# Thermodynamic Unification of Optimal Transport: Thermodynamic Uncertainty
Relation, Minimum Dissipation, and Thermodynamic Speed Limits

Tan Van Vu<EMAIL_ADDRESS>Department of Physics, Keio University,
3-14-1 Hiyoshi, Kohoku-ku, Yokohama 223-8522, Japan  Keiji Saito
<EMAIL_ADDRESS>Department of Physics, Keio University, 3-14-1
Hiyoshi, Kohoku-ku, Yokohama 223-8522, Japan

###### Abstract

Thermodynamics serves as a universal means for studying physical systems from
an energy perspective. In recent years, with the establishment of the field of
stochastic and quantum thermodynamics, the ideas of thermodynamics have been
generalized to small fluctuating systems. Independently developed in
mathematics and statistics, the optimal transport theory concerns the means by
which one can optimally transport a source distribution to a target
distribution, deriving a useful metric between probability distributions,
called the Wasserstein distance. Despite their seemingly unrelated nature, an
intimate connection between these fields has been unveiled in the context of
continuous-state Langevin dynamics, providing several important implications
for nonequilibrium systems. In this study, we elucidate an analogous
connection for discrete cases by developing a thermodynamic framework for
discrete optimal transport. We first introduce a novel quantity called
dynamical state mobility, which significantly improves the thermodynamic
uncertainty relation and provides insights into the precision of currents in
nonequilibrium Markov jump processes. We then derive variational formulas that
connect the discrete Wasserstein distances to stochastic and quantum
thermodynamics of discrete Markovian dynamics described by master equations.
Specifically, we rigorously prove that the Wasserstein distance equals the
minimum product of irreversible entropy production and dynamical state
mobility over all admissible Markovian dynamics. These formulas not only unify
the relationship between thermodynamics and the optimal transport theory for
discrete and continuous cases but also generalize it to the quantum case. In
addition, we demonstrate that the obtained variational formulas lead to
remarkable applications in stochastic and quantum thermodynamics, such as
stringent thermodynamic speed limits and the finite-time Landauer principle.
These bounds are tight and can be saturated for arbitrary temperatures, even
in the zero-temperature limit. Notably, the finite-time Landauer principle can
explain finite dissipation even at extremely low temperatures, which cannot be
explained by the conventional Landauer principle.

##  I Introduction

###  I.1 Background

Thermodynamics, which is built upon several axioms, is one of the most
successful phenomenological theories for studying energy exchanges in
macroscopic systems. Originally developed for the purpose of understanding the
behavior of steam engines, thermodynamics has since been applied to various
fields of science and engineering. The laws of thermodynamics show
extraordinary universality and impose fundamental constraints on physical
systems.

Beyond the macroscopic regime, the past two decades have witnessed substantial
progress in extending the notions of conventional thermodynamics to
microscopic systems, resulting in the frameworks of stochastic and quantum
thermodynamics Sekimoto (2010); Seifert (2012); Vinjanampathy and Anders
(2016); Goold  _et al._ (2016); Deffner and Campbell (2019). These
comprehensive frameworks provide a means of investigating small nonequilibrium
systems subject to significant fluctuations. Various universal relations have
been discovered, including fluctuation theorems Evans  _et al._ (1993);
Gallavotti and Cohen (1995); Crooks (1999); Jarzynski (2000); Esposito  _et
al._ (2009); Campisi  _et al._ (2011), thermodynamic uncertainty relations
Barato and Seifert (2015); Gingrich  _et al._ (2016); Horowitz and Gingrich
(2017, 2020), thermodynamic speed limits Shiraishi  _et al._ (2018); Ito
(2018); Funo  _et al._ (2019); Ito and Dechant (2020); Gupta and Busiello
(2020); Vo  _et al._ (2020); Van Vu and Hasegawa (2021a); Yoshimura and Ito
(2021); Delvenne and Falasco (2021); Salazar (2022); Vo  _et al._ (2022), and
refinements of the Landauer principle Aurell  _et al._ (2011); Goold  _et al._
(2015); Proesmans  _et al._ (2020a); Zhen  _et al._ (2021); Van Vu and Saito
(2022a); Lee  _et al._ (2022). These equalities and inequalities characterize
the fundamental limits of small systems and distinguish the possible from the
impossible in terms of thermodynamics. They are not only theoretically
important but also lead to practical applications in estimating physically
relevant quantities from experimental data, such as free energy Gore  _et al._
(2003) and dissipation Li  _et al._ (2019); Manikandan  _et al._ (2020); Van
Vu  _et al._ (2020); Otsubo  _et al._ (2020); Kim  _et al._ (2020); Skinner
and Dunkel (2021); Dechant and Sasa (2021). In addition, information
manipulations such as measurement, feedback, erasure, and copying have been
incorporated into thermodynamics, leading to significant developments in
several subfields, such as the thermodynamics of information Sagawa (2012);
Parrondo  _et al._ (2015) and computation Bennett (1982); Wolpert (2019). In
parallel, concepts from other fields, such as the resource theory Chitambar
and Gour (2019); Lostaglio (2019) and information geometry Salamon and Berry
(1983); Ruppeiner (1995); Amari and Nagaoka (2000), have also been integrated
into thermodynamics, generating new avenues of research and offering new tools
for analyzing thermodynamic processes Crooks (2007); Feng and Crooks (2008);
Machta (2015); Rotskoff  _et al._ (2017); Nicholson  _et al._ (2018); Scandi
and Perarnau-Llobet (2019); Bryant and Machta (2020); Abiuso  _et al._ (2020);
Brandner and Saito (2020). Accordingly, the integration of thermodynamics with
other disciplines provides new insights into the understanding of
nonequilibrium systems.

Figure 1: (a) Schematic of stochastic and quantum thermodynamics of discrete
Markovian dynamics and the optimal transport problem. This study aims to
reveal the connection between thermodynamics and the optimal transport theory
for discrete cases, including both classical and quantum systems. (b) Our
thermodynamic unification of optimal transport and its consequences, including
minimum dissipation and thermodynamic speed limits. By introducing dynamical
state mobility $m_{t}$, we provide a unified, discrete generalization of the
Benamou–Brenier formula for classical and quantum discrete systems. See the
subsection “Summary of results” for an explanation.

A key quantity in thermodynamics is entropy production, which quantifies the
degree of irreversibility of thermodynamic processes. Entropy production plays
a central role in the fundamental laws of thermodynamics and provides a
quantitative characterization for investigating nonequilibrium processes; a
comprehensive review regarding entropy production can be found in Ref. Landi
and Paternostro (2021). Recently, it has been shown that entropy production
must be increased to achieve a high precision of currents Gingrich  _et al._
(2016) and fast state transformation Shiraishi  _et al._ (2018). However,
minimizing entropy production is also a particularly relevant issue Schmiedl
and Seifert (2007); Chennakesavalu and Rotskof (2022) because it is closely
related to the energy lost to the environment. Owing to the critical role of
entropy production, great efforts have been made to elucidate its properties
and its relationship with other physical quantities Saito and Dhar (2016);
Shiraishi  _et al._ (2016); Neri  _et al._ (2017); Pigolotti  _et al._ (2017);
Pietzonka and Seifert (2018); Manzano  _et al._ (2019); Falasco and Esposito
(2020).

Optimal transport Villani (2008), which is developed independently of
thermodynamics, is a mature field in mathematics and statistics, and its
theory concerns the optimal planning and optimal cost of transporting a
distribution. Specifically, given the individual costs of transporting a unit
weight of a resource from one location to another, the optimal transport
problem is to determine the optimal means of redistributing the distribution
of the resource into the desired distribution to yield the lowest total cost.
Historically, the optimal transport problem was first defined by Monge in 1781
and has since been reformulated in a more general and well-defined form.
Currently, this problem has several theoretical and practical applications in
a variety of scientific fields, including statistics and machine learning
Kolouri  _et al._ (2017), computer vision Haker  _et al._ (2004), linguistics
Huang  _et al._ (2016), classical mechanics Koehl  _et al._ (2019), and
molecular biology Schiebinger  _et al._ (2019). It is noteworthy that the
solution to this problem not only provides an optimal transport plan between
distributions but also defines a useful metric in the space of probability
distributions. Although this metric has been identified by several names in
the literature, such as the Monge–Kantorovich distance or earth mover’s
distance, we refer to it as the Wasserstein distance throughout this paper.

Because the optimal transport theory is concerned with transformations of
probability distributions that are commonly used to characterize the state of
small systems, whether any connection exists between the two disciplines of
optimal transport and stochastic thermodynamics is a natural question. Indeed,
a deep connection between these fields has been elucidated in the context of
overdamped Langevin dynamics, revealing that the problem of minimizing entropy
production can be mapped to the optimal transport problem Jordan  _et al._
(1998); Aurell  _et al._ (2011, 2012); Dechant and Sakurai (2019); Nakazato
and Ito (2021); Chennakesavalu and Rotskof (2022). More specifically, the
minimum entropy production among all processes that transform the initial into
the final distribution can be expressed in terms of the Wasserstein distance
between the two distributions. In addition, the optimal transport plan
provides a feasible solution for the optimal control protocol. The essence of
this connection can be intuitively understood through the Benamou–Brenier
formula Benamou and Brenier (2000), which is given by the following equality:

$\displaystyle
W_{2}(p^{A},p^{B})=\min{\int_{0}^{\tau}\sqrt{D\sigma_{t}}\differential{t}}=\min\sqrt{D\tau\Sigma_{\tau}}.$
(1)

Here, $W_{2}(p^{A},p^{B})$ is the $L^{2}$-Wasserstein distance [cf. Eq. (10)],
$D$ is the diffusion coefficient, $\tau$ is the operational time, $\sigma_{t}$
is the entropy production rate, $\Sigma_{\tau}$ is the total entropy
production, and the minimum is taken over all the overdamped Langevin
processes that transform distribution $p^{A}$ into $p^{B}$. The variational
relation (1) links two apparently unrelated quantities, namely a mathematical
metric and a thermodynamic cost. The metric $W_{2}(p^{A},p^{B})$ on the left-
hand side of Eq. (1) is a static quantity that is determined only by two
distributions, whereas the right-hand side represents a dynamical quantity
that indicates the thermodynamic cost associated with overdamped Langevin
dynamics. This formula leads to remarkable applications for overdamped
Langevin dynamics, such as a thermodynamic speed limit Dechant and Sakurai
(2019) and a finite-time Landauer principle of information erasure for
classical bits modeled by a continuous double-well potential Proesmans  _et
al._ (2020a, b), to name only two. The finite-time correction in the Landauer
principle indicated by the speed limit expression is consistent with
experimental observations Bérut  _et al._ (2012). Moreover, the bounds
obtained from Eq. (1) are tight in the sense that, for any two given
distributions, there always exists an overdamped Langevin dynamics that
transforms the distributions and attains the equality of the bounds.

In contrast to continuous systems, a similar connection between optimal
transport and thermodynamics is yet to be unveiled in discrete systems. We
note that stochastic thermodynamics of discrete systems is highly relevant to
experiments Schuler  _et al._ (2005); Hekking and Pekola (2013); Koski  _et
al._ (2014); Josefsson  _et al._ (2018). Even in continuous systems such as
biological systems, the dynamics can be represented by effective discrete
states Elowitz  _et al._ (2002); Schliwa and Woehlke (2003); Stigler  _et al._
(2011); Cheong  _et al._ (2011). In addition, the Landauer principle of
information erasure problem is a statement for discrete bit operations
Landauer (1961). Therefore, elucidating the thermodynamic interpretation of
the discrete optimal transport problem is essential for an in-depth
understanding of the nonequilibrium thermodynamic structure and, particularly,
for its application to the state transformation speed.

To reveal this type of relationship for discrete systems, two nontrivial
points are worth noting. First, the formula (1) cannot be extended directly to
discrete cases because no exact correspondence to the diffusion coefficient
exists in generic discrete systems. Even if a proper correspondence to the
diffusion coefficient is defined for discrete cases, no guarantee can be given
that the discrete Wasserstein distance can be expressed in the same manner as
in Eq. (1). Second, previous studies have shown that without any additional
constraint, the distribution of Markov jump processes can always be
transformed to the target distribution with arbitrarily small entropy
production Van Vu and Hasegawa (2021b); Remlein and Seifert (2021); Dechant
(2022). This implies that entropy production alone is insufficient to
characterize the transport cost (i.e., the Wasserstein distance), or
equivalently, this implies that another quantity that plays the same role as
the diffusion coefficient in continuous cases must be introduced along with
entropy production. These technical remarks are an obstacle to elucidating the
relationship between optimal transport and thermodynamics in discrete cases.
Simultaneously, overcoming this obstacle is expected to reveal essential and
common thermodynamic structures hidden in nonequilibrium processes.

With this background, we aim to elucidate the deep connection between
thermodynamics and optimal transport in discrete cases (see Fig. 1 for
illustration). Specifically, we develop discrete generalizations of the
Benamou–Brenier formula in the context of Markovian open classical and quantum
dynamics described by the master equations. Our formulas not only unify the
relationship between optimal transport and stochastic thermodynamics for
discrete and continuous cases but also generalize to the quantum case.
Moreover, by developing a thermodynamic framework for discrete optimal
transport, we can derive fundamental bounds for nonequilibrium systems,
including the thermodynamic uncertainty relation, thermodynamic speed limits,
and finite-time Landauer principle for both classical and quantum systems.
These bounds are tight and stronger than previously reported results.
Concerning the Landauer principle, finite-time information erasure should
generate finite heat dissipation even at zero temperature. However, neither
the original Landauer bound Landauer (1961) nor finite-time corrections that
have been proposed thus far for discrete systems Zhen  _et al._ (2021); Van Vu
and Saito (2022a) can explain heat dissipation at extremely low temperatures.
By contrast, our expression for the first time can predict heat dissipation
even at extremely low temperatures.

###  I.2 Summary of results

The main contributions of this study can be summarized as follows.

  * (1)

_Dynamical state mobility and improved thermodynamic uncertainty relation_.—We
define a novel kinetic quantity $m_{t}$ [cf. Eq. (39)], which is essential to
our results. The motivation for this definition is derived from Eq. (1), which
suggests that a kinetic term is relevant for characterizing the Wasserstein
distance. $m_{t}$ is defined by the microscopic Onsager-like coefficients and
reduces to the diffusion coefficient $D$ in the continuous limit; thus, it is
referred to as dynamical state mobility. Similar to dynamical activity Maes
(2020), $m_{t}$ should be measurable in experiments. Using this kinetic term,
we derive an improved thermodynamic uncertainty relation for time-integrated
currents in Markov jump processes, which can be expressed as [cf. Eq. (61)]

$\frac{\expectationvalue{J}^{2}}{{\rm
var}[J]}\leq\eta\frac{\Sigma_{\tau}}{2},$ (2)

where $\expectationvalue{J}$ and ${\rm var}[J]$ denote the mean and variance
of an arbitrary current $J$, respectively, and $\eta\coloneqq
2\mathcal{M}_{\tau}/\mathcal{A}_{\tau}\leq 1$ is an efficiency defined in
terms of dynamical state mobility
$\mathcal{M}_{\tau}\coloneqq\int_{0}^{\tau}m_{t}\differential{t}$ and
dynamical activity $\mathcal{A}_{\tau}$. The inequality (2) indicates that the
precision of currents is constrained by the product of the thermodynamic and
kinetic costs divided by the timescale. Moreover, it provides new insights
into the relationship between precision and cost in Markov jump processes;
that is, increasing only the thermodynamic cost does not guarantee high
precision of currents. Instead, given the same timescale $\mathcal{A}_{\tau}$,
the product of the thermodynamic and kinetic costs must be increased to
achieve high precision. Notably, the relation (2) is tighter than the
conventional thermodynamic uncertainty relation Gingrich  _et al._ (2016);
Horowitz and Gingrich (2017).

  * (2)

_Variational formulas that connect optimal transport to stochastic and quantum
thermodynamics_.—Using the defined state mobility term, we derive variational
formulas that relate the discrete Wasserstein distance to the thermodynamic
and kinetic costs in Markovian dynamics. More specifically, we prove the
following equality for the classical case (cf. Thm. 1):

$\displaystyle\mathcal{W}_{1}(p^{A},p^{B})$
$\displaystyle=\min{\int_{0}^{\tau}\sqrt{\sigma_{t}m_{t}}\differential{t}}=\min\sqrt{\Sigma_{\tau}\mathcal{M}_{\tau}},$
(3)

where $\mathcal{W}_{1}(p^{A},p^{B})$ is the discrete $L^{1}$-Wasserstein
distance between two distributions $p^{A}$ and $p^{B}$ [cf. Eq. (69)], and the
minimum is over all admissible Markov jump processes that transform
distribution $p^{A}$ into $p^{B}$ over a period $\tau$ with a given
connectivity. The relation (3) provides a thermodynamic interpretation for the
Wasserstein distance, implying that the Wasserstein distance is equal to the
minimum product of the thermodynamic and kinetic costs. We also analogously
generalize the formula (3) to the quantum case, in which the classical
Wasserstein distance $\mathcal{W}_{1}(p^{A},p^{B})$ is replaced with a quantum
Wasserstein distance $\mathcal{W}_{q}(\varrho^{A},\varrho^{B})$ between
density matrices $\varrho^{A}$ and $\varrho^{B}$. These formulas can be
considered as discrete generalizations of the Benamou–Brenier formula known in
continuous cases.

