# Efficiency bounds for bipartite information-thermodynamic systems

Shihao Xia1 Shuanglong Han1 Ousi Pan1 Yuzhuo Pan2 Jincan Chen1 Shanhe Su1
<EMAIL_ADDRESS>1Department of Physics, Xiamen University, Xiamen 361005,
People’s Republic of China
2College of Physics and Information Engineering, Quanzhou Normal University,
362000, People’s Republic of China

(August 28, 2024)

###### Abstract

This study introduces a novel approach to derive a lower bound for the entropy
production rate of a subsystem by utilizing the Cauchy-Schwarz inequality. It
extends to establishing comprehensive upper and lower bounds for the
efficiency of two subsystems. These bounds are applicable to a wide range of
Markovian stochastic processes, which enhances the accuracy in depicting the
range of energy conversion efficiency between subsystems. Empirical validation
is conducted using a two-quantum-dot system model, which serves to confirm the
effectiveness of our inequality in refining the boundaries of efficiency.

††preprint: APS/123-QED

##  I introduction

Over the past two decades, the field of stochastic thermodynamics has
witnessed remarkable progress, greatly enriching our comprehension of physical
phenomena occurring in small fluctuating systems. This framework has unveiled
the inherent connection between thermodynamic irreversibility and non-
equilibrium fluctuations.

Simultaneously, the study of stochastic thermodynamics has facilitated
explorations into the interrelationship between information and
thermodynamics, as well as the thermodynamic limits of fluctuating mesoscopic
systems. (Seifert, 2012; Van den Broeck and Esposito, 2015). A pivotal
discovery in this domain is Landauer’s principle (Landauer, 1961), which
quantifies the heat dissipation required for the erasure of information in the
limit of slow quasi-static processes (Dong  _et al._ , 2011). This principle
has been empirically validated in both classical and quantum frameworks (Bérut
_et al._ , 2012; Dago  _et al._ , 2021; Hashimoto and Uchiyama, 2022). In the
era of high-speed computational demands, a noteworthy breakthrough has been
the investigation of the energy consumption associated with finite-time
information erasure (Van Vu and Saito, 2022; Proesmans  _et al._ , 2020; Zhen
_et al._ , 2021). Leveraging the unique synergy between information and
thermodynamics, researchers have devised various heat engines, such as the
Szilard engine (Kim  _et al._ , 2011; Koski  _et al._ , 2014; Malgaretti and
Stark, 2022) and molecular engines (Chen  _et al._ , 2017; Leighton and Sivak,
2023; Barato and Seifert, 2015). These advancements have opened up new avenues
for the design of miniature heat engines.

On the other hand, thermodynamic inequalities have established foundational
limits on entropy production and energy usage (Barato and Seifert, 2015;
Horowitz and Gingrich, 2020; Shiraishi, 2021), deepening our understanding of
thermodynamics and providing valuable insights into microsystems. A multitude
of studies have corroborated the Clausius statement of the second law of
thermodynamics by employing thermodynamic inequalities. This consensus and the
substantial research interests in this field signify its importance and
widespread relevance. (Shiraishi  _et al._ , 2016; Pietzonka and Seifert,
2018; Funo  _et al._ , 2019; Shiraishi  _et al._ , 2018; Van Vu and Hasegawa,
2021a; Aurell  _et al._ , 2011; Van Vu and Hasegawa, 2021b; Tajima and Funo,
2021; Barato and Seifert, 2015; Crooks, 2007; Salamon and Berry, 1983; Sivak
and Crooks, 2012; Scandi and Perarnau-Llobet, 2019; Chen  _et al._ , 2021). In
the pursuit of establishing theoretical bounds, researchers have adopted
various methodologies. Notably, Jensen bounds (Leighton and Sivak, 2023, 2022)
and Cramér-Rao (Tanogami  _et al._ , 2023) bounds represent significant
approaches in delineating these constraints.

While existing researchs provide valuable insights into the boundaries of
energy conversion and entropy production, our investigation adopts an
alternative framework. Specifically, we leverage the Cauchy-Schwarz inequality
(Funo  _et al._ , 2019; Shiraishi  _et al._ , 2018) within the stochastic
thermodynamics context to derive a more refined lower bound for the entropy
production rate in subsystems. This approach enables us to obtain a accurate
approximation to the energy conversion efficiency of these subsystems.

The structure of this paper is organized as follows: Section II delves into
the examination of stochastic thermodynamics in bipartite systems, where the
efficiency bounds for subsystems are derived. Section III focuses on the
double quantum dot system and derives expressions for the associated
thermodynamic quantities by calculating the transition frequency. By building
upon the double quantum dot model, section IV presents our findings in a
detailed discussion. Finally, the paper concludes with section V, which
summarizes the key insights gained from the study.

