Dynamical Gaussian state transfer with quantum error correcting architecture

Transferring a quantum state of a light field to a memory is of particular importance. However, this transfer is usually hampered because the memory system is subjected to some noise and this can limit the performance of the state transfer to a great extent. In this paper, we consider the transfer of a Gaussian state of light to a linear medium memory such as an opto-mechanical oscillator and propose a dynamical feedback controller that suppresses the noise in the memory system. To protect an unknown state, the feedback scheme employs the specific configuration of the quantum error correction; that is, a three-mode Gaussian state having appropriate syndromes is taken as the input. Correspondingly, the memory consists of three independent linear systems. The syndrome errors are estimated continuously in time through the measurement of the output field, and the results are then fed back to control the system. Because the input is Gaussian and the systems are all linear, it is possible to formulate the problem using the framework of the celebrated classical Kalman filtering and linear quadratic Gaussian control. A numerical simulation demonstrates the effectiveness of the control scheme.

In reality, however, the memory performance is essentially limited by environmental noise that inevitably occurs during the transfer process. In a real experiment several state-of-the-art techniques should be, at least implicitly, employed to suppress such noise. However, to the best of our knowledge there exists no systematic method for suppressing the noise. Therefore, control theory for quantum memory needs to be explored.
In working out this subject, a key fact is that due to the dynamical noise the information contained in the memory state is gradually erased, rather than that the noise suddenly vanishes it. Hence, if such an unwanted change of state can be monitored continuously in time, then the measurement result could be fed back to suppress the noise. In fact, in [2], the probe light field for state transfer is continuously measured at the terminal of the optical path to indirectly detect the error during the transfer process. In this paper, we propose a dynamical control scheme that uses such continuous measurement results not only for the error detection but further for * E-mail address: go@z8.keio.jp † E-mail address: yamamoto@appi.keio.ac.jp feedback control to obtain the better performance of the state transfer. Fortunately, measurement-based quantum feedback control theory is well developed [23,24,25] and has many practical uses [26,27,28,29,30,31,32]; hence, it will be also useful for our scheme.
Here we describe two specific features of the proposed system-controller configuration. The first one is that the memory is a linear medium system such as a collective atomic ensemble or an opto-mechanical oscillator. In addition, the input state to be transferred is assumed to be a Gaussian state of light [33], in which case the memory state also becomes Gaussian. Then, the feedback control scheme has the form of celebrated linear quadratic Gaussian (LQG) control with Kalman filtering [24,26,29,30,31]. The second feature is that the controller employes the schematic of quantum error correction (QEC) for the purpose of protecting an unknown input state. The basic idea of QEC is to encode an unknown state into an enlarged codespace so that an appropriate syndrome measurement yields sufficient information about the error, which can then be used to correct the error [34,35,36,37,38,39]. Use of this QEC strategy for quantum memory means that the input is a three-mode unknown Gaussian state having appropriate syndromes, and it is transferred to the memory that is correspondingly enlarged; during the transfer process, the output light fields are measured continuously in time, yielding the estimate of the syndrome errors that can be fed back to control the system.
The effectiveness of this control scheme is actually expected from the fact that the dynamical feedback control has successfully been applied to some QEC problems [40,41,42,43,44,45,46]. It should be noted that the problem considered in this paper is not a QEC problem itself; in QEC, the system state can be protected with the use of ancillary apparatus, whereas in our formalism, the initial state of the memory system cannot be protected, but rather the input state to be transferred to the system is what should be protected. This observation is vital in the sense that our scheme does not violate the no-go theorem for Gaussian QEC [47].
The paper is organized as follows. In Section II, as a preliminary, we describe a single-mode Gaussian state transfer. The dynamical control scheme is addressed in two sections. First, the state preparation and transfer processes are given in Section III. Section IV explains the error detection scheme (Kalman filter) and the feedback controller (LQG controller). Finally, in Section V, we numerically demonstrate the efficiency of our method by taking, as an example of the memory, an opto-mechanical oscillator manipulated under ultra-low temperature.

