# Unified Hierarchical Relationship Between Thermodynamic Tradeoff Relations

Euijoon Kwon  Department of Physics and Astronomy & Center for Theoretical
Physics, Seoul National University, Seoul 08826, Republic of Korea  Jong-Min
Park  Asia Pacific Center for Theoretical Physics, Pohang, 37673, Republic of
Korea  Department of Physics, Postech, Pohang 37673, Republic of Korea  Jae
Sung Lee  School of Physics, Korea Institute for Advanced Study, Seoul 02455,
Republic of Korea  Yongjoo Baek<EMAIL_ADDRESS>Department of Physics and
Astronomy & Center for Theoretical Physics, Seoul National University, Seoul
08826, Republic of Korea

###### Abstract

Recent years have witnessed a surge of discoveries in the studies of
thermodynamic inequalities: the thermodynamic uncertainty relation (TUR) and
the entropic bound (EB) provide a lower bound on the entropy production (EP)
in terms of nonequilibrium currents; the classical speed limit (CSL) expresses
the lower bound on the EP using the geometry of probability distributions; the
power-efficiency (PE) tradeoff dictates the maximum power achievable for a
heat engine given the level of its thermal efficiency. In this study, we show
that there exists a unified hierarchical structure encompassing all of these
bounds, with the fundamental inequality given by a novel extension of the TUR
(XTUR) that incorporates the most general range of current-like and state-
dependent observables. By selecting more specific observables, the TUR and the
EB follow from the XTUR, and the CSL and the PE tradeoff follow from the EB.
Our derivations cover both Langevin and Markov jump systems, with the first
proof of the EB for the Markov jump systems and a more generalized form of the
CSL. We also present concrete examples of the EB for the Markov jump systems
and the generalized CSL.

##  I Introduction

The basis of thermodynamics lies in its fundamental inequalities. At their
origin was the question of how much work a heat engine can do, which led to
the discovery of the Carnot efficiency—a universal cap on the thermal
efficiency for all heat engines [1]. However, deeper explorations resulted in
the discovery of an even more profound inequality, the second law of
thermodynamics, which indicates the inevitable direction of evolution for all
macroscopic thermodynamic systems [2]. This fundamental inequality, besides
incorporating the Carnot bound as a corollary, lays down a versatile framework
for generating a plethora of more specific inequalities, such as Le
Châtelier’s principle
[[Foraformulationoftheprincipleasathermodynamicinequality,
seethediscussioninpages210--214of]Callen1985] and Landauer’s principle [4]. In
this context, clarifying the logical connections among various universal
inequalities has always been a pivotal part of thermodynamics.

In traditional thermodynamics, the scope of inequalities has been limited to
those whose saturation necessitates quasistatic processes. However, in recent
years, researchers have unveiled a range of thermodynamic inequalities that
can be saturated even during general nonequilibrium processes. These newly
discovered inequalities offer nontrivial, tighter bounds on entropy production
(EP) that cannot be derived from the conventional thermodynamic second law.
These novel inequalities can be categorized into four distinct classes.

First, the thermodynamic uncertainty relation (TUR) states that EP is bounded
from below by the ratio of a function of some mean current-like observable to
the variance of the same observable. Initially discovered for biomolecular
processes [5], soon the TUR has been derived for Markov jump systems [6, 7, 8,
9], both overdamped [10, 11, 12, 13] and underdamped Langevin systems [14,
15], and even open quantum systems [16, 17, 18]. In the steady state without
any reversible currents, the lower bound of the TUR has a particularly simple
form (the inverse of the relative fluctuation); but, in other cases, the
inequality typically involves nontrivial derivatives of the mean current whose
empirical measurement is not always straightforward.

Second, the entropic bound (EB) establishes a lower bound for EP based on the
square of the mean value of a current-like observable. While this bound has
been demonstrated for both overdamped and underdamped Langevin systems [19],
its validity has not been proven for Markov jump systems. The EB differs from
the TUR in that the EB’s lower bound is solely expressed in terms of the mean
current without any derivatives applied to it for any nonsteady processes.
This characteristic may make the EB more accessible to empirical verification,
although it comes at the expense of losing an explicit upper bound on the
precision of the current-like observable. As a result, the TUR and the EB have
been treated as independent inequalities, without either being shown to be
more fundamental than the other.

Third, the power–efficiency (PE) tradeoff shows that the power of a heat
engine cannot exceed a quadratic function of its efficiency. This upper bound
reaches zero at both the Carnot and zero efficiency points, precluding the
existence of a finite-size heat engine that sustains positive power at Carnot
efficiency. Initially, the PE tradeoff was proven as an independent
inequality, unrelated to the TUR or the EB [20, 21].

Lastly, the classical speed limit (CSL) states that the EP rate is bounded
from below by the square of the speed at which the system crosses the
“distance” between the initial and the final probability distributions [22,
23]. Initially conceived as the classical counterpart of the quantum speed
limit (QSL) [24], recent studies formulate the CSL using the measure of
distance between probability distributions borrowed from information geometry
and optimal transport theory [25, 26, 27]. Thus, the CSL provides a novel
geometric perspective on the issues of irreversibility and dissipation.

The crucial inquiry pertains to whether these relationships are distinct,
independent inequalities, or if they can be incorporated within a single
unified theoretical framework. Indeed, numerous studies have tried to
establish connections among these tradeoff relations. For instance, in the
case of a steady-state heat engine, the PE tradeoff can be derived using the
TUR [28]. Similarly, for a heat engine described by Langevin equations with a
time-dependent protocol, the PE tradeoff can be viewed as a consequence of the
EB [19]. Efforts have also been made to find a unified framework for the TUR
and the CSL. A recent study derived the CSL from a short-time version of the
unified thermodynamic-kinetic uncertainty relation [29], yet the derivation’s
applicability is limited to step-wise protocols. Other works, such as Refs.
[30, 31], proposed a general information-geometric inequality that encompasses
the TUR and the CSL as its consequence. However, the resulting TUR in Ref.
[30] demonstrates the exponential TUR, while Ref. [31] reveals a kinetic
uncertainty relation (KUR) rather than the conventional TUR. Although Ref.
[32] verified the equivalence between the short-time TUR and the EB, there has
been no discussion concerning the finite-time TUR and the EB. Despite these
attempts, a unified theory that successfully combines all these relationships
remains elusive.

In this study, we establish a unified hierarchical framework encompassing all
tradeoff relations for both Langevin and Markov jump systems. This is achieved
by deriving an extended version of the TUR (XTUR) applicable to both types of
systems. Then, as depicted in Fig. 1, all the inequalities listed above are
connected by a hierarchical structure originating from the XTUR. In this
hierarchy, the higher-level inequalities are applicable to a broader range of
observables, whereas more specific assumptions about the observables are
required to derive the lower-level inequalities. Of particular significance is
the central role of the EB in this hierarchical framework, serving as the
intermediary between the XTUR and the lowest-level tradeoff relations.
Furthermore, this hierarchical structure provides a useful guide for
establishing novel thermodynamic inequalities, such as the EB for Markov jump
systems and a generalized form of the CSL.

The rest of this paper is organized as follows. First, we introduce the
notations used in this paper and summarize the main results (Sec. II). Then,
we prove the XTUR for the most general types of observables (Sec. III). This
is followed by the derivations of the EB from the XTUR (Sec. IV), the
generalized CSL from the EB (Sec. V), and the PE tradeoff from the EB (Sec.
VI). After presenting concrete examples of the EB for Markov jump systems and
the generalized CSL (Sec. VII), we conclude with a discussion of the possible
future works (Sec. VIII).

Figure 1:  The hierarchical relationship among various thermodynamic tradeoff
relations. Each box corresponds to an inequality, and each arrow illustrates
the hierarchical connection between the inequalities it joins, along with the
associated information specifying the conditions for deducing lower-level
bounds.

##  II Notations and the main results

In this section, we present the key thermodynamic inequalities and their
hierarchical relationship applicable to both Langevin and Markov jump systems.
To facilitate this exploration, we begin by introducing the notations utilized
throughout the paper.

###  II.1 Notations for Langevin systems

We consider an $N$-dimensional system whose state is described by the vector
$\mathbf{x}=(x_{1},x_{2},\cdots,x_{N})^{\textsf{T}}$. Given a drift force
$\mathbf{A}=(A_{1},A_{2},\cdots,A_{N})^{\textsf{T}}$ and a Gaussian white
noise $\boldsymbol{\xi}=(\xi_{1},\xi_{2},\cdots,\xi_{N})^{\textsf{T}}$, the
Langevin equation can be written as

$\displaystyle\dot{\mathbf{x}}(t)=\mathbf{A}(\mathbf{x},\omega
t)+\mathbb{B}(\mathbf{x},\omega t)\bullet\boldsymbol{\xi}(t)\;,$ (1)

where $\omega$ is a parameter controlling the speed of the protocol, $\bullet$
denotes the Itô product, the noise satisfies $\langle\xi_{i}(t)\rangle=0$ and
$\langle\xi_{i}(t)\xi_{j}(t^{\prime})\rangle=\delta_{ij}\delta(t-t^{\prime})$,
and $\frac{1}{2}\mathbb{B}(\mathbf{x},\omega
t)\,\mathbb{B}^{\textsf{T}}\\!(\mathbf{x},\omega
t)\equiv\mathbb{D}(\mathbf{x},\omega t)$ is a position-dependent diffusion
matrix. The drift force can be divided into reversible and irreversible parts
as $A_{i}=sf_{i}^{\text{rev}}(\mathbf{x},\omega
t)+f_{i}^{\text{irr}}(\mathbf{x},\omega t)$, where $s$ is a parameter
controlling the magnitude of the reversible drift. The two force components
satisfy

$\displaystyle f_{i}^{\text{rev}}(\boldsymbol{\epsilon}\mathbf{x},\omega t)$
$\displaystyle=-\epsilon_{i}f_{i}^{\text{rev}}(\mathbf{x},\omega t)\;,$
$\displaystyle f_{i}^{\text{irr}}(\boldsymbol{\epsilon}\mathbf{x},\omega t)$
$\displaystyle=\epsilon_{i}f_{i}^{\text{irr}}(\mathbf{x},\omega t)\;,$ (2)

where we use the notation
$\boldsymbol{\epsilon}\mathbf{x}\equiv(\epsilon_{1}x_{1},\epsilon_{2}x_{2},\cdots,\epsilon_{N}x_{N})$,
with $\epsilon_{i}$ being the parity of variable $x_{i}$ under time reversal.
Note that $s$ and $\omega$ are auxiliary variables that can be set to unity in
the final step.

The equivalent Fokker-Planck equation is

$\displaystyle\partial_{t}p(\mathbf{x},t;s,\omega)=-\nabla^{\textsf{T}}\left[\mathbf{J}^{\text{rev}}(\mathbf{x},t;s,\omega)+\mathbf{J}^{\text{irr}}(\mathbf{x},t;s,\omega)\right]\;,$
(3)

where $\mathbf{J}^{\text{rev}}$ and $\mathbf{J}^{\text{irr}}$ are reversible
and irreversible probability currents defined as

$\displaystyle\mathbf{J}^{\text{rev}}=s\boldsymbol{f}^{\text{rev}}(\mathbf{x},\omega
t)\,p(\mathbf{x},t;s,\omega)\;,$
$\displaystyle\mathbf{J}^{\text{irr}}=\left\\{\boldsymbol{f}^{\text{irr}}(\mathbf{x},\omega
t)-\left[\nabla^{\textsf{T}}\mathbb{D}(\mathbf{x},\omega
t)\right]^{\textsf{T}}\right\\}p(\mathbf{x},t;s,\omega)\;,$ (4)

respectively, with the gradient
$\nabla\equiv(\partial_{x_{1}},\partial_{x_{2}},\cdots,\partial_{x_{N}})^{\textsf{T}}$.
The total probability current is then
$\mathbf{J}\equiv\mathbf{J}^{\text{rev}}+\mathbf{J}^{\text{irr}}$. With these,
the mean total EP can be written as

$\Sigma_{\tau}^{\rm tot}=\int_{0}^{\tau}dt\int
d\mathbf{x}\;\frac{{\mathbf{J}^{\rm
irr}}^{\textsf{T}}\mathbb{D}^{-1}(\mathbf{x},\omega t)\mathbf{J}^{\rm
irr}}{p(\mathbf{x},t;s,\omega)}.$ (5)

For the brevity of notation, we shall henceforth omit the function arguments
when there is no risk of confusion.

In this study, we consider both current-like and state-dependent observables,
which take the forms

$\displaystyle\mathcal{J}_{\tau}^{\Lambda}(\Gamma)\equiv\int_{0}^{\tau}dt\,\boldsymbol{\Lambda}^{\textsf{T}}\\!(\mathbf{x}(t),\omega
t;s)\circ\dot{\mathbf{x}}(t)\;,$
$\displaystyle\mathcal{K}_{\tau}^{g}(\Gamma)\equiv\int_{0}^{\tau}dt\,g(\mathbf{x}(t),\omega
t;s)\;,$ (6)

respectively. Here $\Gamma\equiv\\{\mathbf{x}(t)\\}$ denotes a stochastic
trajectory, and $\circ$ stands for the Stratonovich product. In general, one
can measure the composite observable $\Theta_{\tau}^{\Lambda,g}(\Gamma)$
involving these two:

$\Theta_{\tau}^{\Lambda,g}(\Gamma)\equiv\mathcal{J}_{\tau}^{\Lambda}(\Gamma)+\mathcal{K}_{\tau}^{g}(\Gamma)\;.$
(7)

Using the identity
$\langle\boldsymbol{\Lambda}^{\textsf{T}}\circ\dot{\mathbf{x}}\rangle=\int
d\mathbf{x}\,\boldsymbol{\Lambda}^{\textsf{T}}{\textbf{J}}$, the mean current-
like observable can be separated into reversible and irreversible components
as

$\langle\mathcal{J}_{\tau}^{\Lambda}\rangle=\int_{0}^{\tau}dt\int
d\mathbf{x}\,\boldsymbol{\Lambda}^{\textsf{T}}\mathbf{J}=\langle\mathcal{J}_{\tau}^{\Lambda}\rangle^{\rm
rev}+\langle\mathcal{J}_{\tau}^{\Lambda}\rangle^{\rm irr}\;,$ (8)

where $\langle\mathcal{J}_{\tau}^{\Lambda}\rangle^{\rm
rev}\equiv\int_{0}^{\tau}dt\int
d\mathbf{x}\,\boldsymbol{\Lambda}^{\textsf{T}}\mathbf{J}^{\rm rev}$ and
$\langle\mathcal{J}_{\tau}^{\Lambda}\rangle^{\rm
irr}\equiv\int_{0}^{\tau}dt\int
d\mathbf{x}\,\boldsymbol{\Lambda}^{\textsf{T}}\mathbf{J}^{\rm irr}$.

