The Series Product and Its Application to Quantum Feedforward and Feedback Networks
aa r X i v : . [ qu a n t - ph ] A p r The Series Product and Its Application toQuantum Feedforward and Feedback Networks
John Gough ∗ Matthew R. James † September 17, 2018
Abstract
The purpose of this paper is to present simple and general alge-braic methods for describing series connections in quantum networks.These methods build on and generalize existing methods for series (orcascade) connections by allowing for more general interfaces, and byintroducing an efficient algebraic tool, the series product. We alsointroduce another product, which we call the concatenation product,that is useful for assembling and representing systems without neces-sarily having connections. We show how the concatenation and seriesproducts can be used to describe feedforward and feedback networks.A selection of examples from the quantum control literature are ana-lyzed to illustrate the utility of our network modeling methodology.
Keywords:
Quantum control, quantum networks, series, cascade,feedforward, feedback, quantum noise.
Engineers routinely use a wide range of methods and tools to help themanalyze and design control systems. For instance, control engineers often ∗ J. Gough is with the Institute for Mathematical and Physical Sciences, University ofWales, Aberystwyth, Ceredigion, SY23 3BZ, Wales † M.R. James is with the Department of Engineering, Australian National University,Canberra, ACT 0200, Australia. [email protected]. Research supported by theAustralian Research Council. G ( s ) = G ( s ) G ( s ) which is the product of the transfer functionsof the components. The description can also be expressed in the time do-main in terms of the state space parameters G = ( A, B, C, D ) (as we brieflyreview in section 2). The series connection has an algebraic character, andcan be regarded as a product, G = G ⊳ G . Because of new imperativesconcerning quantum network analysis and design, in particular, quantumfeedback control, [24], [25], [18], [23], [26], [4], [17], [12] the purpose of thispaper is to present simple and general algebraic methods for describing seriesconnections in quantum networks. u = y ✲ ✲ ✲ G G G u y Figure 1: Series connection of two (classical, or non-quantum) linear systems,denoted G = G ⊳ G .The types of quantum networks we consider include those arising in quan-tum optics, such as the optical network shown in Figure 2. This networkconsists of a pair of optical cavities (discussed in subsection 3.2) connectedin series by a light beam which serves as an optical interconnect or quantum“wire”. In this paper (section 5) we show how series connections of quantumcomponents such as this may be described as a series product G = G ⊳ G .This product is defined in terms of system parameters G = ( S , L , H ), where H specifies the internal energy of the system, and I = ( S , L ) specifies theinterface of the system to external field channels (as explained in subsection3.4 and section 4). 2 nput beamopticalinterconnect outputbeam mirrorisolator cavity 1 cavity 2 G G G Figure 2: Series connection of two optical cavities via an optical interconnect(light beam) or quantum “wire”, denoted G = G ⊳ G . Each cavity consistsof a pair of mirrors, one of which is perfectly reflecting (shown solid) while theother is partially transmitting (shown unfilled). The partially transmittingmirror enables the light mode inside the cavity to interact with an externallight field, such as a laser beam. The external field is separated into inputand output components by a Faraday isolator. The optical interconnect isformed when light from the output of one cavity is directed into the input ofthe other, here using an additional mirror.Series (also called cascade) connections of quantum optical componentswere first considered in the papers [6], [3], and certain linear feedback net-works were considered in [26]. Our results extend the series connection resultsin these works by including more general interfaces, and by introducing anefficient algebraic tool, the series product. We also introduce another prod-uct, which we call the concatenation product G = G ⊞ G , that is useful forassembling and representing systems without necessarily having connections.Both products may be used to describe a wide range of open quantum phys-ical systems (including those with physical variables that evolve nonlinearly)and networks of such systems (with boson field interconnects such as opticalbeams or phonon vibrations in materials). We believe our modeling frame-work is of fundamental system-theoretic interest. The need for general and3fficient methods for describing networks of quantum components has beenrecognized to some extent and has begun to emerge in the quantum optics andquantum information and computing literature, e.g. [27], [6], [3], [8, Chap-ter 12], [20, Chapter 4], [26], [4]. It is expected that an effective quantumnetwork theory will assist the design of quantum technologies, just as electri-cal network theory and block diagram manipulations help engineers designfilters, control systems, and many other classical electrical systems.Series connections provide the foundation for some important develop-ments in quantum feedback control, e.g. [24], [25], [23], [26], [17], [12], [13].To illustrate the power and utility of our quantum network modeling method-ology, we analyze several examples from this literature. The series and con-catenation products allow us to express these quantum feedback control andquantum filtering examples in a simple, transparent way (there are somesubtle technical issues in some of the examples for which we provide expla-nation and references). We hope this will help open up some of the quantumfeedback control literature to control engineers, which at present is largelyunknown outside the physics community. A number of articles and booksare available to help readers with the background material on which thepresent paper is based. The papers [26] and [22] provide excellent introduc-tions to aspects of the quantum models we use. The paper [2] is a tutorialarticle written to assist control theorists and engineers by providing intro-ductory discussions of quantum mechanics, open quantum stochastic models,and quantum filtering. The book [8] is an invaluable resource for quantumnoise models and quantum optics, while the book [21] provides a detailedmathematical treatment of the Hudson-Parthasarathy theory of the quan-tum stochastic calculus. The book [19] is a standard textbook on quantummechanics.We begin in section 2 by discussing an analog of our results in the contextof classical linear systems theory, elaborating further on the discussion atthe beginning of this section. In section 3 we provide a review of someexample quantum components (including the cavity mentioned above) andconnections. This section includes a brief discussion of quantum mechanics,introduces examples of parametric representations, and provides a glimpseof how the general theory can be used. Open quantum stochastic modelsare described in more detail in section 4. The main definitions and resultsconcerning the concatenation and series products are given in section 5; inparticular, the principle of series connections , Theorem 5.5. In general theseries product is not commutative, but we are able to show how the order4an be interchanged by modifying one of the components, Theorem 5.6. Aselection of examples from the quantum control literature are analyzed insection 6. The appendices contain proofs of some of the results and someadditional technical material. Notation.
