# Tighter thermodynamic bound on speed limit in systems with unidirectional
transitions

Deepak Gupta1 Corresponding author<EMAIL_ADDRESS>Daniel M.
Busiello2<EMAIL_ADDRESS>1Dipartimento di Fisica ‘G. Galilei’, INFN,
Università di Padova, Via Marzolo 8, 35131 Padova, Italy  2Ecole Polytechnique
Fédérale de Lausanne (EPFL), Institute of Physics Laboratory of Statistical
Biophysics, 1015 Lausanne, Switzerland

###### Abstract

We consider a general discrete state-space system with both unidirectional and
bidirectional links. In contrast to bidirectional links, there is no reverse
transition along the unidirectional links. Herein, we first compute the
statistical length and the thermodynamic cost function for transitions in the
probability space, highlighting contributions from total, environmental, and
resetting (unidirectional) entropy production. Then, we derive the
thermodynamic bound on the speed limit to connect two distributions separated
by a finite time, showing the effect of the presence of unidirectional
transitions. Novel uncertainty relationships can be found for the temporal
first and second moments of the average resetting entropy production. We
derive simple expressions in the limit of slow unidirectional transition
rates. Finally, we present a refinement of the thermodynamic bound, by means
of an optimization procedure. We numerically investigate these results on
systems that stochastically reset with constant and periodic resetting rate.

##  I Introduction

Small systems coupled with heat baths evolve under stochastic dynamics. Their
probabilistic description at all time can be understood using the master
equation or the Fokker-Planck equation vK . In this kind of systems, an
increasing interest has been devoted in studying thermodynamic observables,
such as heat dissipated in the environment, work done on the system, entropy
production, probability currents, and so on. The estimation of these
fluctuating quantities requires a description at the level of a single
stochastic trajectory, which can be obtained within the celebrated framework
of stochastic thermodynamics ST-0 ; ST . Interestingly, these stochastic
quantities follow some universal results in the non-equilibrium physics;
namely, fluctuation theorem US , Jarzynski equality JE , Crooks fluctuation
relation CR , and thermodynamic uncertainty relations (TURs) tur-0 .

More recently, among those various stochastic systems, those exhibiting
unidirectional transitions have attracted attention due to broad interest in
several areas of science. Examples include spontaneous decay of atom rahav ,
directed percolation in liquid crystal Masaki , TASEP Ronald ; saha , driven
inelastic Lorentz-gas Chong , and stochastic resetting Evans_2011  (see also
Evans_2020  and references therein). In these systems, there is no reverse
transition along the unidirectional link, in stark contrast with the case of
bidirectional transitions. In this direction, only few attempts have been made
in understanding the thermodynamics of these systems; namely, the first and
second law of thermodynamics Fuchs_2016 ; Busiello_2020 , work fluctuation and
Jarzynski equality Gupta_2020 , integral fluctuation theorem Pal_2017 .

Information theory provides a fascinating language to rephrase most of the
well-known results in thermodynamics, and to shed new light on their meaning.
The connection between information and thermodynamics goes back to the seminal
work by Maxwell and his thought experiment max , and has been recently
properly presented by the work by Parrondo et al. Parrondo . The intense
research activity on this topic has led to numerous remarkable findings, e.g.,
thermodynamic uncertainty relations tur-1 ; tur-2 ; tur-3 ; tur-4 , a
connection of entropy and K-L divergence footprints , the convex property of
relative entropy esposito , the generalized Jarzynski equality sagawa , a
bound on the entropy production due to information flow causal . Several
interesting results in close to equilibrium systems have been also obtained
following differential geometric interpretation of thermodynamics neq-1 ;
neq-2 ; neq-3 ; neq-4 ; crooks ; sivak ; sivak-2 . Arbitrarily far from
equilibrium, Ito ito  found a thermodynamic bound on the time to reach a final
state from an initial state, strengthening the link between information and
thermodynamics. Recently, this result have been used to study the relation
between the cost in the bacterial growth and the evolution time bacteria , and
between adaptation speed and its thermodynamic cost for E. coli ecoli .
Similarly, with different approaches, speed limit has also been investigated
in Refs. Funo ; vo . Here, we aim to extend the formalism and the results
presented in Ref. ito  to systems with unidirectional transitions.