II. SINGLE-MODE GAUSSIAN STATE TRANSFER
FIG. 1: (Color online) Opto-mechanical oscillator as an example of the single-mode linear quantum memory. The right end-mirror of the Fabry-Perot cavity serves as the oscillator, whereâ andb are annihilation operators corresponding to the intra-cavity and oscillator modes, respectively. Adiabatically eliminating the cavity modeâ, Eq. (1) is obtained. F denotes a Faraday isolator that facilitates one-way coupling between the input and cavity modes.
As a preliminary, this section is devoted to describe a single-mode Gaussian state transfer from a light field to a memory, where the former is a coherent CW laser field and the latter is served by a linear system such as an opto-mechanical oscillator shown in Fig. 1 (see e.g., [14,17,48] for a detailed discussion). The input is a coherent state |α in , withâ in the annihilation operator of this coherent light field. The reflected output modê a out could be used to correct the errors occurring during the transfer process. In what follows we describe the dynamics of the memory system and show how much the input state |α in is transferred to the memory.
A general single-mode open linear dynamics is described by the following quantum Langevin equation: whereb denotes the system annihilation operator. Here, ν represents the coupling constant between the system and the input coherent light fieldâ in = α in +â 0 , where α in is the mean andâ 0 is the field annihilation operator. Moreover, we assume that the system couples with an unwanted single-mode noisy environment represented by the annihilation operatorξ with mean zero. Γ is the coupling strength.
Let us now take the white noise approximation on the outer field modesâ 0 andξ; i.e., â 0 (t)â † 0 (t ′ ) = δ(t − t ′ ) and ξ (t)ξ † (t ′ ) = (n + 1)δ(t − t ′ ), where n > 0 represents the strength of the noise. In the case of thermal noise, n is the averaged photon number. This approximation allows us to represent the dynamics ofb in terms of the Ito-type quantum stochastic differential equation (QSDE) [49,50]: whereÂ 0 (t) = t 0â 0 (s)ds andΞ(t) = t 0ξ (s)ds are the so-called quantum Wiener processes. Their infinitesimal increments satisfy the following quantum Ito-rule: In this QSDE representation the input mode is written by dÂ in = α in dt + dÂ 0 . Since the field state is the vacuum, we have dÂ in = α in dt. Also note that dΞ = 0. Now, let us see the steady state of the memory system. Using the above Ito rule we readily obtain the ordinary differential equations of b(t) and b(t) † b(t) , which give b (∞) = −2 √ να in /(ν + Γ) and 2i denote the dimensionless position and momentum operators of the system, respectively. This clearly illustrates that in the long time limit the system certainly acquires the information of the field input state |α in in the mean sense (note that ν and Γ are assumed to be known). However the fluctuation of the state could become much bigger than the vacuum fluctuation 1/2; in this case the state wrote down to the memory is far away from pure and has very small overlap with the input state.

III. PREPARATION AND TRANSFER PROCESSES OF THE INPUT STATE
In this paper, we study the system shown in Fig. 2. First, in the left box, the input state, which is a threemode Gaussian state of light, is prepared. This input state is transferred into the memory shown in the middle box in the figure, that correspondingly consists of three identical linear systems; in the figure, opto-mechanical oscillators are particularly depicted as the memory system. Finally, the output fields are measured, and the results are used for feedback control.
A notable feature of this transfer scheme is that the input state is allowed to contain some unknown parameters. More specifically, the input state is generated by embedding an unknown single-mode fieldÂ in into the three-mode light fields with the use of ancilla squeezed fields and beam splitters. This is no more than the schematic of QEC; actually,Â in corresponds to a source mode to be protected, and for this purpose it is encoded into a larger Hilbert space where we can construct appropriate syndromes that do not depend on the unknown source mode but only contain information about the errors. Those errors could be corrected by feedback control through the syndrome measurement depicted in the lower box of the figure. Note that the three-mode encoding allows us to detect only either the position or the momentum error, but the proposed scheme can be extended to a nine-mode code to detect errors acting on both the position and momentum operators [39].
In this section, we describe the above-mentioned state preparation and transfer processes, and then evaluate in detail the memory performance, when without any error correction.

