Entanglement-assisted quantum feedback control
arXiv:1505.04539v1 [quant-ph] 18 May 2015
Naoki Yamamoto and Tomoaki Mikami
Abstract— The great advantage of quantum metrology relies on the effective use of entanglement, which indeed allows us to achieve strictly better estimation performance over the so-called standard quantum limit. In this paper, we study an analogous strategy utilizing entanglement in a feedback control problem and show that, in the problem of cooling a nano-mechanical oscillator, it indeed can lower the control cost function below the limit attainable by the standard control method.
(a)
(b)
θ
θ
Fig. 1. (a) Standard setup for estimating an unknown parameter θ. (b) Quantum metrology setup using an entangled input.
I. I NTRODUCTION Entanglement is a very special notion appearing only in quantum mechanics, which had been considered as what causes “spooky actions” [1]. However over these three decades it has gained a completely opposite impression, mainly thanks to the fact that it plays a central role in quantum information science [2], [3]. A particularly important application of entanglement in our context is the quantum metrology [4], [5]. Its basic formulation is depicted in Fig. 1. The goal is to estimate an unknown parameter θ contained in the system. A standard estimation method is first to send a known input state and then measure the output state containing the information about θ (Fig. 1 (a)); in this case the estimation error has a strict lower bound called the standard quantum limit (SQL). In the quantum metrology schematic depicted in Fig. 1 (b), on the other hand, an entangled state is chosen as an input and its one portion interacts with the system while the other portion does not; then by measuring the combined output fields, we obtain more information about θ than the standard case (Fig. 1 (a)) and thus can beat the SQL in the estimation error. This schematic was experimentally demonstrated in several optical settings, e.g. [6], [7]. What we learn from the quantum metrology theory is that, in a broad sense, a quantum estimator could have better performance if assisted by entanglement. Therefore it should be a reasonable idea to employ such an entanglementassisted estimator in the measurement-based quantum feedback control [8], [9], which is now well established based on the quantum filtering theory [10], [11]. Actually, if the quantum filter gains more information by constructing a similar configuration depicted in Fig. 1 (b), it is highly expected that it offers a better estimate and as consequence a better control performance. The idea of this entanglementassisted feedback control is briefly mentioned in the recent publication [12], but in fact there have been no study in the literature to examine how much it would be beneficial. N. Yamamoto and T. Mikami are with the Department of Applied Physics and Physico-Informatics, Keio University, Hiyoshi 3-14-1, Kohoku, Yokohama, Japan (e-mail:
[email protected]).
In this paper, for the purpose of evaluating the performance of the entanglement-assisted feedback control, we conduct a case study in the following setup. We consider a general linear quantum system such as an optical cavity and a large atomic ensemble [13], [14], [15], in which case the filter is simplified to the quantum Kalman filter [16]. The input is given by an optical entangled state generated by combining a squeezed field and a coherent field [17]. The control goal is to minimize a quadratic-type cost function. That is, we consider the quantum Linear Quadratic Gaussian (LQG) optimal control problem; fortunately, we can analytically solve this problem by almost the same way as in the classical case [18], [19]. A clear advantage of this formulation is that a strict lower bound of the cost, which is achievable by employing the so-called cheap control [20], is represented by a function of only the estimation error. Hence the ultimate achievable limit of the LQG control cost can be evaluated by only examining the filter. Here we describe two important features of the proposed control configuration. First, we explicitly take into account the amount of usable energy for generating the input field; this allows us to compare the performance of the entanglement-assisted feedback control scheme to that of the standard method with a non-entangled input state yet having the same amount of energy. Second, the model contains an optical loss in the system’s output channel. In fact, for some systems such as a super-conductivity circuit, the system’s output field can easily get diminished along the output waveguide before arriving at the detector. Based on the above-mentioned formulation, we examine a numerical simulation. The system is a nano-mechanical oscillator [21], [22], [23], [24], [25], which is cast as an important platform demonstrating several quantum information processing. For those purposes it is necessary to reduce (i.e. cool) the energy of the oscillator toward its motional ground state. In fact we demonstrate that the entanglement-assisted feedback control has effectiveness in this cooling problem, particularly when the system’s output field is lossy. Finally, we note that the scheme presented in this paper
