A Proof of Theorem 5

Synthesis of linear quantum stochastic systems via quantum feedback networks1

Abstract

Recent theoretical and experimental investigations of coherent feedback control, the feedback control of a quantum system with another quantum system, has raised the important problem of how to synthesize a class of quantum systems, called the class of linear quantum stochastic systems, from basic quantum optical components and devices in a systematic way. The synthesis theory sought in this case can be naturally viewed as a quantum analogue of linear electrical network synthesis theory and as such has potential for applications beyond the realization of coherent feedback controllers. In earlier work, Nurdin, James and Doherty have established 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 perspective and based on current methods and technologies, direct interaction Hamiltonians are challenging to implement for systems with more than just a few degrees of freedom. In order to facilitate more tractable physical realizations of these systems, this paper develops a new synthesis algorithm for linear quantum stochastic systems that relies solely on field-mediated interactions, including in implementation of the direct interaction Hamiltonian. Explicit synthesis examples are provided to illustrate the realization of two degrees of freedom linear quantum stochastic systems using the new algorithm.

1 Introduction

Linear quantum stochastic systems (see, e.g., [1, 2, 3, 4]) arise as an important class of models in quantum optics [5] and in phenomenological models of quantum RLC circuits [6]. They are used, for instance, to model optical cavities driven by coherent laser sources and are of interest for applications in quantum information science. In particular, they have the potential as a platform for realization of entanglement networks [7, 8, 9] and to function as sub-sytems of a continuous variable quantum information system (e.g., the scheme of [10]). Recently, there has also been interest in the possibilities of control of linear quantum stochastic systems with a controller which is a quantum system of the same type [11, 2, 3, 12] (thus involving no measurements of quantum signals), often referred to as “coherent-feedback control”, and an experimental realization of a coherent-feedback control system for broadband disturbance attenuation has been successfully demonstrated by Mabuchi [12]. Linear quantum stochastic systems are a particularly attractive class of quantum systems to study for coherent control because of their simple structure and complete parametrization by a number of matrix parameters.

A natural and important question that arose out of the studies on coherent control is how an arbitrary linear quantum stochastic system can be built or synthesized in a systematic way, in the quantum optical domain, from a bin of quantum optical components like beam splitters, phase shifters, optical cavities, squeezers, etc; this can be viewed as being analogous to the question in electrical network synthesis theory of how to synthesize linear analog circuits from basic electrical components like resistors, capacitors, inductors, op-amps, etc. The synthesis problem is not only of interest for quantum control, but is a timely subject in its own right given the current intense research efforts in quantum information science (see, e.g., [13]). These developments present significant opportunities for investigations of a network synthesis theory (e.g., [14] for linear electrical networks) in the quantum domain as a significant direction for future development of circuit and systems theory. In particular, such a quantum synthesis theory may be especially relevant for the theoretical foundations, development and design of future linear photonic integrated circuits.

As a first step in addressing the quantum synthesis question, Nurdin, James and Doherty [4] have shown that any linear quantum stochastic system can, in principle, be synthesized by a cascade of simpler one degree of freedom quantum harmonic oscillators together with a direct interaction Hamiltonian between the canonical operators of these oscillators. Then they also showed how these one degree of freedom harmonic oscillators and direct bilinear interaction Hamiltonian can be synthesized from various quantum optical components. However, from an experimental point of view, direct bilinear interaction Hamiltonians between independent harmonic oscillators are challenging to implement experimentally with current technology for systems that have more than just a few degrees of freedom (possibly requiring some complex spatial arrangement and orientation of the oscillators) and therefore it becomes important to investigate approximate methods for implementing this kind of interaction. Here we propose such a method by exploiting the recent theory of quantum feedback networks [15]. In our scheme, the direct interaction Hamiltonians are approximately realized by appropriate field interconnections among oscillators. The approximation is based on the assumption that the time delays required in establishing field interconnections are vanishingly small, which is typically the case in quantum optical systems where quantum fields propagate at the speed of light.

