Thermodynamic optimization of finite-time feedback protocols for Markov jump systems

We consider a bipartite classical stochastic system consisting of subsystems $X$ and $Y$, which take discrete states. The entire system is in contact with a heat bath at inverse temperature $\beta$, and its dynamics is described by a Markov jump process. System $X$ is the target of feedback control, while $Y$ plays the role of Maxwell’s demon applying feedback operations to $X$ based on the measurement results on $X$’s state.

The set of possible states for $X$ is $\mathcal{X}\coloneqq\{1,2,\dots,n\}$, and for $Y$ is $\mathcal{Y}\coloneqq\{1,2,\dots,n\}$. Here $y\in\mathcal{Y}$ corresponds to the measurement result on $x\in\mathcal{X}$. The joint state of the entire system $XY$ is represented by the pair $(x,y)\in\mathcal{X}\times\mathcal{Y}$, and the probability to find the system in state $(x,y)$ at time $t$ is denoted as $p_{t}^{XY}(x,y)$. The marginal probabilities for $X$ and $Y$ are defined as $p_{t}^{X}(x)=\sum_{y}p_{t}^{XY}(x,y)$ and $p_{t}^{Y}(y)=\sum_{x}p_{t}^{XY}(x,y)$, respectively.

Since we focus on the feedback process, we assume that $X$ and $Y$ are correlated at the initial time $t=0$, as a consequence of the measurement performed beforehand. Here, we suppose that $Y$ is a controller that stores and uses the information of the target system $X$. Moreover, we make a simple assumption that the measurement is error-free. That is, the initial distribution is given of the form

$\displaystyle p_{0}^{XY}(x,y)=p^{X}(x)\delta_{x,y}$

(1)

with $\delta_{x,y}$ being the Kronecker delta, which guarantees that $x=y$ holds with unit probability. During the feedback, the state $y$ of $Y$ is assumed to remain unchanged, i.e., no transitions occur between different states in $Y$. That is, transitions from $(x^{\prime},y^{\prime})$ to $(x,y)$ are prohibited if $y\neq y^{\prime}$. This assumption is reasonable for our classical stochastic processes, where the feedback operation on $X$ does not influence $Y$’s state itself [37]. Under this assumption, the marginal distribution $p_{t}^{Y}(y)$ is fixed to a certain distribution $p^{Y}(y)$ throughout the process, and the initial joint distribution is given by $p_{0}^{XY}(x,y)=\delta_{x,y}p^{Y}(y)$.

The time evolution of the entire system during the feedback process, from $t=0$ to $t=\tau$, is described by the master equation:

	$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}p_{t}^{XY}(x,y)=\sum_{x^{\prime}:(x^{\prime},x)\in\mathcal{N}_{y}}$	$\displaystyle\Big{[}R_{t}^{X|y}(x,x^{\prime})p_{t}^{XY}(x^{\prime},y)$
		$\displaystyle-R_{t}^{X|y}(x^{\prime},x)p_{t}^{XY}(x,y)\Big{]},$		(2)

where $R_{t}^{X|y}(x,x^{\prime})$ is the transition rate from $(x^{\prime},y)$ to $(x,y)$ at time $t$. This rate describes the feedback operation that $Y$ applies to $x$ when the measurement result is $y$. The set $\mathcal{N}_{y}$ denotes the pairs of different $X$’s states $(x^{\prime},x)$ that are allowed to transition when the measurement result is $y$. These transitions are assumed to be bidirectional, meaning $(x^{\prime},x)\in\mathcal{N}_{y}$ implies $(x,x^{\prime})\in\mathcal{N}_{y}$. The stochastic heat $Q_{t}^{X|y}(x,x^{\prime})$ absorbed by $X$ during a transition from $(x^{\prime},y)$ to $(x,y)$ satisfies the local detailed balance condition:

$$\ln\frac{R_{t}^{X|y}(x,x^{\prime})}{R_{t}^{X|y}(x^{\prime},x)}=-\beta Q_{t}^{X|y}(x,x^{\prime}).$$

(3)

We next introduce mutual information, which represents the amount of information shared between $X$ and $Y$. The mutual information between $X$ and $Y$ at time $t$ is defined as

$\displaystyle I_{t}^{X:Y}\coloneqq S(p_{t}^{X})+S(p_{t}^{Y})-S(p_{t}^{XY}),$

(4)

where $S(p)$ is the Shannon entropy of the probability distribution $p$. If $X$ and $Y$ are uncorrelated, $I_{t}^{X:Y}=0$, while $I_{t}^{X:Y}>0$ otherwise. In our setup, with the assumptiuon (1), $S(p_{0}^{X})=S(p_{0}^{Y})=S(p_{0}^{XY})=S(p^{Y})$, yielding $I_{0}^{X:Y}=S(p^{Y})$, which is the maximum mutual information for fixed $p^{Y}$. This means that $Y$ has fully acquired the information of $X$ at time $t=0$.