differs from that studied in the recent paper [25], which considers the use of system’s internal entanglement to enhance the cooling. Notation: ℜ, ℑ denote the real and imaginary parts, respectively. In : n × n identity matrix. On : n × n zero matrix. 0n×m : n × m zero matrix. II. Q UANTUM K ALMAN
FILTERING AND
LQG
CONTROL
We here review the general theory of quantum Kalman filtering and LQG optimal control. A. Quantum linear systems In this paper we deal with a linear quantum system, whose general form is given as follows. The system variables are collected in a vector of operators x ˆ := [ˆ q1 , pˆ1 , . . . , qˆn , pˆn ]⊤ , where the infinite-dimensional operators qˆi and pˆi have physical meanings of position and momentum. They satisfy the canonical commutation relation (CCR) qˆi pˆj − pˆj qˆi = iδij (we set ~ = 1), which are summarized as 0 1 x ˆx ˆ⊤ − (ˆ xx ˆ⊤ )⊤ = idiag{σ, . . . , σ}, σ = . −1 0 The system variables are governed by the linear dynamics ˆ t, dˆ xt = Aˆ xt dt + F ut dt + BdW
(1)
where ut is a vector of classical (i.e. non-quantum) signal representing the control input. Further, we have an associated measurement signal: ˆ t. dyt = C x ˆt dt + DdW
(2)
Equations (1) and (2) have some special features that do ˆt not appear in the classical case. First, the noise term W ⊤ ˆ ˆ ˆ ˆ ˆ is a vector of operators Wt := [Q1 , P1 , . . . , Qm , Pm ] that ˆ t dW ˆ ⊤ = Θdt. (The mean satisfies the following Ito rule: dW t ˆ is zero: hdWt i = 0.) The correlation matrix Θ is 2m × 2m block-diagonal Hermitian, and their jth block matrix (i.e. the ˆ j and Pˆj ) is in general written as correlation matrix of Q Nj + ℜ(Mj ) + 1/2 ℑ(Mj ) + i/2 Θj = . (3) ℑ(Mj ) − i/2 Nj − ℜ(Mj ) + 1/2 The parameters Nj ∈ R and Mj ∈ C satisfy Nj (Nj + 1) ≥ ˆ j dPˆj − dPˆj dQ ˆ j = idt. Another |Mj |. Also note that dQ feature is that, due to the CCR of xˆt , the system matrices of Eqs. (1) and (2) have specific structure represented by ⊤ A = Σn (G + Σ⊤ n BΣm B Σn /2),
C = DΣm B ⊤ Σn . (4)
Σn is the 2n-dimensional block diagonal matrix defined by Σn := diag{σ, . . . , σ}. G is 2n × 2n real symmetric and B is 2n × 2m real; they are determined by the system’s Hamiltonian. D is an ℓ × 2m real matrix satisfying DΣm D⊤ = 0. F does not have special structure. For more details, see e.g. [15]. Note that, when B = 0, the output is simply a noise ˆ t , implying that the dynamics must be process dyt = DdW subjected to a noise if we want to gain some information about the system.
B. Quantum Kalman filter Let us consider the situation where we want to estimate the system variable x ˆt through the output signal yt . It is indeed possible that, thanks to the quantum filtering theory, we can rigorously define the quantum conditional expectation π(ˆ xt ) := E(ˆ xt |Yt ), where Yt is the σ-algebra composed of the collection of yt up to time t. Note that π(ˆ xt ) is a classical random variable and is the least mean squared estimate of x ˆt . The recursive equation updating π(ˆ xt ) is given by the following quantum Kalman filter: dπ(ˆ xt ) = Aπ(ˆ xt )dt + F ut dt + Kt (dyt − Cπ(ˆ xt )dt), Kt = (Vt C ⊤ + Bℜ(Θ)D⊤ )(Dℜ(Θ)D⊤ )−1 ,
(5)
with π(ˆ x0 ) = hˆ x0 i. Likewise the classical case, dw ¯t := dyt − Cπ(ˆ xt )dt represents the innovation process. Also Vt is the error covariance matrix defined as ⊤ xt ∆ˆ x⊤ xt := x ˆt − π(ˆ xt ). Vt := h∆ˆ xt ∆ˆ x⊤ t ) i/2, ∆ˆ t + (∆ˆ
Vt evolves in time according to the following Riccati differential equation: V˙ t = AVt + Vt A⊤ + Bℜ(Θ)B ⊤ − Kt Dℜ(Θ)D⊤ Kt⊤ . (6) C. Quantum LQG control In the infinite horizon LQG control problem, we consider the following cost function: 1 J[u] = lim T →∞ T
Z
T
(ˆ x⊤ xt t Qˆ
0
+
u⊤ t Rut )dt
,
(7)
where Q ≥ 0 and R > 0 are weighting matrices. The goal is to design the feedback control law of ut that minimizes the cost (7), i.e. u∗ = arg minu J[u]. Now the point is that, due to the tower property hˆ xt i = hπ(ˆ xt )i, the cost function can be represented in terms of the filter variable as 1 T →∞ T
J[u] = lim
Z
0
T
dt π(ˆ xt )⊤ Qπ(ˆ xt ) + u⊤ Ru t t
+ Tr (QV∞ ).