The paper is organized as follows. Section 2 provides a brief overview of linear quantum stochastic systems and the concatenation and series product of such sytems. Section 3 recalls the notion of a model matrix and the concatenation of model matrices, while Section 4 recalls the notion of edges, their elimination, and reduced Markov models. Section 5 reviews a prior synthesis result from [4]. Section 6 presents the main results of this paper and an example to illustrate their application to the realization of a two degrees of freedom linear quantum stochastic system. In Section 7 the special class of passive linear quantum stochastic systems is introduced and it is shown that such systems can be synthesized by using only passive sub-systems and components. Another synthesis example for a passive system is also presented. Finally, Section 8 offers the conclusions of this paper. In order to focus on the results, all proofs are collected together in the Appendix.

2 Linear quantum stochastic systems

This section serves to recall some notions and results on linear quantum stochastic systems that are pertinent for the present paper. A relatively detailed overview of linear quantum stochastic systems can be found in [4] and further discussions in [1, 2, 16, 3], thus they will not be repeated here.

Throughout the paper we shall use the following notations: , will denote the adjoint of a linear operator as well as the conjugate of a complex number, if is a matrix of linear operators or complex numbers then , and is defined as , where denotes matrix transposition. We also define, 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, while denotes a matrix with specified dimension with zero entries. Other useful notations that we shall employ is (with square matrices) to denote a block diagonal matrix with on its diagonal block, and ( a square matrix) denotes a block diagonal matrix with the matrix appearing on its diagonal blocks times.

Let be the canonical position and momentum operators of a many degrees of freedom quantum harmonic oscillator satisfying the canonical commutation relations (CCR) . The integer will be referred to as the degrees of freedom of the oscillator. Letting then these commutation relations can be written compactly as:

with and . Here a linear quantum stochastic system is a quantum system defined by three parameters: (i) A quadratic Hamiltonian with , (ii) a coupling operator where is an complex matrix and (iii) a unitary scattering matrix . We also assume that the system oscillator is in an initial state with density operator . For shorthand, we write or . The time evolution, in the interaction picture, of () is given by the quantum stochastic differential equation (QSDE):

(1)

with

where is a vector of input vacuum bosonic noise fields and is a vector of output fields that results from the interaction of with the harmonic oscillator. Note that the dynamics of and are linear. We refer to as the system matrices of . For the case when , we shall often refer to the linear quantum stochastic system as a one degree of freedom (open quantum harmonic) oscillator.

Elements of and may be partitioned into blocks. For example, may be partitioned as and as , where and are vectors of bosonic input and output field operators of length and , respectively, such that . We refer to and as the multiplicity of and , respectively. It is important to keep in mind that the sum of the multiplicities of all input and output partitions sum up to , the total number of all input and output fields. With this partitioning, a linear quantum stochastic system may be viewed as a quantum device having input ports and output ports as illustrated in Figure 1. The multiplicity of a port is then defined as the multiplicity of the input or output fields coming into or going out of that port.

Figure 1: A linear quantum stochastic system viewed as a device with input ports and output ports.

Let us also recall the notion of the concatenation () and series products () between open quantum systems developed in [17]. The concatenation product between and defines another quantum linear system given by:

while if and have the same number of input (and output) fields the series product defines another linear quantum system given by:

Note that in the definition of the concatenation and series product, it is not required that the elements of and and of and are commuting. That is, and may possibly be sub-components of the same system. Also both operations are associative, but neither operations are commutative. By the associativity, the operations and are unambiguously defined.

The concatenation product corresponds to collecting together the parameters of and to form one larger concatenated system, and the series product is a mathematical abstraction of the physical operation of cascading onto , that is, passing the output fields of as the input fields to . The cascaded system is then another linear quantum stochastic system with parameters given by the series product formula.

3 The model matrix and concatenation of model matrices

The system can also be represented using a so-called model matrix [15]. This representation will be particularly useful for the goals of the present paper. For the model matrix representation is given by (the partitioned matrix):

(2)

with the understanding that if is partitioned as with and and is partitioned accordingly as with and , then the model matrix above can be expressed with respect to this partition as:

(3)

Also, with respect to a particular partitioning of , it is convenient to attach a unique label to each row and column of the partition. For example, for the partitioning (3) we may give the labels for the first, second, …, -th row of , respectively, and for the first, second, …, -th column of , respectively. Moreover, with respect to this labelling (and analogously for any other labelling scheme chosen), elements of the blocks are denoted as:

Since is another representation of a physical system described by , and can be identified directly from the entries of , we will often omit the and for brevity write a model matrix simply as and denote its entries by , with ranging over row labels and ranging over the column labels. Thus, we also refer to the triple in (2) as parameters of the model matrix .