We also introduce entropy production, which represents thermodynamic cost required for the feedback process. The entropy production from time $0$ to $\tau$, denoted as $\Sigma_{\tau}^{XY}$, is defined using the total heat absorbed by $X$ up to time $\tau$, $Q_{\tau}^{X}\coloneqq\int_{0}^{\tau}\sum_{x,x^{\prime},y}Q_{t}^{X|y}(x,x^{\prime})R_{t}^{X|y}(x,x^{\prime})p_{t}^{XY}(x^{\prime},y)\,\mathrm{d}t$, as

$\displaystyle\Sigma_{\tau}^{XY}\coloneqq S(p_{\tau}^{XY})-S(p_{0}^{XY})-\beta Q_{\tau}^{X}.$

(5)

Here, the first two terms represent the entropy change of system $XY$, and the third term represents the entropy change of the heat bath. Since $Y$ does not undergo transitions, $Q_{\tau}^{X}$ equals the total heat absorbed by the system $XY$. Thus, $\Sigma_{\tau}^{XY}$ represents the entropy change of the entire system including the heat bath, and describes dissipation due to irreversibility.

We here introduce the probability of a transition from state $(x^{\prime},y)$ to $(x,y)$ denoted as $j_{t}^{X|y}(x,x^{\prime})\coloneqq R_{t}^{X|y}(x,x^{\prime})p_{t}^{XY}(x^{\prime},y)$, and the probability current from state $(x^{\prime},y)$ to $(x,y)$ defined as $J_{t}^{X|y}(x,x^{\prime})=j_{t}^{X|y}(x,x^{\prime})-j_{t}^{X|y}(x^{\prime},x)$. We also define thermodynamic force for the transition from $(x^{\prime},y)$ to $(x,y)$ as

$$F_{t}^{X|y}(x,x^{\prime})\coloneqq\ln\frac{j_{t}^{X|y}(x,x^{\prime})}{j_{t}^{X|y}(x^{\prime},x)}.$$

(6)

Then, the entropy production rate $\sigma_{t}^{XY}\coloneqq\mathrm{d}\Sigma_{t}^{XY}/\mathrm{d}t$ can be expressed as

$$\sigma_{t}^{XY}=\sum_{x>x^{\prime}}J_{t}^{X|y}(x,x^{\prime})F_{t}^{X|y}(x,x^{\prime}),$$

(7)

which satisfies the second law of thermodynamics $\sigma_{\tau}^{XY}\geq 0$.

entropy production for subsystem $X$, $\Sigma_{\tau}^{X}$, is defined as

$\displaystyle\Sigma_{\tau}^{X}\coloneqq S(p_{\tau}^{X})-S(p_{0}^{X})-\beta Q_{\tau}^{X}.$

(8)

Using this and the change in the mutual information $\Delta I_{\tau}^{X:Y}\coloneqq I_{\tau}^{X:Y}-I_{0}^{X:Y}$, the entropy production can be decomposed as $\Sigma_{\tau}^{XY}=\Sigma_{\tau}^{X}-\Delta I_{\tau}^{X:Y}$ [52]. In the present setup, since the maximum mutual information is stored at the initial time, we have $\Delta I_{\tau}^{X:Y}\leq 0$. Therefore, the decomposition can be rewritten as

$\displaystyle\Sigma_{\tau}^{XY}=\Sigma_{\tau}^{X}+|\Delta I_{\tau}^{X:Y}|.$

(9)

By substituting this decomposition into the second law, we obtain

$\displaystyle\Sigma_{\tau}^{X}\geq-\absolutevalue{\Delta I_{\tau}^{X:Y}}.$

(10)

This indicates that by consuming mutual information, it is possible to achieve a lower entropy production than that determined by the second law for the case where $Y$ is absent, $\Sigma_{\tau}^{X}\geq 0$. We emphasize that the equality in (10) is achieved in the quasi-static limit, which requires infinite time [37].

We next briefly overview optimal transport theory. A key insight of optimal transport theory is that the minimized transport cost, known as the Wasserstein distance, serves as a metric between distributions. Optimal transport theory finds applications in fields such as image processing [53], machine learning [54], and biology [55]. Applying it to thermodynamics reveals that the achievable speed limits for entropy productions can be expressed in terms of the Wasserstein distance between initial and final distributions, enabling the identification of optimal protocols that achieve the equality.