In this paper we use matrices M = { m ij } with entries m ij that are operators on an underlying Hilbert space. The asterisk ∗ is used toindicate the Hilbert space adjoint A ∗ of an operator A , as well as the complexconjugate z ∗ = x − iy of a complex number z = x + iy (here, i = √− x, y are real). Real and imaginary parts are denoted Re( z ) = ( z + z ∗ ) / z ) = − i ( z − z ∗ ) / M † of a matrix M is defined by M † = { m ∗ ji } . Also defined are the conjugate M ♯ = { m ∗ ij } and transpose M T = { m ji } matrices, so that M † = ( M T ) ♯ = ( M ♯ ) T . In thephysics literature, it is common to use the dagger † to indicate the Hilbertspace adjoint. The commutator of two operators A, B is defined by [
A, B ] = AB − BA . δ ( · ) is the Dirac delta function, and δ jk is the Kronecker delta.The tensor product of operators A , B defined on Hilbert spaces H , G is anoperator A ⊗ B defined on the Hilbert space H ⊗ G (tensor product of Hilbertspaces) defined by ( A ⊗ B )( ψ ⊗ φ ) = ( Aψ ) ⊗ ( Bφ ) for ψ ∈ H , φ ∈ G ; weusually follow the standard shorthand and write simply AB = A ⊗ B for thetensor product, and also A = A ⊗ I and B = I ⊗ B . As mentioned in the Introduction (section 1), it is common practice in clas-sical linear control theory to perform manipulations of block diagrams. Suchmanipulations, of course, greatly assist the analysis and design of control sys-tems. To assist readers in interpreting the main quantum results concerningseries and concatenation products (section 5), we describe concatenation andseries products for familiar classical linear systems in algebraic terms.Consider two classical deterministic linear state space models˙ x j = A j x j + B j u j y j = C j x j + D j u j (1)where j = 1 ,
2. As usual, x j , u j and y j are vectors and A j , B j , C j and D j are appropriately sized matrices. These systems are often represented by the5atrix G j = (cid:18) A j B j C j D j (cid:19) , (2)or the transfer function G j ( s ) = C j ( sI − A j ) − B j + D j .In modeling networks of such systems, one may form the concatenationproduct G = G ⊞ G = (cid:18) A A (cid:19) (cid:18) B B (cid:19)(cid:18) C C (cid:19) (cid:18) D D (cid:19) , see Figure 3. In terms of transfer functions, the concatenation of two sys-tems is G ( s ) = diag {G ( s ) , G ( s ) } . The concatenation product simply assem-bles the two components together, without making any connections betweenthem. It is not a parallel connection. u ✲✲ ✲✲ G G G y y u Figure 3: Concatenation product.Of considerable importance is the series connection, described by seriesproduct G = G ⊳ G = (cid:18) A B C A (cid:19) (cid:18) B B D (cid:19)(cid:0) D C C (cid:1) D D , see Figure 1. Here the connection is specified by u = y , and so we requiredim u =dim y . In the frequency domain, the series product is given by the6atrix transfer function product G ( s ) = G ( s ) G ( s ). This product describesa series (or cascade) connection which is fundamental to feedforward andfeedback control.Notice that both products are defined in terms of system parameters(state space parameters or transfer function matrices). Central to quantum mechanics are the notions of observables X , which aremathematical representations of physical quantities that can (in principle)be measured, and state vectors ψ , which summarize the status of physicalsystems and permit the calculation of expectations of observables. Statevectors may be described mathematically as elements of a Hilbert space H ,while observables are self-adjoint operators on H . The expected value of anobservable X when in state ψ is given by the inner product h ψ, Xψ i .A basic example is that of a particle moving in a potential well, [19,Chapter 14]. The position and momentum of the particle are represented byobservables Q and P , respectively, defined by( Qψ )( q ) = qψ ( q ) , ( P ψ )( q ) = − i ~ ddq ψ ( q )for ψ ∈ H = L ( R ). Here, i = √− ~ = h/ π , h is Planck’s constant, and q ∈ R represents position values. In following subsections we use units suchthat ~ = 1, but retain it in our expressions in this subsection. The positionand momentum operators satisfy the commutation relation [ Q, P ] = i ~ . Thedynamics of the particle is given by Schr¨odinger’s equation i ~ ddt V ( t ) = HV ( t ) , with initial condition V (0) = I , where H = P m + mω Q is the Hamiltonian (here, m is the mass of the particle, and ω is the frequency of oscillation). Theoperator V ( t ) is unitary ( V ∗ ( t ) V ( t ) = V ( t ) V ∗ ( t ) = I , where I is the identityoperator, and the asterisk denotes Hilbert space adjoint)—it is analogous tothe transition matrix in classical linear systems theory. State vectors and7bservables evolve according to ψ t = V ( t ) ψ ∈ H , X ( t ) = V ∗ ( t ) XV ( t ) . These expressions provide two equivalent descriptions (dual), the former isreferred to as the
Schr¨odinger picture , while the latter is the
Heisenbergpicture . In this paper we use the Heisenberg picture, which is more closelyrelated to models used in classical control theory and classical probabilitytheory. In the Heisenberg picture, observables (and more generally otheroperators on H ) evolve according to ddt X ( t ) = − i ~ [ X ( t ) , H ( t )] , (3)where H ( t ) = V ∗ ( t ) HV ( t ).Energy eigenvectors ψ n are defined by the equation Hψ n = E n ψ n for realnumbers E n . The system has a discrete energy spectrum E n = ( n + ) ~ ω , n = 0 , , , . . . . The state ψ corresponding to E is called the ground state .The annihilation operator a = r mω ~ ( Q + i P mω )and the creation operator a ∗ lower and raise energy levels, respectively: aψ n = √ nψ n − , and a ∗ ψ n = √ n + 1 ψ n +1 . They satisfy the canonical com-mutation relation [ a, a ∗ ] = 1. In terms of these operators, the Hamiltoniancan be expressed as H = ~ ω ( a ∗ a + ) . Using (3), the annihilation operatorevolves according to ddt a ( t ) = − iωa ( t ) (4)with solution a ( t ) = e − iωt a . Note that also a ∗ ( t ) = e iωt a ∗ , and so commuta-tion relations are preserved by the unitary dynamics: [ a ( t ) , a ∗ ( t )] = [ a, a ∗ ] =1. Because of the oscillatory nature of the dynamics, this system is oftenrefereed to as the quantum harmonic oscillator .It can be seen that the Hamiltonian H is a key “parameter” of the quan-tum physical system, specifying its energy. A diagram of an optical cavity is shown in Figures 4, 5, together with a sim-plified representation. It consists of a pair of mirrors; the left one is partially8ransmitting (shown unfilled), while the right mirror is assumed perfectlyreflecting (shown solid). Between the mirrors a trapped electromagnetic (op-tical) mode is set up, whose frequency depends on the separation between themirrors. This mode is described by a harmonic oscillator with annihilationoperator a (an operator acting on a Hilbert space H (as in subsection 3.1),called the initial space). The partially transmitting mirror affords the oppor-tunity for this mode to interact with an external free field, represented by aquantum stochastic process b ( t ) (to be discussed shortly). When the externalfield is in the vacuum state, energy initially inside the cavity mode may leakout, in which case the cavity system is a damped harmonic oscillator, [8]. isolatorinput beam beamoutput partiallytransmittingmirror reflectingmirrorcavity B ˜ B Figure 4: A cavity consists of a pair of mirrors, one of which is perfectlyreflecting (shown solid) while the other is partially transmitting (shown un-filled). The partially transmitting mirror enables the light mode inside thecavity to interact with an external light field, such as a laser beam. Theexternal field is separated into input and output components by a Faradayisolator.Quantization of a (free) electromagnetic field leads to an expression forthe vector potential A ( x, t ) = Z κ ( ω )[ b ( ω ) e − iωt + iωx/c + b ∗ ( ω ) e iωt − iωx/c ] dω, for a suitable coefficients κ ( ω ), and annihilation operators b ( ω ). Such a fieldcan be considered as an infinite collection of harmonic oscillators, satisfyingthe singular canonical commutation relations[ b ( ω ) , b ∗ ( ω ′ )] = δ ( ω − ω ′ ) , where δ is the Dirac delta function. 