##  II Stochastic thermodynamics of bipartite systems

###  II.1 Stochastic dynamics for a bipartite system

We adopt a conventional stochastic process model for a bipartite system $Z$,
which is described thoroughly and extensively in (Horowitz and Esposito,
2014). The state of $Z$ is described by $z=(x,y)$ with $x$ and $y$
representing the particle counts in subsystems $X$ and $Y$, respectively. The
state $z$’s time-dependent probability, $p(z)$, evolves following the
Markovian master equation

$\dot{p}(z)=\sum_{z^{\prime},v}J_{zz^{\prime}}^{v}.$ (1)

Here, the flux
$J_{zz^{\prime}}^{v}=W_{zz^{\prime}}^{v}p\left(z^{\prime}\right)-W_{z^{\prime}z}^{v}p(z)$
hinges on the time-independent transition rate $W_{zz^{\prime}}^{v}$ from
state $z^{\prime}$ to state $z$ induced by reservoir $v$, and the reverse
transition rate $W_{z^{\prime}z}^{v}$. The transition rate matrix, being non-
negative $W_{zz^{\prime}}^{v}\geq 0$ for $z\neq z^{\prime}$, restricts the
bipartite structure from allowing simultaneous state changes in both
subsystems. Consequently, the form of $W_{zz^{\prime}}^{v}$ is constrained to

$W_{zz^{\prime}}^{v}=\begin{cases}W_{xx^{\prime}\mid y}^{v}&\left(x\neq
x^{\prime},y=y^{\prime}\right)\\\ W_{yy^{\prime}\mid
x}^{v}&\left(x=x^{\prime},y\neq y^{\prime}\right)\\\ 0,&\text{ (otherwise)
}\end{cases}$ (2)

where $W_{xx^{\prime}\mid y}^{v}$ signifies the transition rate for subsystem
$X$ moving from state $x^{\prime}$ to state $x$ given that subsystem $Y$ is in
state $y$, and similarly, $W_{yy^{\prime}\mid x}^{v}$ for transitions within
subsystem $Y$.

###  II.2 Information thermodynamics of the bipartite system

Due to the bipartite nature, the entropy production rate $\dot{\sigma}$ of the
bipartite system can be split into two components:

$\dot{\sigma}=\dot{\sigma}^{X}+\dot{\sigma}^{Y}.$ (3)

Herein, $\dot{\sigma}^{X}$ and $\dot{\sigma}^{Y}$ represent the partial
entropy production rates resulting from transitions in $X$ and $Y$,
respectively, which are expressed as

$\dot{\sigma}^{X}=\frac{1}{2}\sum_{x,x^{\prime},y,v}J_{xx^{\prime}\mid
y}^{v}\ln\frac{p\left(x^{\prime},y\right)W_{xx^{\prime}\mid
y}^{v}}{p(x,y)W_{x^{\prime}x\mid y}^{v}}$ (4)

and

$\dot{\sigma}^{Y}=\frac{1}{2}\sum_{y,y^{\prime},x,v}J_{yy^{\prime}\mid
x}^{v}\ln\frac{p\left(y^{\prime},x\right)W_{yy^{\prime}\mid
x}^{v}}{p(y,x)W_{y^{\prime}y\mid x}^{v}}.$ (5)

The entropy flow towards the environment, $\dot{S}_{r}$, is given by:

$\dot{S}_{r}=\dot{S}_{r}^{X}+\dot{S}_{r}^{Y}$ (6)

where

$\dot{S}_{r}^{X}=\frac{1}{2}\sum_{x,x^{\prime},y,v}J_{xx^{\prime}\mid
y}^{v}\ln\frac{W_{xx^{\prime}\mid y}^{v}}{W_{x^{\prime}x\mid y}^{v}}$ (7)

and

$\dot{S}_{r}^{Y}=\frac{1}{2}\sum_{x,x^{\prime},y,v}J_{yy^{\prime}\mid
x}^{v}\ln\frac{W_{yy^{\prime}\mid x}^{v}}{W_{y^{\prime}y\mid x}^{v}},$ (8)

correspond to the energy flux from $X$ and $Y$ to their respective
environment.

$\dot{I}^{X}=\frac{1}{2}\sum_{x,x^{\prime},y,v}J_{xx^{\prime}\mid
y}^{v}\ln\frac{p(y\mid x)}{p\left(y\mid x^{\prime}\right)}$ (9)

and

$\dot{I}^{Y}=\frac{1}{2}\sum_{x,x^{\prime},y,v}J_{yy^{\prime}\mid
x}^{v}\ln\frac{p(x\mid y)}{p\left(x\mid y^{\prime}\right)}$ (10)

quantify how information exchanges between the two subsystems. When
$\dot{I}^{X}>0$, subsystem $X$ is measuring $Y$; vice versa, $\dot{I}^{X}<0$
signifies that $X$ is consuming information in order to extract energy.

From Eq. (4) to Eq. (10), it is evident that $\dot{\sigma}^{X}$ and
$\dot{\sigma}^{Y}$ must be non-negative, which can be split into three
components as (Su  _et al._ , 2022):

$\dot{\sigma}^{X}=\dot{S}^{X}+\dot{S_{r}}^{X}-\dot{I}^{X}\geq 0$ (11)

$\dot{\sigma}^{Y}=\dot{S}^{Y}+\dot{S_{r}}^{Y}-\dot{I}^{Y}\geq 0.$ (12)

Upon reaching a steady state, the system settles into a stationary probability
distribution $[\dot{p}(x)=0\mathrm{\,and}\dot{p}(y)=0]$, leading to
$\dot{S}^{X(Y)}=0$. Thus, Eqs. (11) and (12) are simplified as follows:

$\dot{\sigma}^{X}=\dot{S}_{r}^{X}-\dot{I}^{X}\geq 0$ (13)

and

$\dot{\sigma}^{Y}=\dot{S_{r}}^{Y}-\dot{I}^{Y}\geq 0.$ (14)

In scenarios where $\dot{I}^{X}>0$, as Eq. (13), the information $\dot{I}^{X}$
is constrained by the entropy flow to the environment $\dot{S}_{r}{}^{X}$,
which represents the dissipation required for $X$ to learn about $Y$.
Information then becomes a resource for energy extraction or work execution
via feedback on $Y$.

