\Delta t$ and $x_n := x(n \Delta t)$. We refer to “time $t$” as “time $t_n := n \Delta t$.” Then, trajectory $\{ x(t') \}_{t' \in [0, t]}$ corresponds to $X_n := (x_0, x_1, \cdots, x_n)$.

Control protocol $\lambda (t)$ can also be discretized. Let $\lambda_n$ be the value of $\lambda$ between $t_n = n\Delta t$ and $t_{n+1} = (n+1)\Delta t$, where it is assumed to be constant during this time interval (see FIG. 1). We denote the trajectory of $\lambda$ from time $0$ to $t_n$ as $\Lambda_n := (\lambda_0, \lambda_1, \cdots, \lambda_{n-1})$. Let $\lambda_{\rm int}$ be the value of parameter $\lambda$ before time $0$, which is not necessarily equal to $\lambda_0$ because we can switch the value of the parameter at time $0$. We also denote the value of $\lambda$ after time $t_N := \tau$ as $\lambda_{\rm fin}$, which is not necessarily equal to $\lambda_N$, either (see also FIG. 1).

Let $P_n[x_n]$ be the probability distribution of $x$ at time $t_n$. In particular, $P_0 [x_0]$ is the initial distribution of $x$. The initial distribution can be chosen as a stationary distribution under external parameters $\lambda_{\rm int}$, as $P_{\rm s}[x_0 | \lambda_{\rm int}]$, which means $P_0 [x_0] = P_{\rm s}[x_0 