9 B ˜ B Figure 5: A simplified representation of the cavity from Figure 4 which omitsthe Faraday isolator. It shows input B and output ˜ B fields and the cavitymode annihilation operator a . This representation will be used for the re-mainder of this paper.An optical signal, such as a laser beam, is a free field with frequency con-tent concentrated at a very high frequency ω ≈ rad/sec. The fluctua-tions about this nominal frequency can be considered as a quantum stochasticprocess consisting of signal plus noise, where the noise is of high bandwidthrelative to the signal. Indeed, a coherent field is a good, approximate, modelof a laser beam, and can be considered as the sum b ( t ) = s ( t ) + b ( t ), where s ( t ) is a signal, and b ( t ) is quantum (vacuum) noise. Such “signal plusnoise” models are of course common in engineering.The cavity mode-free field system has a natural input-output structure,where the free field is decomposed as a superposition of right and left travel-ing fields. The right traveling field component is regarded as the input , whilethe left traveling component is an output , containing information about thecavity mode after interaction. The interaction facilitated by the partiallytransmitting mirror provides a boundary condition for the fields. The twocomponents can be separated in the laboratory using a Faraday isolator. Thisleads to idealized models based on rotating wave and Markovian approxima-tions, where, in the time domain, the input optical field (when in the groundor vacuum state) is described by quantum white noise b ( t ) = b ( t ) [8, Chap-ters 5 and 11], which satisfies the singular canonical commutation relations[ b ( t ) , b ∗ ( t ′ )] = δ ( t − t ′ ) . (5)In order to accommodate such singular processes, rigorous white noise andIt¯o frameworks have been developed, where in the It¯o theory one uses the10ntegrated noise, informally written B ( t ) = Z t b ( s ) ds. The operators B ( t ) are defined on a particular Hilbert space called a Fockspace , F , [21, sec. 19]. When the field is in the vacuum (or ground) state,this is the quantum Wiener process which satisfies the It¯o rule dB ( t ) dB ∗ ( t ) = dt (all other It¯o products are zero). Field quadratures, such as B ( t ) + B ∗ ( t ) and − i ( B ( t ) − B ∗ ( t )) are each equivalent to classical Wiener processes, but do notcommute. A field quadrature can be measured using homodyne detection, [8,Chapter 8].The cavity mode-free field system can be described by the Hamiltonian H = ∆ a ∗ a − i ~ Z k ( ω )( a ∗ b ( ω ) − b ∗ ( ω ) a ) dω, (6)where the first term represents the self-energy of the cavity mode (the number∆ is called the “detuning”, and represents the difference between the nominalexternal field frequency and the cavity mode frequency), while the remainingtwo terms describe the energy flow between the cavity mode and the freefield (a photon in the free field may be created by a loss of a photon from thecavity mode, and vice versa). This Hamiltonian is defined on the compositeHilbert space, the tensor product H ⊗ F ; the tensor product is not writtenexplicitly in the expression (6).The Schr¨odinger equation for the cavity-free field system is derived from(6) under certain assumptions [8], and is given by the It¯o quantum stochasticdifferential equation (QSDE) dV ( t ) = {√ γadB ∗ ( t ) − √ γa ∗ dB ( t ) − γ a ∗ adt − i ∆ a ∗ adt } V ( t ) , (7)with vacuum input and initial condition V (0) = I , so that V ( t ) is unitary.The complete cavity mode-free field system thus has a unitary model. In theHeisenberg picture, cavity mode operators X (operators on the initial space11 ) evolve according the quantum It¯o equation dX ( t ) = − i ∆[ X ( t ) , a ∗ ( t ) a ( t )] dt (8)+ γ a ∗ ( t )[ X ( t ) , a ( t )] + [ a ∗ ( t ) , X ( t )] a ( t )) dt + √ γdB ∗ ( t )[ X ( t ) , a ( t )] + √ γ [ a ∗ ( t ) , X ( t )] dB ( t ) . Here, γ > k ( ω ) in the Hamiltonian (6). In particular,for X = a , the cavity mode annihilation operator, we have da ( t ) = − ( γ i ∆) a ( t ) dt − √ γ dB ( t ); (9)cf. (4). The output field ˜ B ( t ) is given by d ˜ B ( t ) = √ γ a ( t ) dt + dB ( t ) , (10)where one can see the “signal plus noise” form of the field.This is an example of an open quantum system , characterized by theparameters √ γa and ∆ a ∗ a ; the latter being the cavity mode Hamiltonian(specifying internal energy), and the former being the operator coupling thecavity mode to the external field (specifying the interface). These parametersare operators defined on the initial space H . These parameters specify asimpler, idealized model employing quantum noise, in place of the more basicbut complicated Hamiltonian (6). A beamsplitter is a device that effects the interference of incoming opticalfields A , A and produces outgoing optical fields ˜ A , ˜ A , Figure 6. Therelationship between these fields is˜ A ( t ) = βA ( t ) − αA ( t ) , ˜ A ( t ) = αA ( t ) + βA ( t ) , (11)where α and β are complex numbers describing the beamsplitter relations,and they satisfy α ∗ α + β ∗ β = 1, α ∗ β = αβ ∗ (here the asterisk indicates theconjugate of a complex number).The initial space is trivial, H = C , the complex numbers; nevertheless,the Schr¨odinger equation for the beamsplitter is dV ( t ) = { ( S − I ) d Λ } V ( t ) , (12)12 A (cid:0)(cid:0)(cid:0)(cid:0)✲✻✻ A A ˜ A ✲ Figure 6: Diagram of an optical beamsplitter showing inputs A , A andoutputs ˜ A , ˜ A fields.with initial condition V (0) = I , where S is the unitary matrix defined by(14) below, I is the identity matrix, and Λ is the matrix of gauge processes Λ = (cid:18) A A A A (cid:19) . (13)Here, A ij describes the destruction of a photon in channel j and the cre-ation of a photon in channel i . In terms of their formal derivatives, A ij ( t ) = R t a ∗ i ( s ) a j ( s ) ds , where A i ( t ) = R t a i ( s ) ds . The self-adjoint processes A jj areequivalent to classical Poisson processes when the channels are in coherentstates (signal plus quantum noise). These counting processes may be ob-served by a photodetector, [8, Chapters 8 and 11].This open system is characterized by the unitary parameter matrix S = (cid:18) β − αα β (cid:19) , (14)which describes scattering among the field channels. The matrix S specifiesthe interface for the beamsplitter. In general, as we shall explain in more detail in section 4, open quantumsystems with multiple field channels are characterized by the parameter list G = ( S , L , H ) (15)where S is a square matrix with operator entries such that S † S = SS † = I (recall the notational conventions mentioned at the end of section 1), L is a13olumn vector with operator entries, and H is a self-adjoint operator. Thematrix S is called a scattering matrix , the vector L is a coupling vector ;together, these parameters specify the interface between the system and thefields. The parameter H is the Hamiltonian describing the self-energy of thesystem. Thus the parameters describe the system by specifying energies—internal energy, and energy exchanged with the fields. All operators in theparameter list are defined on the initial Hilbert space H for the system.The closed, undamped, harmonic oscillator of subsection 3.1 is specifiedby the parameters H = ( , , ωa ∗ a ) (16)(the blanks indicate the absence of field channels), while the open, dampedoscillator (cavity) of subsection 3.2 has parameters C = ( I, √ γ a, ∆ a ∗ a ) . (17)The beamsplitter, described in subsection 3.3 has parameters M = ( (cid:18) β − αα β (cid:19) , , . (18) cavity A A B = ˜ B = ˜ A ˜ B B = ˜ A a Figure 7: Beam splitter (left) and cavity (right) network.Consider the feedforward network shown in Figure 7, where one of thebeamsplitter output beams is fed into an optical cavity. From the previ-ous subsections, we see that the quantum stochastic differential equations14escribing the network are da ( t ) = ( − γ i ∆) a ( t ) dt − √ γ dB ( t ) (19)˜ A ( t ) = βA ( t ) − αA ( t ) (20)˜ A ( t ) = αA ( t ) + βA ( t ) (21) B ( t ) = ˜ A ( t ) (22) B ( t ) = ˜ A ( t ) (23) d ˜ B ( t ) = √ γa ( t ) dt + dB ( t ) (24) d ˜ B ( t ) = dB ( t ) . (25)It can be seen that algebraic manipulations are required to describe thecomplete system (in general such manipulations may be simple in principle,but complicated in practice). The key motivation for this paper is moreefficient algebraic methods for describing such networks.We now describe how the parameters for the complete network may beobtained. We first assemble the field channels into vectors as follows: A = (cid:18) A A (cid:19) , B = (cid:18) B B (cid:19) , ˜ A = (cid:18) ˜ A ˜ A (cid:19) , ˜ B = (cid:18) ˜ B ˜ B (cid:19) . The beamsplitter acts on the input vector A , and is described by the pa-rameters M given in equation (18)). Now the beamsplitter output has twochannels, while the cavity has one channel (described by the parameters C ,equation (17)), and so we augment the cavity to accept a second channel ina trivial way. This is achieved by forming the concatenation C ⊞ N , where N = (1 , ,
0) represents a trivial component (pass-through). The augmentedcavity C ⊞ N can now accept the output of the beamsplitter, so that thecomplete network is described as a series connection as follows: G = ( C ⊞ N ) ⊳ M . (26)The definition of the concatenation ⊞ and series ⊳ products will be explainedbelow in section 5 (Definitions 5.1 and 5.3, and the principle of series connec-tions, Theorem 5.5). By applying these definitions, we obtain the networkparameters G = (cid:18)(cid:18) β − αα β (cid:19) , (cid:18) √ γ a (cid:19) , ∆ a ∗ a (cid:19) . (27)15 ✲✲ ✲✲ A A ˜ B ˜ B Figure 8: Beam splitter-cavity network representation illustrating the net-work model given by (27).A schematic representation of the network is shown in Figure 8, which illus-trates the important point that components, parts of components, as well asthe complete network, are described by parameters of the form (15).