(8)
The second term is constant, since Vt does not depend on ut . Hence, our problem is to find ut minimizing the first term of Eq. (8) under the condition (5). This is exactly the classical LQG control problem, thus the solution is given by u∗t = −R−1 F ⊤ Pt π(ˆ xt ), (9) P˙ t + Pt A + A⊤ Pt − Pt F R−1 F ⊤ Pt + Q = O. (10) Note that u∗t is a function of the optimal estimate π(ˆ xt ), implying that we can design the optimal estimate and control separately; this is called the separation principle as in the classical case. The minimum value of the cost is given by ⊤ J[u∗ ] = Tr (K∞ Dℜ(Θ)D⊤ K∞ P∞ ) + Tr (QV∞ ).
(11)
W2
D. Lower bound of the minimum cost: The cheap control Let us take R → +0 in the cost function (7); this means that no penalty is imposed on the magnitude of the control input ut , and thus the resulting optimal control is called the cheap control. Then, if the system is minimum phase and right invertible, the first term of the minimum cost (11) can be arbitrarily made small by the cheap control (see [20]). Thus, the strict lower bound of the minimum LQG cost is given by J ∗ [u∗ ] = Tr (QV∞ ). (12) From Eq. (7) this means that the stationary energy hˆ x⊤ x∞ i ∞ Qˆ can be ultimately reduced, by the ideal cheap control, to Tr (QV∞ ), which is a function of only the optimal estimation error and cannot be further suppressed by any control. III. C ONFIGURATION
OF THE ENTANGLEMENT- ASSISTED
FEEDBACK CONTROL
The entanglement-assisted feedback control configuration considered in this paper is depicted in Fig. 2. This model has the following special features. (i) The system is linear and has a single input-output channel. (ii) The entangled optical field is produced by combining ˆ 1 with fixed squeezing level a coherent squeezed field W ˆ r and a coherent field W2 , at a beam splitter (BS1) with reflectivity β12 . It is known that, for all β1 ∈ (0, 1), the output fields of BS1 are entangled. As shown in the figure, one portion of this entangled field couples with the system, while the other portion does not and will be measured directly. We can change the entanglement level by tuning β1 , while maintaining the total amount of energy for creating the squeezed state. Hence we can conduct a fair comparison of the entanglement-assisted method and the standard method without entanglement, given the same amount of ˆ 1 and W ˆ 2 are changed energy. Note that the amplitude of W depending on β1 , so that each output field of BS1 has a constant amplitude. (iii) We assume that the system’s output field is subjected to an optical loss, which can be modeled by introducing a fictitious beam splitter with reflectivity δ 2 ; if δ = 0, ˆ 3 denotes the vacuum field then there is no optical loss. W coming into this fictitious beam splitter. As consequence the ˆ 1, W ˆ 2 , and overall system contains three input channels W ˆ W3 , implying that m = 3 in the system equations (1) and (2), and the correlation matrix (3) is now given by Θ = diag{Θ1 , Θ2 , Θ3 }, 1 1 e−r i , Θ2 = Θ3 = I2 . Θ1 = −i er 2 2 (iv) The system’s output field after subjected to the optical loss again meets the other portion of the entangled input field, at the second beam splitter (BS2) with reflectivity β22 . Then the final output fields are measured by two Homodyne detectors with phase θ1 and θ2 , which generate two output signals y1 and y2 . Note that, if β2 = 0 or β22 = 1, the two optical fields are not combined at BS2 and are measured
β1
W1
W3
δ
System
β2
θ2
Filter & Controller
θ1
Fig. 2. General feedback control configuration for a linear system with the entangled input field. β12 , β22 , and δ2 represent the reflectivity of the beam splitters. (δ represents the optical loss of the system’s output field.) θ1 and θ2 are the phases of Homodyne detectors. The signal path from the controller to the system is not shown.