###  II.3 The thermodynamic efficiency of two subsystems

We now shift our focus to the nonequilibrium steady state, in which each
subsystem exchanges work and heat with their respective external reservoirs.
To articulate thermodynamic quantities, it’s assumed that the transition rates
adhere to the local detailed balance condition specified by the following
equation(Peliti and Pigolotti, 2021; Tanogami  _et al._ , 2023)

$\ln\frac{W_{xx^{\prime}\mid y}^{v}}{W_{x^{\prime}x\mid
y}^{v}}=\beta\left(\epsilon_{x^{\prime}y}-\epsilon_{xy}+\Delta_{xx^{\prime}\mid
y}^{v}\right),$ (15)

where $\beta=\left(k_{\mathrm{B}}T\right)^{-1}$ is the inverse temperature by
assuming that all the reservoirs are at temperature $T$, $\epsilon_{xy}$
represents the energy of state $(x,y)$, and $\Delta_{xx^{\prime}}^{y}$ is the
energy provided by an external agent during the transition
$\left(x^{\prime},y\right)\rightarrow(x,y)$. The parameter
$\Delta_{xx^{\prime}}^{y}$ will affect the heat flow. For the quantum dot
system interacting with an electron reservoir, $-\Delta_{xx^{\prime}}^{y}$
represents the chemical potential of the reservoir, which will be further
illustrated in Sec. III. The average rate of heat absorbed by $X$ from the
environment is then determined by

$\dot{Q}^{X}=-\frac{k_{\mathrm{B}}T}{2}\sum_{x,x^{\prime},y,v}J_{xx^{\prime}\mid
y}^{v}\ln\frac{W_{xx^{\prime}\mid y}^{v}}{W_{x^{\prime}x\mid y}^{v}}.$ (16)

The average rate of work done by the external agent on $X$ is identified as

$\dot{W}^{X}=\frac{1}{2}\sum_{x,x^{\prime},y,v}J_{xx^{\prime}\mid
y}^{v}\Delta_{xx^{\prime}\mid y}^{v}.$ (17)

Subsequently, the average rate of change of internal energy, interpreted as
the work done by $X$ on $Y$, is expressed as

$\dot{W}^{X\rightarrow Y}=\frac{1}{2}\sum_{x,x^{\prime},y,v}J_{xx^{\prime}\mid
y}^{v}\left(\epsilon_{xy}-\epsilon_{x^{\prime}y}\right).$ (18)

The rates $\dot{W}^{Y}$, $\dot{Q}^{Y}$, and $\dot{W}^{Y\rightarrow X}$ are
similarly defined for $Y$. Note that each subsystem adheres to the first law
of thermodynamics that encapsulates local energy conservation:

$\displaystyle\dot{W}^{X\rightarrow Y}$
$\displaystyle=\dot{W}^{X}+\dot{Q}^{X},$ (19)
$\displaystyle\dot{W}^{Y\rightarrow X}$
$\displaystyle=\dot{W}^{Y}+\dot{Q}^{Y}.$

Furthermore, at steady state, the powers and information flows satisfy
$\dot{W}^{X\rightarrow Y}+\dot{W}^{Y\rightarrow X}=0$ and
$\dot{I}^{X}+\dot{I}^{Y}=0$. On the basis of Eqs. (19), (13), and (14), each
subsystem conforms to the second law of thermodynamics at the steady state:

$\displaystyle\dot{\sigma}^{X}$
$\displaystyle=\beta\dot{W}^{X}-\beta\dot{W}^{X\rightarrow Y}-\dot{I}^{X}\geq
0,$ (20) $\displaystyle\dot{\sigma}^{Y}$
$\displaystyle=\beta\dot{W}^{Y}-\beta\dot{W}^{Y\rightarrow X}-\dot{I}^{Y}\geq
0.$

Within this framework, each subsystem satisfies the first and second laws.
Without loss of generality, let $Y$ be the "upstream" subsystem, with the rate
of work input to $Y$ being positive, denoted as $\dot{W}^{Y}>0$. We will limit
our discussion to functional machines that produce work from subsystem $X$,
i.e., $\dot{W}^{X}\leq 0$. They thus each have their own thermodynamic
efficiencies:

$\displaystyle\eta^{Y}$ $\displaystyle:=\frac{\beta\dot{W}^{X\rightarrow
Y}+\dot{I}^{Y}}{\beta\dot{W}^{Y}},$ (21) $\displaystyle\eta^{X}$
$\displaystyle:=\frac{-\beta\dot{W}^{X}}{\beta\dot{W}^{Y\rightarrow
X}+\dot{I}^{Y}}.$

As discussed in Ref. (Leighton and Sivak, 2023) , the efficiency of subsystem
$Y$, denoted as $\eta^{Y}$, reflects how effectively $Y$ transforms the input
work into free energy available for subsystem $X$. Conversely, $\eta^{X}$
measures the efficiency of $X$ in converting this free energy back into output
work. The product of these efficiencies,
$\eta^{Y}\eta^{X}=-\dot{W}^{X}/\dot{W}^{Y}=\eta^{T}$, defines the overall
thermodynamic efficiency of the system. These efficiencies remain valid as
long as both $\dot{W}_{Y}$ and $\beta\dot{W}_{Y\rightarrow X}+\dot{I}_{Y}$ are
strictly positive. This ensures that the conditions $0\leq\eta^{X}\leq 1$ and
$0\leq\eta^{Y}\leq 1$ are satisfied, as derived from Eq. (20).