The Wasserstein distance between two probability distributions $p_{0}^{XY}$ and $p_{\tau}^{XY}$ is defined as

$\displaystyle\mathcal{W}(p_{0}^{XY},p_{\tau}^{XY})\coloneqq\min_{\pi\in\Pi(p_{0}^{XY},p_{\tau}^{XY})}\sum_{x,x^{\prime},y}d_{(x,y),(x^{\prime},y)}\pi_{(x,y),(x^{\prime},y)},$

(11)

where $\pi_{(x,y),(x^{\prime},y)}(\geq 0)$ represents the probability transported from state $(x^{\prime},y)$ to state $(x,y)$. The collection of these probabilities for all states, $\pi=\{\pi_{(x,y),(x^{\prime},y)}\}_{x,x^{\prime},y}$, is referred to as the transport plan. We denote by $\Pi(p_{0}^{XY},p_{\tau}^{XY})$ the set of all transport plans that transform the initial distribution $p_{0}^{XY}$ into the final distribution $p_{\tau}^{XY}$. This set satisfies the probability conservation conditions: $\sum_{x\in\mathcal{X}}\pi_{(x,y),(x^{\prime},y)}=p_{0}^{XY}(x^{\prime},y)$, $\sum_{x^{\prime}\in\mathcal{X}}\pi_{(x,y),(x^{\prime},y)}=p_{\tau}^{XY}(x,y)$. The coefficient $d_{(x,y),(x^{\prime},y)}$ is the minimum number of transitions required for moving from state $(x^{\prime},y)$ to state $(x,y)$ under the Markov jump process defined in Eq. (2), which determines the cost of transport per unit probability. Thus, the definition in Eq. (11) indicates that by fixing the initial distribution $p_{0}$ and the final distribution $p_{\tau}$, the minimum total cost over all possible transport plans $\pi$ can be defined as the distance between the distributions. The transport plan $\pi$ that achieves this minimum cost is referred to as the optimal transport plan.

When probability distribution $p_{0}^{XY}$ evolves into $p_{\tau}^{XY}$ following Eq. (2) from time $t=0$ to $t=\tau$, the entropy production is bounded as [36]

$$\Sigma_{\tau}^{XY}\geq\mathcal{W}\left(p_{0}^{XY},p_{\tau}^{XY}\right)f\left(\frac{\mathcal{W}\left(p_{0}^{XY},p_{\tau}^{XY}\right)}{D\tau}\right),$$

(12)

where $f(x)$ is a function determined by the choice of fixed timescale $D$.

We introduce two quantities representing the timescale: activity and mobility. We define activity $a_{t}$ as

$$a_{t}\coloneqq\sum_{y,\ x\neq x^{\prime}}j_{t}^{X|y}(x,x^{\prime})=\sum_{x>x^{\prime}}a_{t}^{X|y}(x,x^{\prime}).$$

(13)

Here, the local activity between states $(x,y)$ and $(x^{\prime},y)$ is defined as $a_{t}^{X|y}(x,x^{\prime})\coloneqq j_{t}^{X|y}(x,x^{\prime})+j_{t}^{X|y}(x^{\prime},x)$, which corresponds to the number of jumps per unit time at time $t$. Additionally, the time integral representing the total number of jumps from time $0$ to $\tau$ is expressed as $A_{\tau}\coloneqq\int_{0}^{\tau}a_{t}\,\mathrm{d}t$, and its time average is given by $\langle a\rangle_{\tau}\coloneqq A_{\tau}/\tau$. We define mobility $m_{t}$ as

$$m_{t}\coloneqq\sum_{x>x^{\prime}}\frac{J_{t}^{X|y}(x,x^{\prime})}{F_{t}^{X|y}(x,x^{\prime})}.$$

(14)

This represents the probability currents induced by the thermodynamic forces applied to the system. As well as activity, the time integral is expressed as $M_{\tau}\coloneqq\int_{0}^{\tau}m_{t}\,\mathrm{d}t$, and the time average is given by $\langle m\rangle_{\tau}\coloneqq M_{\tau}/\tau$.

Depending on whether $D$ corresponds to either of these two quantities, the function $f$ in the lower bound (12) changes. If $D$ represents the time-averaged mobility $\langle m\rangle_{\tau}$, then $f(x)=x$; if $D$ represents the time-averaged activity $\langle a\rangle_{\tau}$, then $f(x)=2\tanh^{-1}(x)$.