independently; this type of measurement is called the local measurement. In the other case 0 < β22 < 1, it is called the global measurement. The overall system dynamics realizing the above setup is given as follows. First we use the fact that, for a general open linear system interacting with a single optical field, the system-field coupling is represented by an operator of the ˆ = c⊤ x form L ˆ with c the 2n-dimensional complex vector, and this determines the B matrix as follows (see e.g. [15]). That is, by defining the 2 × 2n real matrix √ ℜ(c)⊤ , (13) C¯s = 2 ℑ(c)⊤ we can specify the B matrix in the following form: (14) B = [α1 Σn C¯s⊤ Σ1 , β1 Σn C¯s⊤ Σ1 , 02n×2 ], p where α1 = 1 − β12 . Then the A matrix is determined by Eq. (4), where G is determined from the system Hamiltonian ˆ =x of the form H ˆ⊤ Gˆ x/2. The C matrix is also specified by Eq. (4) and is now given by α1 C¯s (15) C = DΣ3 B ⊤ Σn = D β1 C¯s . 02×2n Here D is of the form
D = [D1 , D2 , O2 ]T2 TL T1 , where D1 =
cos θ1 0
sin θ1 0
,
D2 =
0 cos θ2
0 sin θ2
and p 1 − βk2 I2 p βk I2 O2 Tk = −βk I2 1 − βk2 I2 O2 (k = 1, 2), O2 O2 I2 √ 2 1 − δ I2 O2 δI2 . O2 I2 √ O2 TL = 2 1 − δ I2 −δI2 O2
Note that T1 and T2 represent the scattering process at BS1 and BS2, respectively. Also TL corresponds to the optical
W2
loss in the system’s output field. D1 and D2 represent the Homodyne measurements with phase θ1 and θ2 , respectively. IV. C OOLING
ENHANCEMENT OF MECHANICAL
OSCILLATOR VIA ENTANGLED INPUT FIELD
( q1, p1) Circulator
β1
W1
W3
In this section we conduct a numerical simulation to demonstrate the effectiveness of the proposed entanglementassisted feedback control scheme.
β2
A. System model and control goal The system of interest is a nano-mechanical oscillator connected to an optical cavity shown in Fig. 3. Let (ˆ q1 , pˆ1 ) be the position and momentum operators of the oscillator, and a ˆ2 be the annihilation operator of the cavity. The system Hamiltonian is given by ∆ 2 ˆ = ω (ˆ H q 2 + pˆ21 ) + (ˆ q + pˆ22 ) − λˆ q1 qˆ2 , 2 1 2 2 where the first and second terms represent the oscillator’s and cavity’s free time evolution with frequency ω and ∆, respectively (in the rotating flame of the laser √ frequency). √ a2 − a ˆ†2 )/ 2i. The third Note qˆ2 = (ˆ a2 + a ˆ†2 )/ 2 and pˆ2 = (ˆ term is the linearized radiation pressure force with strength |λ|, representing the interaction between the oscillator and ˆ = xˆ⊤ Gˆ the cavity 1 . From the relation H x/2 we have ω 0 −λ 0 0 ω 0 0 G= −λ 0 ∆ 0 . 0 ∆ 0 0 The system couples to the outer field at a partially reflective end-mirror of the cavity, with strength κ; the cavity-field coupling operator is then given by r √ κ ˆ L = κˆ a2 = (ˆ q2 + iˆ p2 ), 2 p which leads to c = κ/2[0, 0, 1, i]. Thus Eq. (13) yields √ 0 0 1 0 . C¯s = κ 0 0 0 1 This determines the system’s B and C matrices from Eqs. (14) and (15), respectively, and the A matrix from Eq. (4). In addition, we assume that the oscillator is subjected to a thermal environment with its mean photon number n ¯ . This brings modification of some system matrices; particularly in our case we need to change the A matrix to A − γQ/2 and the constant term in the Ricatti equation (6), Bℜ(Θ)B ⊤ , to Bℜ(Θ)B ⊤ + γ(2¯ n + 1)Q, where γ represents the systemenvironment coupling strength and Q = diag{1, 1, 0, 0}. Finally we assume that the oscillator can be directly controlled, for instance by applying an auxiliary laser field, so that the system satisfies the conditions for the cheap control; equivalently, we assume the existence of the matrix F satisfying those conditions. 1 λ depends on the amplitude of the input laser field into the cavity; now, ˆ 1 and W ˆ 2 are as mentioned in part (ii) of Section III, the amplitude of W changed depending on β1 so that the amplitude of the input field into the cavity is kept constant.
a2
δ
θ2
θ1
Fig. 3.
Filter & Controller
Nano-mechanical oscillator coupled to the entangled field. 1.4
1.2
1
E min 0.8
SQL
0.6 0.4 0
0.2
0.4
0.6
0.8
1
Reflectivity of BS1
Fig. 4. The achievable lowest energy of the oscillator, Emin , versus the reflectivity of BS1 in the case δ2 = 0.9. The red solid line shows the case β2 = 0 (local measurement), while the black dotted line shows the case β22 = 0.2 (global measurement).
Our control goal is to cool the oscillator toward its motional ground state; i.e., we want to minimize the stationary energy (more precisely, the effective stationary excitation number) of the oscillator, which is defined by 2 E = (hˆ q1,∞ i + hˆ p21,∞ i − 1)/2.
As described in Sec. II-D, this can be ultimately reduced, by the ideal cheap control, to Emin = (Tr (QV∞ ) − 1)/2,
(16)
where Q is the above-defined diagonal matrix and V∞ is the stationary solution of the Riccati equation (6). In the simulation we evaluate Eq. (16) with several parameters. B. Simulation result The system parameters are set to typical values found in the literature (e.g. [25]); in the unit of ω = 1, we take λ = 0.3, κ = 4 (bad cavity regime), and ∆ = 1 (red detuned driving). The oscillator is subjected to a thermal noise with mean photon number n ¯ = 1 × 105 with coupling strength γ = 1×10−7 , implying that the oscillator has a high Q factor and is subjected to a very low temperature environment. The squeezed coherent field has squeezing level r = 2.3 (10 dB squeezing), which is enough achievable in the current technology.
Figure 4 shows Emin as a function of the reflectivity of the first beam splitter, β12 , in the case δ 2 = 0.9; that is, the system’s output field is very lossy. The reflectivity β12 represents how much the squeezed light field is split into two arms, which determines the amount of entanglement. Note that when β1 = 0 or β12 = 1, the input light fields are not entangled. In particular, β12 = 1 corresponds to the standard case where only the coherent light field is injected to the system; hence the value of Emin in this case has the SQL meaning of SQL, which is now Emin ≈ 0.72 as indicated in the figure. Emin is calculated by solving the Ricatti equation (6), and further, it is minimized with respect to the phases of the two Homodyne detectors (i.e. φ1 and φ2 ) at each β1 . In the figure the red solid line and the black dotted line correspond to the case β2 = 0 (local measurement) and β2 = 0.2 (global measurement), respectively. Notably, in both cases the minimum of Emin is attained at around β12 = 0.65, in which case the input field is entangled. Hence the entanglement-assisted feedback method realizes further cooling below the SQL. Note that controlling with the local measurement (β2 = 0) shows better performance over the global one (β22 = 0.2); in fact in the simulation we found that the local measurement is the best strategy to reduce the energy. In general a global measurement has a meaning of “quantum measurement” in the sense that it can cause non-classical effects such as entanglement; but in our scenario such quantum measurement is not useful. Finally, we should remark that simply injecting the squeezed field into the system without entanglement (i.e. β1 = 0) is the worst case having no benefit in cooling. Next Fig. 5 shows the case δ 2 = 0.1, i.e. we assume that the optical loss in the output channel is relatively small. In this case, we found in the simulation that, for any choice of β2 , the SQL (≈ 0.1) provides almost the lowest energy, as particularly indicated by two lines corresponding to the cases β2 = 0 and β22 = 0.2. This fact states that, if the output channel is well fabricated and the output field becomes nearly noiseless, entanglement does not help to further reduce the energy. As a summary, a clear advantage of the entanglementassisted feedback controller appears in the practical situation where the output field is lossy, since even in such case the controller can gain a non-diminished yet still informative output field through entanglement. V. C ONCLUSION In this paper, we have formulated the entanglementassisted feedback control scheme for general linear quantum systems, which takes into account the amount of resource entanglement, optimized local/global measurement, system noise, and optical loss in the system’s output channel. The control performance is evaluated by the strict lower bound of LQG cost function, which is achievable by the ideal cheap control. In the numerical simulation studying the cooling problem of a nano-mechanical oscillator, it was clarified that the scheme is indeed effective, compared to the standard
0.22 0.2 0.18 0.16
E min 0.14 0.12
SQL
0.1 0.08 0
0.2
0.4
0.6
0.8
1
Reflectivity of BS1
Fig. 5. The achievable lowest energy of the oscillator, Emin , versus the reflectivity of BS1 in the case δ2 = 0.1. The red solid line shows the case β2 = 0 (local measurement), while the black dotted line shows the case β22 = 0.2 (global measurement).
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