  * (3)

_Trade-off between irreversibility and state mobility_.—Through the developed
variational formulas, we reveal a trade-off relation between the
irreversibility and dynamical state mobility in discrete systems, which reads
as follows:

$\Sigma_{\tau}\mathcal{M}_{\tau}\geq\mathcal{W}_{1}(p^{A},p^{B})^{2}.$ (4)

The inequality (4) implies that either the thermodynamic cost $\Sigma_{\tau}$
or kinetic cost $\mathcal{M}_{\tau}$ must be sacrificed (i.e., they cannot be
simultaneously small) to evolve the system state.

  * (4)

_Minimum dissipation and optimal protocol_.—The problem of minimizing entropy
production in discrete systems is trivial if no constraints exist on the
transition rates. Our results shed new light on this issue. More specifically,
the formula (3) implies that fixing the dynamical state mobility
$\mathcal{M}_{\tau}$ is a reasonable constraint from which minimum dissipation
can be immediately determined through the Wasserstein distance, and the
optimal control protocol can be constructed from the optimal transport
problem. When additional constraints exist on system dynamics, we show that a
lower bound on minimum dissipation can be obtained (see Fig. 4 for
illustration).

  * (5)

_Thermodynamic speed limits_.—From the resulting variational formulas, we
derive unified and stringent thermodynamic speed limits that place lower
bounds on the time required for state transformation for both open classical
and quantum systems. The classical bound reads [cf. Eq. (128)]

$\tau\geq\frac{\mathcal{W}_{1}(p^{A},p^{B})}{\expectationvalue{\sqrt{\sigma
m}}_{\tau}}\geq\frac{\mathcal{W}_{1}(p^{A},p^{B})}{\sqrt{\expectationvalue{\sigma}_{\tau}\expectationvalue{m}_{\tau}}},$
(5)

where $\expectationvalue{x}_{\tau}$ denotes the time average of a time-
dependent variable $x_{t}$. The quantum bound has the same form, where
$\mathcal{W}_{1}(p^{A},p^{B})$ is replaced with
$\mathcal{W}_{q}(\varrho^{A},\varrho^{B})$. The inequality (5) implies that
the speed of state transformation is constrained by the time average of the
product of the thermodynamic and kinetic costs. Because we start from the
equality relations, these bounds are tight and can always be saturated for
arbitrary temperatures. In other words, for an arbitrary pair of distributions
or density matrices, there always exist Markovian dynamics that saturate the
bounds. They are also stronger than previously known bounds Shiraishi  _et
al._ (2018); Van Vu and Hasegawa (2021a).

  * (6)

_Finite-time Landauer principle_.—From the variational formulas, we derive
finite-time lower bounds for heat dissipation $Q$ incurred in classical and
quantum information erasure [cf. Eqs. (142) and (150)]. The bounds
characterize both finite-time and finite-error effects on heat dissipation.
Several finite-time Landauer bounds have been derived for discrete systems in
previous studies Zhen  _et al._ (2021); Van Vu and Saito (2022a). However,
these bounds encounter the same problem as the conventional Landauer bound;
that is, they lose the predictive power in the low-temperature regime. By
contrast, our new bounds are tight for _arbitrary_ temperatures, even in the
zero-temperature limit. A further simplified bound including a finite erasure
error $\epsilon\geq 0$ reads [cf. Eq. (146)]

$Q\geq T[\ln
d-h(\epsilon)]+\frac{(1-1/d-\epsilon)^{2}}{\tau\beta\expectationvalue{m}_{\tau}},$
(6)

where $d$ is the system’s dimension, $T$ is the temperature of the heat bath,
$\beta$ is the inverse temperature, and $h(\epsilon)\geq 0$ is a function that
vanishes as $\epsilon\to 0$. In the perfect-erasure ($\epsilon\to 0$) and
quasistatic ($\tau\to\infty$) limits, the above bound reduces to the
conventional Landauer bound $Q\geq T\ln d$.

###  I.3 Relevant literature

Here, we briefly discuss several relevant studies that have attempted to link
the (modified) Wasserstein distances to the thermodynamics of Markov jump
processes.

The Benamou–Brenier formula has two facets. It not only provides a
thermodynamic interpretation but also reveals a geometric structure for the
continuous $L^{2}$-Wasserstein distance. More specifically, $W_{2}$ can be
interpreted as a Riemannian metric on the manifold of probability distribution
functions. Although the discrete $L^{2}$-Wasserstein distance is well defined
and widely used in the literature, unfortunately, it does not possess a
geometric interpretation, and its connection to thermodynamics also remains
unclear. For this reason, many studies have generalized the Wasserstein
distance based on the geometric aspect of the Benamou–Brenier formula for
discrete cases Maas (2011). This modified Wasserstein distance places a lower
bound on irreversible entropy production of Markov jump processes Van Vu and
Hasegawa (2021a); Yoshimura  _et al._ (2022). However, it is system dependent
because the transition rates are concretely used to define this distance.

In contrast to the previous direction, in this study, we consider the
conventional discrete $L^{1}$-Wasserstein distance and focus on its
thermodynamic interpretation. In this regard, Dechant has obtained some
interesting results by relating the discrete $L^{1}$-Wasserstein distance to
the entropy production and dynamical activity of Markov jump processes Dechant
(2022). Here, we consider a different approach by introducing the dynamical
state mobility and obtain discrete generalizations of the Benamou–Brenier
formula. This approach not only unifies the classical discrete and continuous
cases but also extends to the quantum case. Although we focus on the discrete
$L^{1}$-Wasserstein distance, it is noteworthy that the obtained
generalizations of the Benamou–Brenier formula are similar to that for the
continuous $L^{2}$-Wasserstein distance. This suggests that the discrete
$L^{1}$-Wasserstein distance may play the same role as the continuous
$L^{2}$-Wasserstein distance in continuous cases.

The remainder of the paper is organized as follows. Section II presents a
review of the optimal transport problem and the relevant existing results in
the context of continuous-state overdamped Langevin dynamics. We particularly
emphasize the Benamou–Brenier formula of the $L^{2}$-Wasserstein distances and
their connections to stochastic thermodynamics. In Sec. III, we briefly
introduce stochastic thermodynamics of classical Markovian dynamics. Next, we
define the novel kinetic term $m_{t}$ and discuss its relevant properties. We
then derive the improved thermodynamic uncertainty relation for Markov jump
processes and numerically demonstrate it. In Sec. IV, we describe the optimal
transport problem in discrete cases and explain our first theorem that links
the discrete Wasserstein distance to stochastic thermodynamics of Markov jump
processes. The relationship between the obtained and existing results in
continuous cases is also discussed. In Sec. V, we define a quantum Wasserstein
distance and explain our second theorem that generalizes the variational
formula to the quantum case. From the derived variational formulas, in Sec.
VI, we describe two applications: the thermodynamic speed limits and the
finite-time Landauer principle. In Sec. VII, we numerically demonstrate our
findings. Finally, Sec. VIII presents a conclusion with a discussion and
outlook. All detailed mathematical calculations and derivations can be found
in the Appendixes.

##  II Review of continuous optimal transport

In this section, we briefly review the optimal transport problem in continuous
spaces and discuss the Benamou–Brenier formula, which provides a thermodynamic
interpretation of the Wasserstein distances in the context of overdamped
Langevin dynamics.

###  II.1 Optimal transport problem

First, we succinctly introduce the classical transport problem on the
continuous space $\mathbb{R}^{d}$ with $d\geq 1$ (see Ref. Villani (2008) for
details). The problem of optimal transport—that is, determining how a pile of
earth can be optimally transported into another pile of the same volume but
with a different shape—was originally introduced by Monge. The optimality here
is interpreted to mean that the total transport cost is minimized with respect
to a given cost metric. Suppose that the source and target piles of earth are
characterized by probability distribution functions $p^{A}(x)$ and $p^{B}(x)$,
respectively, on the space $\mathbb{R}^{d}$, and the cost metric is given by
$c:\mathbb{R}^{d}\times\mathbb{R}^{d}\mapsto\mathbb{R}_{\geq 0}$. Then, the
Monge optimal transport problem is to identify a one-to-one map
$\varphi:\mathbb{R}^{d}\to\mathbb{R}^{d}$ that minimizes the objective
function

$\min_{\varphi}\int_{\mathbb{R}^{d}}c(x,\varphi(x))p^{A}(x)\differential{x},$
(7)

where the minimum is over all $\varphi$ satisfying
$p^{A}(x)=p^{B}(\varphi(x))|\det(\nabla\varphi(x))|$. However, this
formulation presents an issue regarding the non-existence of a valid transport
map; that is, the map $\varphi$ might not exist in discrete cases because no
mass can be split. Fortunately, this issue was previously resolved by the
relaxation of Kantorovich, which led to a more well-defined problem. Instead
of a transport map $\varphi(x)$, Kantorovich considered a transport plan
$\pi(x,y)$ that is a joint probability distribution function and represents a
coupling of $p^{A}$ and $p^{B}$. This transport plan allows us to split a
single mass and transport it to multiple target locations. The Kantorovich
problem can be formulated as an optimization of the following objective
function:

$\min_{\pi\in\Pi(p^{A},p^{B})}{\int_{\mathbb{R}^{d}\times\mathbb{R}^{d}}c(x,y)\pi(x,y)\differential{x}\differential{y}},$
(8)

where $\Pi(p^{A},p^{B})$ denotes the coupling set of joint probability
distribution functions whose marginal distributions coincide with $p^{A}$ and
$p^{B}$:

$\int_{\mathbb{R}^{d}}\pi(x,y)\differential{y}=p^{A}(x)~{}\text{and}~{}\int_{\mathbb{R}^{d}}\pi(x,y)\differential{x}=p^{B}(y).$
(9)

The concept of optimal transport provides a means for defining useful metrics
on continuous spaces of probability distribution functions. By employing the
cost metric of the Euclidean norm (i.e., $c(x,y)=\|x-y\|^{\alpha}$ for a
positive number $\alpha\geq 1$), the Kantorovich problem reduces exactly to
the $L^{\alpha}$-Wasserstein distance, which is defined as

$W_{\alpha}(p^{A},p^{B})^{\alpha}\coloneqq\min_{\pi\in\Pi(p^{A},p^{B})}{\int_{\mathbb{R}^{d}\times\mathbb{R}^{d}}\|x-y\|^{\alpha}\pi(x,y)\differential{x}\differential{y}}.$
(10)

The $L^{\alpha}$-Wasserstein distance is a genuine metric and satisfies the
triangle inequality. Applying Hölder’s inequality, we can derive a
hierarchical relationship, that is, $W_{\alpha}\leq W_{\alpha^{\prime}}$ for
$\alpha\leq\alpha^{\prime}$. Of the several that exist, the $L^{1}$\- and
$L^{2}$-Wasserstein distances are particularly relevant from the thermodynamic
and geometric perspectives. In the following, we discuss some remarkable
properties of these two distances.

###  II.2 Benamou–Brenier formula

The Wasserstein distance can be expressed in a variational form in several
ways. Interestingly, Benamou and Brenier developed a variational formula for
the $L^{2}$-Wasserstein distance in terms of fluid mechanics Benamou and
Brenier (2000). The Benamou–Brenier formula casts the $L^{2}$-Wasserstein
distance as a minimization problem of a time-integrated cost in terms of
probability distribution functions and velocity fields:

$W_{2}(p^{A},p^{B})^{2}=\min_{v_{t}}{\tau\int_{0}^{\tau}\int_{\mathbb{R}^{d}}\|v_{t}(x)\|^{2}p_{t}(x)\differential{x}\differential{t}},$
(11)

where the minimum is over all smooth paths $\\{v_{t}\\}_{0\leq t\leq\tau}$
subject to the continuity equation

$\dot{p}_{t}(x)+\nabla\dotproduct[v_{t}(x)p_{t}(x)]=0$ (12)

with the initial and final conditions $p_{0}(x)=p^{A}(x)$ and
$p_{\tau}(x)=p^{B}(x)$, respectively. Here, $\nabla$ is the del operator, and
$\dotproduct$ denotes the standard Euclidean inner product between vectors.
Note that given any absolutely continuous curve $\\{p_{t}\\}$, we can always
find a velocity field $\\{v_{t}\\}$ that satisfies Eq. (12). The formulation
(11) not only enables us to find a numerical scheme for computing $W_{2}$ but
also provides the thermodynamic and geometric interpretations of the
$L^{2}$-Wasserstein distance.

Next, we discuss a thermodynamic interpretation of the $L^{2}$-Wasserstein
distance (see Appendix A for a geometric interpretation). Consider an
overdamped system on the continuous space $\mathbb{R}^{d}$, which is
constantly subject to a time-dependent force $F_{t}(x)$ and weakly coupled to
a single heat bath. The system state at time $t$ can be characterized by the
probability distribution $p_{t}(x)$, the time evolution of which is described
by the Fokker–Planck equation:

$\displaystyle\dot{p}_{t}(x)$
$\displaystyle=-\nabla\dotproduct[v_{t}(x)p_{t}(x)],$ (13) $\displaystyle
v_{t}(x)$ $\displaystyle=F_{t}(x)-D\nabla\ln p_{t}(x).$ (14)

Note that the velocity field $v_{t}(x)$ of the system in Eq. (14) is an exact
solution to the continuity equation (12), which drives the source distribution
$p_{0}(x)$ to the target distribution $p_{\tau}(x)$. Based on the framework of
stochastic thermodynamics, the irreversible entropy production during period
$\tau$ can be calculated as Seifert (2012)

$\Sigma_{\tau}=\frac{1}{D}\int_{0}^{\tau}\int_{\mathbb{R}^{d}}\|v_{t}(x)\|^{2}p_{t}(x)\differential{x}\differential{t}.$
(15)

Irreversible entropy production clearly coincides with the time-integrated
cost in the integration in Eq. (11), ignoring the scaling factor. Therefore,
we can rewrite the Benamou–Brenier formula as

$W_{2}(p^{A},p^{B})=\min_{v_{t}}\sqrt{D\tau\Sigma_{\tau}}.$ (16)

From this, the following inequality can be immediately derived:

$\Sigma_{\tau}\geq\frac{W_{2}(p_{0},p_{\tau})^{2}}{D\tau}.$ (17)

Inequality (17) refines the second law of thermodynamics by providing a
stronger bound on irreversible entropy production solely in terms of the
initial and final distributions, given that the operational time and diffusion
coefficient are fixed. The bound can be interpreted as a thermodynamic speed
limit:

$\tau\geq\frac{W_{2}(p^{A},p^{B})}{\sqrt{D\expectationvalue{\sigma}_{\tau}}}.$
(18)

Moreover, it can also be applied to derive a finite-time Landauer principle
Aurell  _et al._ (2012); Proesmans  _et al._ (2020a). It is noteworthy that
the bound can be saturated for any pair of the initial and final distributions
and is tight even in the zero-temperature limit.

Because $v_{t}(x)$ and $F_{t}(x)$ can be considered a one-to-one
correspondence, we can obtain the following equality between irreversible
entropy production and the Wasserstein distance:

$\min_{F_{t}}\Sigma_{\tau}=\frac{W_{2}(p_{0},p_{\tau})^{2}}{D\tau}.$ (19)

This relation implies that the minimum entropy production in all overdamped
processes that transform one distribution into another can be determined
exactly by the Wasserstein distance between the two distributions. In Refs.
Benamou and Brenier (2000); Aurell  _et al._ (2012), it has been demonstrated
that the minimum in Eq. (11) can be achieved with a velocity field of the form
$v_{t}(x)=-\nabla\phi_{t}(x)$, where $\phi_{t}(x)$ is a time-dependent
potential. Thus, the minimum entropy production can always be achieved with a
conservative force $F_{t}(x)=-\nabla V_{t}(x)$, where
$V_{t}(x)=\phi_{t}(x)-D\ln p_{t}(x)$ is a time-dependent potential.

##  III Stochastic thermodynamics of discrete systems

In this section, we first briefly introduce the stochastic thermodynamics of
classical discrete Markovian dynamics described by the master equation; for a
comprehensive review, one can refer to Ref. Seifert (2012). We then define a
novel physical quantity called dynamical state mobility, discuss its relevant
properties, and derive an improved thermodynamic uncertainty relation.

###  III.1 Markov jump processes

We consider a discrete-state system with $N>1$ states, which is weakly
attached to single or multiple thermal reservoirs. Examples of these systems
include diffusive processes on a lattice, biomolecular motors, chemical
reaction networks, and quantum dots. The system can be described in terms of a
time-dependent probability distribution
${p_{t}}\coloneqq[p_{1}(t),\dots,p_{N}(t)]^{\top}$, where $p_{x}(t)$ denotes
the probability of finding the system in state $x$ at time $t$. Assume that
the system is modeled by a time-continuous Markov jump process and that the
transitions from a state $y$ to a state $x$ occur at a nonnegative rate
$w_{xy}(t)$, which can be time dependent according to an external control
protocol. The time evolution of the probability distribution is described by
the master equation:

${\dot{p}_{t}}=\mathsf{W}_{t}{p_{t}},$ (20)

where dot $\cdot$ denotes the time derivative and $\mathsf{W}_{t}=[w_{xy}(t)]$
denotes the matrix of the transition rates with $w_{xx}(t)=-\sum_{y(\neq
x)}w_{yx}(t)$. We consider microscopically reversible dynamics, that is,
$w_{xy}(t)>0$ if and only if $w_{yx}(t)>0$. Hereafter, we assume that the
transition rates satisfy the local detailed balance condition Seifert (2012):

$\ln\frac{w_{xy}(t)}{w_{yx}(t)}=s_{xy}(t),$ (21)

where $s_{xy}(t)$ denotes the entropy change in the environment due to the
jump from state $y$ to $x$ at time $t$. If we fix the transition rates at any
time, the system relaxes toward a stationary state, which may no longer be an
equilibrium state.