In this paper, we consider a discrete state-space system with both
unidirectional and bidirectional transitions. In this system, we first compute
the different contributions of entropy productions. We remark that the
analysis is based on the recent work by Ito ito , and it is carried out in the
case of a system with stochastic resetting and for the general case of
multiple unidirectional transitions. In what follows, we will use the name
resetting entropy production to identify the contribution stemming from
unidirectional links only. Then, we compute the thermodynamic action or the
thermodynamic cost function and the statistical length crooks  for the system
to make transition from an initial state to the final one in a finite time. We
show that the time to connect these two distributions is bounded by a function
which contains the information of thermodynamic cost function and the
statistical length. Here, we obtain three important results: 1) the
thermodynamic cost function and the thermodynamic bound on the speed limit to
connect the initial and final distributions for systems with both
unidirectional and bidirectional transitions, 2) two inequalities and an
uncertainty relation (TUR-like) involving resetting entropy production, and 3)
a novel bound on the speed limit which is stronger than that obtained in ito .
Further, we stress that such a tighter bound is also applicable to systems
with only bidirectional transitions.

The rest of the paper is organized as follows. In Sec. II, we describe a
system with stochastic resetting and compute the entropy productions
associated with it. In order to understand the distance between two
distributions, we discuss K-L divergence in Sec. III. Sec. IV contains the
definition of the statistical length and the thermodynamic action, and their
connection with thermodynamic quantities. Using these ingredients, we discuss
the thermodynamic bound on the speed limit to connect two distributions in
Sec. V, in the presence of unidirectional transition rates. Moreover, we
investigate the effect of small resetting rate on the bound. Sec. VI is
dedicated to novel inequalities and an uncertainty relation (TUR-like) for the
resetting entropy production. In Sec. VII, we propose a derivation of a new
bound on the speed limit, and that can also be made tighter using an
optimization scheme (in Sec. VIII). Finally, we summarize our paper in Sec.
IX. In Appendix A and B, we discuss, respectively, thermodynamics and Fisher
information metric of systems with multiple unidirectional links.

##  II Set up

Consider a system composed of $N$ discrete states. Suppose the system jumps
from a state $n^{\prime}$ to another state $n$ with a non-zero transition rate
$W_{n^{\prime}\to n}>0$, and there also exists a non-zero reverse transition
$W_{n\to n^{\prime}}>0$. In addition to this, the system stochastically resets
to a given state with a rate $\gamma_{n}>0$ from a state $n$. Notice that
unlike $W_{n\to m}$, corresponding to each $\gamma_{n}$ there do not exist a
reverse transition. Thus, the evolution of the probability of the system to be
in state $n$ at time $t$ is described by the following master equation vK :

$\displaystyle\dfrac{\mathrm{d}p_{n}(t)}{\mathrm{d}t}=\sum_{n^{\prime}\neq
n}^{N}[W_{n^{\prime}\to n}~{}p_{n^{\prime}}(t)-W_{n\to
n^{\prime}}~{}p_{n}(t)]~{}+$
$\displaystyle-\gamma_{n}~{}p_{n}(t)+\delta_{n,1}\sum_{m=1}^{N}\gamma_{m}~{}p_{m}(t),\qquad
1\leq n\leq N,$ (1)

where the first term on the right-hand side corresponds to the bidirectional
transitions. The second and third terms on the right-hand side of Eq. (1),
respectively, account for the contributions due to the loss in the probability
of state $n$ and the gain in the probability of state $1$ (i.e., the resetting
state) from all states. In the above equation (1), $\delta_{n,m}$ is the
Kronecker delta defined as $\delta_{n,m}=1$ for $n=m$, and $0$ otherwise.
Since the probability distribution over all states has to be normalized,
$\sum_{n=1}^{N}~{}p_{n}(t)=1$, the sum of the bidirectional transition rates
from a given states to all states is zero: $\sum_{n^{\prime}=1}^{N}~{}W_{n\to
n^{\prime}}=0$. This implies that $W_{n\to n}=-\sum_{n^{\prime}\neq n}W_{n\to
n^{\prime}}<0$. A schematic diagram of a four states system is shown in Fig.
1(upper panel), where solid arrows are the bidirectional links and the dot
dashed ones are the unidirectional (resetting) links.