###  II.2 Notations for Markov jump systems

Here we consider a continuous-time Markov jump process within a discrete state
space $S$. The transition rate from state $m$ to $n$ at time $t$ is denoted by
$R_{nm}(\omega t)$, where $\omega$ is the speed of the protocol. Then, the
master equation of the system is given by

$\displaystyle\dot{p}_{n}(t;\omega)=\sum_{m(\neq n)}j_{nm}(t;\omega)\;,$ (9)

where $p_{n}(t;\omega)$ is the probability of finding the system in state
$n\in S$ at time $t$, and $j_{nm}(t;\omega)=R_{nm}(\omega
t)p_{m}(t;\omega)-R_{mn}(\omega t)p_{n}(t;\omega)$ is the net probability
current from state $m$ to $n$. The mean total EP of this system is given by

$\Sigma_{\tau}^{\rm
tot}=\int_{0}^{\tau}dt\sum_{n>m}j_{nm}\ln\frac{R_{nm}p_{m}}{R_{mn}p_{n}}\;.$
(10)

Moreover, we can define the mean pseudo-EP

$\displaystyle\Sigma_{\tau}^{\text{ps}}\equiv\int_{0}^{\tau}dt\sum_{n\neq
m}\frac{j_{nm}^{2}}{a_{nm}}\;,$ (11)

where $a_{nm}=R_{nm}p_{m}+R_{mn}p_{n}$ is the dynamical activity between
states $m$ and $n$. Using the log-mean inequality, one can prove

$\Sigma^{\text{ps}}_{\tau}\leq\Sigma^{\text{tot}}_{\tau}\,.$ (12)

Consider a trajectory of this Markov jump system, through which a total of $M$
jumps occur during the time interval $[0,\tau]$, with the $i$-th jump taking
place at time $t_{i}$ for $i=1,2,\cdots,M$. For this trajectory, let us define
$N_{nm}(t)$, the number of jumps from state $m$ to $n$ up to time $t$, and
$\eta_{n}(t)\equiv\delta_{x(t),n}$, where $x(t)$ is the state of the system at
time $t$. Note that these two observables satisfy
$\langle\dot{N}_{nm}(t)\rangle=R_{nm}(\omega t)p_{m}(t;\omega)$ and
$\langle\eta_{n}\rangle=p_{n}(t;\omega)$. Then, the current-like and the
state-dependent observables can be written as

$\displaystyle\mathcal{J}_{\tau}^{\Lambda}(\Gamma)\equiv\int_{0}^{\tau}dt\sum_{n\neq
m}\Lambda_{nm}(\omega
t)\dot{N}_{nm}(t)\;,~{}~{}~{}(\Lambda_{nm}=-\Lambda_{mn})$
$\displaystyle\mathcal{K}_{\tau}^{g}(\Gamma)\equiv\int_{0}^{\tau}dt\sum_{n}\eta_{n}(t)g_{n}(\omega
t)\;,$ (13)

respectively. Just as in the case of Langevin systems, one can measure the
composite observable
$\Theta_{\tau}^{\Lambda,g}(\Gamma)\equiv\mathcal{J}_{\tau}^{\Lambda}(\Gamma)+\mathcal{K}_{\tau}^{g}(\Gamma)$.

###  II.3 Main results

Our primary finding is a unified hierarchical framework integrating the four
major thermodynamic inequalities (TUR, EB, CSL, and PE tradeoff) applicable to
both Langevin and Markov jump systems. This hierarchy is depicted in Fig. 1.
Each arrow in the figure, accompanied by its specifics, indicates how the
lower-level inequality is deduced from the the one above it by constraining
the associated observable. The reader is cautioned that, throughout this
study, all expressions are given in units where the Boltzmann constant is
equal to one.

At the apex of the hierarchy is the extended thermodynamic uncertainty
relation (XTUR), which serves as the thermodynamic tradeoff relation for the
most general class of the composite observable
$\Theta_{\tau}^{\Lambda,g}(\Gamma)$. It is formulated as

$\displaystyle\frac{\Omega_{\tau}^{2}}{\text{Var}[\Theta_{\tau}^{\Lambda,g}]}\leq\frac{\Sigma_{\tau}}{2}\;,$
(14)

where $\Sigma_{\tau}$ can be either $\Sigma_{\tau}^{\rm tot}$ or
$\Sigma_{\tau}^{\rm ps}$, and $\Omega_{\tau}$ is defined as

$\displaystyle\Omega_{\tau}\equiv\hat{O}_{\tau}\langle\Theta_{\tau}^{\Lambda,g}\rangle-\langle\mathcal{K}_{\tau}^{g}\rangle+\left\langle\Theta_{\tau}^{s\partial_{s}\Lambda,s\partial_{s}g}\right\rangle\;,$
(15)

with the operator $\hat{O}_{t}\equiv
t\partial_{t}-s\partial_{s}-\omega\partial_{\omega}$ for both Langevin and
Markov jump systems. We note that the terms associated with $s\partial_{s}$ in
Eq. (15) vanish for systems with only even-parity variables since $s=0$ there.
A derivation of Eq. (14) is given in Sec. III.

We stress that the XTUR represents the most comprehensive version of the TUR
in the following sense: it provides an upper bound on the precision of the
general composite observable even when the system involves odd-parity
variables (including both momentum-like and force-like [33] ones) and time-
dependent protocols. In contrast, prior research has predominantly focused on
TURs for only current-like observables. While a handful of studies have
ventured into deriving TURs for composite observables [34, 9, 35], their
outcomes are restricted to the steady state and even-parity variables.

Starting from the XTUR, two other tradeoff relations are derived by
restricting the form of the state-dependent observable. First, the
conventional TUR for only current-like observables is readily obtained by
setting the state-dependent observable to zero ($g=0$) within the XTUR, i.e.,
$\Theta_{\tau}^{\Lambda,g}(\Gamma)=\mathcal{J}_{\tau}^{\Lambda}(\Gamma)$ as
shown in Fig. 1 and Sec. III. Alternatively, by selecting a state-dependent
observable satisfying

$\displaystyle g(\mathbf{x})$
$\displaystyle=-\left[\frac{\boldsymbol{\Lambda}^{\textsf{T}}\mathbf{J}}{p}+\frac{\nabla^{\textsf{T}}(\mathbb{D}\boldsymbol{\Lambda}p)}{p}\right]$
(Langevin), (16) $\displaystyle g_{m}$ $\displaystyle=-\sum_{n\neq
m}\Lambda_{nm}R_{nm}$ (Markov jump), (17)

where $p=p(\mathbf{x},t;s,\omega)$ is the probability density function, then
the XTUR reduces to the EB

$\displaystyle(\langle\mathcal{J}_{\tau}^{\Lambda}\rangle^{\rm
irr})^{2}\leq\Sigma_{\tau}\chi_{\tau}\;,$ (18)

where $\chi_{\tau}=\int_{0}^{\tau}dt\int
d\mathbf{x}\langle\boldsymbol{\Lambda}^{\textsf{T}}\mathbb{D}\boldsymbol{\Lambda}\rangle$
for Langevin systems and
$\chi_{\tau}=\frac{1}{2}\int_{0}^{\tau}dt\sum_{n<m}\Lambda_{nm}^{2}a_{nm}$ for
Markov jump systems. This clarifies the relationship between the TUR and the
EB: they are two different manifestations of a common higher-level tradeoff
relation, the XTUR.

It is worth noting that, compared to the TUR, the EB offers a more
experimentally accessible approach for estimating EP [32]. This can be
attributed to the following two reasons. First, the EB inequality can always
be saturated for arbitrary nonequilibrium processes, whereas this is not
generally true for the TUR, as we discuss in Sec. III. Thus, while the
accurate EP estimation can be made in the EB framework, the same level of
accuracy is not guaranteed for the TUR method. Second, the EB lacks any
differential operators, which the TUR must incorporate in the presence of odd-
parity variables or time-dependent protocols. Empirical measurements of
derivatives are often challenging, if not practically unfeasible. Thus, when
dealing with time-dependent protocols or odd-parity variables, the EB offers a
more feasible experimental framework than the TUR for EP estimation.

Now, proceeding further down the hierarchy, the absence of differential
operators in the EB facilitates connections with the two lower-level
thermodynamic tradeoff relations. The first of these is the CSL. By focusing
on the overdamped Langevin systems with $\boldsymbol{\Lambda}=\nabla
h(\mathbf{x})$ or the Markov jump systems with $\Lambda_{nm}=h_{n}-h_{m}$, and
taking the supremum of Eq. (18) over a function space $\mathcal{H}$ of $h$, we
obtain the following CSL:

$\displaystyle\tau\geq\frac{d_{\mathcal{H}}^{2}(p(0),p(\tau))}{\Sigma_{\tau}\bar{\chi}^{\mathcal{H}}}\;,$
(19)

where $d_{\mathcal{H}}(p,q)$ is the integral probability metric (IPM), which
is a measure of distance between two distributions $p$ and $q$, and
$\bar{\chi}^{\mathcal{H}}$ is the time average of the supremum of
$\chi_{\tau}$. See Sec. V for details. From Eq. (19), we can derive a broad
range of CSLs reported in the literature, such as the original CSL for Markov
jump systems, Eq. (77) [22], the CSL involving the Wasserstein distance, Eq.
(73), and the tighter CSL, Eq. (80) [36].

Finally, the other lower-level corollary of the EB is the PE tradeoff
relation. For the case of Langevin systems, Dechant and Sasa have demonstrated
that the PE tradeoff can be deduced from the EB for both steady-state and
cyclic heat engines [19]. Meanwhile, Pietzonka and Seifert derived the PE
tradeoff from the TUR, but solely for steady-state heat engines [28]. As for
cyclic heat engines, however, establishing a link between the PE tradeoff and
the TUR appears to be unattainable, primarily due to the inclusion of
differential operators in the TUR formulation. In Sec. VI, we show that the EB
makes it possible to derive the PE tradeoff for Markov jump systems,
encompassing both steady-state and cyclic heat engines.

##  III XTUR

In this section, we present a derivation of the XTUR, situated at the apex of
the hierarchy, for both Langevin and Markov jump systems. Following that, we
delve into the equality condition of the XTUR, showing that it is achievable
for the general nonequilibrium processes. Finally, we discuss the connection
between the XTUR and the TUR.

###  III.1 XTUR for Langevin systems

Adapting a previously developed method [11, 14], the XTUR is derived by
applying the Cramér–Rao inequality in conjunction with the perturbed dynamics
described by the following Fokker-Planck equation:

$\displaystyle\partial_{t}p_{\theta}=-\nabla^{\textsf{T}}\left[\mathbf{J}_{\theta}^{\text{rev}}+(1+\theta)\mathbf{J}_{\theta}^{\text{irr}}\right]\;,$
(20)

where $\mathbf{J}_{\theta}^{\text{rev}}$ and
$\mathbf{J}_{\theta}^{\text{irr}}$ are given by

$\displaystyle\mathbf{J}_{\theta}^{\text{rev}}=s\mathbf{f}^{\text{rev}}p_{\theta}\;,\,\quad\mathbf{J}_{\theta}^{\text{irr}}=\left(\mathbf{f}^{\text{irr}}-\left[\nabla^{\textsf{T}}\mathbb{D}\right]^{\textsf{T}}\right)p_{\theta}\,.$
(21)

In other words, the evolution of $p_{\theta}$ is achieved by multiplying the
perturbation factor $1+\theta$ solely to the irreversible component of Eq.
(3). It is straightforward to check that the solution of Eq. (20) is related
to that of Eq. (3) by
$p_{\theta}(\mathbf{x},t;s,\omega)=p(\mathbf{x},t_{\theta};s_{\theta},\omega_{\theta})$,
where $t_{\theta}\equiv(1+\theta)t$, $s_{\theta}\equiv s/(1+\theta)$, and
$\omega_{\theta}\equiv\omega/(1+\theta)$. Note that this scaling perturbation
amounts to changing the drift vector $\mathbf{A}$ to $\mathbf{A}_{\theta}$ in
the original Langevin dynamics described by Eq. (1) as follows:

$\displaystyle\dot{\mathbf{x}}=\mathbf{A}_{\theta}+\mathbb{B}\bullet\boldsymbol{\xi}\;,~{}~{}~{}{\rm
where}~{}~{}\mathbf{A}_{\theta}\equiv\mathbf{A}+\theta\frac{\mathbf{J}_{\theta}^{\text{irr}}}{p_{\theta}}\;.$
(22)

Within this perturbation scheme, the XTUR is derived by taking $\theta=0$ in
the Cramér–Rao inequality

$\displaystyle\frac{(\partial_{\theta}\langle\Theta_{\tau}^{\Lambda,g}\rangle_{\theta})^{2}}{\text{Var}_{\theta}[\Theta_{\tau}^{\Lambda,g}]}\leq\mathcal{I}(\theta)\;,$
(23)

where $\langle\cdots\rangle_{\theta}$ denotes the average over trajectories of
the perturbed dynamics, and
$\mathcal{I}(\theta)\equiv\langle-\partial_{\theta}^{2}\ln\mathcal{P}_{\theta}(\Gamma)\rangle_{\theta}$
represents the Fisher information associated with the path probability
$\mathcal{P}_{\theta}(\Gamma)$ of the perturbed dynamics.

Evaluating the Fisher information at $\theta=0$ leads to

$\displaystyle\mathcal{I}(0)=\frac{1}{2}\left(\Sigma^{\text{tot}}_{\tau}+\mathcal{B}\right)\;,$
(24)

where the term $\mathcal{B}$ is determined by the dependence of the initial
distribution on $s$ and $\omega$ (for details, see Appendix A.1). Opting for
an initial distribution dependent on $s$ or $\omega$ yields a positive
$\mathcal{B}$ [14]. In contrast, choosing an initial distribution independent
of $s$ and $\omega$ leads to $\mathcal{B}=0$. There are some cases where a
positive $\mathcal{B}$ leads to a tighter inequality (for a detailed
explanation, see Appendix B); however, the case of $\mathcal{B}=0$ is more
accessible to empirical measurement. Moreover, as discussed in Sec. IV,
$\mathcal{B}=0$ allows us to derive the EB from the XTUR. Henceforth in this
study, we adopt an initial distribution independent of $s$ and $\omega$, and
thus, $\mathcal{B}=0$.

Evaluation of
$\partial_{\theta}\langle\Theta_{\tau}^{\Lambda,g}\rangle_{\theta}\big{|}_{\theta=0}$
requires the calculations of both
$\partial_{\theta}\langle\mathcal{J}_{\tau}^{\Lambda}\rangle_{\theta}\big{|}_{\theta=0}$
and
$\partial_{\theta}\langle\mathcal{K}_{\tau}^{g}\rangle_{\theta}\big{|}_{\theta=0}$,
which are given by

$\displaystyle\partial_{\theta}\langle\mathcal{J}_{\tau}^{\Lambda}\rangle_{\theta}\big{|}_{\theta=0}$
$\displaystyle=\hat{O}_{\tau}\langle\mathcal{J}_{\tau}^{\Lambda}\rangle+\int_{0}^{\tau}dt~{}\langle(s\partial_{s}\boldsymbol{\Lambda}^{\textsf{T}})\circ\dot{\textbf{x}}\rangle\;,$
$\displaystyle\partial_{\theta}\langle\mathcal{K}_{\tau}^{g}\rangle_{\theta}\big{|}_{\theta=0}$
$\displaystyle=\hat{O}_{\tau}\langle\mathcal{K}_{\tau}^{g}\rangle-\langle\mathcal{K}_{\tau}^{g}\rangle+\int_{0}^{\tau}dt~{}\langle
s\partial_{s}g\rangle\;.$ (25)

Detailed derivations are presented in Appendix A.2. Adding these two equations
results in

$\displaystyle\partial_{\theta}\langle\Theta_{\tau}^{\Lambda,g}\rangle_{\theta}\big{|}_{\theta=0}=\hat{O}_{\tau}\langle\Theta_{\tau}^{\Lambda,g}\rangle-\langle\mathcal{K}_{\tau}^{g}\rangle+\left\langle\Theta_{\tau}^{s\partial_{s}\Lambda,s\partial_{s}g}\right\rangle,$
(26)

where
$\Theta_{\tau}^{s\partial_{s}\Lambda,s\partial_{s}g}=\int_{0}^{\tau}dt\,[(s\partial_{s}\boldsymbol{\Lambda}^{\textsf{T}})\circ\dot{\textbf{x}}+s\partial_{s}g]$.
The right-hand side of Eq. (26) is none other than $\Omega_{\tau}$ written in
Eq. (15). Substituting Eq. (24) with $\mathcal{B}=0$ and Eq. (26) into Eq.
(23), we finally obtain the XTUR (14).