For the purposes of network modeling and design, it can be useful toperform manipulations of the network to yield equivalent networks; this, ofcourse, is common practice in classical electrical circuit theory and controlengineering. For instance, in our example we could move the beam split-ter to the output, but the cavity should be modified (to have two partiallytransmitting mirrors) as follows (see Remark 5.7): G = ( C ⊞ N ) ⊳ M = M ⊳ ( C ′ ⊞ N ′ ) . (28)Here, the modified cavity C ′ ⊞ N ′ (see Figure 9) is described by the subsys-tems C ′ = ( I, β ∗ √ γ a, ∆ a ∗ a ) , N ′ = ( I, − α ∗ √ γ a, . (29) modified cavity B A A A B B Figure 9: Equivalent beam splitter and cavity network.The connections described here so far are unidirectional field mediatedconnections . Components interact indirectly via a quantum field, which actsas a quantum “wire”. One can also consider bidirectional direct connections ,16hich can be accommodated by using interaction Hamiltonian terms in themodels. Our emphasis in this paper will be on field mediated connections,with direct connections readily available in the modeling framework if re-quired. See subsection 5.4.
In this section we describe in more detail the open quantum models of thetype encountered in section 3. Specifically, we consider models specified bythe parameters G = ( S , L , H ) (recall (15)), where S = S . . . S n ... ... ... S n . . . S nn , L = L ... L n , are respectively a scattering matrix with operator entries satisfying S † S = SS † = I , and coupling vector with operator entries, and H is a self-adjointoperator called the Hamiltonian (this parameterization is due to Hudson-Parthasarathy, [15], and is closely related to a standard form of the Lindbladgenerator, given in (33) below). The operators constituting these parametersare assumed to be defined on an underlying Hilbert space H , called the initialspace . These parameters specify an open quantum system coupled to n fieldchannels with corresponding gauge processes: A = A ... A n , Λ = A . . . A n ... ... ... A n . . . A nn . All differentials shall be understood in the It¯o sense - that is, dX ( t ) ≡ X ( t + dt ) − X ( t ). We assume that these processes are canonical , mean-ing that we have the following non-vanishing second order It¯o products: dA j ( t ) dA k ( t ) ∗ = δ jk dt , dA jk ( t ) dA l ( t ) ∗ = δ kl dA j ( t ) ∗ , dA j ( t ) dA kl ( t ) = δ jk dA l ( t ) and dA jk ( t ) dA lm ( t ) = δ kl dA jm ( t ).If we consider the open system specified by G = ( S , L , H ) with canonicalinputs, the Schr¨odinger equation dV ( t ) = { tr[( S − I ) d Λ ] + d A † L (30) − L † S d A − L † L dt − iHdt } V ( t ) ≡ dG ( t ) V ( t )17ith initial condition V (0) = I determines the unitary motion of the system.Equation (30) serves as the definition of the time-dependent generator dG ( t ).Given an operator X defined on the initial space H , its Heisenberg evolutionis defined by X ( t ) = j t ( X ) = V ( t ) ∗ XV ( t ) (31)and satisfies dX ( t ) = ( L L ( t ) ( X ( t )) − i [ X ( t ) , H ( t )]) dt + d A † ( t ) S † ( t )[ X ( t ) , L ( t )] + [ L † ( t ) , X ( t )] S ( t ) d A ( t )+tr[( S † ( t ) X ( t ) S ( t ) − X ( t )) d Λ ( t )] . (32)In this expression, all operators evolve unitarily according to (31) (e.g. L ( t ) = j t ( L )) (commutators of vectors and matrices of operators are defined component-wise), and tr denotes the trace of a matrix. We also employ the notation L L ( X ) = 12 L † [ X, L ] + 12 [ L † , X ] L = n X j =1 ( 12 L ∗ j [ X, L j ] + 12 [ L ∗ j , X ] L j ); (33)this is called the Lindblad superoperator in the physics literature (it is analo-gous to the transition matrix for a classical Markov chain, or the generator ofa classical diffusion process). The dynamics is unitary, and hence preservescommutation relations. The output fields are defined by˜ A ( t ) = V ∗ ( t ) A ( t ) V ( t ) , ˜ Λ ( t ) = V ∗ ( t ) Λ ( t ) V ( t ) , (34)and satisfy the quantum stochastic differential equations d ˜ A ( t ) = S ( t ) d A ( t ) + L ( t ) dtd ˜ Λ ( t ) = S ♯ ( t ) d Λ ( t ) S T ( t ) + S ♯ ( t ) d A ♯ ( t ) L T ( t )+ L ♯ ( t ) d A T ( t ) S T ( t ) + L ♯ ( t ) L T ( t ) dt, where L ( t ) = j t ( L ), etc, as above. The output processes also have canonicalquantum It¯o products.In the physics literature, it is common practice to describe open sys-tems using a master equation (analogous to the Kolmogorov equation for thedensity of a classical diffusion process) for a density operator ρ , a convex18ombination of outer products ψψ ∗ (here ψ is a state vector). Master equa-tions can easily be obtained from the parameters G = ( S , L , H ); indeed, wehave ddt ρ = i [ ρ, H ( t )] + L ′ L ( t ) ( ρ ) , (35)where L ′ L ( ρ ) = L T ρ L ♯ − L ♯ L T ρ − ρ L ♯ L T is the adjoint of the Lindbladian:tr[ ρ ( t ) L L ( X )] = tr[ L ′ L ( ρ ) X ]. Note that while the master equation does notdepend on the scattering matrix S , this matrix plays an important role indescribing the architecture of the input channels, as in subsections 3.5 and6.2. We also mention that if an observable of one or more output channelsis continuously monitored, then a quantum filter (also called a stochasticmaster equation) for the conditional density operator can be written downin terms of the parameters G = ( S , L , H ); an example of this is discussed insubsection 6.3, see [2].Open systems specified by parameters G = ( S , L , H ) preserve the canon-ical nature of the quantum signals. However, if the inputs are not canonical,one will need to modify the equations for the unitary, the Heisenberg dy-namics, and the outputs, etc, to accommodate non-canonical correlations;we do not pursue this matter further here, and in this paper we will alwaysuse canonical quantum signals. This section contains the main results of the paper. The concatenation andseries products are defined in subsection 5.1, and applied to a feedback ar-rangement in Theorem 5.5, the principle of series connections (subsection5.2). This is followed in subsection 5.3 with a specialization to cascade net-works, and a consideration in subsection 5.4 of reducible networks. Theseresults are applied to a range of examples in section 6.
In this subsection we define two products between system parameters. It isassumed that both systems are defined on the same underlying initial Hilbertspace, enlarging if necessary by using a tensor product.19 efinition 5.1 (Concatenation product) Given two systems G = ( S , L , H ) and G = ( S , L , H ) , we define their concatenation to be the system G ⊞ G by G ⊞ G = ( (cid:18) S S (cid:19) , (cid:18) L L (cid:19) , H + H ) . (36)The concatenation product is useful for combining distinct systems, orfor decomposing a given system into subsystems. It does not describe in-terconnections via field channels, but does allow for direct connections viathe Hamiltonian parameters. Systems without field channels are included byemploying blanks; set ( , , H ) ⊞ ( , , H ′ ) := ( , , H + H ′ ) and more generally( , , H ) ⊞ ( S ′ , L ′ , H ′ ) = ( S ′ , L ′ , H ′ ) ⊞ ( , , H ) := ( S ′ , L ′ , H + H ′ ). Definition 5.2 (Reducible system) We say that a system G = ( S , L , H ) is reducible if it can be expressed as G = G ⊞ G (37) for two systems G and G . In particular, the parameters of a reduciblesystem have the form S = (cid:18) S S (cid:19) , L = (cid:18) L L (cid:19) , H = H + H . (38) Such decompositions are not unique. Furthermore, if one or more of thesubsystems is reducible, the reduction process may be iterated to obtain adecomposition G = ⊞ j G j . Definition 5.3 (Series product) Given two systems G = ( S , L , H ) and G = ( S , L , H ) with the same number of field channels, the series product G ⊳ G defined by G ⊳ G = ( S S , L + S L ,H + H + 12 i ( L † S L − L † S † L )) . As will be explained in the following subsection, the series product speci-fies the parameters for a system formed by feeding the output channel of thefirst system into the input channel of the second. Both of these products arepowerful tools for describing quantum networks.20 emark 5.4
Let dG j ( t ) denote the infinitesimal It¯o generators correspond-ing to parameters G j = ( S j , L j , H j ), for j = 1 , G ⊳ G is then dG ( t ) = dG ( t ) + dG ( t ) + dG ( t ) dG ( t ) . (39)The last term is to be computed using the It¯o table for second order productsof differentials. Let us consider a reducible system G = G ⊞ G (recall Definition 5.2),where number of channels in the factors is the same (i.e. dim L = dim L ).The setup is sketched in Figure 10. We investigate what will happen if wefeed one of the outputs, say ˜ A back in as the input A . Either of the twodiagrams in Figure 11 may serve to describe the resulting feedback system.Note that the outputs will be different after the feedback connection has beenmade. ✲ ✲ t ✲ ✲ t A ˜ A A ˜ A Figure 10: Reducible system G ⊞ G with inputs A , A and outputs ˜ A , ˜ A .12 ✲ t ✲ ✲ t ˜ A A A = ˜ A ✲ t ✲ t ˜ A A Figure 11: Direct feedback system G ⊳ G , with input A and output ˜ A .We now state our main result applying the series product to feedback.21 heorem 5.5 (Principle of Series Connections) The parameters G ← for the feedback system obtained from G ⊞ G when the output of the firstsubsystem is fed into the input of the second is given by the series product G ← = G ⊳ G . A proof of this theorem is given in Appendix B.