###  II.4 Bounds on the efficiencies of two subsystems

Equation (20) yields dual equalities:

$\dot{\sigma}^{X}-\beta\dot{W}^{X}=\beta\dot{W}^{Y\rightarrow
X}+\dot{I}^{Y}=\beta\dot{W}^{Y}-\dot{\sigma}^{Y}.$ (22)

Applying any lower bounds $\dot{\sigma}_{LB}^{X}\leq\dot{\sigma}^{X}$ and
$\dot{\sigma}_{LB}^{Y}\leq\dot{\sigma}^{Y}$ to this equation facilitates the
derivation of upper and lower bounds on $\beta\dot{W}^{Y\rightarrow
X}+\dot{I}^{Y}$:

$\dot{\sigma}_{LB}^{X}-\beta\dot{W}^{X}\leq\beta\dot{W}^{Y\rightarrow
X}+\dot{I}^{Y}\leq\beta\dot{W}^{Y}-\dot{\sigma}_{LB}^{Y}.$ (23)

Dividing Eq. (23) by $\dot{W}^{Y}$ results in upper and lower bounds on the
efficiency of $Y$:

$\eta^{T}\left(1+\frac{\dot{\sigma}_{LB}^{X}}{-\beta\dot{W}^{X}}\right)\leq\eta^{Y}\leq
1-\frac{\dot{\sigma}_{LB}^{Y}}{\beta\dot{W}^{Y}}.$ (24)

Conversely, multiplying the reciprocal of Eq. (23) by $-\dot{W}^{X}$ produces
upper and lower bounds on the efficiency of $X$:

$\eta^{T}\left(1-\frac{\dot{\sigma}_{LB}^{Y}}{\beta\dot{W}^{Y}}\right)^{-1}\leq\eta^{X}\leq\left(1+\frac{\dot{\sigma}_{LB}^{X}}{-\beta\dot{W}^{X}}\right)^{-1}.$
(25)

In the context of Markovian dynamics, the recently derived bound (Su  _et al._
, 2022) provides tighter lower bounds of the partial entropy production rates
$\dot{\sigma}^{X}$ and $\dot{\sigma}^{Y}$ for the Markovian dynamics [also see
Appendix. A]:

$\displaystyle\dot{\sigma}^{X}$
$\displaystyle\geq\frac{\left(\dot{S}_{r}^{X}\right)^{2}}{\Theta^{X}},$ (26)
$\displaystyle\dot{\sigma}^{Y}$
$\displaystyle\geq\frac{\left(\dot{S}_{r}^{Y}\right)^{2}}{\Theta^{Y}},$

where

$\Theta^{X}=\frac{1}{2}\sum_{x,x^{\prime},y,v}\left(\ln\frac{W_{xx^{\prime}\mid
y}^{v}}{W_{x^{\prime}x\mid y}^{v}}\right)^{2}W_{xx^{\prime}\mid
y}^{v}p\left(x^{\prime},y\right),$ (27)

and

$\Theta^{Y}=\frac{1}{2}\sum_{y,y^{\prime},x,v}\left(\ln\frac{W_{yy^{\prime}\mid
x}^{v}}{W_{y^{\prime}y\mid x}^{v}}\right)^{2}W_{yy^{\prime}\mid
x}^{v}p\left(y^{\prime},x\right).$ (28)

The inequalities specified in Eq. (26) clearly articulate the dynamic
behaviors within a system interacting with a thermal bath. When bath $v$
triggers a transition from state $\left(x^{\prime},y\right)$ to $(x,y)$,
according to Eq. (15), the energy absorbed by the system during this process
is represented by the equation
$E(x,y)-E\left(x^{\prime},y\right)=-k_{B}T_{v}\ln\frac{W_{xx^{\prime}\mid
y}^{v}}{W_{x^{\prime}x\mid y}^{v}}$. Here, $E(x,y)$ denotes the energy of the
state $(x,y)$, $k_{B}$ is the Boltzmann constant, and $T_{v}$ represents the
temperature of bath $v$. The transition probability coefficient,
$W_{xx^{\prime}\mid y}^{v}p\left(x^{\prime},y\right)$, measures the rate at
which transitions between these states occur. Additionally, the factor
$\Theta^{X/Y}$ can be understood as a dynamic activity factor, quantifying the
level of active transitions influenced by the energy exchanges between the
states and the bath. Inserting these bounds into Eqs. (25) and (24) gives

$\begin{gathered}\eta^{T}\left[1+\frac{\left(\dot{S}_{r}^{X}\right)^{2}}{-\Theta^{X}\dot{W}^{X}}\right]\leq\eta^{Y}\leq
1-\frac{\left(\dot{S}_{r}^{Y}\right)^{2}}{\Theta^{Y}\dot{W}^{Y}},\\\
\eta^{T}\left[1-\frac{\left(\dot{S}_{r}^{Y}\right)^{2}}{\Theta^{Y}\dot{W}^{Y}}\right]^{-1}\leq\eta^{X}\leq\left[1+\frac{\left(\dot{S}_{r}^{X}\right)^{2}}{-\Theta^{X}\dot{W}^{X}}\right]^{-1}.\end{gathered}$
(29)