In a case wherein the system is attached to a single reservoir at inverse
temperature $\beta$ and the transitions between states are induced by the
energy difference, the entropy change reads

$s_{xy}(t)=\beta[\varepsilon_{y}(t)-\varepsilon_{x}(t)],$ (22)

where $\varepsilon_{x}(t)$ denotes the instantaneous energy level of state $x$
at time $t$. Whenever this occurs, we say that the system satisfies the global
detailed balance condition. Notably, the thermal state $p^{\rm
eq}_{x}(t)\propto e^{-\beta\varepsilon_{x}(t)}$ becomes the instantaneous
stationary state of the system (i.e., $\mathsf{W}_{t}{p^{\rm eq}_{t}}=0$).
Hereafter, we consider generic dynamics satisfying the local detailed balance
condition. However, dynamics satisfying Eq. (22) will be used occasionally for
physical interpretation of some quantities.

For convenience, we define the following quantities:

$\displaystyle a_{xy}(t)$ $\displaystyle\coloneqq w_{xy}(t)p_{y}(t),$ (23)
$\displaystyle j_{xy}(t)$ $\displaystyle\coloneqq
w_{xy}(t)p_{y}(t)-w_{yx}(t)p_{x}(t),$ (24)

which quantify the frequency of jumps and the probability current from state
$y$ to $x$ at time $t$, respectively.

###  III.2 Entropy production and dynamical activity

Given the previous setup, we now discuss some relevant thermodynamic
quantities. One central quantity is irreversible entropy production, which
quantifies the degree of irreversibility of the thermodynamic process. Within
the framework of stochastic thermodynamics, total entropy production during a
period $\tau$ can be defined as

$\Sigma_{\tau}\coloneqq\Delta S_{\rm sys}+\Delta S_{\rm env},$ (25)

where $\Delta S_{\rm sys}$ and $\Delta S_{\rm env}$ are the changes in the
entropy of the system and the environment, respectively, expressed as

$\displaystyle\Delta S_{\rm sys}$ $\displaystyle=S(p_{\tau})-S(p_{0}),$ (26)
$\displaystyle\Delta S_{\rm env}$ $\displaystyle=\int_{0}^{\tau}\sum_{x\neq
y}a_{xy}(t)s_{xy}(t)\differential{t}.$ (27)

Here, system entropy production is quantified via the Shannon entropy
$S(p)\coloneqq-\sum_{x}p_{x}\ln p_{x}$, whereas the entropy change of the
environment is defined as the sum of entropic contributions from each
transition between states. Simple calculations show that the entropy
production rate is always nonnegative:

$\sigma_{t}\coloneqq\dot{\Sigma}_{t}=\sum_{x>y}[a_{xy}(t)-a_{yx}(t)]\ln\frac{a_{xy}(t)}{a_{yx}(t)}\geq
0.$ (28)

The non-negativity of irreversible entropy production corresponds to the
second law of thermodynamics. The equality of this zero bound is attained only
when the system is in the instantaneous thermal state at all times.

Another essential quantity in nonequilibrium thermodynamics is dynamical
activity, quantified by the amplitude of transitions between states as

$a_{t}\coloneqq\sum_{x\neq y}a_{xy}(t).$ (29)

The average number of jumps during period $\tau$ can be calculated as

$\mathcal{A}_{\tau}\coloneqq\int_{0}^{\tau}a_{t}\differential{t}.$ (30)

The time average of dynamical activity characterizes the timescale of
thermodynamic processes. The higher the dynamical activity, the stronger the
thermalization. Entropy production and dynamical activity are the time-
antisymmetric and time-symmetric parts, respectively, of the path-integral
action with respect to a time-reversed process Maes (2020). Both quantities
constrain the fluctuation of currents according to the thermodynamic and
kinetic uncertainty relations Gingrich  _et al._ (2016); Terlizzi and Baiesi
(2019).

###  III.3 Dynamical state mobility

####  III.3.1 Definition

Here, we introduce a new quantity called dynamical state mobility, which plays
a crucial role in our results. Before getting into the details, let us briefly
recall the linear response relations Onsager (1931a, b), which express the
equalities between currents and forces in near-equilibrium systems. Consider
an irreversible transport process driven by thermodynamic forces
$F=[F_{x}]^{\top}$, such as affinities in temperatures or chemical potentials.
Let $J=[J_{x}]^{\top}$ be the thermodynamic currents that characterize the
response of the system to the applied forces. In a linear-response regime, the
currents depend only on the thermodynamic forces and can be expressed by the
following linear relations:

$J_{x}=\sum_{y}\mu_{xy}F_{y}~{}\text{or}~{}{J}=\mathsf{L}{F}.$ (31)

These relations (31) are referred to as linear response equations, where the
coefficients $\mu_{xy}$ are known as Onsager kinetic coefficients, and
$\mathsf{L}=[\mu_{xy}]$ is called the Onsager matrix. Onsager reciprocal
relations imply that in the case of time-reversal symmetry, the Onsager matrix
is symmetric (i.e., $\mu_{xy}=\mu_{yx}$). In addition, the entropy production
rate can be expressed in a quadratic form of the forces as

$\sigma=\sum_{x}J_{x}F_{x}=\sum_{x,y}\mu_{xy}F_{x}F_{y}~{}\text{or}~{}\sigma=F^{\top}\mathsf{L}{F}.$
(32)

The non-negativity of the entropy production rate immediately derives that
$\mathsf{L}$ is positive semi-definite.

The Onsager coefficients characterize the response of dynamics close to
equilibrium at the macroscopic level. Nevertheless, they can be mimicked to
dynamics far from equilibrium at the microscopic level. To show this, let us
focus on local transitions between states. The generalized thermodynamic force
associated with each transition from $y$ to $x$ is defined as Gingrich  _et
al._ (2016)

$f_{xy}(t)\coloneqq\ln\frac{a_{xy}(t)}{a_{yx}(t)},$ (33)

which is the sum of the entropy changes in the system and environment derived
from the jump. Since $j_{xy}(t)$ is the probability current associated with
the transition from $y$ to $x$, defining the following coefficient is logical:

$m_{xy}(t)\coloneqq\frac{a_{xy}(t)-a_{yx}(t)}{\ln a_{xy}(t)-\ln
a_{yx}(t)}=\frac{j_{xy}(t)}{f_{xy}(t)}.$ (34)

Intuitively, $\\{m_{xy}(t)\\}$ characterize the responses of the probability
currents against the thermodynamic forces at the transition level. Notice that
$\\{m_{xy}(t)\\}$ are always nonnegative and symmetric (i.e.,
$m_{xy}(t)=m_{yx}(t)\geq 0$). Equation (34) can also be rewritten in the form
of

$j_{xy}(t)=m_{xy}(t)f_{xy}(t),$ (35)

which shows that the currents and thermodynamic forces can be linearly related
in terms of the coefficients $\\{m_{xy}(t)\\}$. Moreover, the entropy
production rate can be expressed in a quadratic form of the thermodynamic
forces $\\{f_{xy}(t)\\}$ as

$\sigma_{t}=\sum_{x>y}m_{xy}(t)f_{xy}(t)^{2}=\sum_{x>y}\sigma_{xy}(t).$ (36)

Here, we define the entropy production rate associated with each transition as
$\sigma_{xy}(t)\coloneqq m_{xy}(t)f_{xy}(t)^{2}$. Equations (35) and (36) have
the same algebraic forms as Eqs. (31) and (32), respectively, which suggests
that the coefficients $\\{m_{xy}(t)\\}$ play similar roles with the Onsager
coefficients for far-from-equilibrium systems.

In a weak-thermodynamic-force limit (i.e., $|f_{xy}(t)|\to 0$), the
coefficient $m_{xy}(t)$ reduces to the average dynamical activity between
states $x$ and $y$:

$m_{xy}(t)\to\frac{a_{xy}(t)+a_{yx}(t)}{2}.$ (37)

This is somewhat analogous to the Einstein relation on mobility in overdamped
Langevin dynamics (see Table 1). Note that the weak-thermodynamic-force limit
is defined at the microscopic state level and can be achieved via two routes:
the completely equilibrium limit in discrete systems and the continuous state
limit (e.g., the limit from the discrete hopping particle system to the
overdamped Fokker–Planck equation). In the continuous state limit, the
difference between neighboring states $x$ and $y$ is infinitesimal, and thus,
the force $f_{xy}(t)$ is also infinitesimal because $a_{xy}(t)/a_{yx}(t)\simeq
1$. We discuss the continuous state limit in the following subsection and show
that the right-hand side in Eq. (37) is proportional to the diffusion
coefficient. In general, the following relation holds for the coefficient
$m_{xy}(t)$:

$\displaystyle\sqrt{a_{xy}(t)a_{yx}(t)}\leq
m_{xy}(t)\leq\frac{a_{xy}(t)+a_{yx}(t)}{2}.$ (38)

It is thus natural to define the sum of $\\{m_{xy}(t)\\}$ over all
transitions:

$m_{t}\coloneqq\sum_{x>y}m_{xy}(t).$ (39)

For convenience, we refer to this term as dynamical state mobility throughout
this paper. This nomenclature derives from the analogy between microscopic
coefficients $\\{m_{xy}(t)\\}$ and the macroscopic mobility (see Table 1).

Table 1: Analogy between the dynamical state mobility and macroscopic mobility. Microscopic level | Macroscopic level
---|---
$j_{xy}=m_{xy}f_{xy}$ | $J=\mu F$
Einstein-like relation $|f_{xy}|\ll 1$ | Einstein relation $|F|\ll 1$
$m_{xy}=(a_{xy}+a_{yx})/2$ | $\mu=\beta D$

To clarify further the identity of $m_{t}$, let us consider a case in which
the system is attached to a single reservoir and satisfies the global detailed
balance condition. In this case, the master equation (20) can be written as
Van Vu and Hasegawa (2021a)

${\dot{p}_{t}}=\mathsf{K}_{t}{f_{t}},$ (40)

where $\mathsf{K}_{t}$ is a symmetric, positive semi-definite matrix given by

$\mathsf{K}_{t}\coloneqq\frac{1}{2}\sum_{x\neq y}m_{xy}(t)\mathsf{E}_{xy}.$
(41)

Here, ${f_{t}}=[f_{1}(t),...,f_{N}(t)]^{\top}$ with $f_{x}(t)=-\ln
p_{x}(t)+\ln p^{\rm eq}_{x}(t)$ and
$\mathsf{E}_{xy}=[e_{uv}]\in\mathbb{R}^{N\times N}$ is a matrix with
$e_{yy}=e_{xx}=1$, $e_{xy}=e_{yx}=-1$, and zeros in all other elements. The
quantities $\\{f_{x}(t)\\}$ are identified as the entropic thermodynamic
forces, which characterize how far the system is driven from the instantaneous
equilibrium state. Equation (40) represents the linear relations between the
rates ${\dot{p}_{t}}$ and forces ${f_{t}}$ through the symmetric matrix
$\mathsf{K}_{t}$. Furthermore, the total entropy production rate can be
written in a quadratic form as Van Vu and Hasegawa (2021a)

$\sigma_{t}=f_{t}^{\top}{\mathsf{K}_{t}}{f_{t}}.$ (42)

Therefore, Eqs. (40) and (42) can be viewed as far-from-equilibrium
counterparts of Eqs. (31) and (32), respectively. The matrix $\mathsf{K}_{t}$
is thus identified as the Onsager-like matrix. Because $\\{m_{xy}(t)\\}$ are
elements of $\mathsf{K}_{t}$, they can be regarded as the Onsager-like kinetic
coefficients for out-of-equilibrium systems. From the definition of
$\mathsf{K}_{t}$, we can easily verify that $m_{t}$ is exactly the sum of
diagonal elements of the Onsager-like matrix:

$m_{t}=\frac{1}{2}\tr{\mathsf{K}_{t}}.$ (43)

Therefore, $m_{t}$ can be identified as a kinetic term. The time integral of
the kinetic term $m_{t}$ can be considered as the kinetic cost of Markov jump
processes, defined by

$\mathcal{M}_{\tau}\coloneqq\int_{0}^{\tau}m_{t}\differential{t}=\tau\expectationvalue{m}_{\tau},$
(44)

where we define the time-averaged quantity for arbitrary time-dependent
quantity $x_{t}$ as

$\expectationvalue{x}_{\tau}\coloneqq\tau^{-1}\int_{0}^{\tau}x_{t}\differential{t}.$
(45)

By summing both sides of Eq. (38) for all $x>y$, we can prove that dynamical
state mobility is upper bounded by dynamical activity as

$m_{t}\leq\frac{a_{t}}{2}.$ (46)

The equality of Eq. (46) can be achieved in the equilibrium limit. It is thus
evident that $\mathcal{M}_{\tau}\leq\mathcal{A}_{\tau}/2$.

####  III.3.2 Continuous limit

Next, we investigate $m_{t}$ in the continuous limit. To this end, we consider
an overdamped Brownian particle trapped in a one-dimensional potential
$V_{t}(x)$. Let $x_{t}$ denote the position of the particle at time $t$. Then,
its dynamics is governed by the Langevin equation:

$\dot{x}_{t}=F_{t}(x_{t})+\sqrt{2D}\xi_{t},$ (47)

where $F_{t}(x)\coloneqq-\partial_{x}V_{t}(x)$ is the total force applied on
the particle, $\xi_{t}$ is a zero-mean Gaussian white noise with variance
$\expectationvalue{\xi_{t}\xi_{t^{\prime}}}=\delta(t-t^{\prime})$, and $D>0$
is the diffusion coefficient. Let $p_{t}(x)$ be the probability distribution
function of finding the particle in state $x$ at time $t$. Then, its time
evolution can be described by the Fokker–Planck equation:

$\dot{p}_{t}(x)=-\partial_{x}\quantity[F_{t}(x)p_{t}(x)-D\partial_{x}p_{t}(x)],$
(48)

where we set $\mu=1$ for simplicity. We now consider the discretization of the
Fokker–Planck equation (48) with space interval $\Delta x>0$ and define
$x_{n}\coloneqq n\Delta x$. By defining the probability distribution and
transition rates as

$\displaystyle p_{n}(t)$ $\displaystyle\coloneqq p_{t}(x_{n})\Delta x,$ (49)
$\displaystyle w_{n+1,n}(t)$ $\displaystyle\coloneqq\frac{D}{(\Delta
x)^{2}}\exp\quantity[\frac{V_{t}(x_{n})-V_{t}(x_{n+1})}{2D}]$
$\displaystyle\simeq\frac{F_{t}(x_{n})}{2\Delta x}+\frac{D}{(\Delta x)^{2}},$
(50) $\displaystyle w_{n-1,n}(t)$ $\displaystyle\coloneqq\frac{D}{(\Delta
x)^{2}}\exp\quantity[\frac{V_{t}(x_{n})-V_{t}(x_{n-1})}{2D}]$
$\displaystyle\simeq\frac{-F_{t}(x_{n})}{2\Delta x}+\frac{D}{(\Delta x)^{2}},$
(51)

we readily obtain the master equation:

$\displaystyle\dot{p}_{n}(t)$
$\displaystyle=w_{n,n-1}(t)p_{n-1}(t)+w_{n,n+1}(t)p_{n+1}(t)$
$\displaystyle-[w_{n+1,n}(t)+w_{n-1,n}(t)]p_{n}(t).$ (52)

We note here from Eqs. (50) and (51) that

$\displaystyle a_{n+1,n}(t)$
$\displaystyle\simeq\quantity[\frac{F_{t}(x_{n})}{2\Delta x}+\frac{D}{(\Delta
x)^{2}}]p_{n}(t),$ (53) $\displaystyle{a_{n+1,n}(t)+a_{n,n+1}(t)\over 2}$
$\displaystyle=Dp_{t}(x_{n})(\Delta x)^{-1}+O(1).$ (54)

The probability currents and thermodynamic forces can be calculated as

$\displaystyle j_{n+1,n}(t)$
$\displaystyle=F_{t}(x_{n})p_{t}(x_{n})-D\partial_{x}p_{t}(x_{n})+{O}(\Delta
x),$ (55) $\displaystyle f_{n+1,n}(t)$
$\displaystyle=\quantity[\frac{F_{t}(x_{n})}{D}-\frac{\partial_{x}p_{t}(x_{n})}{p_{t}(x_{n})}]\Delta
x+{O}(\Delta x^{2}).$ (56)

From these expressions, $m_{n+1,n}(t)$ can be calculated via the definition
(34), which gives

$m_{n+1,n}(t)\to\frac{a_{n+1,n}(t)+a_{n,n+1}(t)}{2}=Dp_{t}(x_{n})(\Delta
x)^{-1}+{O}(1).$ (57)

Equation (57) indicates that $m_{n+1,n}(t)$ converges to the value
$[a_{n+1,n}(t)+a_{n,n+1}(t)]/2$, and these correspond to the product of the
diffusion coefficient and probability distribution (see Table 1). Summing both
sides of Eq. (57) for all $n$, we obtain

$m_{t}\to\frac{a_{t}}{2}\simeq\sum_{n}Dp_{t}(x_{n})(\Delta x)^{-1}=D(\Delta
x)^{-2}.$ (58)

This implies that $m_{t}$ is proportional to $D$ as the scaling factor is
ignored. Thus, $m_{t}$ should play the same role as that of the diffusion
coefficient. It is noteworthy that the diffusion coefficient is exactly the
diagonal Onsager coefficient of overdamped Langevin processes in the linear-
response regime.