For either choice of $D$, the optimal protocol $\{R_{t}^{X|y}(x,x^{\prime})\}$ that achieves the equality in inequality (12) can be constructed by the condition that the probability is transported from $p_{0}^{XY}$ to $p_{\tau}^{XY}$ along the optimal transport plan with a uniform and constant thermodynamic force.

By substituting the decomposition (9) into inequality (12), we obtain the fundamental bound on energy for the fixed initial probability distribution $p_{0}^{XY}$ and the final distribution $p_{\tau}^{XY}$

$\displaystyle\Sigma_{\tau}^{X}\geq-\absolutevalue{\Delta I_{\tau}^{X:Y}}+\mathcal{W}\left(p_{0}^{XY},p_{\tau}^{XY}\right)f\left(\frac{\mathcal{W}\left(p_{0}^{XY},p_{\tau}^{XY}\right)}{D\tau}\right).$

(15)

The equality is achieved by the same protocol that achieves inequality (9).

The minimum entropy production for consuming a certain amount of information via feedback processes is not determined by inequality (15) alone. This is because, for a fixed consumed information $|\Delta I_{\tau}^{X:Y}|$, there exist infinitely many possible final distributions $p_{\tau}^{XY}$, which have different Wasserstein distance $\mathcal{W}\left(p_{0}^{XY},p_{\tau}^{XY}\right)$. Therefore, we can further minimize the right-hand side of (15) which depends on $\mathcal{W}\left(p_{0}^{XY},p_{\tau}^{XY}\right)$ over $p_{\tau}^{XY}$ under fixed $|\Delta I_{\tau}^{X:Y}|$, and determine the truly minimum entropy production for consuming $|\Delta I_{\tau}^{X:Y}|$.

To address the above problem, we first introduce the mathematical tool called Fano’s inequality. It enables us to evaluate the error that arises when a sender $Y$ sends a message to a receiver $X$ through a classical communication channel. Let the set of messages to be transmitted and received be $\{1,2,\dots,n\}$. Due to the presence of errors in the communication channel, when the sender $Y$ sends a message $y$, the receiver $X$ may not always receive $y$ with probability 1. The performance of the communication channel is characterized by the conditional probability distribution $p^{Y|x}(y)\coloneqq p^{XY}(x,y)/p^{X}(x)$, which indicates the ambiguity about the original message $y$ given the received message $x$.

Consider the case where the sender $Y$ stochastically transmits messages according to a distribution $p^{Y}(y)$. Denote the joint probability that $Y$ sends $y$ and $X$ receives $x$ as $p^{XY}(x,y)$. Now we aim to evaluate the error probability $\mathcal{E}\coloneqq\mathrm{Prob}(x\neq y)=\sum_{x\neq y}p^{XY}(x,y)$) based on the channel performance $p^{Y|x}(y)$. This evaluation is provided by Fano’s inequality, and the traditional form is given by

$$S^{Y|X}\leq-\mathcal{E}\ln\mathcal{E}-(1-\mathcal{E})\ln(1-\mathcal{E})+\mathcal{E}\ln(n-1).$$

(16)

Here, the left-hand side represents the conditional Shannon entropy $S^{Y|X}\coloneqq\sum_{x}p^{X}(x)S(p^{Y|x})$, which quantifies the randomness of the communication channel’s performance $p^{Y|x}(y)$. The right-hand side is an increasing function of the error probability $\mathcal{E}$. Thus, Fano’s inequality indicates that the randomness $S^{Y|X}$ of the channel’s performance leads to a higher error probability. The equality holds when the conditional probabilities $p^{Y|x}(y)$ satisfies

$\displaystyle p^{Y|x}(y)=\begin{cases}1-\mathcal{E},&x=y,\\ \mathcal{E}/(n-1),&x\neq y.\end{cases}$

(17)

However, depending on the probability distribution $p^{Y}$, it might not be possible to construct $p^{Y|x}(y)$ in accordance with Eq. (17). Therefore, the traditional Fano inequality (16) is not generally achievable.

On this matter, a previous research [48] has introduced a tighter version of Fano’s inequality that can be achieved for any $p^{Y}$. Applying it to the above setup of communication channel, we can summarize the achievable Fano’s inequality as follows. For fixed $p^{Y}$ and $\mathcal{E}$, there exists a probability distribution $\tilde{p}_{\mathcal{E}}^{Y}$ (whose construction will be explained below) determined by $p^{Y}$ and $\mathcal{E}$, such that

$$S^{Y|X}\leq S\left(\tilde{p}_{\mathcal{E}}^{Y}\right),$$

(18)

which is achievable for any $p^{Y}$.