In general, the transition rates on the right-hand side of Eq. (1) may depend
on time $t$. In what follows, unless specified, the transition rates have an
explicit time-dependence. Note that in Eq. (1), we are considering systems
that resets to only one state. Nevertheless, the generalization for a system
which resets stochastically to any states is straightforward to handle (see
Appendix A), i.e., there can be multiple unidirectional links in the network
Busiello_2020 .

The average entropy of the system is given by

$\displaystyle S^{\mathrm{sys}}=-\sum_{n=1}^{N}p_{n}(t)\ln p_{n}(t),$ (2)

where $p_{n}(t)$ is the solution of master equation (1) subject to an initial
condition $p_{n}(0)$. In the above equation (2), we have set the Boltzmann’s
constant $k_{B}$ equal to 1. For convenience, in the following, we will not
explicitly write the time-dependence in $p_{n}(t)$.

Differentiating Eq. (2) with respect to time $t$, we get the average entropy
production of the system

$\displaystyle\dot{S}^{\mathrm{sys}}=-\sum_{n=1}^{N}\dot{p}_{n}\ln
p_{n}-\sum_{n=1}^{N}\dot{p}_{n},$ (3)

where the dot indicates a time derivative. In the expression above, Eq. (3),
the last term is equal to zero since the probability distribution is
normalized.

Substituting Eq. (1) in Eq. (3), we get

$\displaystyle\dot{S}^{\mathrm{sys}}$ $\displaystyle=-\sum_{n=1}^{N}\ln
p_{n}\bigg{[}\sum_{n^{\prime}\neq n}(W_{n^{\prime}\to
n}~{}p_{n^{\prime}}-W_{n\to n^{\prime}}~{}p_{n})~{}+$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-\gamma_{n}~{}p_{n}+\delta_{n,1}\sum_{m=1}^{N}\gamma_{m}~{}p_{m}\bigg{]}$
$\displaystyle=\sum_{n^{\prime},n}W_{n^{\prime}\to
n}~{}p_{n^{\prime}}\ln\dfrac{p_{n^{\prime}}}{p_{n}}+\sum_{n}\gamma_{n}~{}p_{n}\ln\dfrac{p_{n}}{p_{1}},$
(4)

where $1\leq n,n^{\prime}\leq N$. Here, the second equality is obtained from
the first one using that $W_{n\to n}=-\sum_{n^{\prime}\neq n}W_{n\to
n^{\prime}}$. Following Busiello_2020 , the terms on the right-hand side of
Eq. (4) can be rewritten as following:

$\displaystyle\dot{S}^{\mathrm{sys}}=\overbrace{\dfrac{1}{2}\sum_{n^{\prime},n}J_{n^{\prime}\to
n}~{}F_{n^{\prime}\to
n}}^{\dot{S}^{\mathrm{tot}}}-\overbrace{\dfrac{1}{2}\sum_{n^{\prime},n}J_{n^{\prime}\to
n}\sigma^{\mathrm{bath}}_{n^{\prime}\to n}}^{\dot{S}^{\mathrm{bath}}}~{}+$
$\displaystyle-\overbrace{\sum_{n}J^{\mathrm{reset}}_{n}\sigma_{n}^{\mathrm{reset}}}^{\dot{S}^{\mathrm{reset}}},$
(5)

where $\dot{S}^{\mathrm{tot}}$, $\dot{S}^{\mathrm{bath}}$, and
$\dot{S}^{\mathrm{reset}}$, respectively, are the total entropy production,
the bath entropy production, and the resetting entropy production Fuchs_2016 ;
Busiello_2020 . Herein, the factor $1/2$ avoids the double-counting in the
first two summations. Notice that when the system has multiple unidirectional
links along with bidirectional ones, the resetting entropy production (which
will be recognized as the unidirectional entropy production) can be easily
modified (see Appendix A) Busiello_2020 . In the above equation (5), we
identify the following terms:

$\displaystyle J_{n^{\prime}\to n}$ $\displaystyle=W_{n^{\prime}\to
n}~{}p_{n^{\prime}}-W_{n\to n^{\prime}}~{}p_{n},$ (6) $\displaystyle
F_{n^{\prime}\to n}$ $\displaystyle=\ln\dfrac{W_{n^{\prime}\to
n}~{}p_{n^{\prime}}}{W_{n\to n^{\prime}}~{}p_{n}},$ (7)
$\displaystyle\sigma^{\mathrm{bath}}_{n^{\prime}\to n}$
$\displaystyle=\ln\dfrac{W_{n^{\prime}\to n}}{W_{n\to n^{\prime}}},$ (8)
$\displaystyle\sigma_{n}^{\mathrm{reset}}$
$\displaystyle=\ln\dfrac{p_{1}}{p_{n}}$ (9) $\displaystyle
J^{\mathrm{reset}}_{n}$ $\displaystyle=\gamma_{n}~{}p_{n}.$ (10)

These represent, respectively, the thermodynamic flux (6) for the transition
$n^{\prime}\to n$ and the corresponding thermodynamic force (7), the change in
entropy in the bath due to transition from $n^{\prime}\to n$, the change in
resetting entropy (9) and the flux due to resetting the system from state $n$
to 1 (10). Notice that both the thermodynamic force and the change in entropy
in the bath are anti-symmetric under exchange of indices, i.e.,
$F_{n^{\prime}\to n}=-F_{n\to n^{\prime}}$ and
$\sigma^{\mathrm{bath}}_{n^{\prime}\to n}=-\sigma^{\mathrm{bath}}_{n\to
n^{\prime}}$.

When the system has time-independent rates and there is no resetting mechanism
(i.e., $\gamma_{n}=0$ for all $n$), it eventually relaxes to a stationary
state at large time. This state can be either equilibrium or non-equilibrium
stationary state depending on whether or not, respectively, the transition
rates obey the detailed balance: $W_{n^{\prime}\to n}~{}p_{n^{\prime}}=W_{n\to
n^{\prime}}~{}p_{n}$ for each $(n,n^{\prime})$ link. When the detailed balance
condition is not satisfied, the system is characterized by a total entropy
production, $\dot{S}^{\mathrm{tot}}$. Furthermore, in the presence of time-
independent resetting rate $\gamma_{n}$ (see Fig. 1), the system reaches a
non-equilibrium steady state even when bidirectional transitions obey detailed
balance at time $t=0$: $W_{n^{\prime}\to n}~{}p_{n^{\prime}}(0)=W_{n\to
n^{\prime}}~{}p_{n}(0)$ for each $(n,n^{\prime})$ link.

Figure 1: Thermodynamic flux (6) across $(n,n^{\prime})$ link with respect to
observation time $t$. Herein, we consider a network of four states [panel (a)]
with an initial condition
$(p_{1}(0),p_{2}(0),p_{3}(0),p_{4}(0))=(0.4,0.2,0.25,0.15)$. We have taken
bidirectional transition rates (solid arrows) $W_{2\to 1}=0.5,W_{3\to
2}=0.1,W_{3\to 1}=0.5,W_{4\to 3}=0.5,W_{1\to 4}=0.5$, and the reverse
transition rates are fixed by the detailed balanced condition at time $t=0$.
Moreover, we consider resetting rates $\gamma_{n}=0.1$ for all $n$ (dot-dashed
arrows). Clearly, each thermodynamic flux reaches to a non-zero stationary
value at large time indicating the system is in a non-equilibrium steady state
[panel (b)].

##  III K-L divergence and the Fisher metric

Consider two distributions $\vec{P}=(p_{1},p_{2},\dots,p_{N})$ and
$\vec{Q}=(q_{1},q_{2},\dots,q_{N})$, where $p_{n}\geq 0$ and $q_{n}\geq 0$,
for $n=1,\dots,N$. These distributions are normalized: $\sum_{n}p_{n}=1$ and
$\sum_{n}q_{n}=1$. The distance between $\vec{P}$ and $\vec{Q}$ is given by
the Kullback-Leibler (K-L) divergence Cover :

$\displaystyle
D_{\mathrm{KL}}(\vec{P}||\vec{Q})=\sum_{n=1}^{N}p_{n}\ln\dfrac{p_{n}}{q_{n}}.$
(11)

Let us first summarize the properties of this measure: 1) it is not a
symmetric distance, i.e., $D_{\mathrm{KL}}(\vec{P}||\vec{Q})\neq
D_{\mathrm{KL}}(\vec{Q}||\vec{P})$, 2) it does not follow triangle inequality,
and 3) it is non-negative: $D_{\mathrm{KL}}(\vec{P}||\vec{Q})\geq 0$, where
equality holds if and only if $\vec{P}=\vec{Q}$.