###  III.4 Thermodynamic uncertainty relation:
State mobility is crucial in nonequilibrium

Here, we describe an improved thermodynamic uncertainty relation, showing that
the kinetic cost of dynamical state mobility plays a critical role in
constraining the precision of time-integrated currents. For simplicity, we
consider a steady-state Markov jump process. The generalization to the case of
an arbitrary initial state and arbitrary time-dependent driving is
straightforward.

Let $\Gamma=\quantity{x_{0},(t_{1},x_{1}),\dots,(t_{K},x_{K})}$ be a
stochastic trajectory, in which the system is initially at state $x_{0}$ and
subsequently jumps from state $x_{k-1}$ to $x_{k}$ at time $t_{k}$ for each
$k=1,\dots,K$. For each stochastic trajectory $\Gamma$, we consider a time-
antisymmetric current $J$, defined as

$J(\Gamma)\coloneqq\sum_{k=1}^{K}\Upsilon_{x_{k}x_{k-1}}.$ (59)

Here, $\\{\Upsilon_{xy}\\}$ are arbitrary real coefficients satisfying
$\Upsilon_{xy}=-\Upsilon_{yx}$ for all $x$ and $y$. Examples of currents
include the entropy flux and heat flux to the environment by specific choices
of $\\{\Upsilon_{xy}\\}$. The precision of current $J$ can be quantified by
the square of the current mean divided by its variance
$\expectationvalue{J}^{2}/{\rm var}[J]$. The conventional thermodynamic
uncertainty relation Barato and Seifert (2015); Gingrich  _et al._ (2016) sets
an upper bound on the precision in terms of the total entropy production,
given by the following inequality:

$\frac{\expectationvalue{J}^{2}}{{\rm var}[J]}\leq\frac{\Sigma_{\tau}}{2}.$
(60)

Numerous studies have generalized this relation to other dynamics, from
classical to quantum Proesmans and den Broeck (2017); Brandner  _et al._
(2018); Hasegawa and Van Vu (2019a); Van Vu and Hasegawa (2019); Hasegawa and
Van Vu (2019b); Timpanaro  _et al._ (2019); Guarnieri  _et al._ (2019);
Carollo  _et al._ (2019); Dechant and Sasa (2020); Hasegawa (2020); Van Vu and
Hasegawa (2020a); Liu  _et al._ (2020); Koyuk and Seifert (2020); Wolpert
(2020); Miller  _et al._ (2021); Hasegawa (2021); Lee  _et al._ (2021); Pal
_et al._ (2021); Van Vu and Saito (2022b).

We improve the thermodynamic uncertainty relation by proving that the
precision of currents is upper bounded by the product of the thermodynamic and
kinetic costs, as follows:

$\frac{\expectationvalue{J}^{2}}{{\rm
var}[J]}\leq\eta\frac{\Sigma_{\tau}}{2},$ (61)

where $\eta=2\mathcal{M}_{\tau}/\mathcal{A}_{\tau}\leq 1$ can be regarded as
an efficiency of dynamical activity (see Appendix C.3 for the proof). The new
relation (61) is tighter than the conventional relation (60) and can be
saturated in the case of a one-dimensional random walk. Although the
conventional relation (60) implies that increasing dissipation is necessary to
achieve high precision, it does not ensure the converse; that is, increasing
dissipation is not sufficient for obtaining high precision of currents. This
can be explained through our relation, where the kinetic contribution $\eta$
appears in the bound in addition to the thermodynamic contribution. For
systems far from equilibrium, it is tedious that $\eta\ll 1$, which
equivalently indicates the unattainability of the conventional bound.

For a case in which the system is in an arbitrary initial state and is driven
by a time-dependent protocol, the derived relation can be analogously
generalized as

$\frac{[(\tau\partial_{\tau}-v\partial_{v})\expectationvalue{J}]^{2}}{{\rm
var}[J]}\leq\eta\frac{\Sigma_{\tau}}{2},$ (62)

where $v$ is a speed parameter of the control protocol Koyuk and Seifert
(2020).

Figure 2: Numerical illustration of the thermodynamic uncertainty relations.
(a) Schematic of the five-state Brownian clock and (b) numerical verification.
The current precision $\expectationvalue{J}^{2}/{\rm var}[J]$, new bound
$\eta\Sigma_{\tau}/2$, and conventional bound $\Sigma_{\tau}/2$ are indicated
by the solid, dashed, and dash-dotted lines, respectively. The forward rate
$k_{+}$ is varied, whereas the backward rate $k_{-}$ is fixed at $k_{-}=10$.
The operational time is fixed at $\tau=1$.

In the following, we exemplify the derived thermodynamic uncertainty relation
in a five-state Brownian clock Barato and Seifert (2016). The Brownian clock
is modeled as an inhomogeneous biased random walk on a ring with five states
[see Fig. 2(a)]. The clock’s pointer transits from state $x$ to
$x+1~{}(6\equiv 1)$ at rate $k_{+}>0$, whereas the backward rate is $k_{-}>0$.
The net number of cycles completed by the pointer characterizes the clock’s
time. In other words, time can be counted by a stochastic current $J$ that
increases by $1$ for each transition from state $5$ to $1$ and decreases by
$1$ for the reverse transition from state $1$ to $5$. The stochastic current
$J$ can be defined by setting $\Upsilon_{15}=1=-\Upsilon_{51}$ and
$\Upsilon_{xy}=0$ for others. Thus, the precision of the clock can be
quantified by $\expectationvalue{J}^{2}/{\rm var}[J]$.

We consider the clock operating in the stationary state. To investigate the
quality of the bounds, we fix the backward rate $k_{-}=10$ and vary the
forward rate $k_{+}\in(0,80]$. For each parameter setting, we calculate the
precision of the clock and the bounds of the conventional and new
thermodynamic uncertainty relations using the full counting statistics. The
numerical results are plotted in Fig. 2(b), which verify that the new bound is
always tighter than the conventional bound. In particular, the new bound is
tight even in the far-from-equilibrium regime.

##  IV Results on discrete optimal transport

Thus far, the problem of optimal transport has been discussed in terms of
continuous spaces. In the following, we focus on the case of discrete spaces
and explain the discrete Wasserstein distance. We then state our first
theorem, which connects the discrete Wasserstein distance to stochastic
thermodynamics of Markov jump processes.

###  IV.1 Optimal transport distance

The optimal transport problem in the discrete case is analogous to that in the
continuous case, except that we now deal with discrete $N$-dimensional
distributions. Given two discrete distributions $p^{A}=[p_{x}^{A}]$ and
$p^{B}=[p_{x}^{B}]$, the optimal means of transporting distribution $p^{A}$ to
$p^{B}$ with respect to a cost matrix $C=[c_{xy}]$ becomes the focus. Here,
$c_{xy}\geq 0$ denotes the cost of transporting a unit probability from
$p_{y}^{A}$ to $p_{x}^{B}$.

The transport problem can be formulated using a coupling $\pi=[\pi_{xy}]$
between the probability distributions $p^{A}$ and $p^{B}$. Specifically, $\pi$
is a joint probability distribution such that its marginal distributions
coincide with $p^{A}$ and $p^{B}$ (i.e., the following conditions are
satisfied for all $x$):

$p_{x}^{A}=\sum_{y=1}^{N}\pi_{yx}~{}\text{and}~{}p_{x}^{B}=\sum_{y=1}^{N}\pi_{xy}.$
(63)

Each coupling thus defines a transport plan: for each $x$ and $y$, we
transport an amount $\pi_{xy}$ from $p_{y}^{A}$ to $p_{x}^{B}$. Thus, the
discrete $L^{1}$-Wasserstein distance can be defined as the minimum transport
cost over all admissible couplings:

$W_{1}(p^{A},p^{B})\coloneqq\min_{\pi\in\Pi(p^{A},p^{B})}\sum_{x,y}c_{xy}\pi_{xy},$
(64)

where $\Pi(p^{A},p^{B})$ denotes a set of couplings between $p^{A}$ and
$p^{B}$. Once the cost matrix is provided, the discrete Wasserstein distance
can be efficiently computed using a linear programming method. In addition, as
long as the cost matrix satisfies

$c_{xy}+c_{yz}\geq c_{xz}$ (65)

for any $x$, $y$, and $z$, the resulting distance fulfills the triangle
inequality:

$W_{1}(p^{A},p^{B})+W_{1}(p^{B},p^{C})\geq W_{1}(p^{A},p^{C}).$ (66)

We observe that the definition of the Wasserstein distance depends on the cost
matrix. In other words, each matrix of transport costs induces a
quantitatively different measure of distance. Evidently, an infinite number of
approaches can be used to choose the cost matrix. In the following, we
consider the cost matrix and corresponding Wasserstein distance defined on the
basis of a graph.

Let $\mathcal{G}(V,E)$ denote an undirected graph, where $V$ and $E$ are the
sets of vertices and unordered edges, respectively. Then, any microscopically
reversible Markov jump process can be associated with an undirected graph, in
which $V=\\{1,\dots,N\\}$ is the set of all states of the Markov jump process,
and two vertices $x$ and $y$ are connected by an edge $(x,y)\in E$ if the
transition between $x$ and $y$ is allowed. A jump process that has a unique
steady state can be described by a connected graph; that is, for any unordered
pair $(x,y)$, a sequence of vertices $P=[v_{1},\dots,v_{k}]$ always exists
such that $x=v_{1}$, $y=v_{k}$, and $(v_{i},v_{i+1})\in E$ for all $1\leq
i\leq k-1$. A subgraph $\tilde{\mathcal{G}}$ of a graph $\mathcal{G}$ is one
whose edge set is a subset of that of $\mathcal{G}$. In other words,
$\tilde{\mathcal{G}}$ can be obtained from $\mathcal{G}$ by removing some
edges. This is equivalent to setting some transition rates of the Markov jump
process to zero. For convenience, hereafter, the underlying graph structure of
a Markov jump process is referred to as its topology.

Figure 3: Example of the Wasserstein distance defined based on a graph
consisting of $5$ vertices and $5$ edges. Given the topology
$\mathcal{G}(V,E)$, the shortest-path distances $\\{d_{xy}\\}$ can be
calculated, from which the Wasserstein distance can be defined.

Given the topology of a jump process, we now can define the transport cost
matrix. For each path $P$, let ${\rm len}(P)$ denote its length, which is the
number of edges contained in the path. The shortest-path distance from vertex
$x$ to vertex $y$ can be defined as

$d_{yx}\coloneqq\min_{P}\quantity{{\rm len}(P)},$ (67)

where the minimum is over all paths that connect $x$ to $y$. For undirected
graphs, clearly $d_{xy}=d_{yx}$. From the definition of the distances
$\\{d_{yx}\\}$, we can easily verify that the triangle inequality is
fulfilled, that is,

$d_{xy}+d_{yz}\geq d_{xz}$ (68)

for arbitrary vertices $x$, $y$, and $z$. Employing these shortest-path
distances as the transport costs (i.e., $c_{xy}=d_{xy}$), we hereafter
exclusively focus on the following discrete Wasserstein distance:

$\mathcal{W}_{1}(p^{A},p^{B})\coloneqq\min_{\pi\in\Pi(p^{A},p^{B})}\sum_{x,y}d_{xy}\pi_{xy}.$
(69)

It is noteworthy that only static information (i.e., graph connectivity) is
required to define the Wasserstein distance at this time (see Fig. 3 for
illustration).

In a general case, since $d_{xy}\geq 1$ for all $x\neq y$, the Wasserstein
distance is always lower bounded by the total variation distance:

$\mathcal{W}_{1}(p^{A},p^{B})\geq\mathcal{T}(p^{A},p^{B})\coloneqq\frac{1}{2}\sum_{x}|p_{x}^{A}-p_{x}^{B}|.$
(70)

However, when the underlying graph is fully connected (i.e., the transition
between any two states is admissible), the shortest-path distance becomes

$d_{xy}=1-\delta_{xy},$ (71)

where $\delta_{xy}$ is the Kronecker delta of $x$ and $y$. In this case, the
Wasserstein distance coincides with the total variation distance (see Appendix
D.1 for the proof):

$\mathcal{W}_{1}(p^{A},p^{B})=\mathcal{T}(p^{A},p^{B}).$ (72)

###  IV.2 Thermodynamic interpretation

With the above setup, we can now state the results. Conventionally, the
discrete Wasserstein distance is defined mathematically using the transport
cost matrix based on only the shortest-path distances. However, in the
following theorem, we explicitly show an intimate relationship between the
discrete Wasserstein distance defined in Eq. (69) and the stochastic
thermodynamics of Markov jump processes.

######  Theorem 1.

The Wasserstein distance based on a topology $\mathcal{G}(V,E)$ can be written
in variational forms as

$\displaystyle\mathcal{W}_{1}(p^{A},p^{B})$
$\displaystyle=\min_{\mathsf{W}_{t}}{\int_{0}^{\tau}\sqrt{\sigma_{t}m_{t}}\differential{t}}$
(73)
$\displaystyle=\min_{\mathsf{W}_{t}}\sqrt{\Sigma_{\tau}\mathcal{M}_{\tau}}.$
(74)

Here, the minimum is taken over all transition rate matrices
$\\{\mathsf{W}_{t}\\}_{0\leq t\leq\tau}$ which satisfy the master equation
(20) with the boundary conditions ${p_{0}}={p^{A}}$ and ${p_{\tau}}={p^{B}}$
and induce subgraphs of $\mathcal{G}(V,E)$ for all times.

Theorem 1 is the first central result, and its sketch proof is given in the
following. Note that the minimization is over all transition rate matrices
which are microscopically reversible and induce subgraphs of
$\mathcal{G}(V,E)$ for all times. This means that the transition rate between
two states $x$ and $y$ must be fixed to zero for all times if no edge exists
between vertices $x$ and $y$ in the graph $\mathcal{G}$. Otherwise, as long as
an edge exists between the vertices, the transition rate can be arbitrarily
controlled. Notably, the equality of Eq. (73) can always be ensured with
dynamics that satisfy the global detailed balance condition (see Appendix D.3
for the proof).

###### Proof.

Here, we provide an outline of the proof; see Appendix D.2 for a detailed
derivation. The proof strategy can be mainly divided into the following two
steps. We first prove that

$\mathcal{W}_{1}(p^{A},p^{B})\leq\int_{0}^{\tau}\sqrt{\sigma_{t}m_{t}}\differential{t}\leq\sqrt{\Sigma_{\tau}\mathcal{M}_{\tau}}$
(75)

holds for all admissible Markovian dynamics that transform $p^{A}$ into
$p^{B}$, and we then construct a specific process that attains the equality.
Since the second inequality in Eq. (75) is simply a consequence of the
Cauchy–Schwarz inequality, Eq. (75) can be proved after we verify the first
inequality. This can be done by proving the following relation:

$\mathcal{W}_{1}(p^{A},p^{B})\leq\int_{0}^{\tau}\sum_{x>y}|j_{xy}(t)|\differential{t}\leq\int_{0}^{\tau}\sqrt{\sigma_{t}m_{t}}\differential{t}.$
(76)

The second inequality in Eq. (76) can be derived using the Cauchy–Schwarz
inequality. Thus, we need only show the first inequality in Eq. (76). To this
end, we map the Wasserstein distance to the minimum cost of the minimum cost
flow problem in the field of graph theory. For this problem, we can show that
the Wasserstein distance is exactly the optimal flow cost. Moreover, the
Markov jump process also yields an admissible solution to the flow problem
with the cost $\int_{0}^{\tau}\sum_{x>y}|j_{xy}(t)|\differential{t}$.
Consequently, the first inequality in Eq. (76) is proved. Finally, we
inversely translate the optimal solution of the minimum cost flow problem to
construct a Markov jump process that attains the equality of Eq. (75). ∎

Some remarks regarding Thm. 1 are in order. First, Eqs. (73) and (74) provide
a thermodynamic interpretation of the discrete Wasserstein distance; that is,
$\mathcal{W}_{1}$ equals the minimum product of the thermodynamic and kinetic
costs over all admissible Markovian dynamics that transform the source
distribution into the target one. From a different perspective, it can be
regarded as a trade-off between irreversible entropy production and dynamical
state mobility; that is, to transform a probability distribution into another
one, both $\Sigma_{\tau}$ and $\mathcal{M}_{\tau}$ cannot be simultaneously
small:

$\Sigma_{\tau}\mathcal{M}_{\tau}\geq{\mathcal{W}_{1}(p^{A},p^{B})^{2}}.$ (77)

In other words, either the thermodynamic or kinetic cost must be sacrificed to
achieve a feasible state transformation.