Here, the construction of $\tilde{p}_{\mathcal{E}}^{Y}$ is as follows:

1. (i)

   Let $p_{\downarrow}^{Y}$ be the probability distribution obtained by rearranging $p^{Y}$(Fig. 1(a)) in descending order $p_{\downarrow}^{Y}(1)\geq p_{\downarrow}^{Y}(1)\geq\cdots\geq p_{\downarrow}^{Y}(n)$(Fig. 1(b). If $p_{\downarrow}^{Y}(1)\geq 1-\mathcal{E}$, then $\tilde{p}_{\mathcal{E}}^{Y}$ is defined as $\tilde{p}_{\mathcal{E}}^{Y}=p_{\downarrow}^{Y}$. If $p_{\downarrow}^{Y}(1)<1-\mathcal{E}$, then $\tilde{p}_{\mathcal{E}}^{Y}$ is constructed according to the following steps.

2. (ii)

   Define a function $M_{\mathcal{E}}$ that specifies a state of $Y$ corresponding to $\mathcal{E}$ (the construction of the function $M_{\mathcal{E}}$ will be provided in step (iii)). Then, $\tilde{p}_{\mathcal{E}}^{Y}$ is defined as

   $$\tilde{p}_{\mathcal{E}}^{Y}(y)\coloneqq\begin{cases}1-\mathcal{E},&y=1\\ \frac{\mathcal{E}-\sum_{y=M_{\mathcal{E}}+1}^{n}p_{\downarrow}^{Y}(y)}{M_{\mathcal{E}}-1},&2\leq y\leq M_{\mathcal{E}}\\ p_{\downarrow}^{Y}(y),&M_{\mathcal{E}}<y\leq n.\end{cases}$$

   (19)

   The corresponding histogram is shown in Fig. 1(c).

3. (iii)

   $M_{\mathcal{E}}$ is defined as the largest $m\in\{1,\cdots,n\}$ that satisfies

   $\displaystyle\frac{\mathcal{E}-\sum_{y=m+1}^{n}p_{\downarrow}^{Y}(y)}{m-1}<p_{\downarrow}^{Y}(m).$

   (20)

   When $p_{\downarrow}^{Y}(1)<1-\mathcal{E}$, such $M_{\mathcal{E}}$ is uniquely determined in the range $2\leq M_{\mathcal{E}}\leq n$. This can be interpreted as the largest $M_{\mathcal{E}}$ that ensures the probability distribution $\tilde{p}_{\mathcal{E}}^{Y}$ [defined in Eq. (19)] is in descending order.

The achievable Fano’s inequality (18) can be regarded as an inequality that evaluates the upper bound of the conditional Shannon entropy $S^{X|Y}$ when the error probability $\mathcal{E}$ and the marginal distribution $p^{Y}$ are fixed. Therefore, this inequality can be applied not only to the communication channel setting described above, but also to general joint systems $XY$ characterized by any joint probability distribution $p^{XY}(x,y)$.

To obtain the fundamental bounds on entropy production required to consume information by the feedback processes, we consider the minimization problem of the second term on the right-hand side of inequality (15) for a fixed $|\Delta I_{\tau}^{X:Y}|$. Since this term is a monotonically increasing function of $\mathcal{W}$ for both $f(x)=x$ and $f(x)=2\tanh^{-1}(x)$, it suffices to minimize $\mathcal{W}\left(p_{0}^{XY},p_{\tau}^{XY}\right)$.

Our main theorem is now stated as follows, which will be used to solve the above problem in the next section.

In our setup, we assumed that at $t=0$, $x=y$ holds with probability 1 through an error-free measurement. Therefore, the condition (C) indicates that regardless of the measurement result $y$, the direct transition from the initial state $x=y$ to any other state $x^{\prime}(\neq y)$ is allowed in the feedback control. Inequality (21) provides an upper bound on the mutual information that can be consumed in a finite-time feedback process when the Wasserstein distance is fixed. Since the right-hand side, $S\left(\tilde{p}_{\mathcal{W}}^{Y}\right)$, is an increasing function of $\mathcal{W}$, this implies that the larger the Wasserstein distance between the initial and final distributions, the more mutual information can be consumed by the feedback prosess.

This Theorem comprises two statements: first, that inequality (21) holds for any system $XY$ which satisfies our setup; second, that if the system $XY$ meets the condition (C), the equality in (21) can be achieved by setting $p_{\tau}^{XY}$ to an optimal distribution. We give the proof of these two statements in the two following subsections respectively.