A. State preparation
For discrete variable systems, a series of CNOT gates is often used for encoding. A possible continuous-variable analogue is realized in linear optics circuit shown in the left box of Fig. 2; first, the information source modeÂ in is mixed via a 1 : 2 beam splitter with an ancilla squeezed vacuum field denoted byÂ 2 , and one of the outputs is further combined with the second ancilla squeezed vacuum fieldÂ 3 . This combination of beam splitters is called the tritter [39].
Let us describe the above encoding process in detail. As in the single-mode case discussed in Sec. II, we write the source mode as dÂ in = α in dt + dÂ 1 , where α in and A 1 are the mean amplitude and the quantum fluctuation, respectively. In particular, we assume implying that the momentum element of the mean is known to be zero. As mentioned above, in this case, the three-mode encoding is sufficient to detect the position errors. The quantum fluctuationÂ j (j = 1, 2, 3) is a stochastic process satisfying dÂ j = 0 and the following quantum Ito rule: The fluctuation parameters N 1 and M 1 of the source mode are unknown. For the ancilla modesÂ 2 andÂ 3 , we set which satisfy N j (N j + 1) = M 2 j ; that is, the ancilla modes are the identical pure squeezed vacuum fields with squeezing parameter µ. Let us collect the field quadratures in a single vector aŝ 2i. Then, through the tritter,Ŵ 1 becomes dŴ ′ 1 = βdt + T dŴ 1 , where T is the orthogonal matrix corresponding to the combination of the above-mentioned 1 : 2 and half beam splitters: Using the Ito rule, it is found that the position quadra- This quantity is essentially equivalent to the power spectrum density of (Q ′ i −Q ′ j ) 2 at the center frequency of the career laser field [51]. We then find that Eq. (4) becomes zero for every i, j when taking the limit µ → −∞. This means that, in the Schrodinger picture, this ideal input state generated through the tritter lives in the codespace spanned by {|Q, Q, Q } and is of the following GHZ-like form : where ψ(Q) is a Gaussian wave function of the source state. The tilde means that the state is the ideal one. Next let us focus on the following quantity: which leads to IfÂ 1 ,Â 2 , andÂ 3 are all coherent fields, we have dP fd /dt = 7.5, which thus can be interpreted as the classical limit. This means that a non-classical input state is generated when dP fd /dt < 7.5, which is now equivalent to that µ < 0 and M 1 ≥ 0. Moreover, it is known that a symmetric state such as the one taken here is entangled if dP fd /dt < 6 [52,53]. For instance setting a coherent source state, i.e., N 1 = M 1 = 0, and taking the ideal limit µ → −∞, we obtain dP fd /dt → 4.5; hence in this case the GHZ-like state (5) is indeed entangled.