Second, we show that the discrete Wasserstein distance has analogous
thermodynamic properties with the continuous $L^{2}$-Wasserstein distance. To
this end, we rewrite Eq. (74) in the following form:

$\mathcal{W}_{1}(p^{A},p^{B})=\min_{\mathsf{W}_{t}}\sqrt{\bar{D}\tau\Sigma_{\tau}},$
(78)

where we define time-averaged state mobility
$\bar{D}\coloneqq\expectationvalue{m}_{\tau}$. As previously shown, the
kinetic term $m_{t}$ reduces to the diffusion coefficient in the continuous
limit. Therefore, its time-averaged quantity $\bar{D}$ here plays the same
role as the diffusion coefficient $D$ does in the continuous case.
Consequently, Eq. (78) can be regarded as the discrete analog of the
Benamou–Brenier formula (16) known for the $L^{2}$-Wasserstein distance.
Equation (78) immediately derives a lower bound on irreversible entropy
production:

$\Sigma_{\tau}\geq\frac{\mathcal{W}_{1}(p^{A},p^{B})^{2}}{\bar{D}\tau}.$ (79)

This bound is tight and can always be attained for an arbitrary pair of
distributions. In other words, the minimum entropy production among all
feasible dynamics that have the same value of $\bar{D}$ is given by the
Wasserstein distance:

$\min_{\expectationvalue{m}_{\tau}=\bar{D}}\Sigma_{\tau}=\frac{\mathcal{W}_{1}(p^{A},p^{B})^{2}}{\bar{D}\tau}.$
(80)

Equations (79) and (80) can be considered discrete analogs of Eqs. (17) and
(19), respectively.

Figure 4: Schematic of the thermodynamic structure of minimum dissipation in
the space of transition rates. Solid lines represent dynamics that have the
same value of the average state mobility $\expectationvalue{m}_{\tau}$. Black
circles depict extreme points in which the minimum dissipation is attained,
provided that the average state mobility is fixed. In the presence of other
constraints on transition rates, the minimum dissipation is lower bounded by
the Wasserstein distance and maximum value of $\expectationvalue{m}_{\tau}$.

Third, Thm. 1 provides insights into the problem of minimizing entropy
production in discrete Markovian dynamics. Previous studies have shown that
the irreversible entropy production required to transform the initial into the
final distribution can be arbitrarily small if no constraint is placed on the
transition rates Van Vu and Hasegawa (2021b); Remlein and Seifert (2021);
Dechant (2022). Equation (80) also confirms this, where the minimum entropy
production depends on $\bar{D}$ and can be arbitrarily adjusted. Theorem 1
suggests that fixing $\bar{D}$ is a reasonable constraint under which the
minimum entropy production is determined by the Wasserstein distance, as in
the continuous case. Notably, as Appendix C.1 shows, $\bar{D}$ can be fixed to
an arbitrary positive value, indicating the flexibility of the optimization
problem. The optimal control protocol can also be constructed from optimal
coupling, as shown in our proof of Thm. 1. Moreover, for arbitrary
$\bar{D}>0$, the minimum entropy production in Eq. (80) can always be attained
using a system with conservative forces (see Appendix D.4 for the proof). It
is noteworthy that the discussion thus far has not imposed any other
constraints on the transition rates, except for fixing
$\expectationvalue{m}_{\tau}$. Therefore, if some additional constraints are
placed on the transition rates, such as upper or lower bounds on the magnitude
of transition rates, it may not be the case. Nevertheless, a lower bound can
be derived for the minimum entropy production in this case. Let $\bar{D}_{\rm
max}$ be the maximum of $\expectationvalue{m}_{\tau}$ among all processes that
transform distribution $p^{A}$ into $p^{B}$ under these constraints. Then, the
minimum entropy production is lower bounded by the Wasserstein distance and
$\bar{D}_{\rm max}$ as

$\Sigma_{\tau}\geq\frac{\mathcal{W}_{1}(p^{A},p^{B})^{2}}{\bar{D}_{\rm
max}\tau}.$ (81)

The thermodynamic structure of minimum dissipation is illustrated in Fig. 4.

Finally, as shown in Sec. VI, the variational formulas (73) and (74) have
crucial implications for thermodynamic speed limits and thermodynamic cost of
information erasure at arbitrary temperatures.

Note that each topology induces a different Wasserstein metric. In the
following, we consider a specific topology and discuss the relevance of the
variational formulas (73) and (74). For other common topologies, see Appendix
D.5.

One common topology is one-dimensional nearest-neighbor, in which a jump
between states $x$ and $y$ is admitted if and only if $|x-y|=1$. This topology
is relevant to Brownian random walks and the discretization of a one-
dimensional Langevin system. The shortest-path distances in this topology can
be readily calculated as

$d_{xy}=|x-y|.$ (82)

Because this cost matrix is the discrete analog of the cost function
$c(x,y)=|x-y|$ used in the definition of the continuous $L^{1}$-Wasserstein
distance $W_{1}$, the discrete Wasserstein distance $\mathcal{W}_{1}$ should
be reduced to $W_{1}$ in the continuous limit. Let $\Delta x$ be the space
interval. Then, $\mathcal{W}_{1}$ converges to $W_{1}$ as $\Delta x\to 0$ and
$N\to\infty$:

$\mathcal{W}_{1}(p^{A},p^{B})\Delta x\xrightarrow[\Delta x\to
0]{N\to\infty}W_{1}(p^{A},p^{B}).$ (83)

In addition, as shown in the proof of Thm. 1, $\mathcal{W}_{1}(p^{A},p^{B})$
can be expressed in terms of the probability currents as

$\mathcal{W}_{1}(p^{A},p^{B})=\min_{\mathsf{W}_{t}}{\int_{0}^{\tau}\sum_{x>y}|j_{xy}(t)|\differential{t}}.$
(84)

Equation (84) implies that the discrete Wasserstein distance is equal to the
minimum sum of absolute probability currents. In the case considered here, the
equality (84) reads

$\mathcal{W}_{1}(p^{A},p^{B})=\min_{\mathsf{W}_{t}}{\int_{0}^{\tau}\sum_{x=1}^{N-1}|j_{x+1,x}(t)|\differential{t}}.$
(85)

Noticing that $\sum_{x=1}^{N-1}|j_{x+1,x}(t)|\Delta
x\to\int_{\mathbb{R}}|j_{t}(x)|\differential{x}$ as $\Delta x\to 0$, we obtain
the following limit:

$\mathcal{W}_{1}(p^{A},p^{B})\Delta x\xrightarrow[\Delta x\to
0]{N\to\infty}\min_{j_{t}}\int_{0}^{\tau}\int_{\mathbb{R}}|j_{t}(x)|\differential{x}\differential{t}.$
(86)

Combining Eqs. (83) and (86) gives the following relation:

$\displaystyle W_{1}(p^{A},p^{B})$
$\displaystyle=\min_{j_{t}}{\int_{0}^{\tau}\int_{\mathbb{R}}|j_{t}(x)|\differential{x}\differential{t}},$
(87)

where $j_{t}(x)$ is subject to the equation
$\dot{p}_{t}(x)=-\partial_{x}j_{t}(x)$. Notably, Eq. (87) is exactly the
Benamou–Brenier formula for the continuous $L^{1}$-Wasserstein distance in the
one-dimensional case Chen  _et al._ (2017). Therefore, we can conclude that
Eq. (84) provides a unified generalization of the Benamou–Brenier formula for
the $L^{1}$-Wasserstein distance.

Let us now consider the discretization of one-dimensional Langevin dynamics,
that is, Markov jump processes with transition rates specified as in Eqs. (50)
and (51). For these jump processes, the dynamical state mobility reduces to
the diffusion coefficient in the continuous limit (i.e.,
$\mathcal{M}_{\tau}(\Delta x)^{2}\to D\tau$ as $\Delta x\to 0$). In the
continuous case, the Benamou–Brenier formula (87) can be expressed as

$\displaystyle W_{1}(p^{A},p^{B})$
$\displaystyle=\min_{j_{t}}{\int_{0}^{\tau}\int_{\mathbb{R}}\sqrt{\sigma_{t}(x)m_{t}(x)}\differential{x}\differential{t}}$
$\displaystyle\leq\min_{F_{t}}\int_{0}^{\tau}\sqrt{D\sigma_{t}}\differential{t}=\min_{F_{t}}\sqrt{D\tau\Sigma_{\tau}},$
(88)

where we define $m_{t}(x)\coloneqq Dp_{t}(x)$ and the local entropy production
rate $\sigma_{t}(x)\coloneqq j_{t}(x)^{2}/[Dp_{t}(x)]$. From Thm. 1 and Eq.
(88), we can conclude that in the continuous limit, the equality in Thm. 1
might not be achieved with Markov jump processes whose transition rates are
expressed as in Eqs. (50) and (51). This shows the difference between the
discrete and continuous cases, where the discrete case has more degrees of
freedom than the continuous case.

Theorem 1 characterizes the discrete Wasserstein distance $\mathcal{W}_{1}$ in
terms of the thermodynamic and kinetic costs associated with Markovian
dynamics. In Appendix D.6, we show that Thm. 1 has some useful corollaries
that not only provide alternative expressions for $\mathcal{W}_{1}$ but also
lead to stringent bounds for thermodynamic speed limits. Using other
combinations of irreversible entropy production, pseudo entropy production,
and dynamical activity, the discrete Wasserstein distance $\mathcal{W}_{1}$
can be expressed in similar variational forms.

##  V Quantum generalization

We next generalize our framework to the quantum case. We first briefly
introduce quantum thermodynamics of Markovian open quantum dynamics described
by the Lindblad equations and define a quantum analog of dynamical state
mobility. Then, we define a quantum Wasserstein distance and derive analogous
variational formulas for the quantum Wasserstein distance in terms of
thermodynamic cost.

###  V.1 Markovian open quantum dynamics

We consider a finite-dimensional open quantum system, which is attached to
single or multiple thermal reservoirs. In the weak-coupling limit, the time
evolution of the reduced density matrix can be described by the Lindblad
master equation Lindblad (1976),

$\dot{\varrho}_{t}=\mathcal{L}_{t}(\varrho_{t})\coloneqq-i[H_{t},\varrho_{t}]+\sum_{k}\mathcal{D}[L_{k}(t)]\varrho_{t},$
(89)

where $H_{t}$ is the time-dependent Hamiltonian, $\mathcal{D}$ is the
dissipator given by $\mathcal{D}[L]\varrho\coloneqq L\varrho
L^{\dagger}-\quantity{L^{\dagger}L,\varrho}/2$, and $L_{k}(t)$ are jump
operators. $[\circ,\star]$ and $\\{\circ,\star\\}$ denote the commutator and
anticommutator of the two operators, respectively. Hereafter, we set the
Planck constant to unity $\hbar=1$. To guarantee thermodynamically consistent
dynamics, we assume that the jump operators satisfy the local detailed balance
condition Horowitz and Parrondo (2013); Manzano  _et al._ (2018); that is,
they come in pairs $(k,k^{\prime})$ such that

$L_{k}(t)=e^{s_{k}(t)/2}L_{k^{\prime}}(t)^{\dagger},$ (90)

where $s_{k}(t)=-s_{k^{\prime}}(t)$ denotes the entropy change in the
environment due to the jump operator $L_{k}(t)$. In the case of a single
reservoir at inverse temperature $\beta$, we can write
$s_{k}(t)=\beta\omega_{k}(t)$, where $\omega_{k}(t)$ is the energy change
associated with the $k$th jump.

###  V.2 Entropy production and dynamical activity

Given the previous setup, we can now introduce quantum entropy production and
dynamical activity. Similar to the classical case, irreversible entropy
production can be defined as the sum of entropy changes in the system and
environment as

$\Sigma_{\tau}\coloneqq\Delta S_{\rm sys}+\Delta S_{\rm env},$ (91)

where $\Delta S_{\rm sys}\coloneqq S(\varrho_{\tau})-S(\varrho_{0})$ is the
difference in the von Neumann entropy $S(\varrho)=-\tr{\varrho\ln\varrho}$ of
the system and $\Delta S_{\rm env}$ denotes environmental entropy production,
given by Horowitz and Parrondo (2013); Manzano  _et al._ (2018)

$\Delta S_{\rm
env}\coloneqq\int_{0}^{\tau}\sum_{k}\tr{L_{k}(t)\varrho_{t}L_{k}^{\dagger}(t)}s_{k}(t)\differential{t}.$
(92)

With this definition, we can prove that $\Sigma_{\tau}$ is always nonnegative,
which implies the second law of thermodynamics. For the case of a single
reservoir and the jump operators that characterize transitions between energy
eigenstates (i.e., $[L_{k}(t),H_{t}]=\omega_{k}(t)L_{k}(t)$), the entropy
production of the environment reduces exactly to the conventional form Alicki
(1979),

$\Delta S_{\rm
env}=-\beta\int_{0}^{\tau}\tr{H_{t}\dot{\varrho}_{t}}\differential{t}.$ (93)

Quantum dynamical activity can be analogously defined as in the classical
case. The frequency of jumps at time $t$ can be quantified as

$a_{t}\coloneqq\sum_{k}\tr{L_{k}(t)\varrho_{t}L_{k}^{\dagger}(t)},$ (94)

and the average total number of jumps can be calculated as
$\mathcal{A}_{\tau}\coloneqq\int_{0}^{\tau}a_{t}\differential{t}$. Quantum
dynamical activity characterizes the thermalization rate of thermodynamic
processes. In addition, it has been shown that quantum dynamical activity
constrains the precision of generic counting observables and their first
passage time in quantum jump processes Hasegawa (2020); Van Vu and Saito
(2022b).

It is convenient to alternatively express entropy production and dynamical
activity defined in Eqs. (91) and (94), respectively. Let
$\varrho_{t}=\sum_{x}p_{x}(t)\outerproduct{x_{t}}{x_{t}}$ be the spectral
decomposition of the density matrix $\varrho_{t}$. We then define transition
rates between eigenbasis as
$w_{k}^{xy}(t)\coloneqq|\matrixelement{x_{t}}{L_{k}(t)}{y_{t}}|^{2}\geq 0$.
Notice that $w_{k}^{xy}(t)=e^{s_{k}(t)}w_{k^{\prime}}^{yx}(t)$. Taking the
time derivative of $p_{x}(t)=\matrixelement{x_{t}}{\varrho_{t}}{x_{t}}$, we
obtain the following master equation for the distribution $\\{p_{x}(t)\\}$:

$\dot{p}_{x}(t)=\sum_{k}\sum_{y(\neq
x)}\quantity[w_{k}^{xy}(t)p_{y}(t)-w_{k}^{yx}(t)p_{x}(t)].$ (95)

Analogous to the classical case, we define

$\displaystyle a_{k}^{xy}(t)$ $\displaystyle\coloneqq w_{k}^{xy}(t)p_{y}(t),$
(96) $\displaystyle j_{k}^{xy}(t)$ $\displaystyle\coloneqq
w_{k}^{xy}(t)p_{y}(t)-w_{k^{\prime}}^{yx}(t)p_{x}(t).$ (97)

Using these probability currents, we can write the master equation as

$\dot{p}_{x}(t)=\sum_{k}\sum_{y(\neq x)}j_{k}^{xy}(t).$ (98)

We emphasize that the classical-like master equation (95) is rigorously
derived from Eq. (89). This equation is introduced only for the proof
convenience of several properties that the dynamics (89) possesses. After some
simple manipulations, we can prove that the entropy production rate
${\sigma}_{t}\coloneqq\dot{\Sigma}_{t}$ can be analytically expressed as (see
Appendix E.1 for the proof)

$\sigma_{t}=\frac{1}{2}\sum_{k}\sum_{x,y}j_{k}^{xy}(t)\ln\frac{w_{k}^{xy}(t)p_{y}(t)}{w_{k^{\prime}}^{yx}(t)p_{x}(t)}.$
(99)

Besides, plugging the spectral decomposition of $\varrho_{t}$ and inserting
$\mathbb{1}=\sum_{x}\outerproduct{x_{t}}{x_{t}}$ into Eq. (94), the dynamical
activity rate can also be expressed as

$a_{t}=\sum_{k}\sum_{x,y}w_{k}^{xy}(t)p_{y}(t)=\sum_{k}\sum_{x,y}a_{k}^{xy}(t).$
(100)

Note that both $\sigma_{t}$ and $a_{t}$, which can be written in terms of the
transition rates of the master equation (95), are the entropy production and
dynamical activity rates associated with the Lindblad dynamics (89),
respectively.

###  V.3 Quantum dynamical state mobility

Analogous with the classical case, the quantum analog of dynamical state
mobility can be defined as

$m_{t}\coloneqq\frac{1}{2}\sum_{k}e^{-s_{k}(t)/2}\expectationvalue{L_{k}(t)^{\dagger},\llbracket\varrho_{t}\rrbracket_{s_{k}(t)}(\mathcal{P}_{t}[L_{k}(t)^{\dagger}])},$
(101)

where $\expectationvalue{X,Y}\coloneqq\tr{X^{\dagger}Y}$ denotes the scalar
inner product, $\mathcal{P}_{t}$ is a super-operator given by
$\mathcal{P}_{t}[X]\coloneqq
X-\sum_{x}\matrixelement{x_{t}}{X}{x_{t}}\outerproduct{x_{t}}{x_{t}}$, and the
tilted operator $\llbracket\phi\rrbracket_{\theta}(X)$ is defined for
arbitrary density matrix $\phi$, real number $\theta$, and linear operator $X$
as

$\llbracket\phi\rrbracket_{\theta}(X)\coloneqq
e^{-\theta/2}\int_{0}^{1}e^{\theta u}\phi^{u}X\phi^{1-u}\differential{u}.$
(102)

The quantum kinetic cost can be analogously defined as

$\mathcal{M}_{\tau}\coloneqq\int_{0}^{\tau}m_{t}\differential{t}.$ (103)

From the mathematical definition in Eq. (101), interpreting the term $m_{t}$
as a kinetic term may not be intuitive. In the following, we provide the
physical interpretations of $m_{t}$ from two perspectives.