In our treatment of series connections, we nowhere assumed that the ma-trix entries commuted, and this of course facilitated feedback. However, theprinciple of series connections also applies to the special case where the sub-systems commute, as in a cascade of independent systems, as shown in Figure12. To formulate the cascade arrangement, we first consider the concatenationof the two systems G ⊞ G . The system G = G ⊞ G is reducible withcomponents G j . ✲ ✉ ✲ ✉ G G A ˜ A = A ˜ A Figure 12: Cascade of independent quantum components, G ⊳ G .The notion of cascaded quantum systems goes back to Carmichael [3],who used a quantum trajectory analysis, and Gardiner [6] who used (scalar)quantum noise models of the form G j = (1 , L j , H j ) (no scattering). Asa special case of the series principle, we see that the cascaded generator forthis type of setup is G cascade = G ⊳ G = (1 , L + L , H + H + Im { L ∗ L } ).This is entirely in agreement with Gardiner’s analysis, cf. [8, Chapter 12] with L j = √ γ j c j where we have L ← = √ γ c + √ γ c and H ← = H + H + i √ γ γ ( c ∗ c − c ∗ c ).We now consider cascade arrangements and ask what happens if we try toswap the order of the components. Since the series product is not in general Indeed, the reason we use the term “series” is to indicate that it applies more generallythan to cascades of independent components. parametrically equivalent if their parametersare identical. This implies that, for the same input, they produce the sameinternal dynamics and output. Consider the cascaded systems shown inFigure 13. = ✲ ✉ ✲ ✉ G G ✲ ✉ ✲ ✉ G ′ G Figure 13: Equivalent Systems.We assume that the initial inputs are canonical in both cases and ask, forfixed choices of G and G , what we should take for G ′ so that the setupsare parametrically equivalent. Theorem 5.6
The two cascaded systems shown in Figure 13 are parametri-cally equivalent if and only if G ⊳ G = G ⊳ G ′ . (40) Furthermore, if ( S j , L j , H j ) are the parameters for G and G ( j = 1 , ,then the parameters ( S ′ , L ′ , H ′ ) of G ′ are uniquely determined by S ′ = S † S S , L ′ = S † ( S − I ) L + S † L ,H ′ = H + Im n L † ( S + I ) L − L † S L o . (41)The proof of this theorem is given in Appendix C. Remark 5.7
A useful special case of this result is moving a scattering matrixfrom the input to the output of a modified system:( S , L , H ) = ( I , L , H ) ⊳ ( S , ,
0) = ( S , , ⊳ ( I , S † L , H ) . (42)This is illustrated in subsection 3.5. ✷ .4 Reducible Networks Networks can be formed by combining components with the concatenationand series products. Within this framework, components may interact di-rectly, or indirectly via fields. This framework is useful for modeling existingsystems, as we have seen above, as well as for designing new systems.Let { G j } be a collection of components, which we may combine togetherto form an unconnected system G = ⊞ j G j . The components may interactdirectly via bidirectional exchanges of energy, and this may be specified bya direct connection Hamiltonian K of the form K = i X k ( N ∗ k M k − M ∗ k N k ) , (43)where M k , N k are operators defined on the initial Hilbert space for G . Thecomponents may also interact via field interconnects, specified by a list ofseries connections S = { G j ⊳ G k , . . . , G j n ⊳ G k n } (44)such that (i) the field dimensions of the members of each pair are the same,and (ii) each input and each output (relative to the decomposition G = ⊞ j G j ) has at most one connection.A reducible network N is the system formed from G by implementingthe connections (43) and (44). The parameters of the network N may beobtained as follows. A series chain is a system of the form C = G j l ⊳ G k l ⊳ · · · ⊳ G j m ⊳ G k m . Let C denote the set of maximal-length chains drawn from the list of seriesconnections (44), and let U denote the set of components not involved inany series connection. Then the reducible network is given by N = ( ⊞ G k ∈ U G k ) ⊞ (cid:0) ⊞ C j ∈ C C j (cid:1) ⊞ (1 , , K ) . (45)An example of a reducible network is shown in Figure 14. Remark 5.8
The examples considered in section 6 below are all importantexamples of reducible networks that have appeared in the literature. How-ever, we mention that there are important examples of quantum networks24 t tttttt ✲✲ ✛✲✲✲✲ ✲✲✲✲✛ ✛ G t Figure 14: A reducible network N = G ⊞ ( G ⊳ G ⊳ G ) formed from thecollection G = G ⊞ G ⊞ G ⊞ G of components with connections specifiedby the list of series connections S = { G ⊳ G , G ⊳ G } .that are not reducible. An example of a non-reducible network was consid-ered by Yanagisawa and Kimura, [26, Fig. 4], which consists of two systemsin a feedback arrangement formed by a beam splitter, as occurs if in Figure7 we connect the output ˜ B to the input A (i.e. setting A = ˜ B ). Thefeedback loop formed in this way is “algebraic”, and the resulting in-loopfield is not a free field in general. A general theory of quantum feedbacknetworks, both reducible and non-reducible, is given in [11]. ✷ In this section we look at a number of examples from the literature whichcan be represented by reducible networks.