We now turn to the issue of when the XTUR inequality saturates. Given that the
XTUR is derived from the Cramér–Rao inequality, its equality condition
corresponds directly to that of the Cramér–Rao inequality. As shown in Eq.
(18) of Ref. [11], the equality condition is

$\displaystyle\Theta_{\tau}^{\Lambda,g}(\Gamma)-\langle\Theta_{\tau}^{\Lambda,g}\rangle_{\theta}=k(\theta)\partial_{\theta}\ln\mathcal{P}_{\theta}(\Gamma)\;,$
(27)

where $k(\theta)$ is an arbitrary factor. For $\theta=0$, as detailed in
Appendix A.3, this condition reduces to

$\displaystyle\int_{0}^{\tau}dt\left\\{{\mathbf{C}}_{1}^{\textsf{T}}\circ\dot{\mathbf{x}}(t)+{C}_{2}\right\\}-\int_{0}^{\tau}dt\int
d\mathbf{x}\,{C}_{3}=0$ (28)

where ${\mathbf{C}}_{1}$, ${C}_{2}$, and ${C}_{3}$ are defined as

$\displaystyle{\mathbf{C}}_{1}$
$\displaystyle\equiv\boldsymbol{\Lambda}-\frac{k(0)}{2}\frac{\mathbb{D}^{-1}\mathbf{J}^{\text{irr}}}{p}~{},$
$\displaystyle{C}_{2}$ $\displaystyle\equiv
g+\frac{k(0)}{2}\left(\frac{{\mathbf{J}^{\text{irr}}}^{\mathsf{T}}\mathbb{D}^{-1}\mathbf{J}}{p^{2}}+\frac{\nabla^{\textsf{T}}\mathbf{J}^{\text{irr}}}{p}\right),$
$\displaystyle{C}_{3}$
$\displaystyle\equiv\boldsymbol{\Lambda}^{\textsf{T}}\mathbf{J}+gp~{}.$ (29)

Since Eq. (28) must hold for all trajectories, the equality condition is
satisfied only when $\textbf{C}_{1}={C}_{2}={C}_{3}=0$. The condition for
$\mathbf{\Lambda}$ and $g$ to achieve $\textbf{C}_{1}=0$ and ${C}_{2}=0$ are

$\displaystyle\boldsymbol{\Lambda}^{\rm eq}$
$\displaystyle=\frac{k(0)}{2}\frac{\mathbb{D}^{-1}{\mathbf{J}}^{\text{irr}}}{p}~{},$
$\displaystyle g^{\rm eq}$
$\displaystyle=-\frac{k(0)}{2}\left[\frac{{\mathbf{J}^{\text{irr}}}^{\textsf{T}}\mathbb{D}^{-1}{\mathbf{J}}}{p^{2}}+\frac{\nabla^{\textsf{T}}{\mathbf{J}}^{\text{irr}}}{p}\right]\;.$
(30)

Interestingly, these $\boldsymbol{\Lambda}^{\rm eq}$ and $g^{\rm eq}$ also
make $\int d{\textbf{x}}\,C_{3}$ vanish. Thus, Eq. (III.1) is the equality
condition of the XTUR. It is worth noting that the equality condition cannot
be met in the conventional TUR only with current-like observables (i.e.,
$g=0$), as $g^{\rm eq}$ is generally nonzero.

Finally, we discuss the connection between the XTUR and the previously
discovered TURs. First, for overdamped systems without odd-parity variables,
we can set $s=0$. For these systems, if we consider only current-like
observables, then $g=0$. In this case, the XTUR is reduced to the
Koyuk–Seifert TUR [12]

$\displaystyle\frac{[(\tau\partial_{\tau}-\omega\partial_{\omega})\langle\mathcal{J}_{\tau}^{\Lambda}\rangle]^{2}}{\text{Var}[\mathcal{J}_{\tau}^{\Lambda}]}\leq\frac{\Sigma^{\text{tot}}_{\tau}}{2}\;,$
(31)

which holds for overdamped processes driven by any time-dependent protocol.
Alternatively, if we set $g=-\boldsymbol{\Lambda}^{\textsf{T}}{\mathbf{J}}/p$
for overdamped processes, the XTUR in the steady state becomes the relative
TUR (RTUR) proposed by Dechant and Sasa [34], which is given by

$\displaystyle\frac{\langle\mathcal{J}_{\tau}^{\Lambda}\rangle_{\text{ss}}^{2}}{\text{Var}_{\text{ss}}(\mathcal{J}_{\tau}^{\Lambda}+\mathcal{K}_{\tau}^{g})}\leq\frac{\Sigma^{\text{tot}}_{\tau}}{2}\;.$
(32)

The XTUR can also be simplified to the TURs for underdamped Langevin dynamics.
For the $N$-dimensional underdamped dynamics with position
$\mathbf{x}=(x_{1},\cdots,x_{N})^{\textsf{T}}$ and momentum
$\mathbf{p}=(p_{1},\cdots,p_{N})^{\textsf{T}}$, Eq. (1) can be rewritten as

$\displaystyle\dot{x}_{i}=s\frac{p_{i}}{m_{i}}\;,$
$\displaystyle\dot{p_{i}}=sf_{i}^{\text{rev}}(\mathbf{x},\mathbf{p},\omega
t)+f_{i}^{\text{irr}}(\mathbf{x},\mathbf{p},\omega
t)-\frac{\gamma_{i}}{m_{i}}p_{i}+\xi_{i}\;,$ (33)

where $m_{i}$, $\gamma_{i}$, and $\xi_{i}$ are the mass, the friction
coefficient, and the Gaussian white noise corresponding to component $i$,
respectively, with
$\langle\xi_{i}(t)\xi_{j}(t^{\prime})\rangle=2\gamma_{i}T_{i}\delta_{ij}\delta(t-t^{\prime})$.
We note that, in the underdamped dynamics, the diffusion matrix $\mathbb{D}$
is singular; however, one can directly apply our approach by simply replacing
$\mathbb{D}^{-1}$ with the Moore–Penrose pseudoinverse. For the current-like
observable defined as

$\displaystyle\mathcal{J}_{\tau}^{\Lambda}(\Gamma)=\int_{0}^{\tau}dt\,\boldsymbol{\Lambda}(\mathbf{x},\mathbf{p},\omega
t)\circ\dot{\mathbf{x}}\;,$ (34)

it is straightforward to show that the XTUR reduces to the underdamped TUR

$\displaystyle\frac{\Omega_{\tau}^{2}}{\text{Var}[\mathcal{J}_{\tau}^{\Lambda}]}\leq\frac{\Sigma^{\text{tot}}_{\tau}}{2}\;,$
(35)

where
$\Omega_{\tau}=\hat{O}_{\tau}\langle\mathcal{J}_{\tau}^{\Lambda}\rangle$.
While this expression is analogous to the result of Ref. [14], there are a
couple of notable differences. First, the term $\mathcal{B}$, dependent on the
initial distribution, is absent due to our choice of an initial distribution
independent of $s$ and $\omega$, as discussed earlier in this section. Second,
while $\Omega_{\tau}$ in Ref. [14] involves a partial derivative with respect
to the spatial rescaling parameter, no such term can be found in Eq. (35).
This stems from the rescaling by $s$ of the inertia term $p_{i}/m_{i}$ in Eq.
(III.1), which was not considered in Ref. [14].

###  III.2 XTUR for Markov jump systems

The method for deriving the XTUR presented above can also be adapted to Markov
jump processes. For this purpose, we consider a perturbed Markov jump dynamics
characterized by a modified transition rate $R_{nm}^{\theta}(\omega t)$, with
its corresponding master equation expressed as

$\dot{p}_{n}^{\theta}(t;\omega)=\sum_{m(\neq n)}j_{nm}^{\theta}(t;\omega)\,,$
(36)

where $j_{nm}^{\theta}(t;\omega)=R_{nm}^{\theta}(\omega
t)p_{m}^{\theta}(t;\omega)-R_{mn}^{\theta}(\omega t)p_{n}^{\theta}(t;\omega)$.
Motivated by the approaches of Refs. [9, 29], we set the perturbed transition
rate as $R_{nm}^{\theta}=R_{nm}e^{\theta K_{nm}^{\theta}}$ for $n\neq m$ and
$R^{\theta}_{nn}=-\sum_{m(\neq n)}R^{\theta}_{mn}$ with respect to the
original rate $R_{nm}$, where

$\displaystyle
K_{nm}^{\theta}(t;\omega)\equiv\frac{R_{nm}p_{m}^{\theta}-R_{mn}p_{n}^{\theta}}{R_{nm}p_{m}^{\theta}+R_{mn}p_{n}^{\theta}}\;.$
(37)

Using the antisymmetry $K_{nm}^{\theta}=-K_{mn}^{\theta}$, we can show

$R_{nm}p_{m}^{\theta}K_{nm}^{\theta}-R_{mn}p_{n}^{\theta}K_{mn}^{\theta}=R_{nm}p_{m}^{\theta}-R_{mn}p_{n}^{\theta}.$
(38)

Then, for small $\theta$, Eq. (36) can be expanded as

$\displaystyle\dot{p}_{n}^{\theta}=$ $\displaystyle(1+\theta)\sum_{m(\neq
n)}\left[R_{nm}(\omega t)p_{m}^{\theta}(t;\omega)-R_{mn}(\omega
t)p_{n}^{\theta}(t;\omega)\right]$ $\displaystyle+O(\theta^{2})~{}.$ (39)

One can readily check that the solution of this equation is given by
$p_{n}^{\theta}(t;\omega)=p_{n}(t_{\theta};\omega_{\theta})+O(\theta^{2})$,
where $p_{n}$ is the probability distribution of the original dynamics.

Similar to the approach of the previous section, in order to evaluate the
Cramér–Rao inequality (23) at $\theta=0$, it is necessary to compute the
Fisher information,
$\partial_{\theta}\langle\mathcal{J}_{\tau}^{\Lambda}\rangle_{\theta}\big{|}_{\theta=0}$,
and
$\partial_{\theta}\langle\mathcal{K}_{\tau}^{g}\rangle_{\theta}\big{|}_{\theta=0}$.
First, choosing the $\theta$-independent initial distribution, the Fisher
information at $\theta=0$ is given by

$\displaystyle\mathcal{I}(0)=\frac{1}{2}\Sigma_{\tau}^{\text{ps}}\leq\frac{1}{2}\Sigma_{\tau}^{\text{tot}}\;,$
(40)

whose detailed derivation is presented in Appendix A.4. Note that the
inequality originates from Eq. (12). Next, the mean current-like observable in
the perturbed dynamics is obtained as

$\displaystyle\langle\mathcal{J}_{\tau}^{\Lambda}\rangle_{\theta}$
$\displaystyle=\int_{0}^{\tau}dt\sum_{n\neq m}\Lambda_{nm}(\omega
t)p^{\theta}_{m}(t;\omega)R^{\theta}_{nm}(t;\omega)$
$\displaystyle=\int_{0}^{\tau}dt\sum_{n<m}\Lambda_{nm}(\omega
t)j^{\theta}_{nm}(t;\omega)\;.$ (41)

By expanding the probability current as
$j_{nm}^{\theta}(t;\omega)=(1+\theta)j_{nm}(t_{\theta},\omega_{\theta})+O(\theta^{2})$,
where $j_{nm}$ is the probability current of the original dynamics, we have

$\displaystyle\langle\mathcal{J}_{\tau}^{\Lambda}\rangle_{\theta}$
$\displaystyle=\int_{0}^{\tau_{\theta}}dt\sum_{n<m}\
\Lambda_{nm}\left(\omega_{\theta}t\right)j_{nm}\left(t;\omega_{\theta}\right)+O(\theta^{2})$
$\displaystyle=\langle\mathcal{J}_{\tau_{\theta}}^{\Lambda}\rangle|_{\omega=\omega_{\theta}}+O(\theta^{2})\;.$
(42)

Therefore, we finally arrive at

$\displaystyle\partial_{\theta}\langle\mathcal{J}_{\tau}^{\Lambda}\rangle_{\theta}\big{|}_{\theta=0}=\hat{O}_{\tau}\langle\mathcal{J}_{\tau}^{\Lambda}\rangle\;.$
(43)

Similarly, we can prove that

$\displaystyle\langle\mathcal{K}_{\tau}^{g}\rangle_{\theta}$
$\displaystyle=\int_{0}^{\tau}dt\sum_{m}g_{m}(\omega
t)p^{\theta}_{m}(t;\omega)$
$\displaystyle=\frac{1}{1+\theta}\int_{0}^{\tau_{\theta}}dt\sum_{m}\
g_{m}(\omega_{\theta}t)p_{m}\left(t;\omega_{\theta}\right)+O(\theta^{2})$
$\displaystyle=\left.\frac{1}{1+\theta}\langle\mathcal{K}_{\tau_{\theta}}^{g}\rangle\right|_{\omega=\omega_{\theta}}+O(\theta^{2})\;,$
(44)

which leads to

$\displaystyle\partial_{\theta}\langle\mathcal{K}_{\tau}^{g}\rangle_{\theta}\big{|}_{\theta=0}=(\hat{O}_{\tau}-1)\langle\mathcal{K}_{\tau}^{g}\rangle\;.$
(45)

Plugging Eqs. (40), (43), and (45) into the Cramér–Rao inequality (23) yields
the XTUR (14). Note that the terms associated with the operator
$s\partial_{s}$ in the XTUR for Markov jump processes should vanish due to the
lack of odd-parity variables.

The equality condition of the XTUR with
$\Sigma_{\tau}=\Sigma_{\tau}^{\mathrm{ps}}$ for Markov jump systems is also
identical to that of the Cramér–Rao inequality (27). From Eq. (II.2), the
left-hand side of Eq. (27) at $\theta=0$ can be written as

$\displaystyle\Theta_{\tau}^{\Lambda,g}$
$\displaystyle(\Gamma)-\langle\Theta_{\tau}^{\Lambda,g}\rangle=\int_{0}^{\tau}dt\left(\sum_{n\neq
m}\Lambda_{nm}\dot{N}_{nm}+\sum_{n}\eta_{n}g_{n}\right)$
$\displaystyle-\int_{0}^{\tau}dt\left(\sum_{n\neq
m}\Lambda_{mn}p_{n}R_{mn}+\sum_{n}p_{n}g_{n}\right)\,.$ (46)

By using the expression for the path probability (A.4), the right-hand side of
Eq. (27) at $\theta=0$ can be evaluated as

$\displaystyle\partial_{\theta}\ln\mathcal{P}_{\theta}(\Gamma)\big{|}_{\theta=0}$
$\displaystyle=-\int_{0}^{\tau}dt\sum_{n\neq
m}\eta_{n}R_{mn}K_{mn}^{\theta=0}+\int_{0}^{\tau}dt\sum_{n\neq
m}K_{nm}^{\theta=0}\dot{N}_{nm}\;.$ (47)

Therefore, the equality condition can be rewritten as

$\displaystyle 0$ $\displaystyle=\int_{0}^{\tau}dt\sum_{n\neq
m}\dot{N}_{nm}\left\\{\Lambda_{nm}-k(0)K_{nm}^{\theta=0}\right\\}$
$\displaystyle+\int_{0}^{\tau}dt\sum_{n}\eta_{n}\left\\{g_{n}+k(0)\sum_{m(\neq
n)}R_{mn}K_{mn}^{\theta=0}\right\\}$
$\displaystyle-\int_{0}^{\tau}dt\sum_{n}p_{n}\left(\sum_{m(\neq
n)}\Lambda_{mn}R_{mn}+g_{n}\right)\;.$ (48)

Given that Eq. (III.2) holds for all trajectories, its every integrand must be
identically zero. This condition is met when the observables satisfy
$\Lambda_{nm}=\Lambda_{nm}^{\mathrm{eq}}$ and $g_{n}=g_{n}^{\mathrm{eq}}$,
where

$\displaystyle\Lambda_{nm}^{\rm eq}\equiv
k(0)K_{nm}^{\theta=0},~{}~{}~{}g_{n}^{\rm eq}\equiv-k(0)\sum_{m(\neq
n)}R_{mn}K_{mn}^{\theta=0}.$ (49)

This is not attainable when all observables are current-like. Additionally,
the condition ensures the saturation of the XTUR only when
$\Sigma_{\tau}=\Sigma_{\tau}^{\rm ps}$. For the XTUR with
$\Sigma_{\tau}=\Sigma_{\tau}^{\rm tot}$ to be saturated, the equality
$\Sigma_{\tau}^{\rm ps}=\Sigma_{\tau}^{\rm tot}$ must hold, which is in
general not fulfilled by $\Lambda_{nm}^{\rm eq}$ and $g_{n}^{\rm eq}$.