First, by focusing on the master equation of the distribution
$\\{p_{x}(t)\\}$, we can show that $m_{t}$ is equal to the dynamical state
mobility associated with Markovian jump dynamics (95):

$m_{t}=\sum_{k}\sum_{x>y}\frac{a_{k}^{xy}(t)-a_{k^{\prime}}^{yx}(t)}{\ln
a_{k}^{xy}(t)-\ln a_{k^{\prime}}^{yx}(t)}.$ (104)

Note that by applying the inequality (38) to Eq. (104), we can readily prove
that $m_{t}$ is upper bounded by the dynamical activity,

$\displaystyle m_{t}$
$\displaystyle\leq\sum_{k}\sum_{x>y}\frac{a_{k}^{xy}(t)+a_{k^{\prime}}^{yx}(t)}{2}$
$\displaystyle=\frac{1}{2}\sum_{k}\sum_{x\neq y}a_{k}^{xy}(t)$
$\displaystyle\leq\frac{a_{t}}{2}.$ (105)

Second, let us consider the case of a single reservoir, in which the jump
operators satisfy $[L_{k}(t),H_{t}]=\omega_{k}(t)L_{k}(t)$. In this case, the
thermal state $\varrho_{t}^{\rm eq}\coloneqq e^{-\beta H_{t}}/\tr e^{-\beta
H_{t}}$ is always the instantaneous equilibrium state (i.e.,
$\mathcal{L}_{t}(\varrho_{t}^{\rm eq})=0$). Note that the Lindblad master
equation (89) can be rewritten as Van Vu and Hasegawa (2021a)

$\dot{\varrho}_{t}=\mathcal{U}_{\varrho_{t}}(t,f_{t})+\mathcal{O}_{\varrho_{t}}(t,f_{t}),$
(106)

where $f_{t}\coloneqq-\ln\varrho_{t}+\ln\varrho_{t}^{\rm eq}$ is the quantum
thermodynamic force and $\mathcal{U}_{\phi}(t,X)$ and
$\mathcal{O}_{\phi}(t,X)$ are time-dependent super-operators defined,
respectively, as

$\displaystyle\mathcal{U}_{\phi}(t,X)$ $\displaystyle\coloneqq
i\beta^{-1}[X,\phi],$ (107) $\displaystyle\mathcal{O}_{\phi}(t,X)$
$\displaystyle\coloneqq\frac{1}{2}\sum_{k}e^{-s_{k}(t)/2}[L_{k}(t),\llbracket\phi\rrbracket_{s_{k}(t)}([L_{k}(t)^{\dagger},X])].$
(108)

The super-operators $\mathcal{U}$ and $\mathcal{O}$ characterize the unitary
and dissipative parts of Lindblad dynamics, respectively. They linearly relate
the rate of the density matrix to the thermodynamic force. In addition, the
entropy production rate can be written in a quadratic form of the
thermodynamic force as Van Vu and Hasegawa (2021a)

$\sigma_{t}=\expectationvalue{f_{t},\mathcal{O}_{\varrho_{t}}(t,f_{t})}.$
(109)

Since Eqs. (106) and (109) are analogous to Eqs. (40) and (42) in the
classical case, respectively, the super-operator $\mathcal{O}$ can be regarded
as a quantum Onsager-like super-operator.

We now investigate the relationship between $m_{t}$ and the Onsager-like
super-operator $\mathcal{O}$. To this end, we employ the vectorization of a
linear operator $X$ as

$X=\sum_{i,j}x_{ij}\outerproduct{i}{j}\to|X\rangle\\!\rangle=\sum_{i,j}x_{ij}\ket{i}\otimes\ket{j}.$
(110)

Using this representation, we can rewrite the Lindblad master equation (106)
as

$|\dot{\varrho}_{t}\rangle\\!\rangle=\mathsf{U}_{t}|f_{t}\rangle\\!\rangle+\mathsf{O}_{t}|f_{t}\rangle\\!\rangle,$
(111)

where the linear matrices $\mathsf{U}_{t}$ and $\mathsf{O}_{t}$ are defined as

$\displaystyle\mathsf{U}_{t}$ $\displaystyle\coloneqq
i\beta^{-1}\quantity(\mathbb{1}\otimes\varrho_{t}^{\top}-\varrho_{t}\otimes\mathbb{1}),$
(112) $\displaystyle\mathsf{O}_{t}$
$\displaystyle\coloneqq\frac{1}{2}\sum_{k}e^{-s_{k}(t)}\int_{0}^{1}e^{s_{k}(t)u}\mathsf{O}_{k}(t,u)\differential{u}.$
(113)

Here, $\top$ denotes the matrix transpose and $\mathsf{O}_{k}(t,x)$ is given
by

$\displaystyle\mathsf{O}_{k}(t,u)$ (114) $\displaystyle\coloneqq
L_{k}(t)\varrho_{t}^{u}L_{k}(t)^{\dagger}\otimes(\varrho_{t}^{1-u})^{\top}+\varrho_{t}^{u}\otimes(L_{k}(t)^{\dagger}\varrho_{t}^{1-u}L_{k}(t))^{\top}$
$\displaystyle-
L_{k}(t)\varrho_{t}^{u}\otimes(L_{k}(t)^{\dagger}\varrho_{t}^{1-u})^{\top}-\varrho_{t}^{u}L_{k}(t)^{\dagger}\otimes(\varrho_{t}^{1-u}L_{k}(t))^{\top}.$

Note that $\mathsf{U}_{t}$ and $\mathsf{O}_{t}$ are the matrix representations
of the super-operators $\mathcal{U}$ and $\mathcal{O}$, respectively. Simple
algebraic calculations show that the term $m_{t}$ can be related to the
diagonal elements of the Onsager-like matrix $\mathsf{O}_{t}$ as (see Appendix
E.2 for the proof)

$m_{t}=\frac{1}{2}\sum_{x}\bra{x_{t}}\otimes\ket{x_{t}}^{\top}\mathsf{O}_{t}\ket{x_{t}}\otimes\bra{x_{t}}^{\top}.$
(115)

In this sense, $m_{t}$ can be regarded as a quantum kinetic term.

###  V.4 Quantum optimal transport distance and thermodynamic interpretation

Although the classical Wasserstein distance is well formulated and studied,
its quantum version remains under development. Several quantum generalizations
of the Wasserstein distance have been proposed Carlen and Maas (2014); Chen
_et al._ (2017, 2018); Duvenhage (2020); Palma  _et al._ (2021); Van Vu and
Hasegawa (2021a); Friedland  _et al._ (2021). However, defining the quantum
$L^{1}$-Wasserstein distance unambiguously by directly generalizing the
classical distance has been shown to be impossible Agredo and Fagnola (2017).

By a naive extension using quantum coupling, a quantum optimal transport
distance can be defined as

$W_{q}(\varrho^{A},\varrho^{B})\coloneqq\min_{\varrho^{AB}\in\Pi(\varrho^{A},\varrho^{B})}\tr{C\varrho^{AB}},$
(116)

where the coupling $\Pi(\varrho^{A},\varrho^{B})$ denotes the set of density
matrices $\varrho^{AB}$ defined over the Hilbert space
$\mathcal{H}\otimes\mathcal{H}$ and satisfy
$\tr_{B}{\varrho^{AB}}=\varrho^{A}$ and $\tr_{A}{\varrho^{AB}}=\varrho^{B}$,
and $C$ is a cost matrix that must be properly chosen to guarantee that
$W_{q}$ is a distance. In the classical case, the total variation distance is
a classical Wasserstein distance with an appropriate choice of the cost matrix
$C$. It is thus natural to ask whether a cost matrix $C$ exists such that the
quantum version of the total variation distance (i.e., the trace distance) can
be represented as a quantum Wasserstein distance defined in Eq. (116).
Unfortunately, Ref. Yu  _et al._ (2018) showed that the trace distance could
not be expressed in terms of this type of Wasserstein distance. In other
words, for any choice of the cost matrix $C$, density matrices $\varrho^{A}$
and $\varrho^{B}$ always exist such that the distance $W_{q}$ defined in Eq.
(116) differs from the trace distance:

$W_{q}(\varrho^{A},\varrho^{B})\neq\frac{1}{2}\|\varrho^{A}-\varrho^{B}\|_{1}\eqqcolon\mathcal{T}(\varrho^{A},\varrho^{B}).$
(117)

Our aim is to relate quantum optimal transport distances and dissipation in
Lindblad dynamics. Note that Lindblad dynamics consist of a non-dissipative
unitary part and dissipative Lindblad part. Both parts jointly contribute to
the time evolution of the system’s density matrix. In the vanishing coupling
limit, irreversible entropy production becomes zero, whereas the distance
$W_{q}(\varrho_{0},\varrho_{\tau})$ may be positive since
$\varrho_{0}\neq\varrho_{\tau}$. Therefore, relating dissipation to the
optimal transport distances defined in the current form (116) is impossible.
Inspired by the dissipative structure of Lindblad dynamics, we define the
following distance:

$\mathcal{W}_{q}(\varrho^{A},\varrho^{B})\coloneqq\frac{1}{2}\min_{V^{\dagger}V=\mathbb{1}}\|V\varrho^{A}V^{\dagger}-\varrho^{B}\|_{1}.$
(118)

Here, the minimum is over all possible unitaries $V$. Intuitively, the
distance $\mathcal{W}_{q}$ characterizes the state difference induced by the
dissipative Lindblad part. Thus, it is expected to be relevant to dissipation.
Note that in the zero-dissipation limit (i.e., the system is unitarily
evolved), this distance also vanishes. Although the distance $\mathcal{W}_{q}$
is defined in a variational form, it can be analytically calculated using the
eigenvalues of the density matrices. Interestingly, it becomes exactly the
classical Wasserstein distance between the eigenvalue distributions:

$\mathcal{W}_{q}(\varrho^{A},\varrho^{B})=\frac{1}{2}\sum_{x}|p_{x}^{A}-p_{x}^{B}|=\mathcal{T}(p^{A},p^{B}),$
(119)

where $\\{p_{x}^{A}\\}$ and $\\{p_{x}^{B}\\}$ are increasing eigenvalues of
$\varrho^{A}$ and $\varrho^{B}$, respectively (see Appendix E.3 for the
proof). For this reason, hereafter, $\mathcal{W}_{q}$ is referred to as the
quantum Wasserstein distance. Evidently, this distance satisfies the triangle
inequality. However, it is a pseudo-metric (i.e.,
$\mathcal{W}_{q}(\varrho^{A},\varrho^{B})=0$ for $\varrho^{A}\neq\varrho^{B}$
is possible). This originates from our goal of relating the defined distance
to dissipation in Lindblad dynamics.

For the quantum Wasserstein distance previously defined, we provide the
following thermodynamic interpretation.

######  Theorem 2.

The quantum Wasserstein distance can be written in the following variational
form:

$\displaystyle\mathcal{W}_{q}(\varrho^{A},\varrho^{B})$
$\displaystyle=\min_{\mathcal{L}_{t}}{\int_{0}^{\tau}\sqrt{\sigma_{t}m_{t}}\differential{t}}$
(120)
$\displaystyle=\min_{\mathcal{L}_{t}}\sqrt{\Sigma_{\tau}\mathcal{M}_{\tau}}.$
(121)

Here, the minimum is taken over all super-operators
$\\{\mathcal{L}_{t}\\}_{0\leq t\leq\tau}$ that satisfy the Lindblad master
equation (89) with boundary conditions $\varrho_{0}=\varrho^{A}$ and
$\varrho_{\tau}=\varrho^{B}$.

Theorem 2 is the second central result, and its sketch proof is given in the
following. Interestingly, Thm. 2 has the same structure as Thm. 1 in the
classical case. This implies a universal relationship between the optimal
transport distances and dissipation in classical and quantum discrete systems.

###### Proof.

We briefly describe the proof strategy; for a detailed derivation, see
Appendix E.4. We first prove that the inequalities

$\mathcal{W}_{q}(\varrho^{A},\varrho^{B})\leq\int_{0}^{\tau}\sqrt{\sigma_{t}m_{t}}\differential{t}\leq\sqrt{\Sigma_{\tau}\mathcal{M}_{\tau}}$
(122)

hold for any Markovian open quantum dynamics and then construct a specific
process that simultaneously attains all the equalities of Eq. (122). The
inequalities in Eq. (122) can be proved similarly as in the classical case. To
construct the dynamics that can achieve the equalities, we first construct a
classical Markov jump process that transforms distribution $p^{A}$ into
$p^{B}$ and satisfies

$\mathcal{T}(p^{A},p^{B})=\int_{0}^{\tau}\sqrt{\sigma_{t}m_{t}}\differential{t}=\sqrt{\Sigma_{\tau}\mathcal{M}_{\tau}}.$
(123)

Here, $\\{p_{x}^{A}\\}$ and $\\{p_{x}^{B}\\}$ are increasing eigenvalues of
$\varrho^{A}$ and $\varrho^{B}$, respectively. Subsequently, we construct
Lindblad dynamics based on this classical jump process such that the dynamics
transforms density matrix $\varrho^{A}$ into $\varrho^{B}$, and the quantities
$\sigma_{t}$ and $m_{t}$ are identical to those in the classical jump process.
We can verify that this quantum dynamics attains the equalities of Eq. (122).
∎

Similar to the classical case, the quantum Wasserstein distance can also be
determined through the entropy production and dynamical activity associated
with Markovian quantum dynamics, which is stated in Cor. 9 in Appendix E.5.

##  VI Applications for thermodynamic interpretation of optimal transport

In this section, we present applications for our central results, namely,
Thms. 1 and 2. Specifically, we show that these variational formulas lead to
stringent bounds for thermodynamic speed limits and information erasure at
arbitrary temperatures.

###  VI.1 Classical and quantum thermodynamic speed limits

The speed of state transformation in any system cannot be made arbitrarily
fast because of physical constraints. This fact leads to a natural question:
What is the ultimate limit for state transformation? This question sparked a
lot of research and gave rise to the concept of speed limits.

Precisely speaking, speed limits impose lower bounds on the operational time
required for evolving a system from a given state to a target one. Originally,
speed limits were derived for closed quantum systems, inspired by the
Heisenberg time-energy uncertainty principle Mandelstam and Tamm (1945). One
of the celebrated results is the Mandelstam–Tamm bound, which applies to
closed quantum systems and takes the following form:

$\tau\geq\frac{\mathcal{B}(\varrho_{0},\varrho_{\tau})}{\expectationvalue{\Delta
H}_{\tau}},$ (124)

where
$\mathcal{B}(\varrho,\sigma)\coloneqq\arccos\tr{|\sqrt{\varrho}\sqrt{\sigma}|}$
is the Bures angle and $(\Delta
H_{t})^{2}\coloneqq\tr{H_{t}^{2}\varrho_{t}}-\tr{H_{t}\varrho_{t}}^{2}$ is the
energy fluctuation. Equation (124) implies that the speed of state
transformation in closed quantum systems is constrained by the fluctuation of
energy. Various types of speed limits were subsequently generalized for open
quantum and classical systems Mandelstam and Tamm (1945); Margolus and Levitin
(1998); del Campo  _et al._ (2013); Taddei  _et al._ (2013); Deffner and Lutz
(2013); Pires  _et al._ (2016); Okuyama and Ohzeki (2018); Campaioli  _et al._
(2018); Shanahan  _et al._ (2018); Sun  _et al._ (2021); O’Connor  _et al._
(2021); Hamazaki (2022); Nakajima and Utsumi (2022); García-Pintos  _et al._
(2022); Hasegawa (2022) (see Ref. Deffner and Campbell (2017) for a
comprehensive review).

Although several versions of classical and quantum speed limits exist for open
systems, here we aim to develop thermodynamic bounds that satisfy two
conditions: (i) they should be tight (i.e., for generic initial and final
states, a configuration of the system always exists that transforms these
states and saturates the bounds) and (ii) they should be physically
interpretable (i.e., all quantities appearing in the bound are physically
meaningful). In the following, we derive these thermodynamic speed limits from
the variational formulas for both classical and quantum cases.

####  VI.1.1 Classical case

We consider a discrete classical system modeled by a Markov jump process [Eq.
(20)]. The system is driven by thermodynamic forces and evolves according to
the laws of thermodynamics. Intuitively, to achieve fast transformation, we
must pay some costs. In the following, we derive fundamental bounds on the
operational time that is required to evolve the system’s distribution to the
target one.