These two inequalities encapsulate a thorough formulation of our principal
results, providing a systematic method for establishing efficiency boundaries
for subsystems in relation to their interactions with the environment and
determining minimal rates of entropy production. Then we define
$\eta_{L}^{Y}=\eta^{T}\left[1+\frac{\left(\dot{S}_{r}^{X}\right)^{2}}{-\Theta^{X}\dot{W}^{X}}\right]$,
$\eta_{U}^{Y}=1-\frac{\left(\dot{S}_{r}^{Y}\right)^{2}}{\Theta^{Y}\dot{W}^{Y}}$,
$\eta_{L}^{X}=\eta^{T}\left[1-\frac{\left(\dot{S}_{r}^{Y}\right)^{2}}{\Theta^{Y}\dot{W}^{Y}}\right]^{-1}$
and
$\eta_{U}^{X}=\left[1+\frac{\left(\dot{S}_{r}^{X}\right)^{2}}{-\Theta^{X}\dot{W}^{X}}\right]^{-1}$.
As noted, Eq. (29) in our work bears similarity to Eq. (18) from Ref.
(Leighton and Sivak, 2023). This is because we have employed the same
procedure to determine the bound on the efficiency of the two subsystems.
Specifically, this involves substituting the explicit expression for the lower
bound of the entropy production rate. However, owing to the differing system
descriptions used, we have employed distinct approaches to derive the lower
bound of the entropy production rate. In Ref. (Leighton and Sivak, 2023), the
model is characterized by the overdamped Langevin equation, and the lower
bound on the entropy production rate was derived using Jensen’s inequality as
detailed in (Leighton and Sivak, 2022). Our findings are anchored within the
framework of stochastic thermodynamics. For establishing the lower bound of
the entropy production rate, we utilized the Cauchy-Schwarz inequality. Unlike
Jensen’s inequality, which necessitates convexity of the function, Cauchy-
Schwarz inequality imposes no such restriction. This property renders Cauchy-
Schwarz inequality more versatile within the context of stochastic
thermodynamics.

##  III The double-quantum-dot system

Figure 1: The schematic diagram of a double-quantum-dot system

The double-quantum-dot (DQD) system consists of two quantum dots interacting
via a Coulomb repulsion $U$. The DQD [See Fig. 1] is modeled by the
Hamiltonian

$H_{S}=\varepsilon_{X}d_{X}^{\dagger}d_{X}+\varepsilon_{Y}d_{Y}^{\dagger}d_{Y}+Ud_{X}^{\dagger}d_{X}d_{Y}^{\dagger}d_{Y},$
(30)

where $d_{X}^{\dagger}\left(d_{X}\right)$ creates (annihilates) one electron
on QD $X$ with energy $\varepsilon_{X}$, and
$d_{Y}^{\dagger}\left(d_{Y}\right)$ creates (annihilates) one electron on QD
$Y$ with energy $\varepsilon_{Y}$. QD $X(Y)$ is weakly coupled to two Fermi
reservoirs $XL$ and $XR$ ($YL$ and $YR$). Thus, electrons are transported
through parallel interacting channels. The Hamiltonian of the reservoirs is
given by

$H_{B}=\sum_{k}\sum_{\alpha\in\\{X,Y\\}}\sum_{\beta\in\\{L,R\\}}\varepsilon_{k\alpha\beta}c_{k\alpha\beta}^{\dagger}c_{k\alpha\beta},$
(31)

where $c_{k\alpha\beta}^{\dagger}\left(c_{k\alpha\beta}\right)$ is the
creation (annihilation) operator at energy level $\varepsilon_{k\alpha\beta}$
in bath $\alpha\beta$. The interaction between the DQD and the environment
reads

$H_{I}=\sum_{k}\sum_{\alpha\in\\{X,Y\\}}\sum_{\beta\in\\{L,R\\}}\left(t_{k\alpha\beta}d_{\alpha}c_{k\alpha\beta}^{\dagger}+t_{k\alpha\beta}^{*}c_{k\alpha\beta}d_{\alpha}^{\dagger}\right),$
(32)

where $t_{k\alpha\beta}$ denotes the coupling strength of the transition
between QD $\alpha$ and reservoir $\alpha\beta$ at energy level
$\varepsilon_{k\alpha\beta}$. We use $x_{0}$ and $y_{0}\left(x_{1}\right.$ and
$\left.y_{1}\right)$ to denote that $X$ and $Y$ are in empty (filled) states,
respectively. The energy eigenstates of the DQD coincide with the localized
Fock states
$\left|\left(x_{0},y_{0}\right)\right\rangle,\left|\left(x_{1},y_{0}\right)\right\rangle,\left|\left(x_{0},y_{1}\right)\right\rangle$,
and $\left|\left(x_{1},y_{1}\right)\right\rangle$, where their respective
eigenvalues are $0,\varepsilon_{X},\varepsilon_{Y}$, and
$\varepsilon_{X}+\varepsilon_{Y}+U$.

For a non-degenerate system weakly coupled to different environmental modes,
the dynamics of the populations satisfies Eq. (1). The transition rates follow
from Fermi’s golden rule and are given by

$\displaystyle W_{x_{1}x_{0}\mid y_{i}}^{v}=\Gamma_{y_{i}}^{v}f_{y_{i}}^{v},$
$\displaystyle W_{x_{0}x_{1}\mid
y_{i}}^{v}=\Gamma_{y_{i}}^{v}\left(1-f_{y_{i}}^{v}\right),$ $\displaystyle
W_{y_{1}y_{0}\mid x_{i}}^{v}=\Gamma_{x_{i}}^{v}f_{x_{i}}^{v},$ $\displaystyle
W_{y_{0}y_{1}\mid x_{i}}^{v}=\Gamma_{x_{i}}^{v}\left(1-f_{x_{i}}^{v}\right),$