##  IV EB from XTUR

In this section, we show that the EB is derived from the XTUR by choosing a
specific type of observables for both Langevin and Markov jump systems.

###  IV.1 Langevin systems

Converting the Stratonovich product to the Itô product according to Eq. (107),
we can decompose the current-like observable $\mathcal{J}_{\tau}^{\Lambda}$
into two components:

$\displaystyle\mathcal{J}_{\tau}^{\Lambda}(\Gamma)$
$\displaystyle=\int_{0}^{\tau}dt\,\boldsymbol{\Lambda}^{\textsf{T}}\circ\dot{\mathbf{x}}$
$\displaystyle=\int_{0}^{\tau}dt\,\boldsymbol{\Lambda}^{\textsf{T}}\bullet\dot{\mathbf{x}}+\int_{0}^{\tau}dt\,\text{tr}\,[\mathbb{D}\nabla\boldsymbol{\Lambda}^{\textsf{T}}]\;.$
(50)

If we set $g$ in the state-dependent observable as

$\displaystyle g^{\rm EB}(\mathbf{x},\omega
t;s)=-\boldsymbol{\Lambda}^{\textsf{T}}(s\mathbf{f}^{\text{rev}}+\mathbf{f}^{\text{irr}})-\text{tr}\,[\mathbb{D}\nabla\boldsymbol{\Lambda}^{\textsf{T}}]\;,$
(51)

then the composite observable becomes

$\displaystyle\Theta_{\tau}^{\Lambda,g^{\rm
EB}}(\Gamma)=\int_{0}^{\tau}dt\,\boldsymbol{\Lambda}^{\textsf{T}}\mathbb{B}(\mathbf{x},\omega
t)\bullet\boldsymbol{\xi}(t)\;.$ (52)

Using the Gaussian statistics of $\boldsymbol{\xi}(t)$, it is straightforward
to show that $\langle\Theta_{\tau}^{\Lambda,g^{\rm EB}}\rangle=0$ and

$\displaystyle\text{Var}(\Theta_{\tau}^{\Lambda,g^{\rm
EB}})=\left\langle\left(\Theta_{\tau}^{\Lambda,g^{\rm
EB}}\right)^{2}\right\rangle=2\int_{0}^{\tau}dt\,\langle\boldsymbol{\Lambda}^{\textsf{T}}\mathbb{D}\boldsymbol{\Lambda}\rangle\;.$
(53)

In addition, using $\langle\Theta_{\tau}^{\Lambda,g^{\rm EB}}\rangle=0$ in Eq.
(7), we have

$\langle\mathcal{K}_{\tau}^{g^{\rm
EB}}\rangle=-\langle\mathcal{J}_{\tau}^{\Lambda}\rangle.$ (54)

Meanwhile, using the relations
$s\mathbf{f}^{\text{rev}}+\mathbf{f}^{\text{irr}}=\mathbf{J}/p+(\nabla^{\textsf{T}}\mathbb{D}p)^{\textsf{T}}/p$
and

$\nabla^{\textsf{T}}[\mathbb{D}\boldsymbol{\mathbf{V}}p]=\text{tr}\,[\mathbb{D}\nabla\mathbf{V}^{\textsf{T}}]\,p+\mathbf{V}^{\textsf{T}}(\nabla^{\textsf{T}}\mathbb{D}p)^{\textsf{T}}$
(55)

that holds for any $N$-dimensional vector $\mathbf{V}$, from Eq. (51) we
obtain

$s\partial_{s}g^{\rm
EB}=-\frac{(s\partial_{s}\boldsymbol{\Lambda}^{\textsf{T}})\mathbf{J}}{p}-\frac{1}{p}\nabla^{\textsf{T}}[\mathbb{D}(s\partial_{s}\boldsymbol{\Lambda})p]-\frac{\boldsymbol{\Lambda}^{\textsf{T}}\mathbf{J}^{\rm
rev}}{p}.$ (56)

Then, utilizing Eq. (56), the third term in Eq. (15) is evaluated as

$\displaystyle\left\langle\Theta_{\tau}^{s\partial_{s}\Lambda,s\partial_{s}g^{\rm
EB}}\right\rangle$ $\displaystyle=\int_{0}^{\tau}dt\,\int
d\mathbf{x}\,(s\partial_{s}\boldsymbol{\Lambda}^{\textsf{T}}\mathbf{J}+s\partial_{s}g^{\rm
EB}p)$ $\displaystyle=-\int_{0}^{\tau}dt\,\int
d\mathbf{x}\,\boldsymbol{\Lambda}^{\textsf{T}}\mathbf{J}^{\text{rev}}\;.$ (57)

Combining Eqs. (54) and (IV.1), $\Omega_{\tau}$ in Eq. (15) becomes

$\displaystyle\Omega_{\tau}$ $\displaystyle=\int_{0}^{\tau}dt\int
d\mathbf{x}\,\boldsymbol{\Lambda}^{\textsf{T}}\mathbf{J}-\int_{0}^{\tau}dt\int
d\mathbf{x}\,\boldsymbol{\Lambda}^{\textsf{T}}\mathbf{J}^{\text{rev}}$
$\displaystyle=\int_{0}^{\tau}dt\int
d\mathbf{x}\,\boldsymbol{\Lambda}^{\textsf{T}}\mathbf{J}^{\text{irr}}=\langle\mathcal{J}_{\tau}^{\Lambda}\rangle^{\text{irr}}\;.$
(58)

Finally, putting Eqs. (53) and (IV.1) into the XTUR yields the EB (18). Note
that Eq. (51) for $g^{\rm EB}$ can be transformed into the form shown in Eq.
(16) using Eq. (55).

###  IV.2 Markov jump systems

Noting that the EB for Markov jump systems has not previously been
established, we first prove it without relying on any other thermodynamic
inequalities. Starting with the definition of the current-like observable for
Markov jump systems, shown in Eq. (II.2), we derive the EB as follows:

$\displaystyle\langle\mathcal{J}_{\tau}^{\Lambda}\rangle^{2}$
$\displaystyle=\left(\int_{0}^{\tau}dt\sum_{n<m}\Lambda_{nm}j_{nm}\right)^{2}$
$\displaystyle\leq\frac{1}{2}\int_{0}^{\tau}dt\sum_{n<m}\Lambda_{nm}^{2}a_{nm}\times
2\int_{0}^{\tau}dt\sum_{n<m}\frac{j_{nm}^{2}}{a_{nm}}$
$\displaystyle=\chi_{\tau}\Sigma^{\text{ps}}_{\tau}\leq\chi_{\tau}\Sigma^{\text{tot}}_{\tau}\;,$
(59)

where
$\chi_{\tau}\equiv\frac{1}{2}\int_{0}^{\tau}dt\sum_{n<m}\Lambda_{nm}^{2}a_{nm}$.
For the inequalities in the second and the third lines, the Cauchy–Schwarz
inequality and Eq. (12) are used, respectively.

Now, we show that Eq. (IV.2) can be derived from the XTUR for Markov jump
systems. Towards this aim, we set the function $g$ as

$\displaystyle g_{m}^{\rm EB}=-\sum_{n(\neq m)}\Lambda_{nm}R_{nm}\;.$ (60)

Then, the composite observable $\Theta_{\tau}^{\Lambda,g^{\rm
EB}}=\mathcal{J}_{\tau}^{\Lambda}+\mathcal{K}_{\tau}^{g^{\rm EB}}$ takes the
form

$\displaystyle\Theta_{\tau}^{\Lambda,g^{\rm EB}}=\int_{0}^{\tau}dt\sum_{n\neq
m}\Lambda_{nm}(\omega t)\,[\dot{N}_{nm}(t)-\eta_{m}(t)R_{nm}(\omega t)]\;.$
(61)

Using the relations $\langle\dot{N}_{nm}\rangle=R_{nm}p_{m}$ and
$\langle\eta_{n}\rangle=p_{n}$, we obtain
$\langle\Theta_{\tau}^{\Lambda,g^{\rm EB}}\rangle=0$, which implies
$\langle\mathcal{J}_{\tau}^{\Lambda}\rangle=-\langle\mathcal{K}_{\tau}^{g^{\rm
EB}}\rangle$. Moreover, we can derive

$\displaystyle\text{Var}(\Theta_{\tau}^{\Lambda,g^{\rm
EB}})=\int_{0}^{\tau}dt\sum_{n\neq
m}\Lambda_{nm}^{2}R_{nm}p_{m}=2\chi_{\tau}\,,$ (62)

as detailed in Appendix A.5. Combining these results with the XTUR (14) for
$\Sigma_{\tau}=\Sigma_{\tau}^{\mathrm{ps}}$, we arrive at the EB as given in
Eq. (IV.2).

##  V CSL from EB

In this section, we show how the CSL can be derived from the EB. To achieve
this, we first introduce the integral probability metric (IPM) [37, 38, 39],
which quantifies the statistical difference between two probability
distributions. Then the CSL is derived by applying the IPM within the EB
framework.

###  V.1 Integral Probability Metric (IPM)

The IPM is a metric quantifying the _distance_ between two probability
distributions. For any pair of distributions $p$ and $q$, the IPM is defined
as

$\displaystyle d_{\mathcal{H}}(p,q)=\sup_{h\in\mathcal{H}}\left|\langle
h\rangle_{p}-\langle h\rangle_{q}\right|\;,$ (63)

where $\mathcal{H}$ represents a set of functions with some specific
properties, and $\langle h\rangle_{p}$ and $\langle h\rangle_{q}$ are the
expectation values of the function $h$ with respect to $p$ and $q$,
respectively.

There are some notable examples of the IPM. First, when $\mathcal{H}$ is a set
of functions bounded within the range $[-1,1]$, the IPM is equal to twice the
total variation distance [39], i.e.,
$d_{\mathcal{H}}(p,q)=2d_{\text{TV}}(p,q)$, where

$\displaystyle d_{\text{TV}}(p,q)$
$\displaystyle\equiv\frac{1}{2}\sum_{x\in\Omega}|p(x)-q(x)|$
$\displaystyle=\frac{1}{2}\sup_{h\in[-1,1]}|\langle h\rangle_{p}-\langle
h\rangle_{q}|\;.$ (64)

Second, we consider the case where $\mathcal{H}$ is the set of Lipschitz-$1$
functions, defined as
$\text{Lip}_{1}\equiv\\{h|d(x,y)\geq|h(x)-h(y)|,\,\forall x,y\\}$, where
$d(x,y)$ represents the distance between $x$ and $y$. In this case, due to the
Kantorovich–Rubinstein duality [40, 41], the IPM coincides with the
Wasserstein-$1$ distance

$\displaystyle d_{\text{Lip}_{1}}(p,q)=\mathcal{W}_{1}(p,q)\;,$ (65)

which is widely used in the context of the optimal transport problem [40, 41,
42]. For the continuous random variables, the Wasserstein-$1$ distance is
defined as

$\displaystyle\mathcal{W}_{1}(p,q)\equiv\inf_{\pi(\mathbf{x},\mathbf{y})\in\Phi(p,q)}\int
d\mathbf{x}\,d\mathbf{y}\,d(\mathbf{x},\mathbf{y})\,\pi(\mathbf{x},\mathbf{y})\;,$
(66)

where $\Phi(p,q)$ stands for the set of all joint probability distributions
$\pi(\mathbf{x},\mathbf{y})$ whose marginals are given by $\int
d\mathbf{y}\,\pi=p$ and $\int d\mathbf{x}\,\pi=q$. For Langevin systems,
$d(\mathbf{x},\mathbf{y})$ is typically taken to be the Euclidean distance
$|\mathbf{x}-\mathbf{y}|$. Meanwhile, for discrete random variables, the
definition changes to

$\displaystyle\mathcal{W}_{1}(p,q)\equiv\inf_{\pi_{nm}\in\Phi(p,q)}\sum_{n,m}d_{nm}\pi_{nm}\;,$
(67)

where $\Phi(p,q)$ is the set of all joint probability distributions $\pi_{nm}$
which satisfies $\sum_{m}\pi_{nm}=p_{n}$ and $\sum_{n}\pi_{nm}=q_{m}$. For
Markov jump systems, the distance $d_{nm}$ is typically set to be the length
of the shortest path between states $n$ and $m$ in the graph representation of
possible transitions. See Appendix A.6 for more details about the
Wasserstein-1 distance on graphs.

###  V.2 CSL for Langevin systems

Employing the IPM introduced above, the CSL (19) can be deduced from the EB
(18) by appropriately limiting the associated observable to a specific form.
Since the CSL has been studied for systems without odd-parity variables, here
we focus on the overdamped Langevin systems.

We start by defining $\mathcal{D}$ as the set of functions that are continuous
and almost everywhere differentiable in $\mathbb{R}^{N}$. Let $\mathcal{H}$ be
a subset of $\mathcal{D}$. Then, for any function $h\in\mathcal{H}$, $\nabla
h(\mathbf{x})$ exists everywhere except for a measure-zero subset
$V_{h}\subset\mathbb{R}^{N}$. With this in mind, we choose the current-like
observable satisfying

$\displaystyle\boldsymbol{\Lambda}=\begin{cases}\nabla h(\mathbf{x}),~{}\
&\text{if $\mathbf{x}\in\mathbb{R}^{N}\backslash V_{h}$,}\\\
\boldsymbol{\Lambda}_{0}(\mathbf{x}),~{}\ &\text{if $\mathbf{x}\in
V_{h}$,}\end{cases}$ (68)

where $\boldsymbol{\Lambda}_{0}$ is an arbitrary vector field. Then the mean
value of the current-like observable is given by

$\displaystyle\langle\mathcal{J}_{\tau}^{\Lambda}\rangle$
$\displaystyle=\int_{0}^{\tau}dt\int_{\mathbb{R}^{N}\backslash
V_{h}}d\mathbf{x}\,(\nabla^{\textsf{T}}h)\,\mathbf{J}+\int_{0}^{\tau}dt\int_{V_{h}}d\mathbf{x}\,\boldsymbol{\Lambda}_{0}^{\textsf{T}}\mathbf{J}$
$\displaystyle=\int_{0}^{\tau}dt\int_{\mathbb{R}^{N}}d\mathbf{x}\,h(\mathbf{x})\,\partial_{t}p(\mathbf{x},t)$
$\displaystyle=\langle h\rangle_{p(\tau)}-\langle h\rangle_{p(0)}\;,$ (69)

where $\langle h\rangle_{p(t)}\equiv\int
d\mathbf{x}\,h(\mathbf{x})p(\mathbf{x},t)$. Integration by parts and
$\partial_{t}p=-\nabla\cdot\mathbf{J}$ have been used to obtain the second
equality, along with the fact that $V_{h}$ is a measure-zero set and does not
contribute to the integral. Using this result in Eq. (18) and taking the
supremum over $h\in\mathcal{H}$, the EB transforms into

$\displaystyle
d^{2}_{\mathcal{H}}(p(0),p(\tau))\leq\Sigma^{\text{tot}}_{\tau}\chi^{\mathcal{H}}_{\tau}\;,$
(70)

where
$\chi^{\mathcal{H}}_{\tau}\equiv\sup_{h\in\mathcal{H}}\int_{0}^{\tau}dt\int_{\mathbb{R}^{N}\backslash
V_{h}}d\mathbf{x}\,p\nabla h^{T}\mathbb{D}\nabla h$ is a constant that depends
on the choice of $\mathcal{H}$. Since $\chi^{\mathcal{H}}_{\tau}$ is a time-
extensive quantity, we can define its time average
$\bar{\chi}^{\mathcal{H}}\equiv\chi^{\mathcal{H}}_{\tau}/\tau$. Then, Eq. (70)
can be rewritten in the standard CSL form as

$\displaystyle\tau\geq\frac{d_{\mathcal{H}}^{2}(p(0),p(\tau))}{\Sigma_{\tau}^{\rm
tot}\bar{\chi}^{\mathcal{H}}}\;.$ (71)

By choosing a proper $\mathcal{H}$, various CSLs can be derived from this
inequality. For example, let us take
$\mathcal{H}=\text{Lip}_{1}\subset\mathcal{D}$. Then, as stated in Eq. (65),
$d_{\text{Lip}_{1}}$ corresponds to the Wasserstein-$1$ distance. As for
$\chi_{\tau}^{\text{Lip}_{1}}$, since every $h\in\text{Lip}_{1}$ satisfies
$|\nabla h(\mathbf{x})|\leq 1$, we obtain

$\displaystyle\chi_{\tau}^{\text{Lip}_{1}}$
$\displaystyle=\sup_{h\in\text{Lip}_{1}}\int_{0}^{\tau}dt\int_{\mathbb{R}^{N}\backslash
V_{h}}d\mathbf{x}\,p\nabla h^{T}\mathbb{D}\nabla h$
$\displaystyle\leq\int_{0}^{\tau}dt\int
d\mathbf{x}\,\rho[\mathbb{D}(\mathbf{x},\omega
t)]\,p(\mathbf{x},t)=\int_{0}^{\tau}dt\,\langle\rho[\mathbb{D}]\rangle\;,$
(72)

where $\rho(\mathbb{A})$ is the largest absolute eigenvalue of the matrix
$\mathbb{A}$. Then, the CSL can be rewritten as

$\displaystyle\tau\geq\frac{\mathcal{W}_{1}^{2}(p(0),p(\tau))}{\Sigma_{\tau}^{\rm
tot}\overline{\langle\rho(\mathbb{D})\rangle}}\;,$ (73)

where
$\overline{\langle\rho(\mathbb{D})\rangle}\equiv\frac{1}{\tau}\int_{0}^{\tau}dt\,\langle\rho(\mathbb{D})\rangle$.
If the diffusion matrix is uniform in $\mathbf{x}$ and $t$, we can replace
$\overline{\langle\rho(\mathbb{D})\rangle}$ with $\rho(\mathbb{D})$. Or, if
the diffusion matrix is an identity matrix multiplied by time-dependent
temperature $T(\mathbf{x},t)$, then
$\overline{\langle\rho(\mathbb{D})\rangle}=\frac{1}{\tau}\int_{0}^{\tau}dt\,\langle
T(\mathbf{x},t)\rangle$, where $\langle T(\mathbf{x},t)\rangle=\int
d\mathbf{x}\,T(\mathbf{x},t)\,p(\mathbf{x},t)$.