Let $\mathcal{G}(V,E)$ be the underlying topology of the jump process (i.e.,
the graph connectivity that determines whether the transition between two
states is allowed). Then, we can define the corresponding Wasserstein distance
based on the graph $\mathcal{G}$. According to Thm. 1, we have

$\mathcal{W}_{1}(p_{0},p_{\tau})=\min_{\mathsf{W}_{t}}{\int_{0}^{\tau}\sqrt{\sigma_{t}m_{t}}\differential{t}}=\min_{\mathsf{W}_{t}}\sqrt{\Sigma_{\tau}\mathcal{M}_{\tau}}.$
(125)

Since the system dynamics considered here is one of the admissible dynamics
that transform ${p_{0}}$ into ${p_{\tau}}$, the following inequalities follow
immediately from the equality (125):

$\displaystyle\mathcal{W}_{1}(p_{0},p_{\tau})$
$\displaystyle\leq\int_{0}^{\tau}\sqrt{\sigma_{t}m_{t}}\differential{t}=\tau\expectationvalue{\sqrt{\sigma
m}}_{\tau}$ (126)
$\displaystyle\leq\tau\sqrt{\expectationvalue{\sigma}_{\tau}\expectationvalue{m}_{\tau}}.$
(127)

Consequently, we obtain lower bounds on the operational time in terms of the
Wasserstein distance, thermodynamic cost, and kinetic cost as follows:

$\tau\geq\frac{\mathcal{W}_{1}(p_{0},p_{\tau})}{\expectationvalue{\sqrt{\sigma
m}}_{\tau}}\geq\frac{\mathcal{W}_{1}(p_{0},p_{\tau})}{\sqrt{\expectationvalue{\sigma}_{\tau}\expectationvalue{m}_{\tau}}}.$
(128)

Equation (128) implies that both irreversible entropy production and state
mobility jointly constrain the speed of state transformation. Using Cor. 7 in
Appendix D.6 and following the same procedure, we also obtain similar but
tighter bounds in terms of time-averaged entropy production and dynamical
activity as

$\displaystyle\tau\geq\frac{2\mathcal{W}_{1}(p_{0},p_{\tau})}{\expectationvalue{\sigma\Phi(\sigma/2a)^{-1}}_{\tau}}\geq\frac{2\mathcal{W}_{1}(p_{0},p_{\tau})}{\expectationvalue{\sigma}_{\tau}\Phi({\expectationvalue{\sigma}_{\tau}}/{2\expectationvalue{a}_{\tau}})^{-1}},$
(129)

where $\Phi(x)$ is the inverse function of $x\tanh(x)$. Equations (128) and
(129) are our new thermodynamic speed limits for classical Markov jump
processes.

Some remarks are in order. First, the thermodynamic speed limits in Eqs. (128)
and (129) are tight and saturable. More specifically, for generic initial and
final distributions, we can always construct dynamics that satisfy the global
detailed balance condition and transform the initial distribution into the
final one in a time duration equal to that of the lower bounds.

Second, our bounds are tight for arbitrary temperatures, even in the zero-
temperature limit. Since $\expectationvalue{\sigma}_{\tau}=O(\beta)$,
irreversible entropy production becomes infinite as $\beta\to+\infty$, whereas
dynamical activity remains finite [i.e., $\expectationvalue{a}_{\tau}=O(1)$].
Nevertheless, we show in the following that our bounds remain useful in this
low-temperature limit. Indeed, in the $\beta\to+\infty$ limit, bound (129)
reduces to a nontrivial inequality
$\tau\geq\mathcal{W}_{1}(p_{0},p_{\tau})/\expectationvalue{a}_{\tau}$. In
addition, bound (128) also remains finite because we can prove that
$\beta\expectationvalue{m}_{\tau}$ does not diverge in general. To this end,
we assume that the energy levels are non-degenerate and the system is
typically driven far from the instantaneous equilibrium. Since $m_{xy}(t)$ can
be calculated as

$\displaystyle\beta m_{xy}(t)$
$\displaystyle=\frac{\beta[a_{xy}(t)-a_{yx}(t)]}{\ln a_{xy}(t)-\ln a_{yx}(t)}$
$\displaystyle=\frac{a_{xy}(t)-a_{yx}(t)}{\beta^{-1}[\ln p_{y}(t)-\ln
p_{x}(t)]+\varepsilon_{y}(t)-\varepsilon_{x}(t)},$ (130)

we have

$\beta
m_{xy}(t)\xrightarrow{\beta\to+\infty}\frac{a_{xy}(t)-a_{yx}(t)}{\varepsilon_{y}(t)-\varepsilon_{x}(t)},$
(131)

which remains finite. Therefore, the term $\beta\expectationvalue{m}_{\tau}$
does not diverge in the zero-temperature limit $\beta\to+\infty$.

Third, we compare our results with existing bounds in the literature. In Ref.
Shiraishi  _et al._ (2018), a classical speed limit was obtained for Markov
jump processes, which reads

$\tau\geq\frac{\mathcal{T}(p_{0},p_{\tau})}{\sqrt{\expectationvalue{\sigma}_{\tau}\expectationvalue{a}_{\tau}/2}}.$
(132)

Since $\mathcal{W}_{1}(p_{0},p_{\tau})\geq\mathcal{T}(p_{0},p_{\tau})$ and
$m_{t}\leq a_{t}/2$ for all times, our speed limits in Eq. (128) are stronger
than those in Eq. (132). Our bounds also suggest that the conventional bound
(132) can be asymptomatically saturated only when
$\mathcal{W}_{1}(p_{0},p_{\tau})=\mathcal{T}(p_{0},p_{\tau})$ (e.g., when the
underlying graph is fully connected) and the system is always near the
instantaneous equilibrium. In Refs. Vo  _et al._ (2022); Delvenne and Falasco
(2021), another thermodynamic speed limit, which is tighter than the
conventional bound (132), was derived as

$\tau\geq\frac{2\mathcal{T}(p_{0},p_{\tau})}{\expectationvalue{\sigma}_{\tau}\Phi({\expectationvalue{\sigma}_{\tau}}/{2\expectationvalue{a}_{\tau}})^{-1}}.$
(133)

Since $\mathcal{W}_{1}(p_{0},p_{\tau})\geq\mathcal{T}(p_{0},p_{\tau})$, our
bound (129) is stronger than bound (133). The essential difference is that our
bounds consider the topology of the jump process, whereas the conventional
bounds do not.

####  VI.1.2 Quantum case

Next, we consider an open quantum system described by the Markovian Lindblad
master equation [Eq. (89)]. Following the same procedure as in the classical
case, we derive stringent thermodynamic bounds on the operational time
required to transform the initial density matrix into the final one.

From Thm. 2

$\displaystyle\mathcal{W}_{q}(\varrho_{0},\varrho_{\tau})=\min_{\mathcal{L}_{t}}\quantity{\int_{0}^{\tau}\sqrt{\sigma_{t}m_{t}}\differential{t}}=\min_{\mathcal{L}_{t}}\sqrt{\Sigma_{\tau}\mathcal{M}_{\tau}},$
(134)

we analogously obtain the following inequalities:

$\displaystyle\mathcal{W}_{q}(\varrho_{0},\varrho_{\tau})$
$\displaystyle\leq\int_{0}^{\tau}\sqrt{\sigma_{t}m_{t}}\differential{t}=\tau\expectationvalue{\sqrt{\sigma
m}}_{\tau}$ (135)
$\displaystyle\leq\tau\sqrt{\expectationvalue{\sigma}_{\tau}\expectationvalue{m}_{\tau}}.$
(136)

Consequently, we arrive at the following bounds on the operational time:

$\displaystyle\tau\geq\frac{\mathcal{W}_{q}(\varrho_{0},\varrho_{\tau})}{\expectationvalue{\sqrt{\sigma
m}}_{\tau}}\geq\frac{\mathcal{W}_{q}(\varrho_{0},\varrho_{\tau})}{\sqrt{\expectationvalue{\sigma}_{\tau}\expectationvalue{m}_{\tau}}}.$
(137)

Equation (137) implies that the speed of state transformation in open quantum
systems is constrained by irreversible entropy production and dynamical state
mobility. Notably, it has the same form as the classical bound (128),
indicating the unification of our results. Exploiting Cor. 9 in Appendix E.5
and repeating the same procedure yield other speed limits in terms of entropy
production and dynamical activity, which read

$\displaystyle\tau\geq\frac{2\mathcal{W}_{q}(\varrho_{0},\varrho_{\tau})}{\expectationvalue{{\sigma}\Phi\quantity({\sigma/2a})^{-1}}_{\tau}}\geq\frac{2\mathcal{W}_{q}(\varrho_{0},\varrho_{\tau})}{\expectationvalue{\sigma}_{\tau}\Phi\quantity(\expectationvalue{\sigma}_{\tau}/2\expectationvalue{a}_{\tau})^{-1}}.$
(138)

Equations (137) and (138) are the new quantum thermodynamic speed limits.
Remarkably, these thermodynamic speed limits are tight and saturable. In other
words, for any pair of generic initial and final states, a combination of
Hamiltonian and jump operators always exists that attains the lower bound of
the operational time. Moreover, they are useful even in the zero-temperature
limit. The bounds imply that both the thermodynamic and kinetic costs play a
crucial role in the change speed of open quantum systems.

We discuss the relevance of our results to previous studies. In Ref. Van Vu
and Hasegawa (2021a), a thermodynamic speed limit was derived for Markovian
open quantum dynamics and is given by

$\displaystyle\tau\geq\frac{\mathcal{W}_{q}(\varrho_{0},\varrho_{\tau})}{\sqrt{\expectationvalue{\sigma}_{\tau}\expectationvalue{a}_{\tau}/2}}.$
(139)

According to Eq. (105), we have $m_{t}\leq a_{t}/2$ for all $t$. Therefore,
bound (139) is looser than the new bound (137). In addition, since
$\Phi(x)\geq\sqrt{x}$ for all $x\geq 0$, the new bound (138) is also stronger
than the conventional bound (139). In Ref. Funo  _et al._ (2019), another
thermodynamic speed limit in terms of trace distance was derived for open
quantum systems. Since the metrics used to measure the distance between
quantum states in these bounds are different (i.e., the Wasserstein distance
in our study and the trace distance in Ref. Funo  _et al._ (2019)), they
cannot be directly compared. Nonetheless, by exploiting the dynamical state
mobility introduced in this study, we can derive a similar but tighter speed
limit in terms of the trace distance. The detailed form of this speed limit is
presented in Appendix F.1. However, it is worth noting that the attainability
of this bound is unclear.

###  VI.2 Finite-time Landauer principle

The Landauer principle Landauer (1961) implies that erasing information is
always accompanied by a thermodynamic cost. More specifically, the
thermodynamic cost required to erase a classical bit is at least $T\ln 2$,
where $T$ is the environment temperature. The lower bound $T\ln 2$ (referred
to as the Landauer bound) plays not only a fundamental role in the
thermodynamics of information and computation Bennett (1982); Sagawa (2012);
Parrondo  _et al._ (2015); Goold  _et al._ (2016); Wolpert (2019), it also
resolves the paradox of Maxwell’s demon Maruyama  _et al._ (2009).

Various classical and quantum platforms Bérut  _et al._ (2012); Jun  _et al._
(2014); Yan  _et al._ (2018); Hong  _et al._ (2016); Saira  _et al._ (2020);
Dago  _et al._ (2021) have experimentally confirmed that the Landauer bound
can be achieved in the slow quasistatic limit. However, practical computing
requires fast memory erasure in a short time and thus, in general, consumes a
thermodynamic cost far beyond the Landauer bound. This background strongly
motivates researchers to develop finite-time generalizations of the Landauer
bound, which capture finite-time corrections and can better predict the
erasure cost. Although several finite-time bounds have been developed for both
classical and quantum discrete systems Zhen  _et al._ (2021); Van Vu and Saito
(2022a); Lee  _et al._ (2022), the attainability of these bounds remains
unclear. Moreover, these bounds have looser predictive power in the low-
temperature regime. In the following, we attempt to derive finite-time bounds
that are tight for arbitrary temperatures.

Before presenting the new bounds, we first describe the generic setup of
information erasure for both classical and quantum cases. We consider a
finite-dimensional discrete system attached to a thermal reservoir at
temperature $T$. Information is encoded in the system state and subsequently
erased by controlling the classical energy levels or the quantum Hamiltonian
and driving the system toward its ground state. The erasure protocol should
work for an arbitrary initial state; that is, any initial state should be
reset close to the ground state in a finite time $\tau$. This erasure process
leads to a change in system entropy, which must be compensated for by the heat
dissipated into the reservoir. Because we are interested in the average
thermodynamic cost associated with the erasure protocol, considering the
maximally mixed state as the initial state is convenient. Roughly speaking,
the reasons for this are that the maximally mixed state is sufficient to
understand the average dissipated heat of the erasure process for all initial
states and that if a protocol can reliably reset the system from the maximally
mixed state, then it does so also for an arbitrary state. A detailed
discussion will be given in the following.

####  VI.2.1 Classical case

We consider an information erasure process using a $d$-state classical system,
the dynamics of which is governed by the master equation. The transitions
between states are mediated by a single thermal reservoir at temperature $T$.
The system state is characterized by the probability distribution, which
encodes information we want to erase. The energy levels are controlled
according to a fixed protocol such that the system is always driven toward the
ground state ${p_{*}}=[1,0,\dots,0]^{\top}$, irrespective of the initial
state.

Here, we explain why the initial state should be set to the uniform
distribution ${\overline{p}}=[1/d,\dots,1/d]^{\top}$. First, let
$\mathcal{Q}(p_{0})$ be the heat dissipation of erasure for the initial
distribution ${p_{0}}$. Then, due to the linearity of the master equation and
$\mathcal{Q}(\cdot)$, the average dissipation can be calculated as

$\mathbb{E}[\mathcal{Q}(p_{0})]=\mathcal{Q}(\mathbb{E}[p_{0}])=\mathcal{Q}(\overline{p}),$
(140)

where $\mathbb{E}[\cdot]$ denotes the average over all possible initial
distributions. Equation (140) implies that investigating the case with initial
distribution ${p_{0}}={\overline{p}}$ is sufficient to understand the average
dissipation. Second, let
$\Lambda_{\tau}=\vec{T}\exp\quantity(\int_{0}^{\tau}\mathsf{W}_{t}\differential{t})$
be the map that represents the erasure process, that is,
$\Lambda_{\tau}{p_{0}}={p_{\tau}}$. We can then prove that if the uniform
distribution can be erased within error $\delta>0$ (i.e.,
$\|\Lambda_{\tau}{\overline{p}}-{p_{*}}\|_{F}\leq\delta$), the following
inequality holds for arbitrary initial distribution ${p_{0}}$ (see Appendix
F.2 for the proof):

$\|\Lambda_{\tau}{p_{0}}-{p_{*}}\|_{F}\leq\sqrt{2d\delta}.$ (141)

Equation (141) indicates that if a protocol can erase the uniform
distribution, it can reliably do so also for arbitrary initial states.

We can now present the new bound. Let ${p_{\tau}}$ be the final distribution
for the case ${p_{0}}={\overline{p}}$ and let
$\epsilon\coloneqq\mathcal{T}(p_{\tau},p_{*})=|1-p_{1}(\tau)|$ be the erasure
error, which should be sufficiently small. From Eq. (79), the heat dissipation
is lower bounded by system entropy production and a finite-time correction
term as

$Q\geq-T\Delta S_{\rm
sys}+\frac{\mathcal{W}_{1}(p_{0},p_{\tau})^{2}}{\tau\beta\expectationvalue{m}_{\tau}}.$
(142)

Equation (142) is regarded as the finite-time Landauer principle for classical
systems. The bound is tight and can be saturated for arbitrary temperatures,
even in the zero-temperature limit. As shown in the previous section, the term
$\beta\expectationvalue{m}_{\tau}$ remains finite even when $\beta\to\infty$
(i.e., $T\to 0$). Therefore, bound (142) is useful for arbitrary temperatures.
By contrast, the conventional Landauer bound becomes trivial in the low-
temperature regime (i.e., $Q\geq 0$). We also note that bound (142) is tighter
than the following bound:

$Q\geq-T\Delta S_{\rm
sys}+\frac{\mathcal{T}(p_{0},p_{\tau})^{2}}{\tau\beta\expectationvalue{a}_{\tau}/2},$
(143)

which is obtained from the conventional speed limit (132).

Bound (142) can be simplified by including the erasure error. To this end, we
further bound the terms in Eq. (142) from below as

$\displaystyle-\Delta S_{\rm sys}$ $\displaystyle=\ln d-S(p_{\tau})\geq\ln
d-h(\epsilon),$ (144) $\displaystyle\mathcal{W}_{1}(p_{0},p_{\tau})$
$\displaystyle\geq\mathcal{T}(p_{0},p_{\tau})\geq\quantity|1-1/d-\epsilon|,$
(145)

where
$h(\epsilon)\coloneqq-\epsilon\ln[\epsilon/(d-1)]-(1-\epsilon)\ln(1-\epsilon)\geq
0$ is a function of $\epsilon$ that vanishes as $\epsilon\to 0$. Consequently,
we obtain the following bound on the average heat dissipation:

$Q\geq T\quantity[\ln
d-h(\epsilon)]+\frac{(1-1/d-\epsilon)^{2}}{\tau\beta\expectationvalue{m}_{\tau}}.$
(146)

Equation (146) imposes a lower bound on heat dissipation in terms of the
operational time and erasure error. In the limit of perfect erasure (i.e.,
$\epsilon\to 0$), a simple bound can be derived:

$Q\geq T\ln d+\frac{(1-1/d)^{2}}{\tau\beta\expectationvalue{m}_{\tau}}.$ (147)

For slow erasure (i.e., $\tau\beta\expectationvalue{m}_{\tau}\gg 1$), the
second term in the lower bound vanishes. Thus, Eq. (147) recovers the
conventional Landauer bound for the $d=2$ case. By contrast, in the fast-
erasure limit (i.e., $\tau\beta\expectationvalue{m}_{\tau}\ll 1$), this
correction term becomes dominant, implying that fast erasure is accompanied by
a thermodynamic cost far beyond the Landauer cost.