where
$f_{x_{i}}^{v}=\left\\{1+\exp\left[\beta_{v}\left(\varepsilon_{Y}+iU-\mu_{v}\right)\right]\right\\}^{-1}$
and
$f_{y_{i}}^{v}=\left\\{1+\exp\left[\beta_{v}\left(\varepsilon_{X}+iU-\mu_{v}\right)\right]\right\\}^{-1}(i=0,1)$
are the Fermi distribution functions,
$\beta_{v}=1/\left(k_{\mathrm{B}}T_{v}\right)$, and
$\Gamma_{y_{i}}^{v}\left(\Gamma_{x_{i}}^{v}\right)$ is a positive constant
that characterizes the height of the potential barrier between $X(Y)$ and
reservoir $v$. The potential barrier of $Y$, characterized by
$\Gamma_{x_{i}}^{v}$, depends on the state of $X$, and vice versa. Note that
$\ln\frac{W_{x_{1}x_{0}\mid y_{0}}^{v}}{W_{x_{0}x_{1}\mid
y_{0}}^{v}}=\beta_{v}\left(\varepsilon_{X}-\mu_{v}\right)$ and
$\ln\frac{W_{x_{1}x_{0}\mid y_{1}}^{v}}{W_{x_{0}x_{1}\mid
y_{1}}^{v}}=\beta_{v}\left(\varepsilon_{X}+U-\mu_{v}\right)$. Thus, for the
quantum dot model, the parameter $\Delta_{xx^{\prime}}^{y}$ in Eq. (15) is
equal to $-\mu_{v}$ and will affect the heat flow.

For the purpose of reducing the number of parameters, the energy dependences
of the tunneling rates are parametrized by dimensionless parameters $\delta$
and $\Delta$, i.e.,

$\displaystyle\Gamma_{x_{0}}^{YL}$
$\displaystyle=\Gamma\frac{e^{\Delta}}{\cosh(\Delta)},$
$\displaystyle\Gamma_{x_{1}}^{YL}=\Gamma\frac{e^{-\Delta}}{\cosh(\Delta)},$
(33) $\displaystyle\Gamma_{x_{0}}^{YR}$
$\displaystyle=\Gamma\frac{e^{-\Delta}}{\cosh(\Delta)},$
$\displaystyle\Gamma_{x_{1}}^{YR}=\Gamma\frac{e^{\Delta}}{\cosh(\Delta)},$
$\displaystyle\Gamma_{y_{0}}^{XL}$
$\displaystyle=\Gamma\frac{e^{\delta}}{\cosh(\delta)},$
$\displaystyle\Gamma_{y_{1}}^{XL}=\Gamma\frac{e^{-\delta}}{\cosh(\delta)},$
$\displaystyle\Gamma_{y_{0}}^{XR}$
$\displaystyle=\Gamma\frac{e^{-\delta}}{\cosh(\delta)},$
$\displaystyle\Gamma_{y_{1}}^{XR}=\Gamma\frac{e^{\delta}}{\cosh(\delta)}.$

The dimensionless parameters $\delta$ and $\Delta$ thus allow us to control
the tunneling rates under different conditions. When $\delta=\Delta=0$, this
corresponds to the scenario of completely symmetric and equal tunneling rates.
Conversely, in the limit where $\delta$ approaches infinity, an electron in
channel $X$ can only enter and exit from reservoir $XL$ at energy
$\varepsilon_{X}$, whereas tunneling processes to reservoir $XR$ are permitted
at energy $\varepsilon_{X}+U$. In this limit, transport is possible only
through energy exchange with channel $Y$. Similarly, the parameter $\Delta$
controls the behavior of channel $Y$. According to Eqs. (7)-(10), (27), and
(28), the entropy flows associated with subsystem $X$ and $Y$

$\displaystyle\dot{S}_{r}^{X}$ $\displaystyle=$
$\displaystyle-\beta\Delta\mu_{X}\left(J_{x_{0}x_{1}\mid
y_{0}}^{XR}+J_{x_{0}x_{1}\mid y_{1}}^{XR}\right)$ $\displaystyle-\beta
U\left(J_{y_{1}y_{0}\mid x_{0}}^{YL}+J_{y_{1}y_{0}\mid x_{0}}^{YR}\right),$

$\displaystyle\dot{S}_{r}^{Y}$ $\displaystyle=$
$\displaystyle-\beta\Delta\mu_{Y}\left(J_{y_{0}y_{1}\mid
x_{0}}^{YR}+J_{y_{0}y_{1}\mid x_{1}}^{YR}\right)$ $\displaystyle-\beta
U\left(J_{x_{1}x_{0}\mid y_{0}}^{XL}+J_{x_{1}x_{0}\mid y_{0}}^{XR}\right),$

the rate of informations

$\dot{I}^{X}=\left(J_{y_{1}y_{0}\mid x_{0}}^{YL}+J_{y_{1}y_{0}\mid
x_{0}}^{YR}\right)\ln\frac{p\left(x_{0},y_{1}\right)p\left(x_{1},y_{0}\right)}{p\left(x_{0},y_{0}\right)p\left(x_{1},y_{1}\right)},$
(36)