We here prove inequality (21). By transforming Eq. (4), the mutual information can be rewritten as

$$I_{t}^{X:Y}=S\left(p_{t}^{Y}\right)-S_{t}^{Y|X},$$

where $S_{t}^{Y|X}$ denotes the conditional Shannon entropy of $Y$ given $X$ at time $t$. In the present setting, $p_{t}^{Y}=p^{Y}$ is fixed, and initially, $x=y$ with probability 1. Therefore, $S_{0}^{Y|X}=0$. Thus, we obtain

$\displaystyle\absolutevalue{\Delta I_{\tau}^{X:Y}}$

$\displaystyle=S_{\tau}^{Y|X}.$

(22)

Let the probability that $x\neq y$ holds at time $\tau$ be $\mathcal{E}_{\tau}\coloneqq\sum_{x\neq y}p_{\tau}^{XY}(x,y)$. From achievable Fano’s inequality (18), we obtain

$\displaystyle|\Delta I_{\tau}^{X:Y}|\leq S\left(\tilde{p}_{\mathcal{E}_{\tau}}^{Y}\right).$

(23)

Here, $S\left(\tilde{p}_{\mathcal{E}_{\tau}}^{Y}\right)$ is an increasing function of $\mathcal{E}_{\tau}$, and $\mathcal{E}_{\tau}\leq\mathcal{W}$ holds (the proofs are provided in the Appendix). Substituting these into inequality (23) yields the main Theorem (21).

We prove that the equality in (21) can be achieved by setting $p_{\tau}^{XY}$ to an optimal distribution if the entire system $XY$ satisfies the condition (C). First, we will describe the method for constructing the optimal final distribution $p_{\tau}^{XY}$. Next, we will prove that this final distribution satisfies the constraints $p_{\tau}^{Y}=p^{Y}$ and $\mathcal{W}(p_{0}^{XY},p_{\tau}^{XY})=\mathcal{W}$. Finally, we will show that this final distribution achieves the equality in (21).

The optimal final distribution $p_{\tau}^{XY}$ can be constructed as follows. Since the joint probability distribution of the entire system can be expressed as $p_{\tau}^{XY}(x,y)=p_{\tau}^{X}(x)p_{\tau}^{Y|x}(y)$, it suffices to define $p_{\tau}^{X}(x)$ and $p_{\tau}^{Y|x}(y)$. Here, let $\sigma_{p^{Y}}$ denote the permutation that rearranges the states $y$ in descending order of the probability distribution $p^{Y}$. In other words, $\sigma_{p^{Y}}(y)=n$ means that $y$ corresponds to the $n$-th largest value of $p^{Y}(y)$. If there are states with equal probabilities, their ordering in $\sigma_{p^{Y}}$ can be chosen arbitrarily; once $\sigma_{p^{Y}}$ is defined, it must be fixed throughout the following discussion.

The optimal $p_{\tau}^{X}(x)$ is then defined as

$\displaystyle p_{\tau}^{X}(x)=\begin{cases}\displaystyle\frac{p^{Y}(x)-\tilde{p}_{\mathcal{W}}^{Y}(2)}{\tilde{p}_{\mathcal{W}}^{Y}(1)-\tilde{p}_{\mathcal{W}}^{Y}(2)},\quad&1\leq\sigma_{p^{Y}}(x)\leq M_{\mathcal{W}},\\ 0,&\sigma_{p^{Y}}(x)>M_{\mathcal{W}}.\end{cases}$

(24)

From the definition of $\tilde{p}_{\mathcal{W}}^{Y}$, we have $\tilde{p}_{\mathcal{W}}^{Y}(1)=1-\mathcal{W}$ and $\tilde{p}_{\mathcal{W}}^{Y}(2)=\frac{\mathcal{W}-\sum_{y=M_{\mathcal{W}}+1}^{n}p_{\downarrow}^{Y}(y)}{M_{\mathcal{W}}-1}$. The optimal conditional probability distribution $p_{\tau}^{Y|x}(y)$ is given as

$\displaystyle p_{\tau}^{Y|x}(y)=\begin{cases}\tilde{p}_{\mathcal{W}}^{Y}(1),&y=x,\\ \tilde{p}_{\mathcal{W}}^{Y}(x),&\sigma_{p^{Y}}(y)=1,\\ \tilde{p}_{\mathcal{W}}^{Y}(\sigma_{p^{Y}}(y)),&\mathrm{otherwise.}\end{cases}$

(25)

This procedure for constructing the optimal probability distribution can also be visualized graphically as follows: begin with $\tilde{p}_{\mathcal{W}}^{Y}$ (Fig. 2 (a)), which by definition is a probability distribution arranged in descending order along the states $y$. Apply the permutation $\sigma_{p^{Y}}^{-1}$ to $\tilde{p}_{\mathcal{W}}^{Y}$ (Fig. 2 (b)), which rearranges the states such that the descending order of probabilities matches that of $p^{Y}$. Then, swap the probability of the state $y=x$ with the probability of the state with the largest probability (i.e., the state satisfying $\sigma_{p^{Y}}(y)=1$) (Fig. 2 (c)). From the second operation, the optimal conditional probability distribution $p_{\tau}^{Y|x}$ satisfies $p_{\tau}^{Y|x}(x)=\tilde{p}_{\mathcal{W}}^{Y}(1)=1-\mathcal{W}$.

We next show that its marginal distribution matches the fixed $p^{Y}$. For $y$ such that $\sigma_{p^{Y}}(y)>M_{\mathcal{W}}$ holds, from Eq. (25), $p_{\tau}^{Y|x}(y)=\tilde{p}_{\mathcal{W}}^{Y}(\sigma_{p^{Y}}(y))=p^{Y}(y)$ holds independently of $x$, which yields $\sum_{x}p_{\tau}^{XY}(x,y)=p^{Y}(y)$. For $y$ such that $\sigma_{p^{Y}}(y)\leq M_{\mathcal{W}}$ holds, from Eq. (25) and the definition of $\tilde{p}_{\mathcal{W}}^{Y}$, $p_{\tau}^{Y|x}(y)$ takes the value $\tilde{p}_{\mathcal{W}}^{Y}(1)$ for $y=x$ and $\tilde{p}_{\mathcal{W}}^{Y}(2)$ for $y\neq x$. Therefore,

	$\displaystyle\sum_{x=1}^{n}p_{\tau}^{XY}(x,y)$	$\displaystyle=\sum_{x=1}^{n}p_{\tau}^{X}(x)p_{\tau}^{Y|x}(y)$		(26)
		$\displaystyle=p_{\tau}^{X}(y)\tilde{p}_{\mathcal{W}}^{Y}(1)+[1-p_{\tau}^{X}(y)]\tilde{p}_{\mathcal{W}}^{Y}(2)$		(27)
		$\displaystyle=p^{Y}(y).$		(28)

Here, we used Eq. (24) for the transformation from Eq. (27) to Eq. (28).

We next show that under the condition (C), the Wasserstein distance between the constructed $p_{\tau}^{XY}$ and $p_{0}^{XY}$ equals the fixed value $\mathcal{W}$. When the condition (C) is satisfied, $\mathcal{E}_{\tau}=\mathcal{W}\left(p_{0}^{XY},p_{\tau}^{XY}\right)$ holds (the proof is provided in the Appendix). Therefore, we obtain

	$\displaystyle\mathcal{W}\left(p_{0}^{XY}.p_{\tau}^{XY}\right)$	$\displaystyle=\mathcal{E}_{\tau}$		(29)
		$\displaystyle=1-\sum_{x,y:x=y}p_{\tau}^{XY}(x,y)$		(30)
		$\displaystyle=1-\sum_{x}p_{\tau}^{X}(x)p_{\tau}^{Y|x}(x)$		(31)
		$\displaystyle=1-\tilde{p}_{\mathcal{W}}^{Y}(1)$		(32)
		$\displaystyle=\mathcal{W}.$		(33)

We finally show that the constructed $p_{\tau}^{XY}$ is optimal, i.e., it achieves the equality in (21). From Eq. (25), since $S\left(p_{\tau}^{Y|x}\right)=S\left(\tilde{p}_{\mathcal{W}}^{Y}\right)$ holds regardless of $x$,

	$\displaystyle\absolutevalue{\Delta I_{\tau}^{X:Y}}$	$\displaystyle=S_{\tau}^{Y|X}$		(34)
		$\displaystyle=\sum_{x}p_{\tau}^{X}(x)S\left(p_{\tau}^{Y|x}\right)$		(35)
		$\displaystyle=S\left(\tilde{p}_{\mathcal{W}}^{Y}\right)$		(36)

holds. Therefore, the equality in (21) can be achieved.

In the previous section, we considered the maximization problem for the consumed information $|\Delta I_{\tau}^{X:Y}|$ for fixed $\mathcal{W}\left(p_{0}^{XY},p_{\tau}^{XY}\right)$, whose solution is given by the main Theorem (21). Since the original problem of minimizing $\mathcal{W}\left(p_{0}^{XY},p_{\tau}^{XY}\right)$ for fixed $|\Delta I_{\tau}^{X:Y}|$ corresponds to the dual problem, its solution is directly derived by the main Theorem. From this dual solution, we can obtain the fundamental bound on entropy production required to consume $|\Delta I_{\tau}^{X:Y}|$.