It is worth highlighting that the CSL presented in Eq. (73), a novel
contribution of this study, is generally applicable to systems featuring a
diffusion matrix which is neither diagonal nor uniform in $\mathbf{x}$ and
$t$. This is distinct from the previous CSLs for overdamped Langevin systems,
which focused on systems immersed a single heat bath at fixed temperature [43,
25, 44, 27]. Additionally, we emphasize our use of the Wasserstein-$1$
distance instead of the Wasserstein-$2$ distance employed in previous works
[43, 25, 44, 27]. The difference is rooted in the fact that the
Wasserstein-$1$ distance belongs to the category of the IPM, while the
Wasserstein-$2$ distance does not.

###  V.3 CSL for Markov jump systems

The CSL for Markov jump systems can also be derived from the EB when the
associated current-like observable meets the gradient condition
$\Lambda_{nm}=h_{n}-h_{m}\equiv\nabla h_{nm}$. Then, its ensemble average
satisfies

$\displaystyle\langle\mathcal{J}_{\tau}^{\nabla h}\rangle$
$\displaystyle=\int_{0}^{\tau}dt\sum_{n>m}(h_{n}-h_{m})j_{nm}=\int_{0}^{\tau}dt\sum_{n\neq
m}h_{n}j_{nm}$
$\displaystyle=\int_{0}^{\tau}dt\sum_{n}h_{n}\dot{p}_{n}(t)=\langle
h\rangle_{p(\tau)}-\langle h\rangle_{p(0)}\;,$ (74)

where $\dot{p}_{n}(t)=\sum_{m(\neq n)}j_{nm}$ has been used for the third
equality, and $\langle h\rangle_{p(t)}\equiv\sum_{n}h_{n}p_{n}\\!(t)$. Using
this relation, the EB (18) for $\Sigma_{\tau}=\Sigma_{\tau}^{\mathrm{ps}}$
reduces to

$\displaystyle
d^{2}_{\mathcal{H}}(p(0),p(\tau))\leq\Sigma^{\text{ps}}_{\tau}\chi_{\tau}^{\mathcal{H}}\;,$
(75)

where
$\chi_{\tau}^{\mathcal{H}}\equiv\frac{1}{2}\sup_{h\in\mathcal{H}}\int_{0}^{\tau}dt\sum_{n<m}(h_{n}-h_{m})^{2}a_{nm}$.
Since $\Sigma^{\text{ps}}_{\tau}\leq\Sigma^{\text{tot}}_{\tau}$, once again we
obtain Eq. (71), proving that the CSL for Markov jump systems can also be
formulated in the same form.

For concreteness, let us examine the case of
$\mathcal{H}=\mathcal{H}_{1}\equiv\\{h_{n}|h_{n}\in[-1,1]\\}$. Then, as
discussed above Eq. (V.1), $d_{\mathcal{H}}$ is related to the total variation
distance. Besides, since $|h_{n}-h_{m}|\leq 2$ for all $n$ and $m$, we obtain

$\displaystyle\chi_{\tau}^{\mathcal{H}_{1}}\leq
2\int_{0}^{\tau}dt\sum_{n<m}a_{nm}=2\mathcal{A}_{\tau}\;,$ (76)

where $\mathcal{A}_{\tau}$ is the dynamical activity. Denoting its time
average as $\bar{\mathcal{A}}\equiv\mathcal{A}_{\tau}/\tau$, Eq. (75) implies

$\displaystyle\tau\geq\frac{d_{\text{TV}}^{2}(p(0),p(\tau))}{\Sigma_{\tau}^{\rm
tot}\bar{\mathcal{A}}/2}\;,$ (77)

which reproduces the original CSL derived in Ref. [22].

Another CSL, recently discovered [36] and proven to be tighter than Eq. (77),
can also be derived from Eq. (75). Let $f(x)$ be the inverse function of
$f^{-1}(x)=x\tanh x$. Then, Eqs. (75) and (76) lead to the following
inequalities:

$\displaystyle\frac{d_{\mathcal{H}_{1}}^{2}}{2\mathcal{A}_{\tau}}\leq\frac{d_{\mathcal{H}_{1}}^{2}}{\chi_{\tau}^{\mathcal{H}_{1}}}\leq\Sigma^{\text{ps}}_{\tau}\leq\frac{(\Sigma_{\tau}^{\text{tot}})^{2}}{2\mathcal{A}_{\tau}}f\left(\frac{\Sigma_{\tau}^{\text{tot}}}{2\mathcal{A}_{\tau}}\right)^{-2}\;.$
(78)

The final inequality can be derived using Eq. (10) of Ref. [29]. These yield
$f(\Sigma_{\tau}^{\text{tot}}/2\mathcal{A_{\tau}})\leq\Sigma_{\tau}^{\text{tot}}/d_{\mathcal{H}_{1}}$,
which in turn implies

$\displaystyle\frac{\Sigma_{\tau}^{\text{tot}}}{2\mathcal{A}_{\tau}}\leq
f^{-1}\left(\frac{\Sigma_{\tau}^{\text{tot}}}{d_{\mathcal{H}_{1}}}\right)=\frac{\Sigma_{\tau}^{\text{tot}}}{d_{\mathcal{H}_{1}}}\tanh\left(\frac{\Sigma_{\tau}^{\text{tot}}}{d_{\mathcal{H}_{1}}}\right)\;.$
(79)

Finally, using $d_{\mathcal{H}_{1}}=2d_{\text{TV}}$, we arrive at

$\displaystyle\tau\geq\frac{d_{\text{TV}}}{\bar{\mathcal{A}}\tanh\left(\frac{\Sigma_{\tau}^{\text{tot}}}{2d_{\text{TV}}}\right)}\;,$
(80)

reproducing the tighter CSL proven in Ref. [36].

Yet another application of Eq. (75) is the CSL for Markov jump systems in
terms of the Wasserstein distance. To show this, we select
$\mathcal{H}=\textrm{Lip}_{1}$, which equates $d_{\mathcal{H}}$ to the
Wasserstein-1 distance, as stated in Eq. (65). For Markov jump systems,
$\textrm{Lip}_{1}$ can be expressed as

$\displaystyle\text{Lip}_{1}=\left\\{h_{n}|1\geq|h_{n}-h_{m}|\,\,\text{for all
}(n,m)\in E\right\\}\;,$ (81)

where $E$ denotes the set of all state pairs joined by allowed transitions. A
detailed proof is provided in Appendix A.6. Using the above property of
$\text{Lip}_{1}$, it is straightforward to obtain

$\displaystyle\chi^{\text{Lip}_{1}}_{\tau}\leq\frac{1}{2}\int_{0}^{\tau}dt\sum_{n<m}a_{nm}=\frac{1}{2}\mathcal{A}_{\tau}\;.$
(82)

Combining these results with Eq. (75), we derive

$\displaystyle\tau\geq\frac{\mathcal{W}_{1}^{2}(p(0),p(\tau))}{\Sigma_{\tau}^{\rm
tot}\bar{\mathcal{A}}/2}\;.$ (83)

As was done for the case of the original CSL discussed above, a tighter
version of the above CSL, namely

$\displaystyle\tau\geq\frac{\mathcal{W}_{1}}{\bar{\mathcal{A}}\tanh\left(\frac{\Sigma_{\tau}^{\rm
tot}}{2\mathcal{W}_{1}}\right)}\;,$ (84)

can be derived using inequalities similar to Eq. (78). This result is
equivalent to Eq. (57) of Ref. [26] and Eq. (129) of Ref. [27]. As discussed
in [27], Eq. (84) is generally saturable for any graphs, while Eq. (80) can be
saturated only for fully-connected graphs.

These derivations provide clear evidence that Eq. (75), essentially the EB for
a specific class of current-like observables, is a foundational CSL from which
follows a broad range of CSLs reported in the literature.

##  VI PE tradeoff from EB

Consider a heat engine that operates between hot and cold heat reservoirs
characterized by the temperatures $T_{\rm h}$ and $T_{\rm c}$ $(T_{\rm
h}>T_{\rm c})$, respectively. Let $Q_{\rm h}$ denote the heat absorbed from
the hot reservoir, $Q_{\rm c}$ the heat dissipated into the cold reservoir,
and $W$ the work extracted from the engine. Then, the PE tradeoff relation can
be expressed as [21]

$P\leq\Xi\eta(\eta_{\rm C}-\eta)\,,$ (85)

where $P$ is the power of the engine, $\Xi$ a system-dependent constant,
$\eta\equiv\langle W\rangle/\langle Q_{\rm h}\rangle$ the efficiency of the
engine, and $\eta_{\rm C}=1-T_{\rm c}/T_{\rm h}$ the Carnot efficiency. This
relation dictates that it is impossible to achieve $\eta_{\rm C}$ and finite
power simultaneously, even though an irreversible process can sometimes
achieve $\eta_{\rm C}$ [45, 46]. This tradeoff relation has been proven for
both steady-state and cyclic engines in Langevin systems using the EB [19]. It
has also been derived using the TUR [28], but solely for the steady-state
engines and not for the cyclic engines. In this section, we establish the PE
tradeoff for both steady-state and cyclic engines in Markov jump systems using
the EB.

To achieve this goal, we consider a heat engine consisting of discrete $N$
states in contact with two heat reservoirs, each characterized by a
temperature denoted as $T_{k}$ ($k\in\\{{\rm h},{\rm c}\\}$). The energy
associated with each state is represented as $E_{n}(t)$
($n\in\\{1,2,\ldots,N\\}$). The transition rate from state $n$ to state $m$
under the influence of heat reservoir $k$ is expressed as $R_{mn}^{k}(\omega
t)$, which satisfies the local detailed balance condition:
$R_{mn}^{k}e^{-E_{n}/T_{k}}=R_{nm}^{k}e^{-E_{m}/T_{k}}$. The master equation
governing this system is given by

$\displaystyle\dot{p}_{n}=\sum_{m(\neq n),k}j_{nm}^{k}\;,$ (86)

where $j_{nm}^{k}=R_{nm}^{k}p_{m}-R_{mn}^{k}p_{n}$. We are interested in the
current-like observables expressed as

$\mathcal{J}_{\tau}^{\Delta E,k}=\int_{0}^{\tau}dt\sum_{n\neq m}\Delta
E_{nm}\dot{N}_{nm}^{k}\,,$ (87)

where $\Delta E_{nm}=E_{n}-E_{m}$, and $N_{nm}^{k}$ is the number of jumps
from state $m$ to $n$ under the influence of reservoir $k$ until time $t$.
Then, we can readily identify $\mathcal{J}_{\tau}^{\Delta E,{\rm h}}=Q_{\rm
h}$ and $\mathcal{J}_{\tau}^{\Delta E,{\rm c}}=-Q_{\rm c}$. Choosing $Q_{\rm
h}$ as the observable appearing in the EB, we obtain

$\langle Q_{\rm h}\rangle^{2}\leq\tau\bar{\chi}^{\rm h}\Sigma_{\tau}^{\rm
tot},$ (88)

where $\bar{\chi}^{\rm h}=\frac{1}{2\tau}\int_{0}^{\tau}dt\sum_{n\neq m}\Delta
E_{nm}^{2}a_{nm}^{\rm h}$ with $a_{nm}^{\rm h}=R_{nm}^{\rm h}p_{n}+R_{mn}^{\rm
h}p_{m}$. When the engine is in the periodic state, we choose $\tau$ to be the
period of a cycle. Then, since the energy of the system does not change after
each cycle, we have $\Sigma_{\tau}^{\rm tot}=\langle Q_{\rm c}\rangle/T_{\rm
c}-\langle Q_{\rm h}\rangle/T_{\rm h}=(\eta_{\rm C}-\eta)\langle Q_{\rm
h}\rangle/T_{\rm c}$. Using this expression in Eq. (88) and multiplying both
sides of the inequality by $\eta=\langle W\rangle/\langle Q_{\rm h}\rangle$,
we finally derive

$P\leq\frac{\bar{\chi}^{\rm h}}{T_{\rm c}}\eta(\eta_{\rm C}-\eta),$ (89)

where $P=\langle W\rangle/\tau$. It is straightforward to adapt this proof to
the same inequality for steady-state engines. We stress that Eq. (89) is an
equivalent of the PE tradeoff relation shown in Eq. (85), but extended to both
cyclic and steady-state engines built using Markov jump systems.

For completeness, we note that the PE tradeoff for Langevin systems can also
be derived from the EB by choosing the heat from the hot reservoir as the
observable of interest, as previously shown in Ref. [19]. Thus, the EB
provides a reliable starting point for deducing the PE tradeoff for a broad
range of engines. In contrast, to our knowledge, it seems impossible to derive
the PE tradeoff for cyclic engines starting from the TUR.

##  VII Examples

In this section, we present concrete illustrations of two tradeoff relations
derived in this study: the EB for Markov jumps systems and the CSL for the
Langevin systems subject to an inhomogeneous temperature field.

###  VII.1 EB for two-level batteries

As an example of the EB for Markov jump systems, we consider a charging-
discharging process utilizing a two-level system, as depicted in Fig. 2(a).
The energy of each state is denoted by $E_{i}(t)$ for $i=1,2$. During the
charging process with duration $\tau_{\rm c}$, $E_{1}(t)>E_{2}(t)$ is
maintained so that the particle tends to escape state $1$ to charge state $2$.
Conversely, in the discharging process with duration $\tau_{\rm d}$,
$E_{1}(t)<E_{2}(t)$ is maintained so that the particle is discharged from
state $2$, releasing the energy $E_{2}(t)-E_{1}(t)$ via each transition. For
simplicity, we assume that $E_{2}(t)$ is kept constant at $\Delta$, while
$E_{1}(t)=2\Delta$ during the charging process and $E_{1}(t)=0$ during the
discharging process. Then, keeping the local detailed balance, the transition
rates are given by

$\displaystyle R_{21}(t)$
$\displaystyle=\frac{\mu(t)}{1+e^{-\beta[E_{1}(t)-\Delta]}}\,,$ $\displaystyle
R_{12}(t)$
$\displaystyle=\frac{\mu(t)e^{-\beta[E_{1}(t)-\Delta]}}{1+e^{-\beta[E_{1}(t)-\Delta]}}\;,$
(90)

where $\beta$ is the inverse temperature, and $\mu(t)$ controls the overall
rate of transitions. Suppose we are concerned with quantifying the speed of
the charging-discharging process. This can be quantified by the current-like
observable $\mathcal{J}_{\tau}^{\Lambda}$ satisfying
$\Lambda_{nm}(t)=\text{sgn}[E_{m}(t)-E_{n}(t)]$, which counts the excess
number of charging and discharging transitions compared to the number of
opposite transitions hindering the process.