We consider a simple situation first introduced by Wiseman and Milburn asan example of all-optical feedback, [25, section II.B. A]. Referring to Figure15, vacuum light field A is reflected off mirror 1 to yield an output beam ˜ A which results from interaction with the internal cavity mode a . This beamis reflected onto mirror 2, as shown, where it constitutes the input A . It isassume that both mirrors have the same transmittivity, so that we can modelthe coupling operators for the two field channels as L = L = √ γ a , where γ is the damping rate. We may also assume that the light picks up a phase S = e iθ when reflected by the cavity mirror.25 ptical interconnectmirror 1 mirror 2cavity aA ˜ A A ˜ A Figure 15: All-optical feedback for a cavity. The feedback path is a lightbeam from mirror 1 to mirror 2, both of which are partially transmitting).There is a phase shift θ along the feedback path.Before feedback, the cavity is described by G = ( I , (cid:18) L L (cid:19) ,
0) = (1 , L , ⊞ (1 , L , . The phase shift between the mirrors is described by the system ( S, , L L ✲ t ✲ ✲ t ˜ A A t ( S, , L ′ L ′ ✲ t ✲ ✲ t ˜ A A t ( S, , G cl = (1 , L , ⊳ ( S, , ⊳ (1 , L , S, SL + L , i ( L ∗ SL − L ∗ S ∗ L )) . S, ,
0) to the very end, as shown in the rightdiagram in Figure 16. Then G cl = ( S, , ⊳ (1 , S ∗ L , ⊳ (1 , L , S, SL + L , i ( L ∗ SL − L ∗ S ∗ L )) , as before. Either way, the closed loop feedback system is described by G cl =( S cl , L cl , H cl ) where S cl = S ≡ e iθ ,L cl = SL + L ≡ (cid:0) e iθ (cid:1) √ γa,H cl = Im { L ∗ SL } ≡ γ sin θ a † a. From this we obtain the Heisenberg dynamical equation for the cavity mode da = − (cid:2) a, (cid:0) e iθ (cid:1) √ γa † (cid:3) dA − γ (cid:0) e iθ (cid:1) (cid:0) e − iθ (cid:1) adt − iγ sin θ adt ≡ − (cid:0) e iθ (cid:1) ( √ γdA + γadt ) , and the input/output relation, in agreement with [25, eq. (2.29)], d ˜ A = e iθ dA + (cid:0) e iθ (cid:1) √ γadt. In the paper [24], Wiseman considers two types of measurement feedback, oneinvolving photon counting, and another based on quadrature measurementusing homodyne detection (which is a diffusive limit of photon counts). Inboth cases proportional feedback involving an electrical current was used.We describe these feedback situations in the following subsections using ournetwork theory.Consider the measurement feedback arrangement shown in Figure 17,which shows a vacuum input field A , a control signal c , a photodetector PD,and a proportional feedback gain k . 27 eedback gain ✛✲ ✲✲ PD j ( t )control signal photocurrentinput field output field k quantum system A ( t ) c ( t ) G Figure 17: Direct feedback of photocurrent obtained by photon countingusing a photodetector (PD).Before feedback, the quantum system is described by G = (1 , L, H + F c ) , (46)where H and F are self-adjoint, and c represent a classical control variable.The photocurrent j ( t ) resulting from ideal photodetection of the output fieldis given by “ j ( t ) dt ” = d Λ +
LdA † + L † dA + L † Ldt, (47)where, mathematically, the photocurrent j ( t ) is the formal derivative of afield observable (a self-adjoint commutative jump stochastic process) ˜Λ( t )(the output gauge process) whose It¯o differential is given by the RHS of(47). The feedback is given by c ( t ) = kj ( t ) , (48)where k is a (real, scalar) proportional gain. The feedback gain can beabsorbed into F , and so we assume k = 1 in what follows.An alternative is to again consider the quantum system G given by (46),but replace the photodetector PD in Figure 17 with a homodyne detectorHD. The homodyne detector then produces a photocurrent j ( t ) given by“ j ( t ) dt ” = dJ ( t ) = ( L ( t ) + L ♯ ( t )) dt + dA ( t ) + dA ♯ ( t ) . An ideal homodyne detector HD takes an input field A and produces a quadrature,say A + A ∗ (real quadrature), thus effecting a measurement. This is achieved routinely togood accuracy in optics laboratories, [8, Chapter 8]. F , as above. The measurement result J ( t ) is a field observable (here a self-adjoint commutative diffusive process).In order to describe these types of direct measurement feedback withinour framework, we view the setup before feedback as being described by G = (1 , L, H ) ⊞ ( S fb , L fb , H fb ) ≡ G ⊞ G fb . Here, G describes the internal energy of the system and its coupling tothe input field A . The second term, G fb , describes the way in which theclassical input signal is determined from a second quantum input field (whichwill be replaced by the output ˜ A when the feedback loop is closed). Theidea is that by appropriate choice of the coupling operator L fb , the relevantobservable of the field can be selected. In this way, the photodection andhomodyne detection measurements are accommodated. The singular natureof the feedback signal (which contains white noise in the homodyne case)means that care must be taken to describe it correctly. The correct formof the parameters is given by the Holevo parameterization (Appendix A,equation (55)) rather than the expression arising from the implicit-explicitformalism of [24], since the later does not capture correctly gauge couplings,see Appendix A. We shall interpret the feedback interaction as being due to aHolevo generator K fb ( t ) = H t + H A ( t )+ H A ∗ ( t )+ H Λ( t ), see AppendixA, equation (54). The closed loop system after feedback is given by the seriesconnection G cl = G fb ⊳ G = (cid:0) S fb , L fb + S fb L, H + H fb + Im (cid:0) L ∗ fb S fb L (cid:1)(cid:1) . Here we take K fb ( t ) = F Λ( t ), so that S fb = e − iF , see Appendix A, equa-tion (55). Note that this coupling picks out the required photon numberobservable of the field. We then have G fb = ( e − iF , ,
0) and so G cl = ( e − iF , e − iF L, H ) . This is illustrated in Figure 18. The resulting Heisenberg equation agreeswith the results obtained by Wiseman, [24, eq. (3.44)], which we write inour notation as dX = ( − i [ X, H ] + L e − iF L ( X )) dt + ( e iF Xe − iF − X ) d Λ+ e iF [ X, e − iF L ] dA ∗ + [ L ∗ e iF , X ] e − iF dA. (49)29 Technical aside.
Note that if we set E ( t ) = E Λ( t ), with E self-adjoint,then the Stratonovich equation dV ( t ) = − idE ( t ) ◦ V ( t ) ≡ − idE ( t ) V ( t ) − i dE ( t ) dV ( t ) is equivalent to dV ( t ) = S fb d Λ( t ) V ( t ) where S fb = − i E i E .Therefore the implicit form [24] is not the Stratonovich form [10].) S ✲✲✲ ✲ AC ˜ C ˜ AL t Figure 18: Representation of the direct photocount feedback scheme of Figure17 as a reducible network.
Here we take K fb ( t ) = F ( A ∗ ( t ) + A ( t )) in which case G fb = (1 , − iF, − iF ensures that thecoupling selects the desired field quadrature observable. After feedback, theclosed loop system is G cl = (1 , L − iF, H + 12 ( F L + L ∗ F ))using (39). This is illustrated in Figure 19. The resulting Heisenberg equationthen agrees with [24, eq. (4.21)], which we write as dX = ( − i [ X, H + 12 ( F L + L ∗ F )] + L L − iF ( X )) dt +[ X, ( L − iF )] dA ∗ + [( L − iF ) ∗ , X ]) dA. (50)( Technical aside.
Note that for diffusions (that is, no gauge terms) theHolevo generator and Stratonovich generator coincide: that is, dV ( t ) =( e − idK fb ( t ) − V ( t ) is the same as dV ( t ) = − idK fb ( t ) ◦ V ( t ), Appendix A.)30 t ✲✲ ✲✲ AC ˜ C ˜ AL t Figure 19: Representation of the direct homodyne feedback scheme (Figure17 with HD replacing PD) as a reducible network.
Consider a quantum system G q continuously monitored by observing thereal quadrature ˜ B + ˜ B ∗ of an output field ˜ B . This measurement can ideallybe carried out by homodyne detection, but due to finite bandwidth of theelectronics and electrical noise, this measurement could be more accuratelymodeled by introducing a classical system (low pass filter) and additive noiseas shown in Figure 20, as analyzed in [23]. Here, B is a vacuum field, I is theoutput of the ideal homodyne detector (HD), v is a standard Wiener process,and Y is the (integral of) the electric current providing the measurementinformation.We wish to derive a filter to estimate quantum system variables X q fromthe information available in the measurement Y . classical systemquantum system HD detection system G q ++ YIB v G c Figure 20: Model of a realistic detection scheme for a quantum system,showing ideal homodyne detection followed by a classical system (e.g. lowpass filter) and additive classical noise.The quantum system is given by G q = (1 , L q , H q ), and the classical de-31ection system is given by the classical stochastic equations dx ( t ) = ˜ f ( x ( t )) dt + g ( x ( t )) dw ( t ) ,dY ( t ) = h ( x ( t )) dt + dv ( t ) , (51)where x ( t ) ∈ R n , y ( t ) ∈ R , ˜ f , g are smooth vector fields, h is a smoothreal-valued function, and w and v are independent standard classical Wienerprocesses. As described in the Appendix D, this classical system is equivalentto a commutative subsystem of G c = (1 , L c , H c ) ⊞ (1 , L c , L c = − ig T p − ∇ T g , L c = h and H c = ( f T p + p T f ). We represent the systemof Figure 20 as a redicible network, as shown in Figure 21. Y s s ✲✲ ✲✲✲ ✲ classical system A A = ˜ B quantum system B L c L c G q G c L q ˜ A ˜ A HD s Figure 21: Representation of the realistic detection scheme of Figure 20 as areducible network.Here, the classical noises are represented as real quadratures w = A + A ∗ , v = A + A ∗ . Note that since L c is skew-symmetric, only the real quadrature w = A + A ∗ = ˜ B + ˜ B ∗ affects the classical system (this captures the idealhomodyne detection). The complete cascade system is G = ((1 , L c , H c ) ⊳ (1 , L q , H q )) ⊞ (1 , L c ,
0) (52)= ( I , (cid:18) L + L c L c (cid:19) , H q + H c + 12 i ( L ∗ c L q − L ∗ q L c ))Applying quantum filtering [1], [2], the unnormalized quantum filter forthe cascade system G is dσ t ( X ) = σ t ( − i [ X, H q + H c + 12 i ( L ∗ c L q − L ∗ q L c )]+ L L + L c L c ( X )) dt + σ t ( L ∗ c X + XL c ) dy. (53)32ere, X is any operator defined on the quantum-classical cascade system.For instance, X = X q ⊗ ϕ , where ϕ is a smooth real valued function on R n .In particular, if X = X q is a quantum system operator, one can compute thedesired estimate of X q from π t ( X q ) = σ t ( X q ) /σ t (1).Equation (53) can be normalized, and compared with [23, eq. (17)]. Inthe case that the quantum system is a linear gaussian system, and the filteris a linear system, the complete filter reduces to a Kalman filter from whichthe desired quantum system variables can be estimated. In this paper we have presented algebraic tools for modeling quantum net-works. The tools include a parametric representation for open quantum sys-tems, and the concatenation and series products. The concatenation prod-uct allows us to form a larger system from components, without necessarilyincluding connections. The series product, through the principle of seriesconnections (Theorem 5.5), provides a mechanism for combining systems viafield mediated connections. We demonstrated how to model a class of quan-tum networks, called reducible networks, using our theory and we illustratedour results by examining some examples from the literature.Future work will involve further development of the network theory de-scribed here, and applying the theory to develop control engineering toolsand to applications in quantum technology, e.g. [16].