Several model matrices can be concatenated to form a larger model matrix. Such a concatenation corresponds to collecting together the model parameters of the individual matrices in a larger model matrix and is again denoted by the symbol . If and then the concatenation is defined as:

4 Edges, Elimination of Edges, and Reduced Markov Models

Following [15], a particular row partition labelled with in a model matrix can be associated with an output port (having multiplicity ) while a particular column partition with can be associated with an input port (having multiplicity ). In a system which is the concatenation of several sub-systems, it is possible to connect an output port from one sub-system to an input port of another sub-system (possibly the same sub-system to which the output port belongs) to form what is called an internal edge denoted by . For this connection to be possible, the ports and must have the same multiplicity. Such an edge then represents a channel from port to port . All ports which are connected to other ports to form an internal edge or channel are referred to as internal ports and fields coming into or leaving such ports are called internal fields. All other input and output ports that are not connected in this way are viewed as having semi-infinite edges (since they do not terminate at some input or output port, as appropriate) and are referred to as external ports and the associated semi-infinite edges are referred to as external edges. Fields coming into or leaving external ports are called external fields. From a point of view in line with circuit theory, one may think of a linear quantum stochastic system as being a “node” on a network and quantum fields as quantum “wires” that can connect different nodes.

In any internal edge , there is a finite delay present due to the time which is required for the signal from port to travel to port . As a consequence of these finite time delays, concatenated systems with internal edges cannot be represented by a Markov model such as presented in Section 2 (see [4] and the references therein for an overview of Markov models). However, as shown in [15], the non-Markov model converges to a reduced Markov model in the limit that the time delay on all internal edges go to zero. That is, for negligibly small time delays, the reduced Markov model acts as an approximation of the non-Markov model. In particular, such a reduced model serves as a powerful approximation of quantum optical networks in which signals travel at the speed of light and the time delay can be considered to be practically zero if the internal input and output ports are not extremely far apart. We recall the following results:

Theorem 1

[15, Theorem 12 and Lemma 16] Let , , be the time delay for an internal edge and assume that is invertible. Then in the limit that , with the edge connected reduces to a simplified model matrix with input ports labelled and output ports labelled (i.e., the connected ports and are removed from the labelling and the associated row and column removed from ). The block entries of are given by:

with and . is the model matrix of a linear quantum stochastic system with parameters:

for all and .

Several internal edges may be eliminated one at a time in a sequence leading to a corresponding sequence of reduced model matrices. The following result shows that such a procedure is unambiguous:

Theorem 2

[15, Lemma 17] The reduced model matrix obtained after eliminating all the internal edges in a set of internal edges one at a time is independent of the sequence in which these edges are eliminated.

Suppose that can be partitioned as:

(4)

where the subscript refers to “internal” and to “external”. That is, parameters with subscript or pertain to internal ports, those with subscript or pertain to external ports, while and pertain to scattering of internal fields to external fields and vice-versa, respectively. Interconnection among internal input and output ports can then be conveniently encoded by a so-called adjacency matrix. Let denote the total multiplicity of internal input and output ports and let us view a port with multiplicity as distinct ports of multiplicity . Suppose that these multiplicity 1 ports are numbered consecutively starting from 1, then an adjacency matrix is an square matrix whose entries are either or with ( only if the -th output port and the -th input port form a channel or internal edge. Note that at most only a single element in any row or column of can take the value 1. Internal edges can be simultaneously eliminated as follows:

Theorem 3

[15, Section 5] Suppose that has a partitioning based on internal and external components as in (4) and that connections between internal ports have been encoded in an adjacency matrix . If exists, then the reduced model matrix after simultaneous elimination of all internal edges has the parameters:

5 Prior work

Let be a linear quantum stochastic system with and (). Let so that and satisfies the commutation relations of Section 2. Then the following result holds:

Theorem 4

[4] Let . Then

where and .

