identity matrix. Similarly, denotes a matrix with zero entries whose dimensions can be determined from context. denotes a block diagonal matrix with square matrices on its diagonal block, and a block diagonal matrix with the square matrix appearing on its diagonal blocks times.
2.2 The class of linear quantum stochastic systems
In this paper, we will be concerned with a class of quantum stochastic models of open (i.e., quantum systems that can interact with an environment) Markov quantum systems that are widely used and are standard in quantum optics. Such models have been in the physics and mathematical physics literature since the 1980’s, see, e.g., [11, 12, 13, 14, 15]. In particular, we focus on the special sub-class of linear quantum stochastic models, see, e.g., [15, section 6.6], [14, sections 3, 3.4.3, 5.3, chapters 7 and 10], [16, section 4], [17, section 5], [18, 2, 3, 7, 19, 4, 20, 21]. These linear quantum stochastic models describe such quantum optical devices as optical cavities [22, section 5.3.6][23, chapter 7], linear quantum amplifiers [14, chapter 7], and finite bandwidth squeezers [14, chapter 10]. Following the terminology in [2, 3, 7], we shall refer to this class of models as linear quantum stochastic systems.
Suppose we have independent quantum harmonic oscillators labelled . Each oscillator has position and momentum operators and , respectively. The position and momentum operators satisfy the canonical commutation relations , , and , where denotes the Kronecker delta that takes on the value 1 only if , but is otherwise 0. Equivalently, we may describe them in terms of the annihilation and creation operators , with , satisfying the canonical commutation relations , and . The independent oscillators can be coupled to one or more external independent quantum fields, say of them. In a Markov quantum system, the independent fields are essentially quantum noises modelled by bosonic annihilation field operators that can be defined on a separate Fock space (over ) for each field operator [11, 13, 24]. For each there is a corresponding creation field operator that is defined on the same Fock space and is the operator adjoint of , i.e., . The field operators are adapted quantum stochastic processes with forward differentials and that have the quantum Itô products [11, 13, 24]:
More informally, as in the quantum Langevin formalism, we can express and , where for are independent quantum white noise processes satisfying the informal commutation relations and , where , and denotes the Dirac delta function.
Let us collect the position and momentum operators in the column vector defined as . Note that in terms of , we may write the canonical commutation relations as with . We take the composite system of quantum harmonic oscillators to have a quadratic Hamiltonian given by , where is a real symmetric matrix. The oscillators are coupled to the quantum field via the informal singular interaction Hamiltonian [12, 14], where with is a linear coupling operator of the oscillator position and momentum operators to . Collect the coupling operators together in one linear coupling vector , with , and the field operators together as . Then the joint evolution of the oscillators and the quantum fields is given by a unitary adapted process satisfying the Hudson-Parthasarathy quantum stochastic differential equation (QSDE) [11, 13, 24, 25]:
where is a complex unitary matrix (i.e., ) called the scattering matrix, and . The processes for are adapted quantum stochastic processes that are referred to as gauge processes, and the forward differentials have the quantum Itô products:
with all other remaining cross products between and either of , or being zero. Informally, we may express .
For any adapted processes and satisfying a quantum Ito stochastic differential equation, we have the quantum Ito rule . Using the quantum Ito rule and the quantum Ito products given above, as well as exploiting the canonical commutation relations between the operators in , the Heisenberg evolution of the canonical operators in the vector satisfies the quantum stochastic differential equation, see [16, section 4], [17, section 5], [2, 7]:
with , , , and , where is a vector of output fields that results from the interaction of the quantum harmonic oscillators and the incoming quantum fields . Note that the dynamics of is linear, while depends linearly on and . We refer to as the degrees of freedom of the oscillators. If , we shall often refer to the linear quantum stochastic system as a one degree of freedom (open quantum harmonic) oscillator.
Following , we denote a linear quantum stochastic system with Hamiltonian , coupling vector and scattering matrix simply as or . We also recall the concatenation product and series product for open Markov quantum systems  defined by , and . Since both products are associative, the products and are unambiguously defined.
3 Synthesis of linear quantum stochastic systems
The network synthesis problem for linear quantum stochastic systems can be stated (in a strict sense, as explained below) as the problem of how to systematically realize a given linear quantum stochastic system with a given fixed set of matrix parameters from a bin of certain basic quantum optical components; see  for details of these basic components. A particular solution was proposed to the synthesis problem, see [7, Theorem 5.1]: Any linear quantum stochastic system with degrees of freedom can be synthesized via a quantum network consisting of a cascade connection of one degree of freedom harmonic oscillators together with a direct interaction Hamiltonian that is bilinear in the canonical operators of the oscillators. Partition as with and as with . Then according to [7, Theorem 5.1] a system can be decomposed as , where the ’s are (simpler) one degree of freedom open harmonic oscillators () with parameter values specified by the theorem, and is a direct bilinear interaction Hamiltonian of the form . The work  then shows how each of the ’s can be synthesized from the bin of given components and how can be realized. However, in current practical experiments, implementation of can be challenging for systems that have more than just a few degrees of freedom. Therefore, it is of interest to characterize the class of systems that can be synthesized by pure cascade connection alone, that is, with .
As alluded to at the beginning of this section, we emphasize that  considers a strict type of realization problem, that is, it deals with how to synthesize a given and fixed triplet that describes a linear quantum stochastic system . This type of strict realizability is relevant, for instance, in cases where the internal dynamics may represent some (continuous time) quantum information processing algorithm and thus needs to be realized as given. However, for some linear quantum control problems such as robust disturbance attenuation  and LQG synthesis , the internal dynamics are inconsequential. In this case there is freedom to modify/transform these dynamics and what is important is the associated (classical) complex transfer function associated with the system matrices 222As in , here we shall not define the transfer function of quantum systems, but associate to a quantum system with system matrices a classical, doubled-up , transfer function . However, we also remark that can actually be properly interpreted as a genuine transfer function for the quantum system following [26, 21, 27, 1], this being a common practice in the physics community via Fourier transform methods [23, 14]. In any case, we are dealing with the same object and thus the particular interpretation attached to it becomes immaterial for our purpose.. As is well known, a transfer function is invariant under a similarity transformation of the system matrices for any invertible matrix . However, for linear quantum systems the transformation matrix for a similarity transformation is restricted in that it has to be a symplectic matrix: is real and satisfies the condition . This ensures that the transformed variable satisfies the required canonical commutation relations (CCR) of quantum mechanics: , so the system remains physical. Note that the set of all symplectic matrices of a fixed dimension form a group and in particular is again a symplectic matrix. Such a similarity transformation in quantum systems corresponds to replacing with . This motivates us to introduce the following definition:
Let and be two linear quantum stochastic systems. Then is said to be transfer function equivalent to or is a transfer function realization of if and there exists a symplectic matrix such that , (or, equivalently, and ). is then said to be transfer function realizable by , and vice-versa.
It is important to note that two transfer function equivalent systems and will not necessarily generate the same input-output dynamics for all . This is because although they can have different parameters, they always have the same initial value , whilst for quantum systems clearly cannot be zero due to the CCR condition. If the matrix of is Hurwitz then the input-output dynamics of and converge in the limit . However, as remarked earlier, for some linear quantum control design objectives internal dynamics and initial conditions do not play an essential role, only the transfer function does.
4 Conditions for realizability by a pure cascade connection
In this section we state and prove a theorem that characterizes the class of linear quantum stochastic systems that can be realized simply by a cascade connection of one degree of freedom (open quantum harmonic) oscillators. Let us first introduce the following notation: for all , and , and let for so that , where . Moreover, we introduce the following terminology: A square matrix is said to be lower block triangular if it has a lower block triangular form when partitioned into blocks:
where , , is of dimension . We start with the following lemma:
The cascade connection of one degree of freedom harmonic oscillators realizes a linear quantum stochastic system with , where , whenever and whenever . In particular, is lower block triangular.
Proof. The proof proceeds along the lines of the proof of [7, Theorem 5.1]. By the series product formula (cf. section 2.2) for the cascade of two one degree of freedom oscillators and , we get the oscillator . Letting , the latter may be compactly written as: with , and . Repeating the computation for iteratively for and writing and at each iteration , we arrive at the desired result with , , and as stated in the lemma.
To see that is lower block triangular, we note that may be expressed as follows:
Note that since is by definition a Hermitian matrix, the block elements above the diagonal blocks are the Hermitian transpose of the corresponding elements below the diagonal blocks. It follows therefore that the imaginary part of the block at block row and block column must satisfy the relation: . However, from the expression for derived above and its symmetry, we already have that if :
Therefore, the off-diagonal upper block elements of cancel those of when they are summed and we conclude that the matrix is a lower block triangular matrix.
Recall again the partitioning of as with and of as with . We may now state the following result:
A linear quantum stochastic system with degrees of freedom is realizable by a pure cascade of one degree of freedom harmonic oscillators (without a direct interaction Hamiltonian) if and only if the matrix given by is a lower block triangular matrix with blocks of size . If this condition is satisfied then can be explicitly constructed as the cascade connection with , and for .
Proof. The proof of the only if part follows directly from Lemma 3, as follows. If can be realized by a pure cascade connection of one degree of freedom harmonic oscillators then by the lemma, is a lower block triangular matrix. However, since is block diagonal, it follows that the matrix is also a lower block triangular matrix.
Conversely, the if part of the proof can be shown by explicitly constructing a pure cascade connection of one degree oscillators that realizes . If the matrix associated with is lower block triangular then so is the matrix . As we already saw in the proof of Lemma 3, this structure implies that whenever and if . Now, using the notation of Lemma 3, let us define the one degree of freedom harmonic oscillators for as , and for . It follows from Lemma 3 that . That is, this cascade connection realizes .
Theorem 4 has a direct consequence on the weaker notion of transfer function realization of a linear quantum system. The main result is the following corollary:
A linear quantum system is transfer function realizable by a pure cascade connection of one degree of freedom harmonic oscillators if and only if there is a symplectic transformation matrix such that the linear quantum stochastic system has an matrix which is lower block triangular.
5 Passive linear quantum stochastic systems
In this section it will be shown that the class of passive linear quantum stochastic systems (as defined below) are transfer function realizable by a cascade connection. In  it has been shown by a constructive algorithm that a “generic” sub-class of such systems are transfer function realizable by pure cascading, the generic systems being required to satisfy assumptions on the distinctness of the eigenvalues and invertibility of certain matrices. In this section we remove such assumptions, and show by exploiting the algebraic structure of passive systems that the result is valid in general for all passive systems.
For , let be the annihilation operators for mode and define . Then satisfies the CCR Moreover, note that with
We also make note that and from the relation we have the identities:
Therefore, we also have
A system is said to be passive if we can write and for some complex Hermitian matrix , a complex (here again denotes the number of input and output fields in and out of ) matrix , and some real constant . As discussed in , here the term passive for such systems is physically motivated since they can be implemented using only passive components like optical cavities, mirrors, beamsplitters and phase shifters; this follows from Theorem 5.1 of  and the constructions shown in section 6 of that paper. Also shown in , we can express and in the form and . Therefore, we may set and . Note also from  that the block diagonal elements is of the form for some for all . Now we shall derive some properties of and show that there exists a unitary and symplectic matrix that transforms it into a lower block triangular matrix.
, where .
Proof. For the proof, we exploit the identities (2) as well as the following easily verified identities: . Using these identities, we have the following:
Then we have the following theorem:
Let be the complex unitary matrix in a Schur decomposition of the matrix of Lemma 6: , where is a lower triangular matrix. Then the matrix
is a real, unitary, and symplectic matrix that transforms into a lower block triangular matrix: , where is a real lower block triangular matrix. Therefore, every passive linear quantum system has a transfer function realization by pure cascading and such a realization is obtained by applying the construction of Theorem 4 to . Moreover, each of the one degree of freedom oscillator in the cascade will also be passive.
Proof. The existence of is guaranteed by the well known result that every complex matrix has a Schur decomposition of the form with lower triangular. Note then that we also have . Let be as defined in the theorem. Then by Lemma 6 the following is true:
Now, since is a lower triangular matrix, it follows by inspection (using the special structure of ) that is a lower block diagonal matrix, as claimed. That is real follows from the fact that we may write . That it is unitary follows from the observation that and are unitary (as a consequence of (2)) and that is also unitary. To see that is also symplectic define and . By the unitarity of we have that and satisfy the same the commutation relations as and (i.e., is again an annihilation operator). Then we have
Now, this implies that consists of the canonical position and momentum operators associated with the modes in and satisfies the same CCR as . But since and is real, preservation of the CCR implies that is necessarily a symplectic matrix (this is standard knowledge in quantum mechanics; see, e.g., [28, section III]).
Using the fact that established above, it follows from Theorem 4 that the passive quantum system is transfer function equivalent to whose matrix is lower block triangular. Let and . By Theorem 4 we have with , and for . We now show that each is passive. Recall that and write with . Using (2) we have that . By expanding both sides of the equality and collecting and equating terms of the same index, it follows that , where is the -th block component of . On the other hand, since is passive we have that