Figure 2:  (a) Schematic illustration of the charging-discharging process.
During the charging phase, the energy level of state 1 is increased, charging
the battery by prompting the particle to jump to state 2. In contrast, during
the discharging phase, the energy level of state 1 is decreased below that of
state 2, resulting in battery discharge. In each phase, $\Delta$ represents
the energy gap between the two states. (b) Verification of the EB (91) for the
charging-discharging process. The black solid line depicts the mean charging-
discharging speed. The dashed curves represent the EBs set by the mean EP and
the mean pseudo-EP.

Since $\Lambda_{nm}^{2}=1$ within this setup, $\chi_{\tau}$ in the EB (IV.2)
reduces to the dynamical activity $\mathcal{A}_{\tau}$. Then, the EB is
rewritten as
$\langle\mathcal{J}_{\tau}\rangle\leq\sqrt{\mathcal{A}_{\tau}\Sigma_{\tau}/2}$.
In the periodic state, we can define the mean charging-discharging speed as
$\overline{\langle J\rangle}\equiv\langle\mathcal{J}_{\tau_{\rm c}+\tau_{\rm
d}}^{\Lambda}\rangle/(\tau_{\rm c}+\tau_{\rm d})$, the mean EP as
$\overline{\langle\Sigma^{\text{tot}}\rangle}\equiv\langle\Sigma_{\tau_{\rm
c}+\tau_{\rm d}}^{\text{tot}}\rangle/(\tau_{\rm c}+\tau_{\rm d})$, the mean
pseudo-EP as
$\overline{\langle\Sigma^{\text{ps}}\rangle}\equiv\langle\Sigma_{\tau_{\rm
c}+\tau_{\rm d}}^{\text{ps}}\rangle/(\tau_{\rm c}+\tau_{\rm d})$, and the mean
dynamical activity as $\bar{\mathcal{A}}\equiv\mathcal{A}_{\tau_{\rm
c}+\tau_{\rm d}}/(\tau_{\rm c}+\tau_{\rm d})$. Using these definitions, we
obtain the upper bounds on the charging-discharging speed

$\displaystyle\overline{\langle
J\rangle}\leq\sqrt{\overline{\langle\Sigma^{\text{ps}}\rangle}\bar{\mathcal{A}}/2}\leq\sqrt{\overline{\langle\Sigma^{\text{tot}}\rangle}\bar{\mathcal{A}}/2}\;.$
(91)

The behaviors of $\overline{\langle J\rangle}$,
$\sqrt{\overline{\langle\Sigma^{\text{ps}}\rangle}\bar{\mathcal{A}}/2}$, and
$\sqrt{\overline{\langle\Sigma^{\text{tot}}\rangle}\bar{\mathcal{A}}/2}$ as
functions of the energy gap $\Delta$ are shown in Fig. 2(b) for $\tau_{\rm
c}=\tau_{\rm d}=10$, $\mu(t)=1$, and $\beta=1$. Besides their consistency with
the inequalities of Eq. (91), two points merit attention. First, the
difference between the two upper bounds is negligible for small $\Delta$,
which corresponds to the near-equilibrium regime. Second, the charging-
discharging speed saturates to the tighter upper bound set by the pseudo-EP
for large $\Delta$. This behavior stems from the equality condition of the
tighter upper bound, which is given by

$\displaystyle\Lambda_{nm}^{\rm
eq}=\frac{p_{m}R_{nm}-p_{n}R_{mn}}{p_{m}R_{nm}+p_{n}R_{mn}}\;.$ (92)

In the limit of large $\Delta$, since the transitions become almost totally
irreversible, this quantity becomes either $1$ or $-1$ for every transition
until the occupation probability of the higher energy level reaches zero. For
transitions occurring afterwards, the quantity may have different values, but
such transitions negligibly contribute to the physical quantities. Hence, the
observable with $\Lambda_{nm}(t)=\Lambda_{nm}^{\rm eq}$ is practically equal
to our chosen observable with $\Lambda_{nm}(t)=\text{sgn}[E_{m}(t)-E_{n}(t)]$,
thus ensuring the saturation of the tighter bound. In contrast, the weaker
bound by the total EP cannot be saturated in this regime, as the discrepancy
between the EP and the pseudo-EP increases as the system moves farther away
from equilibrium.

Figure 3:  (a) Illustration of the tilted double-well potential and the
nonuniform temperature field. The dashed and the solid lines represent the
potential for $t<0$ and $t\geq 0$, respectively. The colormap below the
potential indicates the temperature field, which is colder (hotter) on the
left (right). (b) Verification of the CSL (73) for the process depicted in
(a). (c) Illustration of the probability distribution (dashed curves)
broadening within a harmonic potential (solid curve) as temperature increases
with time. (d) Verification of the CSL (73) for the process depicted in (c).

###  VII.2 CSL in Langevin systems subjected to inhomogeneous temperature
field

To illustrate the CSL in a nonuniform temperature field, we consider a one-
dimensional Brownian particle trapped in either a single-well or a double-well
potential [36]. Its equation of motion is given by

$\displaystyle\gamma\dot{x}=-\frac{\partial V(x,t)}{\partial x}+\sqrt{2\gamma
T(x,t)}\bullet\xi(t)\;,$ (93)

where $\xi(t)$ is a Gaussian white noise satisfying $\langle\xi(t)\rangle=0$
and $\langle\xi(t)\xi(t^{\prime})\rangle=\delta(t-t^{\prime})$. We note that
the spatial inhomogeneity in the noise amplitude stems only from the
temperature field $T(x,t)$ and not from the constant friction coefficient
$\gamma$, in which case the Itô description of the multiplicative noise is
indeed appropriate in the overdamped limit, as discussed in Ref. [47].

We verify the CSL for two different processes of the system. The first example
concerns the relaxation of the Brownian particle when the potential $V(x)$ and
the temperature field $T(x)$ are abruptly changed. More specifically, we
consider the potential

$\displaystyle V(x)$
$\displaystyle=V_{\text{DW}}(x)+\Theta_{\text{tilt}}\frac{x}{x_{m}}\,,$ (94)

where
$V_{\text{DW}}(x)=E_{b}[\left(x/x_{m}\right)^{4}-2\left(x/x_{m}\right)^{2}]$
is a double-well potential, and $\Theta_{\text{tilt}}$ controls the magnitude
of the linear tilting potential. This potential has widely been used to
investigate the finite-time Landauer principle [48, 36] governing the entropic
cost of erasing one bit of information. Meanwhile, the temperature field is
given by

$\displaystyle T(x)$
$\displaystyle=T_{0}\left(1+\delta\tanh\frac{x}{x_{T}}\right)\,,$ (95)

where $\delta$ and $x_{T}$ control the magnitude and the length scale of
temperature change across the system, respectively. For $t<0$, both
$\Theta_{\textrm{tilt}}$ and $\delta$ are set to zero, letting the particle
equilibrate in the bare double-well potential $V_{\text{DW}}(x)$ to the
uniform temperature field $T(x)=T_{0}=1$. Then, at $t=0$, we set $\Theta_{\rm
tilt}=2$ and $\delta=0.5$, switching on the tilting potential and the
nonuniform temperature field, as illustrated in Fig. 3(a). This protocol
amounts to erasing one bit of information encoded by the Brownian particle by
tuning both the potential and the temperature field. Given the other
parameters fixed at $\gamma=1$, $x_{m}=1$, $x_{T}=0.2$, and $E_{b}=1$, Fig.
3(b) verifies the CSL stated in Eq. (73), where $\overline{\langle
T\rangle}\equiv\frac{1}{\tau}\int_{0}^{\tau}dt\langle T(x,t)\rangle$ plays the
role of $\overline{\langle\rho(\mathbb{D})\rangle}$. Thus, Eq. (73) provides a
way to study the finite-time Landauer principle involving the local heating as
well as the potential manipulation.

The next example pertains to the Brownian particle confined in a harmonic
potential $V(x)=\frac{1}{2}kx^{2}$ with the temperature of the heat bath
changing in time as

$\displaystyle T(t)=\begin{cases}T_{0}&\text{for $t<0$}\;,\\\
T_{0}(1+t/\tau_{\rm T})&\text{for $t\geq 0$}\;.\end{cases}$ (96)

This means that the particle distribution becomes broader as time goes by, as
illustrated in Fig. 3(c). Choosing $k=1$ and $\tau_{\rm T}=1/4$, in Fig. 3(d)
we demonstrate the validity of the CSL stated in Eq. (73).

##  VIII Conclusion

Through this study, we have unveiled the hierarchical connections among
various thermodynamic tradeoff relations, encompassing the TUR, the EB, the
CSL, and the PE tradeoff. This hierarchy commences with the XTUR, which deals
with the most general types of observables. All the other tradeoff relations
stem from the XTUR by imposing specific constraints upon the associated
observable. When state-dependent observables are excluded, the XTUR simplifies
to the TUR. On the other hand, choosing state-dependent observables as
described in Eqs. (51) and (60) transforms the XTUR into the EB. Further down
the hierarchy, focusing on the systems without odd-parity variables, the CSL
and the PE tradeoff follow from the EB by employing the heat current and the
IPM as the associated current-like observable, respectively.

This hierarchy offers a fundamental and comprehensive perspective on the
landscape of significant thermodynamic inequalities that have emerged in the
field of nonequilibrium thermodynamics over the past decade. In particular,
the structure indicates that the TUR, the EB, the CSL, and the PE tradeoff can
all be categorized under a single class of thermodynamic relations, which we
might call the XTUR class. Hence, if future research uncovers another
thermodynamic inequality, one critical task will be to ascertain whether this
new relation falls within the XTUR class or deviates from it. The latter case
would suggest that the inequality is truly new and may open up the avenue
towards a hitherto unexplored realm of thermodynamics.

For the future research, it is highly desirable to extend our theory to open
quantum systems [49, 50, 51, 52, 53, 54, 16, 17, 23, 18, 27]. Over the past
decade, numerous thermodynamic tradeoff relations have also emerged in the
quantum domain, such as the quantum TURs [16, 17, 18, 31] and the quantum
speed limits [54, 23, 27, 31]. However, these relations have largely been
investigated as independent properties. Therefore, a critical forthcoming task
is to establish a unified hierarchical framework for these tradeoff relations
governing open quantum systems. This will facilitate a deeper understanding of
quantum thermodynamics and its applications.

Acknowledgments. — The authors thank Su-Chan Park for fruitful discussions.
This research has been supported by the POSCO Science Fellowship of POSCO TJ
Park Foundation (E.K. and Y.B.), KIAS Individual Grant No. PG064901 (J.S.L.),
and an appointment to the JRG Program at the APCTP through the Science and
Technology Promotion Fund and Lottery Fund of the Korean Government (J.-M.P.).
This was also supported by the Korean Local Governments - Gyeongsangbuk-do
Province and Pohang City (J.-M.P.).

##  Appendix A Detailed derivations

###  A.1 Fisher information in Langevin systems, Eq. (24)

When the system follows the modified Langevin dynamics governed by Eq. (22),
its path probability satisfies [55]

$\displaystyle\mathcal{P}_{\theta}[\Gamma]=\mathcal{N}p(0)\,e^{-\frac{1}{4}\int_{0}^{\tau}dt\,(\dot{\mathbf{x}}-\mathbf{A}_{\theta})^{\textsf{T}}\bullet\mathbb{D}^{-1}\bullet(\dot{\mathbf{x}}-\mathbf{A}_{\theta})},$
(97)

where $\mathcal{N}$ is the normalization constant, and $p(0)\equiv
p(\textbf{x},0;s,\omega)$ is the initial probability distribution. Then, we
have

$\displaystyle\partial^{2}_{\theta}\ln\mathcal{P}_{\theta}=\frac{1}{2}\int_{0}^{\tau}dt\,\partial^{2}_{\theta}\mathbf{A}_{\theta}^{\textsf{T}}\mathbb{D}^{-1}\bullet(\dot{\mathbf{x}}-\mathbf{A}_{\theta})$
$\displaystyle\,\,\,\,\,\,\,\,-\frac{1}{2}\int_{0}^{\tau}dt\,\partial_{\theta}\mathbf{A}_{\theta}^{\textsf{T}}\mathbb{D}^{-1}\partial_{\theta}\mathbf{A}_{\theta}+\frac{\partial_{\theta}^{2}p(0)}{p(0)}-\frac{\\{\partial_{\theta}p(0)\\}^{2}}{p(0)^{2}}.$
(98)

Using
$\partial_{\theta}\mathbf{A}_{\theta}\big{|}_{\theta=0}=\mathbf{J}^{\text{irr}}/p$
and $\langle\dot{\textbf{x}}-\textbf{A}_{\theta}\rangle_{\theta}=0$, the
Fisher information at $\theta=0$ is obtained as

$\displaystyle\mathcal{I}(0)$
$\displaystyle=\langle-\partial^{2}_{\theta}\ln\mathcal{P}_{\theta}\rangle_{\theta=0}$
$\displaystyle=\frac{1}{2}\int_{0}^{\tau}dt\int
d\mathbf{x}\,\frac{{\mathbf{J}^{\text{irr}}}^{\textsf{T}}\mathbb{D}^{-1}\mathbf{J}^{\text{irr}}}{p}$
$\displaystyle\qquad+\left\langle\left[(s\partial_{s}+\omega\partial_{\omega})\ln
p(0)\right]^{2}\right\rangle=\frac{1}{2}\left(\Sigma^{\text{tot}}_{\tau}+\mathcal{B}\right)\;,$
(99)

where $\mathcal{B}=2\langle\left[(s\partial_{s}+\omega\partial_{\omega})\ln
p(0)\right]^{2}\rangle$. If we choose an initial distribution independent of
$s$ and $\omega$, then $\mathcal{B}=0$.