$\dot{I}^{Y}=\left(J_{x_{1}x_{0}\mid y_{0}}^{XL}+J_{x_{1}x_{0}\mid
y_{0}}^{XR}\right)\ln\frac{p\left(y_{0},x_{1}\right)p\left(y_{1},x_{0}\right)}{p\left(y_{0},x_{0}\right)p\left(y_{1},x_{1}\right)},$
(37)

and the coefficients

$\displaystyle\Theta^{X}$
$\displaystyle=\frac{1}{2}\left(\varepsilon_{X}-\mu_{XL}\right)^{2}\left[W_{10\mid
0}^{XL}p(0,0)+W_{01\mid 0}^{XL}p(1,0)\right]$ (38)
$\displaystyle+\frac{1}{2}\left(\varepsilon_{X}-\mu_{XR}\right)^{2}\left[W_{10\mid
0}^{XR}p(0,0)+W_{01\mid 0}^{XR}p(1,0)\right]$
$\displaystyle+\frac{1}{2}\left(\varepsilon_{X}+U-\mu_{XL}\right)^{2}\left[W_{10\mid
1}^{XL}p(0,1)+W_{01\mid 1}^{XL}p(1,1)\right]$
$\displaystyle+\frac{1}{2}\left(\varepsilon_{X}+U-\mu_{XR}\right)^{2}\left[W_{10\mid
1}^{XR}p(0,1)+W_{01\mid 1}^{XR}p(1,1)\right],$

$\displaystyle\Theta^{Y}$
$\displaystyle=\frac{1}{2}\left(\varepsilon_{Y}-\mu_{YL}\right)^{2}\left[W_{10\mid
0}^{YL}p(0,0)+W_{01\mid 0}^{YL}p(1,0)\right]$ (39)
$\displaystyle+\frac{1}{2}\left(\varepsilon_{Y}-\mu_{YR}\right)^{2}\left[W_{10\mid
0}^{YR}p(0,0)+W_{01\mid 0}^{YR}p(1,0)\right]$
$\displaystyle+\frac{1}{2}\left(\varepsilon_{Y}+U-\mu_{YL}\right)^{2}\left[W_{10\mid
1}^{YL}p(0,1)+W_{01\mid 1}^{YL}p(1,1)\right]$
$\displaystyle+\frac{1}{2}\left(\varepsilon_{Y}+U-\mu_{YR}\right)^{2}\left[W_{10\mid
1}^{YR}p(0,1)+W_{01\mid 1}^{YR}p(1,1)\right].$

##  IV results and discussion

Figure 2: The efficiencies $\eta^{X}$ and $\eta^{Y}$, upper bounds
$\eta_{U}^{X}$ and $\eta_{U}^{Y}$, and lower bounds $\eta_{L}^{X}$ and
$\eta_{L}^{Y}$ for the double-quantum-dot system vary with the dimensionless
Coulomb interaction strength $\beta U$ and the parameters $\delta$ and
$\Delta$ associated with the tunneling rate, where
$\beta\left(\mu_{XR}-\mu_{XL}\right)=1$,
$\beta\left(\mu_{YR}-\mu_{YL}\right)=2,\Gamma=1,\varepsilon_{X}=\frac{\left(\mu_{XR}+\mu_{XL}\right)}{2}-\frac{U}{2}$,
and $\varepsilon_{Y}=\frac{\left(\mu_{YR}+\mu_{YL}\right)}{2}-\frac{U}{2}$.
For (a) and (b), the variations are with $\beta U$ and $\delta$, where
$\Delta=1.5$. For (c) and (d), the variations are with $\beta U$ and $\Delta$,
where $\delta=1.5$.

Figure 2 plots the efficiencies $\eta^{X}$ and $\eta^{Y}$, upper bounds
$\eta_{U}^{X}$ and $\eta_{U}^{Y}$, and lower bounds $\eta_{L}^{X}$ and
$\eta_{L}^{Y}$ for subsystems $X$ and $Y$ varying with the dimensionless
Coulomb interaction strength $\beta U$ and the parameter $\delta$ and $\Delta$
associated with the tunneling rate.

From Figure 2(a), it is evident that for subsystem $X$, the discrepancy
between the upper bound $\eta_{U}^{X}$ and the actual efficiency $\eta^{X}$ is
very minimal, and this difference decreases significantly as the value of
$\delta$ increases. Furthermore, when $\delta$ is held constant, the lower
bound $\eta_{L}^{X}$ becomes progressively tighter as $\beta U$ decreases. It
is worth emphasizing that when both $\beta U$ and $\delta$ are small, the
three efficiencies—upper bound, actual efficiency, and lower bound—converge
closely to one another.

On the other hand, Figure 2(b) presents the results for subsystem $Y$. These
results indicate that the upper bound $\eta_{U}^{Y}$ becomes tighter primarily
when $\beta U$ is small. As $\beta U$ increases, this upper bound approaches
the general result, namely $\eta^{Y}\leq 1$. Therefore, the discussion mainly
pertains to the lower bound $\eta_{L}^{Y}$, which becomes very tight under low
limits. When $\beta U$ is fixed, a larger $\delta$ improves the effectiveness
of the lower bound as a limit. Similarly, when $\delta$ is fixed, a larger
lower limit of $\beta U$ enhances the effectiveness of this boundary. When
both $\beta U$ and $\delta$ are large, the lower bound is notably stringent.
When we adjust $\beta U$ and $\Delta$, the results are shown in Figures 2(c)
and (d). It can be observed that the overall trend is consistent with the
adjustments of $\beta U$ and $\delta$. These show that our boundaries are
reliable.

Additionally, it is important to note that in the definition of efficiency in
Eq. (21), we assume $\dot{W}^{X}\leq 0$ and $\dot{W}^{Y}>0$ for this model.
Therefore, for the quantum dot model described in Sec. III, there exists a
certain range of parameter values that enable the proper operation of the
bipartite system. For example, as shown in Figure 2(a), when both $\beta U$
and $\delta$ are small, the three efficiencies converge. This convergence
occurs because a small value of $\beta U$ indicates a very weak interaction
between the two subsystems. Simultaneously, within the range of small values
of $\delta$, the work $\dot{W}^{X}$ tends to zero. This results in the
bipartite system failing to operate normally. Consequently, all three
efficiencies converge towards zero under these conditions.

##  V conclusions

This paper successfully establishes a foundational lower bound on the entropy
production rate of subsystems by employing the Cauchy-Schwarz inequality,
leading to a significant advancement in understanding the efficiency
boundaries of dual subsystems. The derivation of these efficiency bounds,
grounded in the lower limit of entropy production, ensures applicability to
both finely and coarsely-grained systems. Our empirical investigations,
utilizing the double-quantum-dot system, not only validate the proposed
inequalities but also demonstrate their capability to significantly tighten
the efficiency boundaries. This work, deeply rooted in the principles of
stochastic thermodynamics, offers invaluable insights applicable to a wide
range of quantum-dot systems.

###### Acknowledgements.

This work has been supported by the National Natural Science Foundation
(Grants No. 12075197 and 12364008), Natural Science Foundation of Fujian
Province (Grant No. 2023J01006), Fundamental Research Fund for the Central
Universities (No. 20720240145), Guiding Project of Fujian Provincial
Department of Science and Technology (2021H0051), and Talent Project of
Quanzhou Science and Technology Bureau (2020C029R).

##  Appendix A Derivation of the bound of the entropy production rate of
subsystem $X$

In this section, we provide a brief derivation of the efficiency limit for
learning. Firstly, we evaluate the upper bound of
$\left|\dot{S}_{r}^{X}\right|$ as follows

$\displaystyle\left|\dot{S}_{r}^{X}\right|$
$\displaystyle=\left|\frac{1}{2}\sum_{x,x^{\prime},y,v}J_{xx^{\prime}\mid
y}^{v}\ln\frac{W_{xx^{\prime}\mid y}^{v}}{W_{x^{\prime}x\mid y}^{v}}\right|$
(40)
$\displaystyle=\left|\frac{1}{2}\sum_{x,x^{\prime},y,v}\ln\frac{W_{xx^{\prime}\mid
y}^{v}}{W_{x^{\prime}x\mid y}^{v}}\sqrt{W_{xx^{\prime}\mid
y}^{v}p\left(x^{\prime},y\right)+W_{x^{\prime}x\mid
y}^{v}p(x,y)}\frac{J_{xx^{\prime}\mid y}^{v}}{\sqrt{W_{xx^{\prime}\mid
y}^{v}p\left(x^{\prime},y\right)+W_{x^{\prime}x\mid y}^{v}p(x,y)}}\right|$
$\displaystyle\stackrel{{\scriptstyle(1)}}{{\leq}}\frac{1}{2}\sqrt{\sum_{x,x^{\prime},y,v}\left(\ln\frac{W_{xx^{\prime}\mid
y}^{v}}{W_{x^{\prime}x\mid y}^{v}}\right)^{2}\left[W_{xx^{\prime}\mid
y}^{v}p\left(x^{\prime},y\right)+W_{x^{\prime}x\mid
y}^{v}p(x,y)\right]}\sqrt{\sum_{x,x^{\prime},y,v}\frac{\left(J_{xx^{\prime}\mid
y}^{v}\right)^{2}}{W_{xx^{\prime}\mid
y}^{v}p\left(x^{\prime},y\right)+W_{x^{\prime}x\mid y}^{v}p(x,y)}}$
$\displaystyle\stackrel{{\scriptstyle(2)}}{{\leq}}\frac{1}{2}\sqrt{\sum_{x,x^{\prime},y,v}\left(\ln\frac{W_{xx^{\prime}\mid
y}^{v}}{W_{x^{\prime}x\mid y}^{v}}\right)^{2}\left[W_{xx^{\prime}\mid
y}^{v}p\left(x^{\prime},y\right)+W_{x^{\prime}x\mid
y}^{v}p(x,y)\right]}\sqrt{\frac{1}{2}\sum_{x,x^{\prime},y,v}J_{xx^{\prime}\mid
y}^{v}\ln\frac{W_{xx^{\prime}\mid
y}^{v}p\left(x^{\prime},y\right)}{W_{x^{\prime}x\mid y}^{v}p(x,y)}}$
$\displaystyle=\frac{1}{2}\sqrt{\sum_{x,x^{\prime},y,v}\left(\ln\frac{W_{xx^{\prime}\mid
y}^{v}}{W_{x^{\prime}x\mid y}^{v}}\right)^{2}\left[W_{xx^{\prime}\mid
y}^{v}p\left(x^{\prime},y\right)+W_{x^{\prime}x\mid
y}^{v}p(x,y)\right]}\sqrt{\dot{\sigma}^{X}}$
$\displaystyle=\sqrt{\frac{1}{2}\sum_{x,x^{\prime},y,v}\left(\ln\frac{W_{xx^{\prime}\mid
y}^{v}}{W_{x^{\prime}x\mid y}^{v}}\right)^{2}W_{xx^{\prime}\mid
y}^{v}p\left(x^{\prime},y\right)\sqrt{\dot{\sigma}^{X}}}$
$\displaystyle=\sqrt{\Theta^{X}\dot{\sigma}^{X}}.$

This evaluation incorporates the Cauchy-Schwartz inequality (Funo  _et al._ ,
2019; Shiraishi  _et al._ , 2018) and the inequality
$\frac{(x-y)^{2}}{x+y}\leq\frac{x-y}{2}\log\frac{x}{y}$ for nonnegative $x$
and $y$ in step $(1)$ and $(2)$, respectively. With the application of Eq.
(40), we naturally derive the lower limit on the entropy-production rate of
subsystem $X$, as presented in Eq. (26) in the main text.

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