B. Transfer process
Let us next describe the transfer process of the modê W ′ 1 to the memory served by the three identical linear systems. The dynamics of the memory is given by the combination of Eq. (2) with the input field replaced by dŴ ′ 1 ; that is, the vector of quadratures, 2i, satisfies the following QSDE: whereŴ = (Ŵ 1 ,Ŵ 2 ) ⊤ and The fluctuation vectorŴ 1 is given by Eq. (3) and is the vector of noise processes to which the linear systems are subjected; henceΞ j (j = 1, 2, 3) satisfies Again, n > 0 represents the noise strength. Finally, u ∈ R 6 represents the control input for the memory. Note here it is assumed that both the quadratures of each linear system can be controlled. In what follows of this section we set u = 0. The state transfer can be evaluated in terms of the mean vector and the covariance matrix, as discussed in Sec. II. First, the mean vector x = ( q 1 , . . . , . That is, the system state becomes Gaussian with mean vector parallel to β, implying that the memory certainly acquires information of the input state in the mean sense (note that ν and Γ are assumed to be known). Next let us consider the covariance matrix of the system: Using the Ito rule, the time evolution of V is found in the following form called the Lyapunov differential equation: where Note Λ 2 = Λ 3 by assumption. Now, to explicitly evaluate the state transfer, let us take the source state to be coherent, in which case M 1 = N 1 = 0; then the steady solution of Eq. (8) is obtained as Note V 2 = V 3 . Then we have the explicit form of the fidelity between the input state |ψ in and the steady state of the memory with its mean appropriately displaced: where V in = T ΛT ⊤ is the covariance matrix of the input state |ψ in . Note that Eq. (5) is the ideal limit of |ψ in .
Eq. (9) shows that, when Γ = 0, we have F = 1 without respect to the value of µ. This means that the input state is perfectly transferred into the memory when it is not coupled to the noisy environment. But in reality the system must be subjected to some noise; QEC is expected to overcome this issue. Motivated by some continuous variable QEC protocols found in the literature [36,37,38,39], let us take the operatorq i −q j (i, j = 1, 2, 3) as a syndrome. Actually, this has desirable properties shown as follows. First, it can be seen from the structure of the system dynamics (7) that the syndrome does not contain the unknown parameters α in , N 1 , and M 1 . Second, in the ideal limit µ → −∞, the variance of each syndrome is given by which takes zero only when the system does not couple to the noisy environment. This means that, if in the small interval [t, t + dt) only one of the three linear systems is subjected to the noise, simultaneous "measurement" of the syndrome operators can detect in which system that error has occurred and how much it is. (The reason why the double quotation is taken is that the syndrome operator cannot be directly measured in our setting, but it needs to be estimated; see the next section.) Because of these two properties, in the limit µ → −∞, the operatorŝ q i −q j certainly plays a role of a syndrome. Here it should be noted that F takes the maximum value when µ = 0; that is, without error correction the encoding process merely degrades the performance of the state transfer [54]. Hence we really need the correction process. Before closing this section, let us evaluate the entanglement of the system state, using essentially the same quantity as Eq. (6). Again in the case N 1 = M 1 = 0, we have (4.5 + 3e µ )ν ν + Γ + (7.5 + 15n)Γ ν + Γ .
Hence, together with ∆(q i −q j ) 2 = 0 in this case, the memory state becomes an entangled GHZ-like state. But in the realistic situation with Γ > 0 such entanglement can vanish. Particularly in the ideal case µ → −∞ the sufficient condition for entanglement, P sys < 6, leads to n < 0.1(ν/Γ) − 0.1; that is, in the case of thermal noise, the averaged photon number of the environment field must be less than about one order of magnitude below the S/N rate ν/Γ.

IV. SYNDROME FILTER AND DYNAMICAL FEEDBACK CONTROL
In the previous section it was found thatq i −q j serves as the syndrome through which the error can be detected. Since this observable cannot be measured directly in our setting, it should be appropriately estimated through indirect measurement. In this section we first describe a continuous-time estimator, i.e., the filter, for the syndromes. We then present the dynamical feedback controller that is based on the syndrome filter.