###  A.2 Evaluation of
$\partial_{\theta}\langle\mathcal{J}_{\tau}^{\Lambda}\rangle|_{\theta=0}$ and
$\partial_{\theta}\langle\mathcal{K}_{\tau}^{g}\rangle|_{\theta=0}$, Eq.
(III.1)

We begin by evaluating
$\partial_{\theta}\langle\mathcal{J}_{\tau}^{\Lambda}\rangle|_{\theta=0}$.
Through direct substitution, we can show that the total probability current
$\mathbf{J}_{\theta}\equiv\mathbf{J}_{\theta}^{\rm
rev}+(1+\theta)\mathbf{J}_{\theta}^{\rm irr}$ of the modified dynamics
governed by Eq. (22) is related to the original current by
$\mathbf{J}_{\theta}({\textbf{x}},t;s,\omega)=(1+\theta)\mathbf{J}({\textbf{x}},t_{\theta};s_{\theta},\omega_{\theta})$.
Using this relation, the mean of $\mathcal{J}_{\tau}^{\Lambda}$ in the
modified dynamics is obtained as

$\displaystyle\langle\mathcal{J}_{\tau}^{\Lambda}\rangle_{\theta}$
$\displaystyle=\int_{0}^{\tau}dt\int
d\mathbf{x}\,\boldsymbol{\Lambda}^{\textsf{T}}({\textbf{x}},\omega
t;s)\mathbf{J}_{\theta}({\textbf{x}},t;s,\omega)$
$\displaystyle=\int_{0}^{\tau}dt\int
d\mathbf{x}\,(1+\theta)\boldsymbol{\Lambda}^{\textsf{T}}({\textbf{x}},\omega
t;s)\mathbf{J}({\textbf{x}},t_{\theta};s_{\theta},\omega_{\theta})$
$\displaystyle=\int_{0}^{\tau_{\theta}}dt\int
d\mathbf{x}\,\boldsymbol{\Lambda}^{\textsf{T}}\left({\textbf{x}},\omega_{\theta}t;s\right)\mathbf{J}({\textbf{x}},t;s_{\theta},\omega_{\theta})\;.$
(100)

The Taylor expansion
$\boldsymbol{\Lambda}(\mathbf{x},\omega_{\theta}t;s)=\boldsymbol{\Lambda}(\mathbf{x},\omega_{\theta}t;s_{\theta})+\theta
s_{\theta}\partial_{s}\boldsymbol{\Lambda}(\mathbf{x},\omega_{\theta}t;s)|_{s=s_{\theta}}+O(\theta^{2})$
yields

$\displaystyle\langle\mathcal{J}_{\tau}^{\Lambda}\rangle_{\theta}$
$\displaystyle=\langle\mathcal{J}_{\tau_{\theta}}^{\Lambda}\rangle|_{s=s_{\theta},\omega=\omega_{\theta}}$
$\displaystyle+\theta\int_{0}^{\tau_{\theta}}dt\,s_{\theta}\partial_{s}\boldsymbol{\Lambda}^{\textsf{T}}|_{s=s_{\theta}}\mathbf{J}({\textbf{x}},t;s_{\theta},\omega_{\theta})+O(\theta^{2}).$
(101)

Differentiating both sides of this equation with respect to $\theta$ and
setting $\theta=0$, we arrive at

$\displaystyle\partial_{\theta}\langle\mathcal{J}_{\tau}^{\Lambda}\rangle_{\theta}\big{|}_{\theta=0}$
$\displaystyle=\hat{O}_{\tau}\langle\mathcal{J}_{\tau}^{\Lambda}\rangle+\int_{0}^{\tau}dt~{}\langle(s\partial_{s}\boldsymbol{\Lambda}^{\textsf{T}})\circ\dot{\textbf{x}}\rangle\,.$
(102)

Now, we turn to the evaluation of
$\partial_{\theta}\langle\mathcal{K}_{\tau}^{g}\rangle|_{\theta=0}$. The mean
of $\mathcal{K}_{\tau}^{g}$ in the modified dynamics satisfies

$\displaystyle\langle\mathcal{K}_{\tau}^{g}\rangle_{\theta}$
$\displaystyle=\int_{0}^{\tau}dt\int d\mathbf{x}\,g(\mathbf{x},\omega
t;s)p_{\theta}(\mathbf{x},t;s,\omega)$
$\displaystyle=\frac{1}{1+\theta}\int_{0}^{\tau_{\theta}}dt\int
d\mathbf{x}\,g(\omega_{\theta}t;s)p(t;s_{\theta},\omega_{\theta})\,.$ (103)

Proceeding with the Taylor expansion
$g(\mathbf{x},\omega_{\theta}t;s)=g(\mathbf{x},\omega_{\theta}t;s_{\theta})+\theta
s_{\theta}\partial_{s}g(\mathbf{x},\omega_{\theta}t;s)|_{s=s_{\theta}}+O(\theta^{2})$,
we obtain

$\displaystyle\langle\mathcal{K}_{\tau}^{g}\rangle_{\theta}$
$\displaystyle=\frac{1}{1+\theta}\langle\mathcal{K}_{\tau_{\theta}}\rangle|_{s=s_{\theta},\omega=\omega_{\theta}}$
$\displaystyle+\theta\int_{0}^{\tau}dt\,s_{\theta}\partial_{s}g|_{s=s_{\theta}}p(t;s_{\theta},\omega_{\theta})+O(\theta^{2})\,.$
(104)

Differentiating both sides with respect to $\theta$, we finally derive

$\displaystyle\partial_{\theta}\langle\mathcal{K}_{\tau}^{g}\rangle_{\theta}\big{|}_{\theta=0}$
$\displaystyle=\hat{O}_{\tau}\langle\mathcal{K}_{\tau}^{g}\rangle-\langle\mathcal{K}_{\tau}^{g}\rangle+\int_{0}^{\tau}dt~{}\langle
s\partial_{s}g\rangle\;.$ (105)

###  A.3 XTUR saturation for Langevin systems, Eq. (28)

From the definition of $\Theta_{\tau}^{\Lambda,g}$, the left-hand side of Eq.
(27) at $\theta=0$ can be written as

$\displaystyle\Theta_{\tau}^{\Lambda,g}(\Gamma)-\langle\Theta_{\tau}^{\Lambda,g}\rangle=$
$\displaystyle\int_{0}^{\tau}dt\left(\boldsymbol{\Lambda}^{\textsf{T}}\circ\dot{{\textbf{x}}}+g\right)$
$\displaystyle-\int_{0}^{\tau}dt\int
d{\textbf{x}}\left(\boldsymbol{\Lambda}^{\textsf{T}}{\textbf{J}}+gp\right)\,.$
(106)

To compute the right-hand side of Eq. (27), we must use the conversion
relationship between the Stratonovich and the Itô products

$\mathbf{h}^{\textsf{T}}(\mathbf{x},t)\circ
d\mathbf{x}=\mathbf{h}^{\textsf{T}}(\mathbf{x},t)\bullet
d\mathbf{x}+\text{tr}[\mathbb{D}\nabla\mathbf{h}^{\textsf{T}}(\mathbf{x},t)]\,dt\;,$
(107)

which holds for an arbitrary function $\mathbf{h}(\mathbf{x},t)$. Here, we
have used the relation $d\mathbf{x}=\mathbf{A}dt+\mathbb{B}\bullet
d\mathbf{W}$, where $\mathbf{W}$ stands for the Wiener process. To derive this
formula, one starts from the definition of the Stratonovich product

$\displaystyle h_{i}(\mathbf{x},t)\circ
dx_{i}=\frac{1}{2}\left[h_{i}(\mathbf{x}+d\mathbf{x},t+dt)+h_{i}(\mathbf{x},t)\right]dx_{i}\;.$
(108)

Keeping the terms up to the order of $dt$, the right-hand side can be expanded
as

$\displaystyle h_{i}(\mathbf{x},t)\circ dx_{i}=h_{i}(\mathbf{x},t)\bullet
dx_{i}+\frac{1}{2}\sum_{j}\partial_{x_{j}}h_{i}(\mathbf{x},t)dx_{i}dx_{j}\;.$
(109)

Using
$dx_{i}dx_{j}=\sum_{k,l}\mathbb{B}_{ik}\mathbb{B}_{jl}dW_{k}dW_{l}+O(dt^{3/2})$
and the heuristic Itô rule $dW_{k}dW_{l}=\delta_{kl}dt$, this simplifies to

$\displaystyle h_{i}(\mathbf{x},t)\circ dx_{i}=h_{i}(\mathbf{x},t)\bullet
dx_{i}+\sum_{j}\mathbb{D}_{ij}\partial_{x_{j}}h_{i}(\mathbf{x},t)dt\;.$ (110)

Summing this equation side by side over $i$, we derive the conversion formula
shown in Eq. (107).

Using Eqs. (97), (107), and (22), we can rewrite Eq. (27) at $\theta=0$ as

$\displaystyle\partial_{\theta}\log\mathcal{P}_{\theta}(\Gamma)\big{|}_{\theta=0}$
$\displaystyle=\frac{1}{2}\int_{0}^{\tau}dt~{}\frac{{\mathbf{J}^{\text{irr}}}^{\textsf{T}}}{p}\mathbb{D}^{-1}\bullet(\dot{\mathbf{x}}-\mathbf{A})$
$\displaystyle=\frac{1}{2}\int_{0}^{\tau}dt~{}\frac{{\mathbf{J}^{\text{irr}}}^{\textsf{T}}}{p}\mathbb{D}^{-1}\circ\dot{\mathbf{x}}-\frac{1}{2}\int_{0}^{\tau}dt~{}\text{tr}\left[\mathbb{D}\nabla\left(\frac{{\mathbf{J}^{\text{irr}}}^{\textsf{T}}}{p}\mathbb{D}^{-1}\right)\right]$
$\displaystyle~{}~{}~{}-\frac{1}{2}\int_{0}^{\tau}dt\left[\frac{{\mathbf{J}^{\text{irr}}}^{\textsf{T}}\mathbb{D}^{-1}\\{\mathbf{J}+(\nabla^{\textsf{T}}\mathbb{D}p)^{\textsf{T}})\\}}{p^{2}}\right]\;.$
(111)

Here we note that

$\displaystyle\textrm{tr}\left[\mathbb{D}\nabla\left(\frac{{\mathbf{J}^{\text{irr}}}^{\textsf{T}}}{p}\mathbb{D}^{-1}\right)\right]=\mathbb{D}_{ij}\partial_{x_{j}}\\!\left[\frac{J^{\text{irr}}_{k}(\mathbb{D}^{-1})_{ki}}{p}\right]$
$\displaystyle=\left[\partial_{x_{j}}(\mathbb{D}^{-1})_{ki}\right]\mathbb{D}_{ij}\frac{J^{\text{irr}}_{k}}{p}+\nabla^{\textsf{T}}\left(\frac{\mathbf{J}^{\text{irr}}}{p}\right)$
$\displaystyle=-\frac{{\mathbf{J}^{\text{irr}}}^{\textsf{T}}\mathbb{D}^{-1}}{p}(\nabla^{\textsf{T}}\mathbb{D})^{\textsf{T}}-\frac{\nabla
p^{\textsf{T}}\mathbf{J}^{\text{irr}}}{p^{2}}+\frac{\nabla^{\textsf{T}}\mathbf{J}^{\text{irr}}}{p}$
$\displaystyle=-\frac{{\mathbf{J}^{\text{irr}}}^{\textsf{T}}\mathbb{D}^{-1}(\nabla^{\textsf{T}}\mathbb{D}p)^{\textsf{T}}}{p^{2}}+\frac{\nabla^{\textsf{T}}\mathbf{J}^{\text{irr}}}{p}\;.$
(112)

For the third equality in Eq. (A.3),
$\mathbb{D}_{ij}\partial_{x_{j}}(\mathbb{D}^{-1})_{ki}=-(\mathbb{D}^{-1})_{ki}\partial_{x_{j}}\mathbb{D}_{ij}$
has been used since
$\partial_{x_{j}}[(\mathbb{D}^{-1})_{ki}\mathbb{D}_{ij}]=\partial_{x_{j}}\delta_{kj}=0$.
By substituting the outcome of Eq. (A.3) into Eq. (A.3), we arrive at

$\displaystyle\partial_{\theta}\log\mathcal{P}_{\theta}(\Gamma)\big{|}_{\theta=0}$
$\displaystyle=\frac{1}{2}\int_{0}^{\tau}dt\left(\frac{{\mathbf{J}^{\text{irr}}}^{\textsf{T}}}{p}\mathbb{D}^{-1}\circ\dot{\mathbf{x}}-\frac{{\mathbf{J}^{\text{irr}}}^{\textsf{T}}\mathbb{D}^{-1}\mathbf{J}}{p^{2}}-\frac{\nabla^{\textsf{T}}\mathbf{J}^{\text{irr}}}{p}\right)\;.$
(113)

Finally, plugging Eqs. (A.3) and (A.3) into Eq. (27), we obtain the equality
condition stated in Eq. (28).

###  A.4 Fisher information in Markov jump systems, Eq. (40)

When the system follows the modified Markov jump process satisfying Eq. (36),
then its path probability is given by [56, 57]

$\displaystyle\mathcal{P}[\Gamma]$
$\displaystyle=p_{n_{0}}(0)\exp\bigg{[}\int_{0}^{\tau}dt\sum_{n}\eta_{n}(t)R_{nn}^{\theta}(\omega
t)$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+\int_{0}^{\tau}dt\sum_{n\neq
m}\ln R^{\theta}_{nm}(\omega t)\dot{N}_{nm}\bigg{]}\;,$ (114)

where $p_{n_{0}}(0)$ represents the initial-state distribution, set to be
independent of the perturbing parameters. Note that the first and the second
terms in the exponent correspond to the sums of the staying and the transition
probabilities, respectively. Then, the Fisher information satisfies

$\displaystyle\mathcal{I}(\theta)=-\langle\partial^{2}_{\theta}\log\mathcal{P}[\Gamma]\rangle_{\theta}$
$\displaystyle=-\int_{0}^{\tau}dt\sum_{n}p^{\theta}_{n}\partial^{2}_{\theta}R^{\theta}_{nn}-\int_{0}^{\tau}dt\sum_{n\neq
m}p^{\theta}_{m}R^{\theta}_{nm}\partial^{2}\ln R^{\theta}_{nm}$
$\displaystyle=\int_{0}^{\tau}dt\sum_{n\neq
m}\left(p_{m}^{\theta}\partial^{2}_{\theta}R^{\theta}_{nm}-p^{\theta}_{m}R^{\theta}_{nm}\partial^{2}_{\theta}\ln
R^{\theta}_{nm}\right)$ $\displaystyle=\int_{0}^{\tau}dt\sum_{n\neq
m}p^{\theta}_{m}R^{\theta}_{nm}(\partial_{\theta}\ln R^{\theta}_{nm})^{2}\;.$
(115)

Hence, the Fisher information at $\theta=0$ is obtained as

$\displaystyle\mathcal{I}(0)$ $\displaystyle=\int_{0}^{\tau}dt\sum_{n\neq
m}p_{m}R_{nm}(K_{nm}^{\theta=0})^{2}$
$\displaystyle=\frac{1}{2}\int_{0}^{\tau}dt\sum_{n\neq
m}a_{nm}(K_{nm}^{\theta=0})^{2}=\frac{1}{2}\Sigma_{\tau}^{\text{ps}}\;.$ (116)

###  A.5 Variance of the current-like observable
in Markov jump processes, Eq. (62)

The number of jumps from state $m$ to state $n$ during an infinitesimal time
interval $dt$, namely $\dot{N}_{nm}dt=N_{nm}(t+dt)-N_{nm}(t)$, can be
factorized as $\eta_{m}(t)T_{nm}(t)dt$, where $T_{nm}(t)dt$ denotes the number
of jump from state $m$ to state $n$ provided that the system is in the state
$m$ at time $t$. Here, $T_{nm}(t)dt=1$ with probability $R_{nm}(t)dt$ and
$T_{nm}dt=0$ with probability $1-R_{nm}(t)dt$, which implies $\langle
T_{nm}(t)dt\rangle=\langle[T_{nm}(t)dt]^{2}\rangle=R_{nm}(t)dt$. Since the
jumps are independent of each other, we have

$\displaystyle\langle[T_{nm}(t)-R_{nm}(t)][T_{n^{\prime}m^{\prime}}(t^{\prime})-R_{n^{\prime}m^{\prime}}(t^{\prime})]\rangle$
$\displaystyle=R_{nm}(t)\delta_{nn^{\prime}}\delta_{mm^{\prime}}\delta(t-t^{\prime}).$
(117)

Therefore, using Eq. (61), the variance of the composite observable
$\Theta_{\tau}^{\Lambda,g^{\rm EB}}$ is calculated as

$\displaystyle\text{Var}(\Theta_{\tau}^{\Lambda,g^{\rm
EB}})=\langle(\Theta_{\tau}^{\Lambda,g^{\rm EB}})^{2}\rangle$
$\displaystyle=\int_{0}^{\tau}dt\int_{0}^{\tau}dt^{\prime}\sum_{n\neq
m}\sum_{n^{\prime}\neq m^{\prime}}\Lambda_{nm}(\omega
t)\Lambda_{n^{\prime}m^{\prime}}(\omega t^{\prime})$
$\displaystyle\times\langle\eta_{m}(t)\eta_{m^{\prime}}(t^{\prime})[T_{nm}(t)-R_{nm}(t)][T_{n^{\prime}m^{\prime}}(t^{\prime})-R_{n^{\prime}m^{\prime}}(t^{\prime})]\rangle.$
(118)

Note that $\eta_{m}(t)$ and $T_{nm}(t)$ are independent of each other,
allowing the factorization of the ensemble average to
$\langle\eta_{m}(t)\eta_{m^{\prime}}(t^{\prime})\rangle\langle[T_{nm}(t)-R_{nm}(t)][T_{n^{\prime}m^{\prime}}(t^{\prime})-R_{n^{\prime}m^{\prime}}(t^{\prime})]\rangle$.
Therefore, using Eq.(A.5), we obtain

$\displaystyle\text{Var}(\Theta_{\tau}^{\Lambda,g^{\rm EB}})$
$\displaystyle=\int_{0}^{\tau}dt\sum_{n\neq m}\Lambda_{nm}(\omega
t)^{2}\langle\eta_{m}(t)^{2}\rangle R_{nm}(\omega t)$
$\displaystyle=\int_{0}^{\tau}dt\sum_{n\neq m}\Lambda_{nm}(\omega
t)^{2}p_{m}(t)R_{nm}(\omega t).$ (119)

###  A.6 Wasserstein-1 distance on graphs
and derivation of Eq. (81)

We start with a brief overview of the theory of optimal transport on graphs
[27]. Let $\mathcal{G}(V,E)$ represent an undirected graph, where $V$ denotes
the set of all states, and $E$ the set of all state pairs joined by allowed
transitions. For any given path $P$, $\text{len}(P)$ denotes its length,
defined as the number of edges belonging to the path. Using this notation, the
shortest-path distance between states $n$ and $m$, denoted as $d_{nm}$, can be
expressed as $d_{nm}\equiv\min_{P}{\text{len}(P)}$. Notably, $d_{nm}$ serves
as a metric applicable to the graph $\mathcal{G}(V,E)$. In this context, the
Wasserstein-1 distance between two distributions, $p$ and $q$, on the graph
$\mathcal{G}(V,E)$ is defined as

$\displaystyle\mathcal{W}_{1}(p,q)\equiv\min_{\pi_{nm}\in\Phi(p,q)}\sum_{n,m}d_{nm}\pi_{nm}\;,$
(120)

where $\Phi(p,q)$ represents a set of joint probability distributions
$\pi_{nm}$ that adhere to the constraints $\sum_{m}\pi_{nm}=p_{n}$ and
$\sum_{n}\pi_{nm}=q_{m}$. From the Kantorovich-Rubinstein duality [40, 41], we
can recast the Wasserstein-1 distance in a dual formulation as

$\displaystyle\mathcal{W}_{1}(p,q)=\sup_{h\in\text{Lip}_{1}}\sum_{n}h_{n}(p_{n}-q_{n})\;,$
(121)

where $\text{Lip}_{1}$ denotes the set of Lipschitz functions on the graph
$\mathcal{G}(V,E)$ defined by

$\displaystyle\text{Lip}_{1}\equiv\left\\{h_{n}|d_{nm}\geq|h_{n}-h_{m}|\,\,\text{for
all }n,m\in V\right\\}\;.$ (122)

With this knowledge, we aim to prove Eq. (81), which states that the above is
equivalent to

$\displaystyle\text{Lip}_{1}=\left\\{h_{n}|1\geq|h_{n}-h_{m}|\,\,\text{for all
}(n,m)\in E\right\\}\;.$ (123)

To facilitate our discussion, we introduce the notations

$\displaystyle L\equiv\left\\{h_{n}|d_{nm}\geq|h_{n}-h_{m}|\,\,\text{for all
}n,m\in V\right\\}\;,$ $\displaystyle
L^{\prime}\equiv\left\\{h_{n}|1\geq|h_{n}-h_{m}|\,\,\text{for all }(n,m)\in
E\right\\}\;.$ (124)

If $h\in L^{\prime}$, then $|h_{x}-h_{y}|\leq 1$ for all $(x,y)\in E$.
Therefore,

$\displaystyle|h_{n}-h_{m}|$
$\displaystyle\leq|h_{n}-h_{x_{1}}|+|h_{x_{1}}-h_{x_{2}}|+\cdots+|h_{x_{k}}-h_{m}|$
$\displaystyle\leq\text{len}(P)\;,$ (125)

where $P=(n,x_{1},\cdots,x_{k},m)$ denotes a path connecting $n$ and $m$.
Minimizing the path length over all paths connecting $n$ and $m$ in Eq. (A.6)
leads to the inequality $|h_{n}-h_{m}|\leq d_{nm}$, which imples $h\in L$.
This proves $L^{\prime}\subseteq L$. Conversely, if $h\in L$, it is
straightforward to see that $|h_{x}-h_{y}|\leq 1$ for all $(x,y)\in E$. Thus
$h\in L^{\prime}$, proving $L\subseteq L^{\prime}$. Consequently,
$L=L^{\prime}$, proving Eq. (81).

##  Appendix B Effect of $\mathcal{B}$ on TUR tightness

Throughout our discussions, we selected an initial distribution independent of
both $s$ and $\omega$. For Langevin systems, this means fixing $\mathcal{B}=0$
in Eq. (24). Nevertheless, it is in principle possible to make the initial
distribution depend on $s$ or $\omega$, which allows nonzero $\mathcal{B}$.
Following this scheme, the XTUR (14) is modified to

$\displaystyle\frac{\Omega_{\tau}^{2}}{\text{Var}[\Theta_{\tau}^{\Lambda,g}]}\leq\frac{\Sigma_{\tau}^{\rm
tot}+\mathcal{B}}{2}\;.$ (126)

Setting $g=0$, thus $\Theta_{\tau}^{\Lambda,g}=\mathcal{J}^{\Lambda}_{\tau}$,
yields the modified version of the underdamped TUR (35)

$\displaystyle\frac{\left[(t\partial_{t}-s\partial_{s}-\omega\partial_{\omega})\langle\mathcal{J}_{\tau}^{\Lambda}\rangle\right]^{2}}{\text{Var}[\mathcal{J}_{\tau}^{\Lambda}]}\leq\frac{\Sigma^{\text{tot}}_{\tau}+\mathcal{B}}{2}\;.$
(127)

Then, it is natural to ask how the presence or absence of $\mathcal{B}$
affects the tightness of the bound set by the TUR. To put it more precisely,
we define

$\displaystyle\mathcal{Q}_{a}\equiv\frac{\text{Var}(\mathcal{J}^{\Lambda}_{\tau})}{\left[(t\partial_{t}-s\partial_{s}-\omega\partial_{\omega})\langle\mathcal{J}_{\tau}^{\Lambda}\rangle\right]^{2}}(\Sigma_{\tau}^{\rm
tot}+\mathcal{B})$ (128)

for the initial condition that depends on $s$ or $\omega$, and

$\displaystyle\mathcal{Q}_{b}\equiv\frac{\text{Var}(\mathcal{J}^{\Lambda}_{\tau})}{\left[(t\partial_{t}-s\partial_{s}-\omega\partial_{\omega})\langle\mathcal{J}_{\tau}^{\Lambda}\rangle\right]^{2}}\Sigma_{\tau}^{\rm
tot}$ (129)

for the initial condition that does not depend on $s$ and $\omega$. According
to the TUR, these two $\mathcal{Q}$ factors both satisfy
$\mathcal{Q}_{a,b}\geq 2$. Thus, our question boils down to which
$\mathcal{Q}$ factor comes closer to the lower bound $2$.

To explore this issue on a concrete basis, we set up an analytically tractable
model. More specifically, we consider an underdamped Brownian particle on a
one-dimensional ring subjected to a constant driving force, as described by
the Langevin equation

$\displaystyle\dot{x}$
$\displaystyle=s\frac{p}{m}\,,~{}~{}~{}~{}\dot{p}=sf-\frac{\gamma}{m}p+\xi\;,$
(130)

where $\langle\xi(t)\rangle=0$ and
$\langle\xi(t)\xi(t^{\prime})\rangle=2T\delta(t-t^{\prime})$. We focus on the
steady state, so that the initial distribution must be given in the form

$\displaystyle P^{\text{ss}}(p;s)=\frac{1}{\sqrt{2\pi
mT}}\exp\left[{-\frac{1}{2mT}\left(p-\frac{smf}{\gamma}\right)^{2}}\right]\;.$
(131)

Now, let us specify two initial distributions that lead to different forms of
the TUR. The first is the $s$-dependent initial distribution $P_{a}\equiv
P^{\text{ss}}(p;s)$, and the second is the $s$-independent counterpart
$P_{b}\equiv P^{\text{ss}}(p;s=1)$. Given these, let us denote by
$\langle\cdots\rangle_{a}$ and $\langle\cdots\rangle_{b}$ the ensemble
averages with respect to the initial distributions $P_{a}$ and $P_{b}$,
respectively. Then, taking $s=1$ in the end for fair comparison, the
$\mathcal{Q}$ factors are given by

$\displaystyle\mathcal{Q}_{a,b}\equiv\left.\frac{\text{Var}(\mathcal{J}_{\tau}^{\Lambda})}{\left[(\tau\partial_{\tau}-s\partial_{s})\langle\mathcal{J}_{\tau}^{\Lambda}\rangle_{a,b}\right]^{2}}\left(\Sigma_{\tau}^{\rm
tot}+\mathcal{B}_{a,b}\right)\right|_{s=1}\;,$ (132)

where

$\displaystyle\mathcal{B}_{a}$
$\displaystyle=2\left\langle\frac{f^{2}}{\gamma^{2}T^{2}}\left(p-\frac{smf}{\gamma}\right)^{2}\right\rangle_{a}=\frac{2s^{2}mf^{2}}{\gamma^{2}T}\;,$
$\displaystyle\mathcal{B}_{b}$ $\displaystyle=0\;.$ (133)

To proceed further, we choose the particle displacement
$\mathcal{J}_{\tau}^{1}=\int_{0}^{\tau}dt\,\dot{x}=\int_{0}^{\tau}dt\,s\frac{p}{m}=x(\tau)-x(0)$
as the current-like observable of interest. Using the formal solution for the
particle’s momentum

$\displaystyle p(t)$
$\displaystyle=\frac{smf}{\gamma}+\left[p(0)-\frac{smf}{\gamma}\right]e^{-\frac{\gamma}{m}t}$
$\displaystyle\qquad+\sqrt{2\gamma
T}\int_{0}^{t}dt^{\prime}\,e^{-\frac{\gamma}{m}(t-t^{\prime})}\xi\;,$ (134)

the mean of the observable for each initial distribution is computed as

$\displaystyle\langle\mathcal{J}_{\tau}^{1}\rangle_{a}=\frac{s^{2}\tau
f}{\gamma}\,,$
$\displaystyle\langle\mathcal{J}_{\tau}^{1}\rangle_{b}=\frac{s^{2}\tau
f}{\gamma}+s(1-s)\left(1-e^{-\gamma\tau/m}\right)\frac{fm}{\gamma^{2}}\;,$
(135)

from which follow

$\displaystyle(\tau\partial_{\tau}-s\partial_{s})\langle\mathcal{J}_{\tau}^{1}\rangle_{a}\big{|}_{s=1}=-\frac{\tau
f}{\gamma}\,,$
$\displaystyle(\tau\partial_{\tau}-s\partial_{s})\langle\mathcal{J}_{\tau}^{1}\rangle_{b}\big{|}_{s=1}=-\frac{\tau
f}{\gamma}+\left(1-e^{-\gamma\tau/m}\right)\frac{fm}{\gamma^{2}}\;.$ (136)

Meanwhile, as detailed in Ref. [14], we have

$\displaystyle\left.\Sigma_{\tau}^{\text{tot}}\right|_{s=1}$
$\displaystyle=\tau f^{2}/T\gamma,$
$\displaystyle\left.\text{Var}(\mathcal{J}^{1}_{\tau})\right|_{s=1}$
$\displaystyle=\frac{2\tau
T}{\gamma}\left[1-\frac{m}{\tau\gamma}\left(1-e^{-\gamma\tau/m}\right)\right]\;.$
(137)

Using Eqs. (B)–(B) in Eq. (132), the $\mathcal{Q}$ factors are obtained as

$\displaystyle\mathcal{Q}_{a}=2\left(1+\frac{2\tau_{0}}{\tau}\right)\left[1-\frac{\tau_{0}}{\tau}(1-e^{-\tau/\tau_{0}})\right]\,,$
$\displaystyle\mathcal{Q}_{b}=\frac{2}{1-\frac{\tau_{0}}{\tau}(1-e^{-\tau/\tau_{0}})}\;,$
(138)

where $\tau_{0}\equiv\gamma/m$. With these, it is straightforward to show
$\mathcal{Q}_{b}\geq\mathcal{Q}_{a}\geq 2$. These inequalities are verified in
Fig. 4, which plots the $\mathcal{Q}$ factors as the observation time $\tau$
is varied. In the long-time regime, both $\mathcal{Q}$ factors approach each
other and converge to $2$. In the short-time regime, however, their behaviors
differ significantly: while $Q_{b}$ diverges to infinity, $Q_{a}$ converges
again to $2$, thus saturating the TUR. This demonstrates that the
$s$-dependent initial distribution results in a tighter bound compared to the
$s$-independent initial distribution.

Figure 4:  Behaviors of $\mathcal{Q}_{a}$ and $\mathcal{Q}_{b}$, obtained in
Eq. (B), as the observation time $\tau$ is varied. The dashed line at
$\mathcal{Q}=2$ corresponds to the TUR saturation.

Beyond this specific example, it is difficult to prove in general whether the
same inequalities hold for the general Langevin systems. But at least, when
the observation time is short ($\tau\ll 1$), we can show that
$\mathcal{Q}_{b}\geq\mathcal{Q}_{a}\geq 2$ holds for more general cases
involving current-like observables of the form
$\mathcal{J}_{\tau}^{\Lambda}(\Gamma)=\int_{0}^{\tau}dt\,\Lambda(x,p)\circ\dot{x}$.
Towards this end, we first compute

$\displaystyle\langle\mathcal{J}_{\tau}^{\Lambda}\rangle=\int_{0}^{\tau}dt\int
dxdp\,\frac{s}{m}p\Lambda(x,p)P(x,p,t;s,\omega)\;.$ (139)

When $\tau\ll 1$, we can use the expansion

$\displaystyle(\tau\partial_{\tau}-s\partial_{s}-\omega\partial_{\omega})\langle\mathcal{J}_{\tau}^{\Lambda}\rangle$
$\displaystyle\quad=\partial_{\tau}[(\tau\partial_{\tau}-s\partial_{s}-\omega\partial_{\omega})\langle\mathcal{J}_{\tau}^{\Lambda}\rangle]\big{|}_{\tau=0}\tau+O(\tau^{2})\;.$
(140)

Using these, the denominator of the $\mathcal{Q}$ factor satisfies

$\displaystyle\partial_{\tau}[(\tau\partial_{\tau}-s\partial_{s}-\omega\partial_{\omega})\langle\mathcal{J}_{\tau}^{\Lambda}\rangle]\big{|}_{\tau=0}$
$\displaystyle\quad=-\int
dxdp\,\Lambda(x,p)\,s(s\partial_{s}+\omega\partial_{\omega})P(x,p,0;s,\omega)\;.$
(141)

This vanishes when the initial distribution is independent of $s$ and
$\omega$; otherwise, it is generally finite. This indicates that the
denominator of the $\mathcal{Q}$ factor is $O(\tau^{4})$ in the former case
and $O(\tau^{2})$ in the latter. The other observables in the $\mathcal{Q}$
factor typically scale as
$\text{Var}(\mathcal{J}_{\tau}^{\Lambda})\sim\tau^{2}$,
$\Sigma_{\tau}^{\text{tot}}\sim\tau$, and $\mathcal{B}\sim\tau^{0}$ for
$\tau\ll 1$. Combining these scaling behaviors, we obtain
$\mathcal{Q}_{a}\sim\tau^{0}$ for nonzero $\mathcal{B}$ and
$\mathcal{Q}_{b}\sim\tau^{-1}$ for vanishing $\mathcal{B}$. Thus, when
$\tau\ll 1$, we conclude that the TUR with nonzero $\mathcal{B}$ is tighter
than the TUR with vanishing $\mathcal{B}$.

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