Figure 2: Realization of via a cascade connection of one degree of freedom quantum harmonic oscillators and bilinear direct interaction Hamiltonians between every pair of oscillators

The theorem says that can in principle be constructed as the cascade connection of the one degree of freedom oscillators , together with a direct bilinear interaction Hamiltonian which is the sum of bilinear interaction Hamiltonians between each pair of oscillator and . This construction is depicted in Figure 2. It was then shown how each can be constructed from certain basic quantum optical components and how can be implemented between any pair of oscillators. However, a drawback of this approach, based on what is feasible with current technology, is the challenging nature of implementing . This may possibly involve complex positioning and orientation of the oscillators and thus practically challenging for systems with more than just a few degrees of freedom. Although advances in experimental methods and emergence of new technologies may eventually alleviate this difficulty, it is naturally of immediate interest to also explore alternative methods of implementing this interaction Hamiltonian, at least approximately. In the next section, we propose such an alternative synthesis by exploiting the theory of quantum feedback networks that has been elaborated upon in preceding sections of the paper.

6 Main synthesis results

For , let for , and , with , and , , and with . Here is as defined in the previous section. Let for , and note that with , , and as already defined.

Figure 3: The concatenation decomposition of as .

Consider now the model matrix for the concatenated system , see Figure 3. With respect to the natural partitioning of induced by the ’s (via their concatenation), we label the first rows of as , the next rows as , the rows after that and so on until the last rows are labelled . Similarly, we label the first columns of as , the next columns as , the columns after that and so on until the last columns . On occasions, we will need to write a bracket around one of both of the subscripts of or (e.g., as in or ).

Theorem 5

Let the output port be connected to the input port to form an internal edge/channel for all . Assuming that is invertible , then the reduced model matrix obtained by allowing the delays in all internal edges go to zero has parameters given by:

The proof of the theorem is given in Appendix A. We will also exploit the following lemma:

Lemma 6

For any real matrix and unitary complex numbers and satisfying , there exist complex matrices and such that . In fact, a pair and satisfying this is given by with an arbitrary non-zero real number and , where . Or, alternatively, and
.

See Appendix B for a proof of the lemma. As a consequence of the above theorem and lemma, we have the following result:

Corollary 7

Let whenever , and for all . Also, let , and with satisfying , and ( and given), where , and the pair () be given by:

(5)

or

(6)

where , and is an arbitrary non-zero real constant for all . Then the reduced Markov model has the decomposition with and for . Moreover, the network formed by forming the series product of within the concatenated system and defined by is a linear quantum stochastic system with parameters given by:

In other words, realizes a linear quantum stochastic system with the above parameters.

Remark 8

Note that the series connection can be viewed as forming and futher eliminating the internal edges in or and, hence, is in essence a special case of a reduced Markov model [15].

The proof of the corollary is given in Appendix C. The corollary shows that an arbitrary linear quantum stochastic system can be realized by a quantum network constructed according to Theorem 5 and the corollary, with an appropriate choice of the parameters , and (). From here, any system can then be easily obtained as . That is, as a cascade of a static network that realizes the unitary scattering matrix and the system (see [17, 4]).

Example 9

Consider the realization problem of a two degrees of freedom linear quantum stochastic system () with

Let and . Define and , with certain parameters still to be determined. Choose and , so that as required in Corollary 7. Then set , , and . Compute and set . Then by Corollary 7 we set and compute , and . Thus, we have determined all the parameters of and . Labelling the ports of and according to the convention adopted in this section, can be implemented by concatenating and and eliminating the internal edges and to form and then eliminating the edge (cf. Remark 8) to obtain as an approximate realization of . This realization is illustrated in Figure 4. and can then be physical realized in the quantum optics domain following the constructions proposed in [4]. Using the schematic symbols of [4], a quantum optical circuit that is a physical realization of is depicted in Figure 5.

Figure 4: Realization of via a quantum feedback network. In the top figure the internal edges and are formed and eliminated to obtain a reduced Markov model . Then in the bottom figure a series (cascade) connection is formed by the eliminating the internal edge to realize
Figure 5: A quantum optical circuit that realizes according to the quantum feedback network in Figure 4; the schematic symbols are taken from [4]. Here is cavity mode of the optical cavity around which the physical realization of the oscillator is based, . The dashed box labelled is the part of the circuit realizing , and the input and output ports of are indicated by their respective labels. The value of the parameters of devices in this circuit can be determined according to [4].

7 Synthesis of passive systems

Let us now consider a special class of linear quantum stochastic systems that we shall refer to as passive linear quantum stochastic systems, for a reason that is explained below, and show that any such system can be built from passive components. This class of systems has also been considered in, e.g., [18, 19].

For let be the annihilation operators for mode and define . Then satisfies the CCR

Moreover, note that: