Coherent versus measurement feedback: Linear systems theory for quantum information

To control a quantum system via feedback, we generally have two options in choosing control scheme. One is the coherent feedback, which feeds the output field of the system, through a fully quantum device, back to manipulate the system without involving any measurement process. The other one is the measurement-based feedback, which measures the output field and performs a real-time manipulation on the system based on the measurement results. Both schemes have advantages/disadvantages, depending on the system and the control goal, hence their comparison in several situation is important. This paper considers a general open linear quantum system with the following specific control goals; back-action evasion (BAE), generation of a quantum non-demolished (QND) variable, and generation of a decoherence-free subsystem (DFS), all of which have important roles in quantum information science. Then some no-go theorems are proven, clarifying that those goals cannot be achieved by any measurement-based feedback control. On the other hand it is shown that, for each control goal, there exists a coherent feedback controller accomplishing the task. The key idea to obtain all the results is system theoretic characterizations of BAE, QND, and DFS in terms of controllability and observability properties or transfer functions of linear systems, which are consistent with their standard definitions.


I. INTRODUCTION
Should we perform measurement or not? This question appears to be critical in quantum physics, particularly in quantum information science. For quantum computation, for instance, it is of essential importance to study differences between the conventional closed-system approach and the measurement-based one (i.e. the so-called one-way computation). This paper focuses on a specific aspect of this abstract and broad question; we will consider feedback control problems. That is, for a given open system (plant), we want to engineer another system (controller) connected to the plant so that the plant or the whole system behaves in a desirable way. The fundamental question is then, in our case, as follows; should we measure the plant or not, for engineering a closedloop system? More precisely, in the former case, we measure the plant's output and engineer a classical controller that manipulates the plant using the measurement result -this is called the measurement-based feedback (MF) approach. In the latter case, we do not measure it, but rather connect a fully quantum controller directly to the plant system in a feedback manner -this is called the coherent feedback (CF) approach.
A typical example is shown in Fig. 1; the plant is an open mechanical oscillator coupled to a ring-type optical cavity, and the control goal is to minimize the energy of the oscillator, or equivalently to cool the oscillator towards its motional ground state. As mentioned above, there are two feedback control strategies. One is the MF controller ( Fig. 1 (a)) that measures the output * yamamoto@appi.keio.ac.jp fieldŴ out 1 by for instance a homodyne detector; then, using the continuous-time measurement results y(t), it produces the control signal u(t) for modulating the input fieldŴ 2 . The other option is the CF control ( Fig. 1 (b)), where we construct another fully quantum system that feeds the output fieldŴ out 1 back to the input fieldŴ 2 , without involving any measurement component. The question is then about how to design a MF/CF controller that cools the oscillator most effectively.
Controller synthesis for a quantum system is in general non-trivial, but researchers' longstanding efforts have built a solid mathematical framework for dealing with those problems. For the MF case, actually there exists a beautiful quantum feedback control theory [1, 2, 3] that was developed based on the quantum filtering [4,5,6] together with the classical control theory [7,8,9]. In fact, the above-described cooling problem can be formulated as a quantum Linear Quadratic Gaussian (LQG) feedback control problem and explicitly solved. Also the theory has been applied to various control problems in quantum information science such as error correction [10,11,12]. Notably, experiment of MF control is now within the reach of current technologies [13,14,15,16]. The CF control, on the other hand, has still a relatively young history though its initial concept was found in [17] back in 1994; but recently it has attracted increasing attention, leading as a result development of the basic control theory [18,19,20,21] and applications [22,23,24,25]. Some experimental demonstrations of CF control [26,27,28,29] also warrant special mention; in fact, one of the main advantages of CF is in its experimental feasibility compared to the MF approach.
Let us return to our question; which controller, MF or CF, is better? Now note that a CF controller is a fully quantum system whose random variables are in general represented by non-commutative operators, while a MF controller is a classical system with commutative random variables. Hence from a mathematical viewpoint the class of MF controllers is completely included in that of CF controllers. Thus our question is as follows; in what situation is a CF controller better than a MF controller? Actually there have been several studies exploring answers to this question [17,30,31,32,33]; most of these studies discussed problems of minimizing a certain cost function such as energy of an oscillator or the time required for state transfer. In particular in [32,33], the authors studied the problem discussed in the second paragraph and clarified that an optimal CF controller outperforms any MF controller only when the total mean phonon number of the oscillator is in the quantum regime; in other words, the two types of controllers do not show a clear difference in their performance for cooling, in a classical situation. This in more broad sense implies that a CF controller would outperform a MF controller only in a purely quantum regime. Consequently, our question can be regarded as a special case of the fundamental problem in physics asking in what situation a fully quantum device (such as a quantum computer) outperforms any classical one (such as a classical computer).
Towards shedding a new light on the above-mentioned fundamental problem, this paper attempts to clarify a boundary between the CF and MF controls for specific control problems. The problems are not what aim to min- imize a cost function, but we will consider the following three; (i) realization of a back-action evasion (BAE) measurement, (ii) generation of a quantum non-demolished (QND) variable, and (iii) generation of a decoherence-free subsystem (DFS). The followings are brief descriptions of these notions in the input-output formalism [34,35]. First, a BAE measurement is such that only a single noise quadrature (shot noise) appears in the output measurement process and its conjugate (back-action noise) is evaded [36,37]; as a result BAE enables high-precision detection for a tiny signal such as a gravitational wave force, below the so-called standard quantum limit (SQL). Next, a QND variable is a physical quantity that can be measured without being disturbed [38]; more precisely, it is not affected by an input probe field but still appears in the output field, which can be thus measured repeatedly. Lastly, a DFS is a subsystem that is completely isolated from surrounding environment, so it can be used for instance in quantum computation or memory [39,40]; that is, a DFS is a subsystem whose variables are not affected by any input probe/environment field, and further, they do not appear in the corresponding output fields. These three notions play crucial roles especially in quantum information science, thus their realizations are of essential importance. Indeed we find in the literature some feedback-based approaches realizing BAE [41,42,43], QND [44], and DFS [45,46,47].
Another feature of this paper is that we focus on general open linear quantum systems [1, 34,35]; this is a wide class of systems containing for instance optical devices [48], mechanical oscillators [32,33,41,42,43,49,50,51,52,53,54,55], and large atomic ensembles [56,57,58,59,60]. Linear systems are typical continuous-variables (CV) systems [61,62], which are applicable to several CV quantum information processing both in Gaussian case [63,64] and non-Gaussian case [65,66]. In both classical and quantum cases, for linear systems, the so-called controllability and observability properties can be well defined with an explicit geometric picture; further, those properties have equivalent representations in terms of a transfer function, which explicitly describes the relation between input and output. In fact a main advantage of focusing on linear systems is that we can have systematic characterizations of BAE, QND, and DFS in terms of the controllability and observability properties or transfer functions, which are consistent with the standard definitions found in the literature. Figure 2 is an at a glance overview of those characterizations, showing unification of the notions. Indeed this is the key idea to obtain all the results. Therefore our problem is that, for a given open linear system, we aim to design a CF/MF controller to realize BAE, QND, or DFS. For this problem, the results summarized in Table I are obtained. That is, no MF controller can achieve any of the control goals for general linear systems (there are two kinds of general configurations for feedback control, as indicated by "type" in Table I).
In contrast to these no-go theorems, for every category in the table we can find an example of CF controller achieving the goal. From the viewpoint of the above-mentioned fundamental question asking differences of the ability of quantum and classical devices, therefore, these results imply that BAE, QND, and DFS are the properties that can only be realized in a fully quantum device. This paper is organized as follows. Section II reviews some useful facts in classical linear systems theory and describes a general linear quantum system with some examples. In Sec. III we discuss the three control goals, BAE, QND, and DFS, in the general input-output formalism and give their systematic characterizations in terms of the controllability-observability properties and also transfer functions; again, these new characterizations are special feature of this paper. Then the proofs of the no-go theorems are given in Secs. IV and V, each of which are devoted to the proofs for the type-1 and the type-2 MF control configuration, respectively. Sections VI and VII demonstrate systematic engineering of a CF controller achieving the control goal. In particular, in the type-2 case, we will study a Michelson's interferometer composed of two mechanical oscillators, which is used for gravitational wave detection.
Notations: For a matrix A, the kernel and the range are defined by Ker(A) = {x | Ax = 0} and Range(A) = {y | y = Ax, ∀x}, respectively. The complement of a linear space X is denoted by X c . ∅ means the null space. In this paper we do not use the terminology "observable" to represent a measurable physical quantity (i.e. a self adjoint operator), because it has a different meaning in control theory; a physical quantity is called a "variable", e.g. a QND variable rather than a QND observable.

II. PRELIMINARIES: LINEAR SYSTEMS THEORY AND LINEAR QUANTUM SYSTEMS
A. Linear systems theory A standard form of classical linear systems is given by (1) x(t) ∈ R n is a vector of n c-number variables. u(t) and y(t) are vectors of real-valued input and output signals, respectively. A, B, and C are real matrices with appropriate dimensions. For this system, especially in this paper, the following three problems are important; (i) which components of x can be controlled by the input u, (ii) which components of x can be observed from y, and (iii) in what condition the input u does not appear in the output y? The answers are briefly described below. See [7,8,9] for more detailed discussion. The first problem can be explicitly solved by examining the following controllability matrix: (2) Indeed this matrix fully characterizes the controllable and uncontrollable variables with respect to (w.r.t.) u(t).
To see this fact, suppose m = dim Range(C u ) < n and let {d (1) i } be independent vectors spanning Range(C u ) and Range(C u ) c , respectively. Further let us Then, as AC u is spanned by {d (1) i }, there exists a matrix A 11 satisfying AT 1 = T 1 A 11 . On the other hand AT 2 is in general spanned by all the vectors; i.e. AT 2 = T 1 A 12 + T 2 A 22 . Note also that there exists a matrix B 1 satisfying B = T 1 B 1 . These relations are summarized in terms of the invertible square matrix T = [T 1 , T 2 ] as Thus the dynamics of x ′ = T −1 x is given by Hence we call these sets the uncontrollable subspace and the controllable subspace, respectively [77]. The following fact is especially useful in this paper: the system has an uncontrollable variable The answer to the second question is obtained in a similar fashion. Let us define the observability matrix Assume dimKer(O y ) = ℓ < n. Then, there exists a linear transformation 2 ∈ R ℓ such that the system equations are of the following form: Thus x ′ 1 and x ′ 2 constitute the observable and unobservable subsystems w.r.t. y, respectively. The variables are represented by as in the above case, we call these subspaces the observable subspace and unobservable subspace, respectively. In particular, there always exists a coordinate transformation The above two facts readily leads to the answer to the third question; that is, there is no subsystem that is controllable w.r.t. u and observable w.r.t. y, which is algebraically represented by Note that this is further equivalent to Range(C u ) ⊆ Ker(O y ), which particularly implies CT 1 = 0 with T 1 defined below Eq. (2). Hence we have , we now see that u acts only on x ′ 1 = V ⊤ 1 x while x ′ 1 does not appear in y; accordingly, u does not appear in y.
The above conditions (4), (7), and (8) can be represented in terms of a transfer function; let us define the Laplace transformation of a time-varying signal z(t) by  In this paper, we consider a general open system composed of n oscillators with canonical conjugate pairsq i andp i (i = 1, . . . , n). Let us collect them into a single vector asx = [q 1 ,p 1 , . . . ,q n ,p n ] ⊤ . Then, the CCR q ipj −p jqi = iδ ij (we assume = 1) is represented bŷ Σ n is a 2n × 2n block diagonal matrix; we often omit the subscript n. The system is driven by the Hamiltonian . Further, it couples to environment/probe fields through the HamiltonianĤ int = i j (L jÂ * j −L * jÂ j ), whereL j = c ⊤ jx (c j ∈ C 2n , j = 1, . . . , m). AlsoÂ j is the annihilation operator on the jth field, which under the Markovian approximation satisfies [Â i (s),Â * j (t)] = δ ij δ(s − t); i.e. it is the white noise operator. Then, the Heisenberg equations ofq j andp j are summarized to the following linear equation [1, 34,35]: The coefficient matrices are given by A = Σ n (G + C ⊤ Σ m C/2) ∈ R 2n×2n (note that the second term is the Ito-correction term) and Also we have definedŴ = [Q 1 ,P 1 , . . . ,Q m ,P m ] ⊤ , wherê Further, the field variables change tô The set of equations (12) and (14) is the most general form of open linear quantum systems.
All the 2m elements of the vectorŴ out in Eq. (14) cannot be measured simultaneously, because they do not commute with each other. In fact, without introducing additional noise fields as explained just later, we can measure only at most half of them; that is, the output equation associated with a linear measurement, which is realized by a Homodyne detector, is of the form where M 1 is a m×2m real matrix satisfying M 1 Σ m M ⊤ 1 = 0 and M 1 M ⊤ 1 = I. Actually, all the elements of y(t) are classical signals commuting with each other as well as with those of y(s) for all time s, t; [y i (s), y j (t)] = 0, ∀i, j, ∀s, t.

Let us further introduceȳ
is a symplectic and orthogonal matrix, which as a result leads to The elements ofȳ correspond to the canonical conjugate operators to those of Eq. (15); i.e. the CCR y(s)ȳ ⊤ (t) − (ȳ(t)y ⊤ (s)) ⊤ = iδ(s − t)I holds.
If we want to measure all the quadratures ofŴ out , it is still possible by introducing additional noise fieldŝ V = [Q ′ 1 ,P ′ 1 , . . . ,Q ′ m ,P ′ m ] ⊤ and performing Homodyne measurement on the joint fields composed ofŴ out and V; that is, the output equation is given by where in this case M 1 is with the size 2m × 4m and it satisfies M 1 Σ 2m M ⊤ 1 = 0, etc. We thus have 2m measurement outcomes, though they are subjected to the additional noise. Note that, by simply replacing C andŴ by [C ⊤ , 0] ⊤ and [Ŵ ⊤ ,V ⊤ ] ⊤ , this dual Homodyne detection scheme can be represented by Eqs. (12) and (15). Hence in what follows, without loss of generality, we use Eq. (15) to represent the most general linear measurement.

C. Examples
(i) A simple open linear system is an empty optical cavity with two input and output fields, depicted in Fig. 3 (a). The system equations are given by a is the annihilation operator of the cavity mode.Â i and A out i are the white noise operators of the ith incoming and the outgoing optical fields, respectively. κ i is the coupling strength betweenâ and the ith field, which is proportional to the transmissivity of the coupling mirror. In this paper we express the variables in the quadrature form, which in this case are defined asx =  (13). Then, the above system equations are rewritten as Typically this system works as a low-pass filter [48]; that is, for a noisy input fieldŴ 1 , the corresponding modecleaned output fieldŴ out 2 is generated, which will be used later for e.g. some quantum information processing. To attain this goal,Ŵ out 1 is measured to detect the error signal for locking the optical path length in the cavity. AlsoŴ 2 is a vacuum field. That is, in this case, the two input-output fields have different roles.
(ii) The mechanical oscillator shown in Fig. 3 (b) can also be modeled as a linear system. This system is composed of a mechanical oscillator with mode (q 1 ,p 1 ) and a cavity with modeâ 2 = (q 2 + ip 2 )/ √ 2. The cavity couples to a probe fieldŴ = [Q,P ] ⊤ . After linearization, the system equation ofx = [q 1 ,p 1 ,q 2 ,p 2 ] ⊤ is obtained as m and ω are the mass and the resonant frequency of the oscillator. κ is the coupling constant between the oscillator and the cavity field, which is proportional to the strength of radiation pressure force. γ is the coupling constant between the cavity and the probe fields. As indicated from the equations, it is possible to extract some information about the oscillator's behavior by measuring the probe output fieldŴ out . A typical situation is that the oscillator is pushed by an external forceF with unknown strength; we attempt to estimate this value, by measuringŴ out . The oscillator's motion is usually much slower than that of the cavity field, thus we can adiabatically eliminate the cavity mode and have a reduced dynamical equation of only the oscillator: where λ = 2κ 2 /γ represents the strength of the direct coupling between the oscillator and the probe field. This equation clearly shows that onlyP out contains the information about the oscillator and accordinglyF ; thusP out should be measured, implying M 1 = [0, 1] in Eq. (15).
(iii) The last example is the Michelson's interferometer composed of two identical mechanical oscillators with mass m and resonant frequency ω, depictd in Fig. 3 (c). This is a simplest configuration among various schemes that are expected to have capability of direct detection of a gravitational wave (GW) [36,37,54,55]. A basic detection mechanism is as follows. A coherent light field W 1 is injected into the left input port (bright port), while in the other port (dark port) the inputŴ 2 is set to be a vacuum. If a gravitational wave comes, one arm shrinks while the other one extends, thereby the oscillators experience tiny force along opposite directions,F and −F . As a result the dynamics of the two oscillators can be modeled by the combination of Eq. (17): Let us rewrite this equation in terms of the common modesq ′ 1 = (q 1 +q 2 )/ √ 2,p ′ 1 = (p 1 +p 2 )/ √ 2 and the differential modesq ′ 2 = (q 1 −q 2 )/ √ 2,p ′ 2 = (p 1 −p 2 )/ √ 2. Then these two modes are decoupled and the forceF appears only in the dynamics ofx ′ 2 = [q ′ 2 ,p ′ 2 ] ⊤ , which is exactly the same as Eq. (17): Thus, ideally, by measuringP out 2 we can detectF .

III. SYSTEM THEORETIC CHARACTERIZATION OF BAE, QND, AND DFS
The problem considered in this paper is to design a MF/CF controller connected to the plant system so that the plant or the whole closed-loop system achieves a certain control goal. We consider the following three goals: realization of back-action evasion (BAE) measurement, generation of a quantum non-demolished (QND) variable, and generation of a decoherence-free subsystem (DFS). Actually there are a lot of works investigating their mathematical characterizations, physical realizations, and applications especially in quantum information science. This section shows system theoretic characterizations of these notions in terms of controllability and observability properties or transfer functions, in a consistent way with the standard definitions.

A. BAE
The idea of BAE originally comes from the research for GW detection. The Michelson's interferometer described in Sec. II-C is a simplest system for this purpose, and we now know from Eq. (19) that the measurement output y =P out 2 = √ λq ′ 2 +P 2 would offer some information aboutF . The issue is that, in addition to the unavoidable noiseP 2 called the shot noise, the output y contains its conjugateQ 2 , which is called the back-action (BA) noise, as seen explicitly in the Laplace domain: The slight change of the oscillator's position due to the GW effect,ĝ, is defined in the Fourier domain s = iΩ aŝ , where L is the optical path length in the interferometer. Hence under the assumption Ω ≫ ω, the normalized signal containingĝ is given bỹ The noise power ofỹ is bounded from below by the following standard quantum limit (SQL): The last inequality is due to the Heisenberg uncertainty relation |Q 2 | 2 |P 2 | 2 ≥ 1/4. (For the simple notation, the power spectrum is defined without involving the delta function.) The SQL appears because the output contains the BA noiseQ 2 in addition to the shot noiseP 2 . Thus, towards high-precision detection ofĝ by beating the SQL, a special system configuration should be devised so that the measurement output is free fromQ 2 . That is, we need BAE. In fact, if BAE is realized, then by injecting â P 2 -squeezed light field into the dark port, we can reduce the noise power below the SQL and have chance to detect g; for some specific configurations achieving BAE, see [36,37,54,55].
The above discussion can be generalized for the system (12) and (15). Let us assume that the signal to be detected is contained in the output (15): That is,Q = M 1Ŵ is the shot noise, which must appear in y. The BA noise is then given by the conjugatê P = M 2Ŵ . Note that these are vectors of operators: BAE is realized, if the output (21) does not contain the BA noiseP. (We will not consider the so-called variational measurement approach, in which case M 1 is frequency dependent.) In the language of linear systems theory, as stated in Eq. (8), this condition means that there is no subsystem that is controllable w.r.t.P and observable w.r.t. y; i.e.
where CP is the controllability matrix generated from (A, Σ n C ⊤ Σ m M ⊤ 2 ) and O y is the observability matrix generated from (A, M 1 C). Further, again as described in Eq. (8), the condition (23) is equivalent to Under this condition, the system equations (21) and (22) are given in a transformed coordinate by d dt showing that actually there is no signal flow fromP to y. Finally, similar to the classical case (9)

B. QND
Next to see the idea of QND variables, let us here study the atomic ensemble trapped in a cavity [1, 16,67,68] shown in Fig. 4. The atoms couple with a probe polarized light field, via the Faraday interaction. In terms of the total energy operatorĴ z and its conjugatesĴ x andĴ y , which satisfy the CCRs e.g.Ĵ yĴz −Ĵ zĴy = iĴ x , the ideal dynamics of atomic ensemble is described by P is the phase quadrature of the input field's noise operator corresponding to the polarization, and M represents the coupling strength between the atoms and the field. In this setting, the amplitude quadrature of the output field should be measured, giving the following measurement output equation: From these two equations, we find that, through the Faraday interaction, the polarization of the probe field rotates depending on the total energyĴ z , butĴ z itself does not change; that is,Ĵ z is a QND variable that can be measured without being disturbed. Typically M is relatively small, and then the system variables obey a skew-Hermitian dynamics, implying that they preservê J 2 x +Ĵ 2 y +Ĵ 2 z . Hence, in the large ensemble limit and in the short time period, the dynamics is constrained in the tangent space of this super-sphere with radius J = N/2 (N is the number of atoms). In particular let us setĴ x to be a constant J rather than the operator-valued variable.
Then the system variables are given by the usual CCR pairsq =Ĵ y / √ J andp =Ĵ z / √ J satisfyingqp −pq = i, and the above system dynamics can be simplified to the following linear equation: where µ = 2M J. Clearly,p is not disturbed by the noise while it appears in the output signal, thusp is a QND variable. A merit of QND measurement is in the application to state preparation; if a QND variable exists, it is sometimes possible to deterministically stabilize its eigenstate by feedback [1], which can be highly non-classical such as a spin-squeezed state [16,67].
As in the BAE case, we have a general characterization of the linear system (12) and (15) having a QND variable. Letr = v ⊤x be a QND variable with v ∈ R 2n . Then, by definition,r must not be affected by the input fieldŴ, while it appears in the output signal (15), y = M Cx + MŴ. This means that, in the language of linear systems theory,r = v ⊤x is uncontrollable w.r.t. W and observable w.r.t. y. Thus, the iff condition for a QND variable to exist is given by and the vector v lives in this intersection. Here, CŴ and O y are the controllability and observability matrices of the system (12) and (15). Note that the condition v ∈ Ker(C ⊤ W ) can be explicitly represented by Now let us collect QND variables into a single vector x ′ 2 . Then, as described in Sec. II-A,x ′ 2 constitutes an uncontrollable subsystem w.r.t.Ŵ, which can be clearly seen in the transformed coordinate: Note C 2 = 0 due to the observability condition. Hence, x ′ 2 is free fromŴ, while it appears in y. Remarkably, is a generalization of a standard QND variable, which is usually considered to be static (i.e.x ′ 2 (t) = x ′ 2 (0), ∀t); see [69] for further detailed discussion. The above equation now enables us to obtain the equivalent condition to Eq. (26) in terms of the transfer functions: The idea of the third control goal, generation of a DFS, can be clearly seen from the work [59], which studies a quantum memory served by an atomic ensemble in a cavity. Each atom has Λ-type energy levels, constituted by two metastable ground states (|s , |g ) and an excited state |e . The state transition between |e and |g is naturally coupled to the cavity modeâ 1 with strength g √ N (N denotes the number of atoms), while the |s ↔ |e transition is induced by a classical magnetic field with time-varying Rabi frequency ω(t). The system variables are the polarization operatorâ 2 =σ ge / √ N and the spinwave operatorâ 3 =σ gs / √ N , whereσ • is the collective lowering operator; in a large ensemble limit, they can be well approximated by annihilation operators. Consequently the system dynamics is given by where κ denotes the cavity decay rate and δ is the detuning between the cavity center frequency and the |s ↔ |e transition frequency. This system works as a quantum memory in the following way. First, a state to be stored is carried by an appropriately shaped optical pulse on the input fieldÂ, and it is transferred to the metastable state |s ; the Rabi frequency ω(t) is suitably designed throughout this writing process. In the storage stage, the classical magnetic field is turned off, i.e. ω(t) = 0. It is seen from Eq. (29) that the spin-wave operatorâ 3 is then completely decoupled from the fieldsÂ andÂ out ; that is,â 3 constitutes a linear DFS, and ideally its state is perfectly preserved. In the language of systems theory, this DFS is uncontrollable w.r.t.Â and unobservable w.r.t.Â out . Note thatâ 3 is not a variable on the socalled decoherence-free subspace, which though has the same abbreviation. In general, if the system's Hilbert space can be decomposed to (H 1 ⊗ H 2 ) ⊕ H 3 and H 1 is free from external noise, then it is called the DF subsystem and particularly when dimH 2 = 1 it is called the DF subspace [39,40]; now we are dealing with the case whereâ 3 and (â 1 ,â 2 ) live in H 1 and H 2 , respectively, while dimH 3 = 0. For other examples of such an infinite dimensional DFS, see [51,52,53,70,71,72,73]. The above fact reasonably leads to a general characterization of a linear DFS contained in the system (12) and (14). By definition, a DFS is completely decoupled from the probe/environment field, so it is not affected byŴ and also it does not appear inŴ out . In the language of systems theory, a variable contained in the DFS is uncontrollable w.r.t.Ŵ and unobservable w.r.t.Ŵ out . Thus the iff condition for a DFS to exist is given by where CŴ and OŴ out are the controllability and observability matrices of the system (12) and (14). In particular, as seen in Eqs. (4) and (7), there always exists a coordinate transformation such thatr = v ⊤x is a variable of the DFS iff the vector v ∈ R 2n is contained in the intersection Ker(C ⊤ W ) ∩ Ker(OŴ out ); that is, it satisfies (A convenient method to construct such v is given in [72].) Then, as in the QND case, by collecting all variables in the DFS into a single vectorx ′ 2 , we find that the system equations can be transformed to d dt In this paper, we study a general linear system having multi input and multi output fields (it is called a MIMO system). The first essential question is about which input and output fields should be used for feedback. We define the type-1 control as a configuration where at most all the input and output fields can be used for this purpose. Note that, if the system has single input-output channel such as the one shown in Sec. II-C (ii), the control configuration must be of type-1. Figure 5 illustrates the general configuration of type-1 MF control. That is, at most all the plant's output fields can be measured, and the measurement results y(t) are then processed in a classical system (controller) that produces a control signal u(t). From the standpoint comparing MF and CF, we assume that the control is carried out by modulating the input probe fields, which can be physically implemented using an electric optical modulator on the optical field; in the type-1 case, hence, at most all the plant's input fields can be modulated using the control signal u(t). This section examines the type-1 MF control configuration and shows the no-go theorems given in the left column of Table I. A. The closed-loop system with type-1 MF As described above, the MF control is carried out by modulating the input probe fields. This mathematically means that the input field is replaced byŴ + u, where u = [u 1 , . . . , u 2m ] ⊤ is a vector of classical control signals representing the modulation. Hence our plant system is now given by Note that the output field is directly controlled. (In what follows we omit the subscript of Σ • for notational simplicity.) The output signal is obtained by measuringŴ out : whereQ = M 1Ŵ with M 1 the symplectic matrix defined in Sec. II-B. Also the conjugate noise operator is given bŷ P = M 2Ŵ ; as mentioned before, these matrices satisfy M 1 ΣM ⊤ 2 = I, etc. The controller is a classical system that processes the measurement result y(t) and produces the control signal u(t). The dynamical equation of this system can be generally represented by where (A K , B K , C K ) are the parameter matrices to be designed. x K is the vector of controller's variables, and its dimension is also a parameter; hence there is a large freedom in engineering the controller. Note that the matrices are not necessarily of full rank, meaning that in this case some output fields are not measured or some input fields are not modulated. Combining all the above equations, we have the closed-loop (quantum-classical hybrid) dynamics ofx e = [x ⊤ , x ⊤ K ] ⊤ as follows; Hence,Q is the shot noise. Equation (37) can be expressed in terms of the quadraturesQ andP as: due toŴ = M ⊤ 1Q +M ⊤ 2P . We aim to find a set of matrices (A K , B K , C K , M 1 , M 2 ) that achieves the control goal described in Sec. III; but as shown below, it is impossible to accomplish this task.

B. BAE
Suppose that BAE holds for the closed-loop dynamics (39) with output (38); that is, the condition (23) holds for this system, which is now Ker(C ⊤ P ) c ∩ Range(O ⊤ y ) = ∅. (Equivalently, the transfer function of this closed-loop system satisfies Ξ P→y [s] = 0, ∀s.) This is further equivalent, as implied by Eq. (24), to First, the case k = 0 leads to M 1 CΣC ⊤ ΣM ⊤ 2 = 0. Then, using this condition, we find that the case k = 1 yields M 1 CAΣC ⊤ ΣM ⊤ 2 = 0. This further allows us from the case k = 2 to have M 1 CA 2 ΣC ⊤ ΣM ⊤ 2 = 0. Repeating this procedure we eventually obtain This is exactly the BAE condition for the original plant system (21) and (22), i.e.
Equivalently, the transfer function of the original plant system satisfies Ξ Theorem 1: If the original plant system does not have the BAE property, then, any type-1 MF control cannot realize BAE for the closed-loop system.

C. QND
First of all, let us consider the case where the closedloop system (37) and (38) has a QND variabler. This should be "purely quantum", meaning thatr is composed of only the quantum variablesx = [q 1 ,p 1 , . . . ,q n ,p n ] ⊤ ; hence it is of the formr = v ⊤x =ṽ ⊤x e withṽ = [v ⊤ , 0 ⊤ ] ⊤ . As described in Eq. (26), this meansṽ ∈ Ker(C ⊤ W ) ∩ Range(O ⊤ y ), with CŴ and O y the controllability and observability matrices of the system (37) and (38). To prove the no-go theorem, the following two facts are useful. First,ṽ ∈ Ker(C ⊤ W ) means that for all k ≥ 0. It follows from a similar procedure as in the BAE case that this is equivalent to v ⊤ A k ΣC ⊤ Σ = 0, ∀k ≥ 0; i.e. v ∈ Ker(C ⊤ W ) with CŴ the controllability matrix of the original plant system (12) and (15). Second, v ∈ Ker(O y ) is expressed by for all k ≥ 0. This is equivalent to M 1 CA k v = 0, ∀k ≥ 0, meaning that v ∈ Ker(O y ) for the original plant system. Now we prove the theorem. Suppose that the original plant system (12) and (15) does not have a QND variable; hence for any variabler = v ⊤x , the vector v satisfies v ∈ Ker(C ⊤ W ) c or v ∈ Range(O ⊤ y ) c for the original plant system. In particular, since the unobservability property does not depend on the choice of a specific coordinate, the latter condition is equivalently converted to v ∈ Ker(O y ). But as proven above, these two conditions are equivalent toṽ ∈ Ker(C ⊤ W ) c orṽ ∈ Ker(O y ) for the closed-loop system; that is, the closed-loop system does not have a QND variable of the formr = v ⊤x =ṽ ⊤x e . Thus the following result is obtained.
Theorem 2: If the original plant system does not have a QND variable, then, any type-1 MF control cannot generate a QND variable in the closed-loop system.

D. DFS
Finally we prove the no-go theorem for generating a DFS via the type-1 MF control. Let us assume that the closed-loop dynamics (37) with the output field contains a DFS composed of "purely quantum" variables of the formr = v ⊤x =ṽ ⊤x e . Then, it follows from the statement below Eq. (30) thatṽ ∈ Ker(C ⊤ W ) and v ∈ Ker(OŴ out ) hold. As proven in the QND case, the first condition equivalently leads to v ∈ Ker(C ⊤ W ) for the original plant system (12) and (14). Also in almost the same way we can prove that the second condition is equivalent to v ∈ Ker(OŴ out ) for the original plant system. These two conditions on v mean that the original plant system (12) and (14) has a DFS, thus we have the following theorem.
Theorem 3: If the original plant system does not have a DFS, then, any type-1 MF control cannot generate a DFS in the closed-loop system.

V. THE NO-GO THEOREMS: TYPE-2 CASE
In the type-1 case, it is assumed that at most all the plant's output fields can be used for feedback control and they are equally evaluated. For example, in the type-1 BAE case, the BA noiseP must not appear in all the elements of y. But it is sometimes more reasonable to give different roles to the output fields; such a control schematic in the MF case is illustrated in Fig. 6, which we call the type-2 control configuration. In this case, at most all the components ofŴ out 1 can be used for feedback control, while those ofŴ out 2 are for evaluation; that is, they will be measured to extract some information about the system or will be kept untouched for later use. For instance, we attempt to design a MF control based on the measurement ofŴ out 1 , so that the BA noise does not appear in the measurement output ofŴ out 2 . However, we will see that such a MF control strategy does not work to achieve any of the control goals. That is, in this section, the type-2 no-go theorems in Table I will be proven.
A. The closed-loop system with type-2 MF As in the case of type-1 control, we study the situation where the feedback control is performed by modulating the input fields. The plant system driven by the modulated fields obeys the following dynamical equation: u 1 and u 2 are the vectors of control signals that represent the time-varying amplitude of the input fieldsŴ 1 and W 2 , respectively. Note that in general the size of C 1 and C 2 need not to be equal. The output fieldŴ out 1 is measured by a set of dyne detectors, which yield M is the symplectic matrix, representing which quadratures ofŴ out 1 is measured. The measurement result y(t) is sent to a classical feedback controller of the form Note that u 2 is allowed to contain the direct term from y, i.e. u 2 = C K2 x K + D K y; but this modification does not change the results shown below, thus for simplicity we assume D K = 0. Combining all the above equations, we end up with the closed-loop dynamics ofx There are two kinds of output signals of the system. One is y(t), which is used for feedback control. Due to the direct control term, it is now of the form The other one is used for evaluation, which is obtained by measuring the second output fieldŴ out 2 : where we have definedQ = M 1Ŵ2 .

B. BAE
The goal of BAE is to evade the BA noise so that it does not appear in the output signal (43). NowQ = M 1Ŵ2 is the unavoidable shot noise andP := M 2Ŵ2 is the BA noise, where the matrices satisfy e.g. M 1 ΣM ⊤ 2 = I. Note that the noise term of the closed-loop system (41) can be expressed by noise term of Eq. (41) Also the original system without control is given by We start with the assumption that BAE holds for the closed-loop system (41) and (43). In terms of the transfer function, this means that Ξ Let us now focus on the Laplace transform of y(t): But Eq. (43) clearly indicates that Ξ P→y [s] = 0, ∀s. This equivalently leads to the following set of equalities: Likewise the proof in the type-1 case, we have which implies that the original system (44) satisfies Ξ Then, using Eq. (45), we deduce Hence, for the original system (44), the transfer function fromP to z is zero; i.e., Ξ

C. QND
The idea for the proof is the same as that taken in the type-1 case. Again, a QND variable is of the formr = v ⊤x =ṽ ⊤x e withṽ = [v ⊤ , 0 ⊤ ] ⊤ . Now the closed-loop system is given by Eqs. (41), (42), and (43), showing that it is subjected to the input noise field [Ŵ ⊤ 1 ,Ŵ ⊤ 2 ] ⊤ and it generates the measurement outputs [y ⊤ , z ⊤ ] ⊤ . Thus by definitionr is a QND variable iffṽ ∈ Ker( The former condition means that ) holds for the original plant system (44).
Related to the latter one, let us consider the conditionṽ ∈ Ker(O y ) ∩ Ker(O z ). This is expressed by for all k ≥ 0, which equivalently leads to Thus v ∈ Ker(O y ) ∩ Ker(O z ) holds for the original plant system (44). From the same discussion as that in Sec. IV-C together with the above results, we obtain the following no-go theorem: Theorem 5: If the original plant system does not have a QND variable, then, any type-2 MF control cannot generate a QND variable in the closed-loop system.

D. DFS
Let us assume that the closed-loop system (41) with the output fieldsŴ out 1 andŴ out 2 , which now satisfy contains a DFS. Equivalently, it contains a subsystem that is uncontrollable w.r.t.Ŵ 1 andŴ 2 and unobservable w.r.t.Ŵ out 1 andŴ out 2 . As before, a variable contained in the DFS is of the formr = v ⊤x =ṽ ⊤x e . Then, first, the uncontrollability condition leads to the same results as in the QND case, i.e. v ∈ Ker(C ⊤ W1 ) ∩ Ker(C ⊤ W2 ) holds for the original plant system (44). Further, it is immediate to see that the unobservability condition yields ) hold for the original plant system. Thus we have the following result.
Theorem 6: If the original plant system does not have a DFS, then, any type-2 MF control cannot generate a DFS in the closed-loop system.

VI. COHERENT FEEDBACK REALIZATIONS: TYPE-1 CASE
Here we turn our attention to the CF control and in what follows will see that, as shown in Table I, it has a capability of achieving the control goals, BAE, QND, and DFS. That is, as mentioned in Sec. I, these are situations where a quantum device has a clear advantage over a classical one. This section is devoted to prove the results in the type-1 CF case.
A. The closed-loop system with type-1 CF The plant system is given by Eqs. (12) and (14) with inputŴ and outputŴ out . In the type-1 control configuration, as described in Sec. IV, at most all the components ofŴ out can be used for feedback, and also at most all the components ofŴ can be controlled. A CF controller is constructed by directly connecting another fully quantum system to the plant system by a feedback way. This means that, in the type-1 CF case,Ŵ out is connected to the controller's input and the controller's output is connected toŴ, without involving any measurement process. The CF control configuration satisfying this setting, which avoids self-interaction of the fields, is depicted in Fig. 7. The controller has two kinds of input-output fields, and its system equation is given by where This condition imposes the size of C 1 and C 2 to be equal, although they are not necessarily of full rank. Note that more generally a scattering process from e.g.Ŵ out toŴ 2 can be introduced, but here it is not necessary. Combining Eqs. (12), (14), (46), and (47), we obtain the dynamical equation of the closed-loop system: wherex e = [x ⊤ ,x ⊤ K ] ⊤ , A e = Σ(G e + C ⊤ e ΣC e /2), C e = [C, C 1 + C 2 ], and ⋆ denotes the symmetric elements of G e .

B. BAE
Let us assume that we can engineer a CF controller satisfying C 1 + C 2 = 0. Then the closed-loop system (48) takes the following form: The structure of this equation shows that, notably, the controller is directly coupled to the plant, yet there is no direct interaction between the field and the controller. This system configuration is called the direct interaction, meaning that an additional quantum device is prepared and is directly coupled to the plant system, not through input/output fields; hence the system (49) is a CF-based realization of the direct interaction.
Here we study the opto-mechanical oscillator described in Sec. II-C (ii), as a plant system. Since this system has one input-output field, the control configuration must be of type-1. Also it is easy to verify that this system does not satisfy BAE, and further, it does not have a QND variable. The goal is to design a CF controller such that BAE is realized for the closed-loop system toward high-precision detection of the unknown forceF . For this purpose, we take the CF scheme described above, leading to Eq. (49). The controller is single mode with variablex K = [q 3 ,p 3 ] ⊤ , and it has two input fieldsŴ 1 = [Q 1 ,P 1 ] ⊤ andŴ 2 = [Q 2 ,P 2 ] ⊤ . The controller's system matrices are chosen so that they satisfy which leads to Physical implementation of the controller specified by these matrices will be discussed in the end of this subsection. Together with the termF , which directly acts on p 1 , the dynamics of the closed-loop system is given by where SinceQ out 2 does not contain any information aboutF , we need to measureP out 2 , implying that the output signal is given by y = MŴ out The set of equations (51) and (52) is exactly the same as that of the modified opto-mechanical oscillator proposed by Tsang and Caves [50], which is shown in Fig. 8 (a). Notably, this system realizes BAE measurement for de-tectingF ; in fact, with the choice g = κ/ √ mω the transfer function from the BA noiseQ 1 to the output y =P out 2 takes zero: Thus by injecting aP 1 -squeezed light field (i.e. by reducing the noise ofP 1 ), in principle we can detectF with better accuracy compared to the case without BAE. A detailed investigation of this BAE scheme in a practical setting was recently reported in [74]. Recall now that the system (51) and (52) is constructed by a CF control. That is, by a constructive method, we have proven that the type-1 CF control can realize BAE.
Lastly, let us consider an optical implementation of the above CF controller. The form of C 1 (or C 2 ) in Eq. (50) represents the so-called QND interaction of the controller and the fieldŴ 1 (orŴ 2 ), which can be physically implemented though in a nontrivial way [75]. The controller's Hamiltonian specified by G K in Eq. (50) simply expresses the optical phase shift. Consequently, a detuned optical cavity coupled to two input-output fields via QND interactions, illustrated in Fig. 8 (b), is one possible physical realization of the CF controller proposed here. Note that its practical implementation is harder than that of the system given in [50]. But apart from such difficulty, again, what should be emphasized here is the fact that the type-1 CF control is capable of realizing BAE.

C. QND
Let us continue to examine the above CF-controlled opto-mechanical oscillator (51) and (52); actually we here show that this system contains QND variables, by proving Eq. (26), which is now Ker(CŴ √ mω, the range of the controllability matrix CŴ 1 = [B e , A e B e , A 2 e B e ] is spanned by the following independent vectors: Note that A k e B e (k ≥ 3) does not anymore produce an independent vector. Clearly, But they are orthogonal to both v 1 and v 2 , meaning that v 1 and v 2 are contained in Range(O ⊤ y ). Consequently, are uncontrollable w.r.t.Ŵ 1 and observable w.r.t. y (see the discussion around Eq. (26)); that is,q ′ andp ′ are QND variables generated by the CF control. Indeed, they are subjected to the dynamical equation of the form which clearly shows that (q ′ ,p ′ ) are free fromŴ 1 .
Here an interesting by-product is obtained. It is easy to see [q ′ (t),p ′ (t)] = 0, ∀t. Together with the fact that (q ′ ,p ′ ) are independent from other variables, this means that they are essentially classical variables which are detectable from the output field. In general, if a quantum system contains a subsystem whose variables are all commutative, then it is called a classical subsystem [69]; thus we now found that the CF-controlled opto-mechanical system (51) contains a classical subsystem (53).

D. DFS
To show that the type-1 CF control has capability of generating a DFS, let us return to the general closed-loop system (48). Suppose now that the original plant system (12) and (14) does not have a DFS, and further that a quantum controller with parameters C 1 = C 2 = C/2 and G K = G can be engineered. Hence, the plant and the controller have the same number of modes. Then Eq. (48) takes the following form: Now we prove that this system contains a DFS, i.e. a subsystem that is uncontrollable w.r.t.Ŵ 1 and unobservable w. ). This means, as discussed above Eq. (31), that v ⊤ exe = v ⊤x − v ⊤x K is uncontrollable and unobservable, hence this is the variable of a DFS generated by the CF control. Note that 2n independent vectors (v 1 , . . . v 2n ) can be taken to construct v

VII. COHERENT FEEDBACK REALIZATIONS: TYPE-2 CASE
In this section we study the type-2 CF control for realizing BAE, QND, and DFS. As in the type-1 case, a specific system achieving each control goal will be shown.
A. The closed-loop system with type-2 CF As explained in Sec. V, the type-2 control means that two roles are given to the output fields of the plant system; one is for feedback control, and the other one is for evaluation. Hence the system to be controlled is For designing a CF controller, there are some variation in its structure. Here we particularly consider the CF control configuration illustrated in Fig. 9; that is, the controller has a single kind of input-output fields that are directly connected to the plant's input and output fields. For a general type-2 CF control configuration, see [18]. Note also that, in our case, C 1 and C 2 are of the same size, although they are not necessarily of full rank. Hence the dynamics of the CF controller is given by where A K = Σ(G K + C ⊤ K ΣC K /2), and the feedback connection is realized bŷ Here S is an orthogonal and symplectic matrix representing a scatting process fromŴ out 1 toŴ 3 . Combining the above equations yields the dynamical equation of the closed-loop system; ⋆ denotes the symmetric elements of G e .

B. BAE
To demonstrate that the type-2 CF is capable of realizing BAE, we here study the Michelson's interferometer as a plant system, which is described in Sec. II-C (iii) with Fig. 3 (c). The system is composed of two oscillators driven by an unknown forceF along opposite directions. The oscillators' dynamical motion is described by Eq. (18), which is specified by the following system matrices: G = diag{mω 2 , 1/m, mω 2 , 1/m} and This system works as a sensor for detecting the forceF ; but as explained before, the noise power of the output signal is bounded from below by the SQL (20). Hence the purpose here is to design a CF controller that realizes BAE and as a result beats the SQL. Actually, the plant system has two input-output ports, hence it can be treated within the type-2 CF control framework.
Here we consider the CF configuration described in the previous subsection. That is,Ŵ out 1 andŴ 2 are optically connected through CF. In particular, as a CF controller, let us take a single input-output optical cavity, whose dynamical equation is specified by the following matrices: where ǫ is the coupling constant between the field and the cavity mode. Later we will set α = β, which thus represents the detuning. S represents a phase shift acting on the input optical field in the formÂ 3 = −iÂ out 1 . Thus the closed-loop system is a 3-modes single input-output linear system, depicted in Fig. 10.
With the above setup, the closed-loop system (56) takes the following form: Let us seek the parameters (α, β, ǫ) that achieve BAE. First, it is easy to see c ⊤ 1 A k e b f = 0, ∀k ≥ 0, or equiva- does not contain any information aboutF . Thus we measure implying thatQ 1 is the shot noise whileP 1 is the BA noise. Thus the parameters should be chosen so that the BAE condition (23) i.e. Ker(C ⊤ P1 ) c ∩ Range(O ⊤ y ) = ∅ is satisfied, which is carried out by examining the equivalent condition (24): c ⊤ 2 A k e b 1 = 0, ∀k ≥ 0. The case k = 0 is already satisfied. To see the case k ≥ 1, let us focus on Then, the condition is satisfied if we impose c ⊤ 2 A e b 1 = 0 and A 2 e b 1 ∝ b 1 , which yield λ m + αǫ = 0, ω 2 + 2ǫ 2 = αβ + ǫ 2 .
Let us especially take the parameter α = β < 0, implying that the CF controller is an optical cavity with negative detuning α. The parameters are then explicitly given by When ω ≪ 1, they are approximated by λ/m and − λ/m, respectively. Actually under the condition (59), the output is described in the Laplace domain by which is free from the BA noiseP 1 [s]. As expected, this BAE measurement beats the SQL and enables highprecision detection ofF . To see this fact, let us evaluate the power spectrum density of the noise. As seen before, F induces the oscillators's position shiftĝ in the Fourier Then, under the assumption ω ≪ Ω, the normalized signal is given bỹ Using ǫ = λ/m we obtain S[iΩ] = |ỹ −ĝ| 2 = λ m 2 L 2 Ω 4 + 1 4λL 2 |Q 1 | 2 , which has the same form as that of the non-controlled scheme in Eq. (20), except that the BA noise is replaced by the shot noise. Therefore, by injecting aQ 1 -squeezed light field into the first input port (i.e. the bright port), we can realize a broadband noise reduction below the SQL (20) in the output noise power.
Note that, while we have found a CF controller achieving BAE for high-precision detection ofF below the SQL, the result obtained here does not mean to emphasize that the proposed schematic is an alternative configuration for gravitational wave detection. Actually, the schematic is very different from several effective methods, particularly in that the second output port is not anymore a dark port. Taking into account the input laser noise and the fact that a squeezed light field is in general fragile, the above-described ideal detection ofĝ below the SQL would be a difficult task in a practical situation. Rather the main purpose here is to prove the capability of a type-2 CF controller for realizing BAE. Also, as demonstrated above, it is remarkable that the problem for designing BAE can be solved, by a system theoretic approach based on the controllability/observability notion; this approach might shed a new light on the problem for designing a system for gravitational wave detection.

C. QND
We here see that the closed-loop system studied in the previous subsection contains QND variables. Note that the original interferometer does not have a QND variable.
First let us calculate the controllability matrix CŴ ′ 1 = [B e , A e B e , A 2 e B e , . . .] with A e and B e given in Eq. (57). It was already seen that b 1 generates two dimensional subspace spanned by b 1 and A e b 1 , under the condition (59). Now, by further imposing α = β, we have A 2 Hence dimRange(CŴ ′ 1 ) = 4. Let us take two independent vectors v 1 and v 2 spanning Ker(C ⊤ W ′ 1 ); then v ⊤ 1xe and v ⊤ 2xe are not affected by the input fieldŴ ′ 1 . Moreover, these variables appear in the output signal (58) as shown below. Actually we can prove that c 2 and A ⊤ e c 2 are both independent to the above four vectors, implying Thus Range(CŴ ′ 1 ) ∪ Range(O ⊤ y ) = R 6 holds, which further leads to Range(CŴ ′ 1 ) c ⊆ Range(O ⊤ y ). Consequently, we find v 1 , v 2 ∈ Range(O ⊤ y ), meaning that v ⊤ 1x e and v ⊤ 2xe appear in y and thus they are QND variables. That is, the type-2 CF controller described in Sec. VII-B has capability of generating QND variables.

D. DFS
Lastly we again study a general plant system (55); suppose that it satisfies C 1 = C 2 = C/2 and does not contain a DFS. For this system, let us choose a type-2 CF controller with system matrices G K = G and C K = C, which is directly connected to the plant (i.e. S = I).
Then the closed-loop system (56) takes exactly the same form as Eq. (54), which contains a DFS. Therefore, this type-2 CF controller has ability to generate a DFS.

VIII. CONCLUSION
This paper has given some general answers to the question about whether or not measurement should be involved in the feedback structure for controlling a quantum system. That is, for a general linear quantum system, we have obtained the no-go theorems stating that the control goal, realization of BAE, QND, or DFS, cannot be achieved by any MF control; on the other hand, for each control goal, we have found an example of CF control accomplishing the task. From the viewpoint that MF is essentially a classical operation on the system while CF is a fully quantum one, these results imply that BAE, QND, and DFS are genuine quantum objectives that cannot be realized by any feedback-based classical operation.
The key idea to obtain all the results is the following system theoretic characterizations of BAE, QND, and DFS, which are also summarized in Fig [s] = 0, ∀s & Ξx′ 2 →Ŵ out [s] = 0, ∀s. Although in this paper these characterizations are not fully used except Sec. V-B, they will serve as powerful tools in quantum device engineering in a practical situation. In fact, in reality due to several experimental imperfections, it is often the case that the controllability/observability matrix becomes of full rank, and thus the perfect achievement of the above geometric conditions cannot be expected. Nonetheless, the functional approach based on the transfer function allows us to obtain an approximate solution of those problems. For instance for the BAE case, even if Ker(C ⊤ P ) c ∩ Range(O ⊤ y ) = ∅ or equivalently ΞP →y [s] = 0, ∀s is never satisfied, an approximate BAE measurement can be engineered by solving a minimization problem ΞP →y [s] → min. Actually, in the history of classical control, the so-called geometric control theory was first deeply investigated [8], pursuing e.g. ideal disturbance decoupling. Later, towards wider applicability of the control theory, several functional approaches were developed [9]; the linear quadratic Gaussian (LQG) control and H ∞ control, which are respectively based on the minimization of the H 2 norm · 2 and the H ∞ norm · ∞ of a transfer function, are typical successful results. A notable fact is that, as mentioned in Sec. I, recently quantum versions of those classical feedback control methods have been deeply developed. Therefore combination of the geometric and functional approaches will constitute a new methodology in the field of quantum control and information. Of course, we should note that, under the evaluation of minimizing a norm of a transfer function, comparing MF and CF controls again becomes an open problem.
In this paper, from the standpoint comparing CF and MF, we assumed that a MF controller is given by a dynamical one with internal variable x K and that the control is carried out by modulating the plant's input fields. However, the control configuration is not limited to the dynamical one; the direct measurement feedback proposed by Wiseman [76] is indeed the first scheme applying the classical feedback control in the quantum domain. The control configuration of direct MF is different from the dynamical one in several points; what is most notable here is the fact obtained in [44], clarifying that a direct MF can produce a QND variable, unlike the dynamical one. Let us here review this result.
The plant system is an optical cavity containing a χ 2 nonlinear crystal, and further, the cavity mode can be directly controlled by a modulator. The output signal is obtained by measuring the amplitude quadrature of the output field. The system equations are then given by dx dt = −κ 0 0 0 x + 1 0 u − √ κ Q P , y = √ κq +Q, wherex = [q,p] ⊤ is the cavity mode quadratures, u(t) is the control signal representing the amplitude modulation, and κ is the coupling strength between the cavity and the probe fields. Note that this modulation effect does not appear in the output. The direct feedback considered in [44] is of the form u = √ κy, which ideally enables us to modify the system dynamics so thatx evolves in time with the following linear equation: Clearly,q is not disturbed by the noise while it appears in the output signal, implying that we can measureq without disturbing it. That is,q is a QND variable. The above result means that the type-1 no-go theorem does not hold, if a direct MF is allowed. The other cases will be addressed in future works.