A Time-Ordered Exponentials in the sense ofHolevo
Holevo [14] developed a parameterization of open system dynamics that isdifferent to the Hudson-Parthasarathy parameters G = ( S, L, H ). Holevo’sparameterization is defined as follows. Let K ( t ) = H t + H A ( t ) + H A ∗ ( t ) + H Λ( t ) , (54)where { H αβ } consists of bounded operators with H αβ = H βα , and the indices α, β range from 0 to 1 (here we are considering a single field channel forsimplicity). The time-ordered exponential with Holevo generator { H αβ } is33he unitary adapted process U satisfying the quantum stochastic differentialequation dU ( t ) = (cid:0) e − idK ( t ) − (cid:1) U ( t )with U (0) = 1, [14], [9]. Expanding the differential e − idK ( t ) − dU ( t ) = X n ≥ ( − i ) n n ! ( dK ) n U ( t ) . Now for a system with parameters G = ( S, L, H ) we have dU ( t ) = { ( S − I ) d Λ( t ) + LdA ∗ ( t ) − L ∗ dA − ( iH + 12 L ∗ L ) dt } U ( t ) . Comparing these expressions, we find that S = exp ( − iH ) , L = exp ( − iH ) − H H ,H = H − H H − sin( H )( H ) H . (55)The relationship between the generating coefficients H αβ and the param-eters G = ( S, L, H ) are exactly as occur in the implicit-explicit formalismof [24], however, this formalism only coincides with the Stratonovich-It¯o cor-respondence in the case where H = 0 [10]. B Proof of Theorem 5.5
There are a number of independent derivations of the series product. Forinstance it can be derived from a purely Hamiltonian formalism for quantumnetworks [11], alternatively Gardiner’s arguments in the Heisenberg picturecan be extended to include the scattering terms [12]. Here we present adiscretization argument for the input/output fields based on [9]. Ratherthan considering a continuous noise source, we take a beam consisting ofqubits (spin one-half particles) with a rate of one qubit every τ seconds.A qubit has the Hilbert space H = C spanned by a pair of orthogonalvectors e and e . We define raising/lowering operators σ ± for each qubit by σ + ( αe + βe ) = αe and σ − ( αe + βe ) = βe . In our model of the interaction34f a qubit with a given plant, we shall assume that the interaction is muchshorter than τ so that at most one qubit may interacting with a given plantat any instant of time. For two plants in cascade, we shall take them to beseparated so that the time of flight of the qubits is exactly τ seconds. This ispurely for convenience and can be easily relaxed. For definiteness, we assumethat each qubit is prepared independently in the “ground state” e and wedenote by σ ± k the raising/lowering operators for the k th qubit: the operatorscorresponding to different qubits commute, while we have σ − k σ + k + σ + k σ − k = 1, (cid:0) σ + k (cid:1) = 0 = (cid:0) σ − k (cid:1) . At time t k = kτ ( k ∈ N ), we take the most recent qubitto interact with the first system to be the k th qubit, and the most recent tointeract with the second to be the ( k − x > ⌊ x ⌋ and set σ αβτ ( k ) := (cid:20) σ + k √ τ (cid:21) α (cid:20) σ − k √ τ (cid:21) β where α, β may take the values zero and one and where [ B ] = 1, [ B ] = B forany operator B . In the following, we shall denote by O ( τ n ) any expressionwhich is norm-convergent to zero as τ → τ n . The identity τ σ α τ ( k ) σ βτ ( k ) = σ αβτ ( k ) + O ( τ ) will be important in what follows and willcorrespond to the discrete version of the second order It¯o products. For t > A αβτ ( t ) := τ ⌊ t/τ ⌋ X k =1 σ αβτ ( k )are well-known approximations to the fundamental processes A αβ ( t ) in thelimit τ → + , [9].We shall fix bounded operators H αβj on the j th system such that H αβ † j = H βαj and set H ( j ) τ ( k ) = H αβj ⊗ σ αβτ ( k ) . We shall first recall some well knownresults [9] for the situation where the qubits interact with only the firstsystem (that is, set H αβ = 0). The discrete time evolution is describedby unitary kicks every τ seconds according to U τ ( t ) = U ⌊ t/τ ⌋ · · · U U where U k = exp n − i τ H (1) τ ( k ) o . Expanding the exponential yields U k = 1 + τ G αβ ⊗ σ αβτ ( k ) + O ( τ ) with the G αβ forming the coefficients of the unitary QSDEwith parameters G related to H = n H (1) αβ o as in Appendix A.35n the limit τ → + , the discrete time process U τ ( t ) converges weakly inmatrix elements to the solution of the QSDE dU ( t ) = G αβ ⊗ dA αβ ( t ) U ( t ) . We now turn to the case of a cascaded system. This time the discrete timedynamics is given by V τ ( t ) = V ⌊ t/τ ⌋ · · · V V where V k = exp n − i τ H (1) τ ( k ) − i τ H (2) τ ( k − o .Expanding the exponential now yields V k = 1 + τ G αβ ⊗ σ αβτ ( k ) + τ G αβ ⊗ σ αβτ ( k −
1) + O (cid:0) τ (cid:1) . with the G αβ forming the coefficients of the unitary QSDE with parameters G related to H as in Appendix A.To better understand what is going on, we compute V k V k − = 1 + τ G αβ ⊗ σ αβτ ( k )+ τ n G αβ + G αβ + G α G β o ⊗ σ αβτ ( k − τ G αβ ⊗ σ αβτ ( k −
2) + O (cid:0) τ (cid:1) . This may be iterated to give V k V k − · · · V l =1 + τ n G αβ + G αβ + G α G β o ⊗ k − X j = l σ αβτ ( k − τ G αβ ⊗ σ αβτ ( k ) + τ G αβ ⊗ σ αβτ ( l −
1) + O (cid:0) τ (cid:1) . Under the same mode of convergence as before, we obtain the limit QSDE dV t = G (2 ← αβ ⊗ dA αβ ( t ) V ( t )where we recognize G αβ (2 ← = G αβ + G αβ + G α G β as the coefficients the uni-tary QSDE with the series product parameters G ⊳ G , see (39). Therefore G ← ≡ G ⊳ G . The generalization to multi-dimensional noise is straight-forward. 36 Proof of Theorem 5.6
Clearly, if (40) is satisfied, then both cascade systems are described by thesame parameters, which implies that they are equivalent. Now suppose thetwo systems are parametrically equivalent, with S ′ undetermined. Now byDefinition 5.3 we may obtain expressions for G ⊳ G and G ⊳ G ′ . Equatingthe first terms, we have S S = S S ′ , and solving for S ′ one obtains S ′ = S † S S , as in (41). Next, equating the second terms gives L + S L = L + S L ′ . This expression can be solved for L ′ , as in (41). Similarly, theHamiltonian term H ′ in (41) can be found by equating the third terms. D Classical Systems as Commutative Quan-tum Subsystems
In this subsection we explain how to model the classical system (51), shownin Figure 22, as a commutative subsystem of a larger quantum system. Thisrepresentation is used in subsection 6.3. In equation (51), x ( t ) ∈ R n , y ( t ) ∈ R , ˜ f , g are smooth vector fields, h is a smooth real-valued function, and w and v are independent standard classical Wiener processes. classical system ✲ ✲✲ ❄ ++ v yw ✒✑✓✏ Figure 22: Block diagram of the classical system (51).To model this classical system, we take the underlying Hilbert space ofthe system to be h = L ( R n ) with q j , p j being the usual canonical positionand momentum observables: q j ψ ( x ) = X j ψ ( x ) and p j ψ ( x ) = − i∂ j ψ ( ~x ).We write q = ( q , . . . , q n ) T , p = ( p , . . . , p n ) T , and ∇ = ( ∂ , . . . , ∂ n ) T . If ϕ isa smooth function of x , then we find that, by It¯o’s rule, for ϕ t = ϕ ( x ( t )), dϕ = L classical ( ϕ ) dt + g T ∇ ϕ dw, (56)where L classical ( ϕ ) = f T ∇ ϕ + g T ∇ (cid:0) g T ∇ ϕ (cid:1) is the (classical) generator of thediffusion process x ( t ) in (51). 37e seek a quantum network representation G c , as shown in Figure 23. L c t ✲✲ ✲✲ ✲ classical system A A G c HD y ˜ A ˜ A L c t Figure 23: Network representation of the classical system (51) shown inFigure 22.The classical noises are viewed as real quadratures of quantum noises w = A + A ∗ , v = A + A ∗ . Now define port operators L c = − ig T p − ∇ T g , L c = h and internal Hamiltonian H c = (cid:0) f T p + p T f (cid:1) , where f = ˜ f − [ ∇ g ] g (the Stratonovich drift) and g are n -vectors whose components are viewed asfunctions of q and h = h ( q ) is viewed as a self-adjoint observable function of q .We claim that the classical system (51) behaves as an invariant commutativesubsystem of the open quantum system G c = (1 , L c , H c ) ⊞ (1 , L c , dX c = ( − i [ X c , H c ] + L L c ( X c ) + L L c ( X c )) dt +[ X c , L c ]( dA ∗ + dA ) + [ X c , L c ]( dA ∗ − dA ) (57)Now set X c = ϕ = ϕ ( q ), a smooth function of the position operator. Then(57) gives dϕ = ( − i [ ϕ, H c ] + L L c ( ϕ ) + L L c ( ϕ )) dt +[ ϕ, L c ]( dA ∗ + dA ) + [ ϕ, L c ]( dA ∗ − dA )= ( f T ∇ ϕ + 12 g T ∇ (cid:0) g T ∇ ϕ (cid:1) ) dt + g T ∇ ϕ dw, (58)where, we have used − i [ ϕ, H c ] = f T ∇ ϕ , L L c ( ϕ ) = g T ∇ ( g T ∇ ϕ ), L L c ( ϕ ) =0, [ ϕ, L c ] = g T ∇ ϕ , and [ ϕ, L c ] = 0. Hence the classical dynamics (56) isembedded in the dynamics of the position observable q only in the quantumsystem G q (independent of momentum dynamics). Note that only the realquadrature of the input field affects these dynamics, and they are unaffectedby the field A . 38ext we look at the outputs. The first output is not of interest, so wefocus on the second one. The output y ( t ) of the homodyne detector HD inFigure 23 is dy = d ˜ A + d ˜ A ∗ = ( L c + L ∗ c ) dt + dA + dA ∗ = hdt + dv (59)which agrees with (51), as required. The unnormalized quantum filter for G c is dσ t ( X c ) = σ t ( − i [ X c , H c ] + L L c ( X c ) + L L c ( X c )) dt + σ t ( L ∗ c X c + X c L c ) dy. (60)When X c = ϕ , this reduces to dσ t ( φ ) = σ t ( L classical ( ϕ )) dt + σ t ( hϕ ) dy, (61)which is the usual Duncan-Mortensen-Zakai equation of classical nonlinearfiltering, [5, Chapter 18]. References [1] V.P. Belavkin. Quantum stochastic calculus and quantum nonlinearfiltering.
J. Multivariate Analysis , 42:171–201, 1992.[2] L. Bouten, R. Van Handel, and M.R. James. An introduction to quan-tum filtering.
SIAM J. Control and Optimization , 46(6):2199–2241,2007.[3] H.J. Carmichael. Quantum trajectory theory for cascaded open systems.
Phys. Rev. Lett. , 70(15):2273–2276, 1993.[4] C. D’Helon and M.R. James. Stability, gain, and robustness in quantumfeedback networks.
Phys. Rev. A. , 73:053803, 2006.[5] R.J. Elliott.
Stochastic Calculus and Applications . Springer Verlag, NewYork, 1982.[6] C.W. Gardiner. Driving a quantum system with the output field fromanother driven quantum system.
Phys. Rev. Lett. , 70(15):2269–2272,1993. 397] C.W. Gardiner and M.J. Collett. Input and output in damped quan-tum systems: Quantum stochastic differential equations and the masterequation.
Phys. Rev. A , 31(6):3761–3774, 1985.[8] C.W. Gardiner and P. Zoller.
Quantum Noise . Springer, Berlin, 2000.[9] J. Gough. Holevo-ordering and the continuous-time limit for open Flo-quet dynamics.
Letters in Math. Physics , 67:207–221, 2004.[10] J. Gough. Quantum Stratonovich calculus and the quantum Wong-Zakaitheorem.
Journ. Math. Phys. , 47, 113509, 2006.[11] J. Gough and M.R. James, Quantum Feedback Networks: HamiltonianFormulation.
Communications in Mathematical Physics , to appear, DOI10.1007/s00220-008-0698-8, quant-ph/0804.3442, 2008.[12] J. Gough, Construction of bilinear control Hamiltonians using the seriesproduct,
Phys. Rev. A , 78, 052311, 2008.[13] J. Gough, R. Gohm, M. Yanagisawa, Linear quantum feedback net-works,
Phys.Rev. A , 78, 062104, 2008.[14] A.S. Holevo. Time-ordered exponentials in quantum stochastic calculus.In
Quantum Probability and Related Topics , volume 7, pages 175–202.World Scientific, 1992.[15] R.L. Hudson and K.R. Parthasarathy. Quantum Ito’s formula andstochastic evolutions.
Commun. Math. Phys. , 93:301–323, 1984.[16] M.R. James and J. Gough. Quantum dissipative systems and feedbackcontrol design by interconnection. arxiv.org/quant-ph/0707.1074 2007.[17] M.R. James, H. Nurdin, and I.R. Petersen. H ∞ control of linear quan-tum systems. IEEE Trans Auto. Control , 53(8), 1787-1803, 2008.[18] S. Lloyd. Coherent quantum feedback.
Phys. Rev. A , 62:022108, 2000.[19] E. Merzbacher.
Quantum Mechanics . Wiley, New York, third edition,1998.[20] M.A. Nielsen and I.L. Chuang.
Quantum Computation and QuantumInformation . Cambridge University Press, Cambridge, 2000.4021] K.R. Parthasarathy.
An Introduction to Quantum Stochastic Calculus .Birkhauser, Berlin, 1992.[22] R. van Handel, J. Stockton, and H. Mabuchi. Feedback control of quan-tum state reduction.
IEEE Trans. Automatic Control , 50:768–780, 2005.[23] P. Warszawski, H.M. Wiseman, and H. Mabuchi. Quantum trajectoriesfor realistic detection.
Phys. Rev. A , 65:023802, 2002.[24] H. Wiseman. Quantum theory of continuous feedback.
Phys. Rev. A ,49(3):2133–2150, 1994.[25] H. M. Wiseman and G. J. Milburn. All-optical versus electro-opticalquantum-limited feedback.
Phys. Rev. A , 49(5):4110–4125, 1994.[26] M. Yanagisawa and H. Kimura. Transfer function approach to quantumcontrol-part I: Dynamics of quantum feedback systems.
IEEE Trans.Automatic Control , (48):2107–2120, 2003.[27] B. Yurke and J.S. Denker. Quantum network theory.