A. Syndrome Filter
Let us focus on the following two kinds of operator vectors:ŝ HereB 1 := Z 1 T ⊤ andB 2 := Z 2 T ⊤ are isometric matrices with If only one of the three linear systems is subjected to the noise,ŝ 2 can detect in which system that error has occurred. Hence,ŝ 2 serves as the syndrome operator. The first element ofŝ 1 is used to evaluate entanglement of the memory state. Our task is to estimate the operatorŝ 1 orŝ 2 through certain indirect measurement. The lower box in Fig. 2 depicts an optical configuration that achieves this goal for the case ofŝ 1 ; that is, we construct the tritter acting on the output fields that are reflected at the field-system coupler. The observables to be measured by homodyne detectors with appropriate LO phase are then given by Hence measuringŶ 1 actually implies the indirect measurement ofŝ 1 . Note this contains the source fluctuation dP 1 , hence in this case only the mean value of the input state, β, can be kept unknown. On the other hand, for the case ofŝ 2 , two of the three output fields are measured; Y 2 does not contain both the mean and covariance of the source state, thus we can estimate the syndromes without respect to the unknown parameters. Note here that the position squeezing of the ancilla fields reduces the fluctuation dQ 2 and dQ 3 ; in particular taking the ideal limit µ → −∞ we have dŶ 2 = √ 2νŝ 2 dt, thus this is no more than the syndrome measurement.
We now describe the filter ofŝ i , which is constructed with the measurement results ofŶ i (i = 1, 2). Let us write Eq. (13) or (14) in the following general form: For the case of estimatingŝ 1 , the matrices correspond to ). For the simple notation, we do not put the index i onŶ , C, and D. The continuous measurement ofŶ enables us to construct the filter for the dynamics (7): where Z implies Z 1 or Z 2 . The set of above equations is called the quantum Kalman filter [24,26,29,30,31]. The 6-dimensional c-number vector π(x t ) = E(x t |Y t ) represents the quantum conditional expectation ofx t conditioned on the set of measurement data Y t = {y s | 0 ≤ s ≤ t} with y s the measurement result ofŶ s . Note π(x t ) is the least mean square estimate ofx t . V c is the conditional covariance matrix V c = π(∆x∆x ⊤ +(∆x∆x ⊤ ) ⊤ )/2 with ∆x =x − π(x), and it obeys the following Riccati differential equation: where K is given by Eq. (17). Note this is not a stochastic process. The filtering equation for the syndromeŝ 1 orŝ 2 can be directly obtained from Eq. (16) as follows: Due to the coupling to the noisy environment, the system state must escape from the codespace spanned by {|q, q, q }, but this error can be detected by estimating the syndrome operators. Therefore, it is expected that a feedback controller minimizing the estimated value of the syndromes corrects that error. Note that the controller must be a dynamical one so that it can deal with the dynamical noise; hence, the minimization should be carried out through the whole transfer process. Fortunately, because of the linearity of both the dynamics and the output equation, this requirement is satisfied when employing the LQG control strategy [24,26,29,30,31]. That is, we can construct an optimal control law u * that minimizes the following quadratic-type cost function: where R = R ⊤ > 0 represents the penalty for the control input. Corresponding to the syndrome operatorsŝ 1 or s 2 , the weighting matrix Q is set to: It is immediately seen that Therefore, for both cases the optimal control tries to minimize the syndrome errors. In the case ofŝ 1 , the optimal control further takes into account the effect of entanglement and tries to keep the GHZ-like form of the state. The LQG control theory gives the explicit form of the (stationary) optimal controller: where P is the solution to the following algebraic Riccati equation:Ã Let us here assume that R = rI 6 > 0. Then, corresponding toŝ 1 andŝ 2 , the matrix P is respectively obtained in the following form: where We here give two brief remarks. (i) The first one is about the structure of the optimal feedback control input u * . In the case ofŝ 1 , it is given by The first element can be regarded as a linearization to the following sign function (λ = f 2 /3r): λ sgn(π(q 2 −q 1 )) + sgn(π(q 3 −q 1 )) , where sgn(x) = +1 if x > 0 and −1 otherwise. Hence, in the small interval [t, t + dt), if the position shift ofq 1 due to the thermal noise is larger than the others, the controller adds the largest inverse shift −2λdt on dq 1 so as to cancel out the error. The other elements of u * have similar meanings. It should be maintained that the optimal controller obtained in the LQG control setup has this kind of natural structure found in the usual QEC scheme. (ii) The second remark is related to the nogo theorem [47] mentioned in Sec. I. The QEC problem corresponding to our setting is that the system's initial state is to be protected by feedback control. However, this goal is never accomplished, because, in general, a stable Kalman filter forgets the initial values of π(x 0 ) and V c (0); that is, the initial Gaussian state with mean π(x 0 ) and covariance V c (0) cannot be protected. This is a way to understand the no-go theorem for Gaussian QEC from the dynamical control viewpoint.

V. SIMULATION
In this section, a numerical simulation is provided to demonstrate effectiveness of the proposed control scheme. As an example of the memory we here take an optomechanical oscillator shown in FIG. 1. The oscillator must be subjected to thermal noise even at ultra-low temperature; in this case the noise strength n corresponds to the averaged photon number n = (e ωm/kB T − 1) −1 , where k B , T , and ω m denote the Boltzmann constant, the temperature, and the career frequency of the thermal channel, respectively.
The parameters are taken as follows. The oscillator couples with the input field with strength ν/2π = 30 kHz, while for the thermal channel it is Γ/2π = 1 Hz. The thermal channel is with frequecy ω m /2π = 10 MHz and with temperature T = 4 K, which leads to n = 8.8 × 10 3 . The mean value of the input state is taken as α in = −230 Hz 1/2 , which leads to − √ νβ = [100, 0, 100, 0, 100, 0]. Note that |α in | 2 is the mean photon number per unit time.

A. Coherent source state
First let us consider the case where the source state is taken as a coherent state |α in . This means that the input fluctuation is set to N 1 = M 1 = 0, implying that, in addition to the syndrome operators,p 1 +p 2 +p 3 can be estimated. Hence, we can now apply the feedback control scheme based on π(ŝ 1 ). Figure 3 shows how much the encoding and the feedback control improve the fidelity between the three-mode input state and the oscillators' steady state. They are quantified as follows. (i) Encoding strength is directly evaluated by the squeezing parameter µ; actually, as indicated by Eq. (4), making −µ bigger means that the input state is going to approximate the GHZ-like state (5). (ii) The LQG optimal control input u * can take possibly a bigger value by tuning the penalty parameter r smaller (Recall that we have set R = rI 6 in Eq. (21).) Hence, −log 2 r is interpreted as the control strength. (iii) The fidelity is given by the same form as Eq. (9) with V ∞ now replaced by the covariance matrix of the ensemble average of the controlled system state, say V ′ ∞ ; that is, F = 1/ det(V ′ ∞ + V in ). Note that the controlled system variablex t stochastically changes in time according to the QSDE (7) with the input u * a function of the estimate π(ŝ 1 ), i.e., Eq. (23). Hence V ′ ∞ corresponds to the first 6 × 6 block matrix of the 9 × 9 covariance matrix of (x ⊤ , π(ŝ 1 ) ⊤ ) ⊤ that satisfies the controlled Lyapunov equation (A1); see Appendix for the detailed calculation.
Let us now discuss the performance of the LQG optimal feedback control. It is observed from Fig. 3 that bigger control strength indicates bigger fidelity for all value of µ. This means that the feedback control can always reduce the excess fluctuation brought from the thermal environment, without respect to how much the source state is encoded. Regarding the encoding strength, however, care should be taken in its choice. Actually, while larger squeezing enables us to detect a bigger error signal which can induce more efficient feedback control, at the same time the memory state becomes more sensitive to the thermal noise. In other words, too much squeezing makes the input state fragile to the noise; consequently there exists an optimal value of the squeezing parameter, and it is about µ * = −0.4. Note this value can be reached within the current technology. In this case, the amount of improvement of the fidelity via the feedback control compared to the case without encoding and control is about 0.05; but this is not a big improvement. One reason to this limitation is that the error taken in this paper is the worst one in the sense that each system is subjected to the thermal noise for all time; this kind of error is not fully tractable via the standard QEC protocol. Therefore, it is expected that the proposed control scheme could show possibly much better performance of the state transfer against some weak noise such as simple displacement acting only one of the oscillators. and (b) π(q1). For both plots, the less (red) and larger (blue) fluctuating lines correspond to the cases where the optimal feedback control is performed or not, respectively.
Next, Fig. 4 shows the time evolutions of (a) the syndrome π(q 2 −q 3 ) and (b) the first oscillator's position π(q 1 ). For both plots, the less (red) and larger (blue) fluctuating lines correspond to the cases where the optimal feedback control is performed or not, respectively. The error bars show the standard deviation of the estimation error, which is calculated from the solution to the Riccati equation (19). The parameters are taken as µ = −0.4 and r = 10 −9 , which were shown in Fig. 3 to be the optimal values attaining the maximum fidelity.
Note that, because α in is unknown,q 1 cannot be exactly estimated; but we here plot the exact value of π(q 1 ) just for demonstration. Figure 4 (a) demonstrates that the fluctuation of the syndrome is certainly suppressed by the feedback control; this is the result that should be expected, since the controller is designed to minimize the syndrome errors. On the other hand, it is not straightforwardly expected that controllingq 1 would actually work well, but Fig. 4 (b) demonstrates that the feedback control based on the syndrome estimation is fairly effective for correcting the error acting on each position of the oscillator.

B. Squeezed source state
Here we are concerned with the situation where the source state is squeezed. The squeezing parameter is taken as µ 1 = log(2M 1 + 2N 1 + 1) with M 1 real, and N 1 and M 1 are both unknown in addition to α in . We then have to use the filter forŝ 2 , which does not contain N 1 and M 1 . But for comparison we further consider a squeezed source state with known covariance and unknown mean value; in this case the Kalman filter forŝ 1 can be used for feedback control. This comparison will reveal how much the additional information (i.e., the estimate ofp 1 +p 2 +p 3 ) improves the control performance at the expense of limiting the class of input states. Figure 5 shows the fidelity between the input state and the controlled oscillators' steady state, versus the squeezing parameters µ and µ 1 . The control penalty is r = 10 −9 . In the figure the upper and lower surfaces correspond to the filter forŝ 1 andŝ 2 , respectively. First, a notable fact observed from the lower surface is that, when aiming to transfer the input state with the completely unknown source state, the maximum fidelity is attained when µ 1 = 0; that is, squeezing the source state always decreases the fidelity. This is consistent with the standard understanding that an unknown squeezed state is in general fragile to the thermal noise, because it randomly rotates the phase of the state, which as a result brings that state into a mixed state. On the other hand, the upper surface in the figure shows that, when transferring the input state with known fluctuation, which means thatŝ 1 can be used for estimation and control, squeezing the source state improves the fidelity. Note that the improvement is observed if µ 1 > 0 and µ < 0; that is, the source state is now momentum-squeezed while the ancilla states are position-squeezed. Actually in this case the input state more closely approximates the GHZ-like state, which is an eigenstate ofp 1 +p 2 +p 3 . Therefore, the additional information π(p 1 +p 2 +p 3 ) should certainly improves the control performance of the state transfer. However, as in the previous case, too much squeezing degrades the fidelity, hence µ 1 must be optimized.

VI. CONCLUSION
In this paper we proposed a new feedback control scheme for state transfer, which has the form of QEC.
In particular, due to the Gaussianity of the input state and the linearity of the memory dynamics, the celebrated Kalman filtering and LQG feedback control were employed. We have considered a specific system, but the proposed control strategy is applicable to any linear quantum system, as long as the output process can be properly defined. Although it was shown in the example that the controller is not a very effective one, it is expected that it can show much bigger improvement for some linear systems having relatively weak noise. Also possible combination with a feedforward control scheme [21,22] could be helpful to attain better performance for Gaussian state transfer.

(A1)
Now, the covariance matrix of the controlled memory state, V ′ , corresponds to the first 6 × 6 block matrix of V z . In this case, the steady solution V ′ ∞ can be explicitly obtained as follows: Note that as r → 0 (i.e., cheap control), we have V ′ ∞ → V c (∞), which is the steady solution to Eq. (19). That is, in this limit, the quantum fluctuation of the controlled memory state is maximally reduced down to the level of estimation error.