First, the right-hand side of inequality (21), $S\left(\tilde{p}_{\mathcal{W}}^{Y}\right)$, is a strictly increasing function of $\mathcal{W}$ in the range $0\leq\mathcal{W}\leq 1-p_{\downarrow}^{Y}(1)$, and takes a constant value $S\left(p^{Y}\right)$ for $\mathcal{W}\geq 1-p_{\downarrow}^{Y}(1)$ (the proof is provided in the Appendix). Therefore, given a fixed $p^{Y}$, the inverse function $\mathcal{W}_{p^{Y}}^{\min}:[0,S\left(p^{Y}\right)]\to[0,1-p_{\downarrow}^{Y}(1)]$ can be defined as $S\left(\tilde{p}_{\mathcal{W}_{p^{Y}}^{\min}(I)}\right)=I$. This function provides the minimum Wasserstein distance $\mathcal{W}\left(p_{0}^{XY},p_{\tau}^{XY}\right)$ for the fixed amount of consumed information , $|\Delta I_{\tau}^{X:Y}|$.

Here, considering Eq. (15) with $D$ chosen as the time-averaged mobility $\langle m\rangle_{\tau}$, the right-hand side becomes an increasing function of $\mathcal{W}\left(p_{0}^{XY},p_{\tau}^{XY}\right)$. Therefore, minimizing $\mathcal{W}\left(p_{0}^{XY},p_{\tau}^{XY}\right)$ is equivalent to minimizing entropy production $\Sigma_{\tau}$. This allows us to derive the speed limit in the feedback process as

$\displaystyle\Sigma_{\tau}^{X}\geq-\absolutevalue{\Delta I_{\tau}^{X:Y}}+\frac{{\mathcal{W}_{p^{Y}}^{\min}\left(\absolutevalue{\Delta I_{\tau}^{X:Y}}\right)}^{2}}{\tau\langle m\rangle_{\tau}},$

(37)

which provides the minimum entropy production required to consume information $|\Delta I_{\tau}^{X:Y}|$ through feedback processes keeping time-averaged mobility $\langle m\rangle_{\tau}$ fixed.

The equality in inequality (37) is achieved by simultaneously satisfying the equalities in inequalities (12) and (21). First, the equality in (21) is achieved by setting the final distribution $p_{\tau}^{XY}$ to the form determined by Eqs. (24) and (25), while satisfying condition (C). Next, the equality in (12) is achieved by constructing a protocol $\{R_{t}^{X|y}(x,x^{\prime})\}_{t=0}^{\tau}$ that transports the probability between $p_{0}^{XY}$ and $p_{\tau}^{XY}$ along the optimal transport plan, with uniform and constant thermodynamic force. Consequently, inequality (37) is satisfied. This highlights the thermodynamic implication of our main Theorem.

When we choose time-averaged activity $\langle a\rangle_{\tau}$ as $D$, the second term on the right-hand side of inequality (12) becomes $2\mathcal{W}\tanh^{-1}[\mathcal{W}/(\tau\langle a\rangle_{\tau})]$. Similarly to the case where mobility is fixed, this term is an increasing function of $\mathcal{W}$ for any fixed $\tau$ and $\langle a\rangle_{\tau}$. Therefore, by applying the main Theorem, we can derive another speed limit

$\displaystyle\Sigma_{\tau}^{X}\geq-\absolutevalue{\Delta I_{\tau}^{X:Y}}+2\mathcal{W}_{p^{Y}}^{\min}\left(\absolutevalue{\Delta I_{\tau}^{X:Y}}\right)\tanh^{-1}\frac{\mathcal{W}_{p^{Y}}^{\min}\left(\absolutevalue{\Delta I_{\tau}^{X:Y}}\right)}{\tau\langle a\rangle_{\tau}},$

(38)

which provides the minimum entropy production required to consume the information $\absolutevalue{\Delta I_{\tau}^{X:Y}}$ in feedback control when time-averaged activity $\langle a\rangle_{\tau}$ is fixed.

The equality in inequality (38) is achieved by simultaneously achieving the equalities in inequalities (12) and (21). First, the equality in (21) is realized by setting the final distribution $p_{\tau}^{XY}$ to the form determined by Eqs. (24) and (25), while satisfying condition (C). Next, the equality in (12) is achieved by constructing a protocol $\{R_{t}^{X|y}(x,x^{\prime})\}_{t=0}^{\tau}$ that transports the probability between the specified initial and final distributions $p_{0}^{XY}$ and $p_{\tau}^{XY}$ along the optimal transport plan, with uniform and constant thermodynamic force. Consequently, inequality (38) is achieved. This again highlights the thermodynamic implication of the main Theorem.