If we consider dynamical activity instead of dynamical state mobility, we can
obtain another finite-time Landauer bound. By transforming the speed limit
(129), we can show that heat dissipation is lower bounded by the Wasserstein
distance and dynamical activity as

$\frac{Q}{T}\geq-\Delta S_{\rm
sys}+2\mathcal{W}_{1}(p_{0},p_{\tau})\tanh^{-1}\quantity(\frac{\mathcal{W}_{1}(p_{0},p_{\tau})}{\tau\expectationvalue{a}_{\tau}}).$
(148)

This new bound is always tighter than the bound reported in Ref. Lee  _et al._
(2022), which uses the total variation distance to quantify the distance
between probability distributions. In general, bound (148) can be either
stronger or looser than bound (142).

####  VI.2.2 Quantum case

Here, we consider a quantum process of erasing information. The erasure
process is implemented using a controllable $d$-dimensional qudit system,
which is attached to a thermal reservoir at temperature $T$. The density
matrix of the qudit encodes the information we want to erase and then is
driven toward the ground state $\varrho_{*}=\outerproduct{0}{0}$ by
controlling the Hamiltonian.

Analogous to the classical case, the initial state is conveniently set to the
maximally mixed state $\overline{\varrho}=\mathbb{1}/d$. This is because
assigning the maximally mixed state to the initial state is sufficient to
understand the average heat dissipation in the quantum case. In addition, we
can show that if an erasure protocol can erase the maximally mixed state, it
can reliably do so also for arbitrary initial states. More specifically, let
$\Lambda_{\tau}(\cdot)=\vec{T}\exp\quantity(\int_{0}^{\tau}\mathcal{L}_{t}\differential{t})(\cdot)$
be the quantum map that describes the erasure process [i.e.,
$\Lambda_{\tau}(\varrho_{0})=\varrho_{\tau}$]. Then, if the maximally mixed
state can be erased within error $\delta>0$ (i.e.,
$\|\Lambda_{\tau}(\overline{\varrho})-\varrho_{*}\|_{F}\leq\delta$), the
erasure error for arbitrary initial state $\varrho_{0}$ can be upper bounded
as follows Van Vu and Saito (2022a):

$\|\Lambda_{\tau}(\varrho_{0})-\varrho_{*}\|_{F}\leq\sqrt{2d\delta}.$ (149)

Equation (149) provides insight into the reliability of the erasure protocol
by verifying the $\varrho_{0}=\overline{\varrho}$ case.

Next, we present the finite-time bound for quantum information erasure.
Because $\varrho_{0}=\mathbb{1}/d$, the Wasserstein distance coincides with
the trace distance [i.e.,
$\mathcal{W}_{q}(\varrho_{0},\varrho_{\tau})=\mathcal{T}(\varrho_{0},\varrho_{\tau})$].
Consequently, a finite-time bound on heat dissipation can be obtained as

$Q\geq-T\Delta S_{\rm
sys}+\frac{\mathcal{T}(\varrho_{0},\varrho_{\tau})^{2}}{\tau\beta\expectationvalue{m}_{\tau}}.$
(150)

Equation (150) is the finite-time quantum Landauer principle, which is tight
and can be saturated for arbitrary temperatures. In addition to the
conventional Landauer term $-T\Delta S_{\rm sys}$, a finite-time correction
term exists in the lower bound, which does not vanish even in the zero-
temperature limit. Therefore, the inequality (150) provides a stringent bound
on heat dissipation for information erasure in both fast-driving and low-
temperature regimes.

We compare bound (150) with an existing bound derived in Ref. Van Vu and Saito
(2022a), which reads

$Q\geq-T\Delta S_{\rm
sys}+\frac{\mathcal{T}(\varrho_{0},\varrho_{\tau})^{2}}{\tau\beta\expectationvalue{a}_{\tau}/2}.$
(151)

Since $m_{t}\leq a_{t}/2$ for all times, it is immediately clear that bound
(150) is always stronger than bound (151).

Next, we derive a simplified bound which includes the erasure error. Let
$\epsilon\coloneqq\mathcal{T}(\varrho_{\tau},\varrho_{*})$ be the erasure
error, which should be small. We can then analogously bound the terms in Eq.
(150) from below as

$\displaystyle-\Delta S_{\rm sys}$ $\displaystyle=\ln
d-S(\varrho_{\tau})\geq\ln d-h(\epsilon),$ (152)
$\displaystyle\mathcal{T}(\varrho_{0},\varrho_{\tau})$
$\displaystyle\geq\quantity|\mathcal{T}(\varrho_{0},\varrho_{*})-\mathcal{T}(\varrho_{\tau},\varrho_{*})|=\quantity|1-1/d-\epsilon|,$
(153)

where we use an inequality relating the entropy difference between two quantum
states to their trace distance in the first line Audenaert (2007). Inserting
Eqs. (152) and (153) into Eq. (150), we arrive at the following simple bound:

$Q\geq T[\ln
d-h(\epsilon)]+\frac{(1-1/d-\epsilon)^{2}}{\tau\beta\expectationvalue{m}_{\tau}}.$
(154)

Equation (154) is the simplified Landauer bound that includes finite-time and
finite-error corrections. Remarkably, it has the same structure as the
classical bound (142). In the limit of perfect and slow erasure, bound (154)
reduces to the conventional Landauer bound as $d=2$.

In analogy to the classical case, a finite-time Landauer bound in terms of the
quantum Wasserstein distance and dynamical activity can also be obtained. By
rearranging the speed limit (138), we can prove that

$\frac{Q}{T}\geq-\Delta S_{\rm
sys}+2\mathcal{W}_{q}(\varrho_{0},\varrho_{\tau})\tanh^{-1}\quantity(\frac{\mathcal{W}_{q}(\varrho_{0},\varrho_{\tau})}{\tau\expectationvalue{a}_{\tau}}).$
(155)

This finite-time bound can be considered a quantum analog of the classical
bound (148).

##  VII Numerical demonstrations

Next, we numerically illustrate the applications of our results, the
thermodynamic uncertainty relation, speed limits, and finite-time Landauer
principles in several classical and quantum systems.

Figure 5: Numerical illustration of the classical thermodynamic speed limits
and finite-time Landauer principle for both optimal and nonoptimal protocols.
(a) Schematic of the two-level system. (b)(e) Time variations of the control
parameters of the optimal and nonoptimal protocols. (c)(f) Numerical
verification of the speed limits $t\geq t_{c,1}\geq t_{c,2}$ and (d)(g)
finite-time Landauer principle $Q\geq Q_{c,1}\geq Q_{c,2}$. The other
parameters are fixed as $\beta=10$ and $\tau=10$. The weighting factor
$\lambda$ is set such that $(1-\lambda)/\lambda=10^{5}$.

###  VII.1 Illustration of classical thermodynamic speed limits and finite-
time Landauer principle

First, we illustrate the classical speed limits and finite-time Landauer
principle in a two-level system. The system dynamics can be described by a
Markov jump process with transition rates given by

$w_{xy}(t)=\gamma_{xy}(t)\frac{e^{\beta\varepsilon_{y}(t)}}{e^{\beta\varepsilon_{y}(t)}+e^{\beta\varepsilon_{x}(t)}},$
(156)

where $\gamma_{xy}(t)=\gamma_{yx}(t)$ are tunable parameters and
$\varepsilon_{x}(t)$ denotes the instantaneous energy level of state $x$. For
simplicity, we set
$\varepsilon_{t}\coloneqq\varepsilon_{2}(t)-\varepsilon_{1}(t)$ and
$\gamma_{xy}(t)=1$ for all transitions. The parameter $\\{\varepsilon_{t}\\}$
thus defines a time-dependent control protocol.

To illustrate the bounds, we consider an information erasure process in which
an arbitrary initial distribution is always reset to the ground state within a
finite error. We examine two control protocols, namely, optimal and
nonoptimal. The optimality here refers to the dissipation of the least amount
of heat while achieving the predetermined error.

The optimal protocol is numerically obtained by solving the minimization
problem with the following objective functional:

$\mathcal{F}_{c}[\\{\varepsilon_{t}\\}]\coloneqq\lambda
Q+(1-\lambda)\mathcal{T}(p_{\tau},p_{*}),$ (157)

where $\lambda\in[0,1]$ is a weighting factor. The functional
$\mathcal{F}_{c}$ consists of two incompatible objectives, namely, heat
dissipation and erasure error, which cannot be simultaneously small. To reduce
the erasure error, we must pay the price of dissipation. Conversely, reducing
dissipation could enhance the error between the final and ground states. As
$\lambda$ is fixed, the solution of the optimization problem corresponds to a
Pareto-optimal protocol, in which heat dissipation cannot be further minimized
without increasing the error. Imposing constraints on the control parameters
is physically reasonable. Hereafter, we consider the constraint
$\varepsilon_{t}\in[10^{-2},10^{1}]$. To solve the problem (157) under both
equality and inequality constraints, we discretize the control parameters into
$1000$ points and optimize the functional $\mathcal{F}$ with the aid of
nonlinear programming solvers. We determine the weighting factor $\lambda$
such that both the optimal and nonoptimal protocols reset the uniform
distribution $\overline{p}=[1/2,1/2]^{\top}$ to the ground state within the
same error. The time variation of the protocol is plotted in Fig. 5(b).
Notably, the increase in the energy gap between the two levels
$\varepsilon_{1}$ and $\varepsilon_{2}$ is constant in the intermediate period
but tends to slow down in the late period.

The nonoptimal protocol simply lifts the energy level $\varepsilon_{2}(t)$,
forcing the system to descend to the ground state. The time-dependent control
parameter $\varepsilon_{t}$ is specified as

$\varepsilon_{t}=0.422\times\exp\quantity(\frac{\tau+t}{2\tau-t}),$ (158)

which is illustrated in Fig. 5(e). We can observe that, unlike the optimal
protocol, the energy gap in the nonoptimal protocol rapidly increases in the
late period. In the final time, this naive protocol should dissipate more heat
than the optimal protocol.

The process of information erasure is performed within the period $\tau$. At
each time $t\leq\tau$, according to Eqs. (128) and (132), the operational time
is lower bounded as

$t\geq\frac{\mathcal{W}_{1}(p_{0},p_{t})}{\sqrt{\expectationvalue{\sigma}_{t}\expectationvalue{m}_{t}}}\eqqcolon
t_{c,1}\geq\frac{\mathcal{T}(p_{0},p_{t})}{\sqrt{\expectationvalue{\sigma}_{t}\expectationvalue{a}_{t}/2}}\eqqcolon
t_{c,2}.$ (159)

These bounds are numerically verified for the optimal and nonoptimal protocols
in Figs. 5(c) and 5(f), respectively. As shown, the derived bound $t\geq
t_{c,1}$ is tight and stronger than the existing bound $t\geq t_{c,2}$ for all
times.

Likewise, the dissipated heat is lower bounded by the entropy change and
finite-time correction term as

$\displaystyle Q$ $\displaystyle\geq-T\Delta S_{\rm
sys}+\frac{\mathcal{W}_{1}(p_{0},p_{t})^{2}}{t\beta\expectationvalue{m}_{t}}\eqqcolon
Q_{c,1}$ $\displaystyle\geq-T\Delta S_{\rm
sys}+\frac{\mathcal{T}(p_{0},p_{t})^{2}}{t\beta\expectationvalue{a}_{t}/2}\eqqcolon
Q_{c,2}.$ (160)

The numerical results are plotted in Figs. 5(d) and 5(g) for the optimal and
nonoptimal protocols, respectively. As seen, the new lower bound $Q_{c,1}$
tightly bounds the dissipated heat $Q$ in both protocols, whereas the existing
lower bound $Q_{c,2}$ is loose and does not provide a good prediction for heat
dissipation. Notice that the average heat dissipation at the final time
$\tau=10$ is approximately $0.33$, which is far beyond the conventional
Landauer bound $\beta^{-1}\ln 2\approx 0.069$. This implies that the finite-
time correction is dominant over the entropy change in this case.

###  VII.2 Illustration of quantum thermodynamic speed limits and finite-time
Landauer principle

We next exemplify the quantum speed limits and finite-time Landauer principle
with a simple model of information erasure using a spin-$1/2$ qubit. The qubit
is weakly attached to a heat bath at inverse temperature $\beta$. The time
evolution of the reduced density matrix can be described by the Lindblad
equation with the Hamiltonian

$H_{t}=\frac{\varepsilon_{t}}{2}\quantity[\cos(\theta_{t})\sigma_{z}+\sin(\theta_{t})\sigma_{x}]$
(161)

and jump operators

$\displaystyle L_{1}(t)$
$\displaystyle=\sqrt{\gamma\varepsilon_{t}(n_{t}+1)}\outerproduct{0_{t}}{1_{t}},$
(162) $\displaystyle L_{2}(t)$
$\displaystyle=\sqrt{\gamma\varepsilon_{t}n_{t}}\outerproduct{1_{t}}{0_{t}}.$
(163)

Here, $\\{\ket{0_{t}},\ket{1_{t}}\\}$ are the instantaneous energy
eigenstates, $\sigma_{x,y,z}$ are the Pauli matrices, $\gamma$ is the coupling
strength, $n_{t}\coloneqq 1/(e^{\beta\varepsilon_{t}}-1)$, and
$\varepsilon_{t}$ and $\theta_{t}$ are time-dependent control parameters. More
specifically, $E_{t}$ characterizes the energy gap between the energy
eigenstates, whereas $\theta_{t}$ quantifies the relative strength of coherent
tunneling to energy bias Leggett  _et al._ (1987). The qubit is initially
prepared in the state $\varrho_{0}=\mathbb{1}/2$ and subsequently driven
toward the ground state $\varrho_{*}=\outerproduct{0}{0}$ of $\sigma_{z}$. If
$\theta_{t}$ is time invariant, quantum coherence in the energy eigenstates
cannot be created, and the protocol is thus classical. Otherwise, it becomes a
quantum protocol.

Figure 6: Numerical illustration of the quantum thermodynamic speed limits and
finite-time Landauer principle for both optimal and nonoptimal protocols. (a)
Schematic of the spin-$1/2$ qubit, which is coupled to a heat bath at inverse
temperature $\beta$. (b)(e) Time variations of the control parameters of the
optimal and nonoptimal protocols. (c)(f) Numerical verification of the speed
limits $t\geq t_{q,1}\geq t_{q,2}$ and (d)(g) finite-time Landauer principle
$Q\geq Q_{q,1}\geq Q_{q,2}$ as well as the amount of quantum coherence
$\mathcal{C}_{t}$. The other parameters are fixed as $\beta=10$, $\gamma=0.1$,
and $\tau=10$. The weighting factor $\lambda$ is set such that
$(1-\lambda)/\lambda=10^{3}$.

An infinite number of approaches can be used to reset the qubit with a
probability close to $1$. As in the classical case, two protocols are
considered, namely, Pareto-optimal and nonoptimal. Both protocols are designed
to erase information with the same error.

The optimal protocol minimizes two incompatible objectives, the average
dissipated heat and erasure error. Specifically, the protocol can be achieved
by solving the minimization problem with the following multi-objective
functional Van Vu and Saito (2022a):

$\mathcal{F}_{q}[\\{\varepsilon_{t},\theta_{t}\\}]\coloneqq\lambda
Q-(1-\lambda)F(\varrho_{\tau},\varrho_{*}),$ (164)

where $\lambda\in[0,1)$ is a weighting factor and
$F(\varrho,\sigma)\coloneqq(\tr|\sqrt{\varrho}\sqrt{\sigma}|)^{2}$ is the
fidelity of the two quantum states $\varrho$ and $\sigma$ Jozsa (1994).
Because of the physical limitations, placing constraints on the control
parameters is natural. Hereafter, we impose the following lower and upper
bounds on the parameters: $\varepsilon_{t}\in[10^{-1},10^{2}]$ and
$\theta_{t}\in[-\pi,\pi]$. By numerically solving the nonlinear optimization
problem (164), we can obtain the optimal protocol, as plotted in Fig. 6(b). As
seen, the parameter $\theta_{t}$ is fixed to $0$ for all times, implying that
the optimal protocol is classical and does not generate any amount of
energetic coherence. Furthermore, the energy gap increases gradually in the
intermediate period and changes rapidly in the final stage.

The nonoptimal protocol is defined as

$\varepsilon_{t}=2.04\times\exp\quantity(\frac{\tau+t}{2\tau-t}),~{}\theta_{t}=\pi\quantity(\frac{t}{\tau}-1),$
(165)

which is plotted in Fig. 6(e). This protocol naively increases the energy gap
while varying the coherent parameter. We also observe that the increase in the
energy gap is different from that of the optimal case.

We first demonstrate the quantum thermodynamic speed limits. According to Eqs.
(137) and (139), the operational time is lower bounded as follows:

$t\geq\frac{\mathcal{W}_{q}(\varrho_{0},\varrho_{t})}{\sqrt{\expectationvalue{\sigma}_{t}\expectationvalue{m}_{t}}}\eqqcolon
t_{q,1}\geq\frac{\mathcal{W}_{q}(\varrho_{0},\varrho_{t})}{\sqrt{\expectationvalue{\sigma}_{t}\expectationvalue{a}_{t}/2}}\eqqcolon
t_{q,2}.$ (166)

We illustrate these bounds for both the optimal and nonoptimal protocols in
Figs. 6(c) and 6(f), respectively. As shown, the derived bound $t\geq t_{q,1}$
is tight for all times and is stronger than the existing bound $t\geq
t_{q,2}$.

Next, we verify the finite-time quantum Landauer principle. The lower bounds
on the average heat dissipation are given by Eqs. (150) and (151) as

$\displaystyle Q$ $\displaystyle\geq-T\Delta S_{\rm