Langevin equation in the Itô form: $$\eta (x_{n+1} - x_n) = - K x_n \Delta t + \lambda_n \Delta t + \sqrt{2 \eta k_{\rm B}T} \Delta W_n,  
\label{linear_Langevin}$$ where $K$ is a positive constant and $\lambda_n$ is a control parameter. If the initial distribution of $x_0$ is Gaussian, the distribution of $x_n$ remains Gaussian with Eq. (\[linear\_Langevin\]). In this case, the obtained mutual information by measurement (\[measurement\_Langevin\]) at time $t_n$ is given by $$\langle I_{\rm c} [x_n : y_n] \rangle = \frac{1}{2} \ln \left(1 + \frac{S_n}{R}\Delta t \right) = \frac{S_n}{2R}\Delta t + o(\Delta t),$$ where $S_n := \langle x_n^2 \rangle - \langle x_n \rangle^2$. Therefore, the total mutual information $$\langle I_{\rm c} \rangle = \lim_{N \to \infty} \sum_{n=0}^{N} \frac{S_n}{2R}\Delta t$$ can converge, while the measurement is continuous.

We consider the Kalman filter on Eq. (\[linear\_Langevin\]) with measurement (\[measurement\_Langevin\]). The Kalman filter is a standard method to construct the optimal estimator of $x_n$, denoted as $\hat x_n$, in terms of the mean square error. From measurement outcomes $Y_n$, $\hat x_n$ is obtained as the solution to the following simultaneous differential equations [@Welch]: $$\hat x_{n+1} - \hat x_n = - \frac{K \hat x_n + \lambda_n}{\eta}  \Delta t + \frac{A_n}{R} (y_n - \hat 