Now suppose that the distributions $\vec{P}$ and $\vec{Q}$ differ by a small
amount, i.e., $\vec{Q}=\vec{P}+\vec{\mathrm{d}P}$. Therefore, the distance
between these two distributions would be

$\displaystyle
D_{\mathrm{KL}}(\vec{P}||\vec{P}+\vec{\mathrm{d}P})=\sum_{n=1}^{N}p_{n}\ln\dfrac{p_{n}}{p_{n}+\mathrm{d}p_{n}}.$
(12)

Expanding the right-hand side of the above equation up to second order, we get

$\displaystyle
D_{\mathrm{KL}}(\vec{P}||\vec{P}+\vec{\mathrm{d}P})\approx\dfrac{1}{2}\sum_{n=1}^{N}\dfrac{(\mathrm{d}p_{n})^{2}}{p_{n}},$
(13)

where we have used the condition $\sum_{n=1}^{N}\mathrm{d}p_{n}=0$. From the
right-hand side of the above equation, we can identify the square of a line
element ito :

$\displaystyle\mathrm{d}s^{2}=\sum_{n=1}^{N}\dfrac{(\mathrm{d}p_{n})^{2}}{p_{n}}\approx
2~{}D_{\mathrm{KL}}(\vec{P}||\vec{P}+\vec{\mathrm{d}P}),$ (14)

If the system can be externally modulated using a set of protocols
$\vec{\lambda}=(\lambda_{1},\lambda_{2},\dots,\lambda_{m})$, then we write the
above equation as

$\displaystyle\mathrm{d}s^{2}=\sum_{n=1}^{N}~{}p_{n}\bigg{(}\sum_{i=1}^{m}\dfrac{\mathrm{\partial}\ln
p_{n}}{\mathrm{\partial}\lambda_{i}}~{}\mathrm{d}\lambda_{i}\bigg{)}^{2}$
$\displaystyle=\sum_{i,j}\mathrm{d}\lambda_{i}~{}\mathrm{d}\lambda_{j}~{}g_{i,j},$
(15)

where $g_{i,j}=\sum_{n=1}^{N}p_{n}\frac{\partial\ln
p_{n}}{\partial\lambda_{i}}\frac{\partial\ln p_{n}}{\partial\lambda_{j}}$ is
the Fisher’s information metric Cover . In the following, we use the quantity
${\rm d}s$ to compute statistical length and thermodynamic cost function ito .

##  IV Statistical length and thermodynamic cost function

Let us consider a system whose evolution is characterized by only one
parameter, that is the time over which we observe its evolution, i.e.,
$\lambda=t$. Our interest is to compute how far is the final distribution at
$t=\tau$ from the initial distribution at $t=0$, given that the evolution of
the system is governed by the Eq. (1) subjected to initial condition
$\vec{P}(t=0)=[p_{1}(0),p_{2}(0),\dots,p_{N}(0)]$. A measure of this distance
that takes into account the whole dynamics, not just initial and final states,
is provided by the statistical length $\mathcal{L}$, defined as crooks ; ito

$\displaystyle\mathcal{L}=\int~{}\mathrm{d}s=\int_{0}^{\tau}~{}\mathrm{d}t~{}\dfrac{\mathrm{d}s}{\mathrm{d}t}.$
(16)

Here, $\frac{\mathrm{d}s}{\mathrm{d}t}$ is the intrinsic speed which the
probability vector traverses the path with PRX . Stated differently, the
statistical length $\mathcal{L}$ measures the total distance of the path
covered by the probability vector from time $t=0$ to $t=\tau$, i.e., from an
initial distribution to a final distribution, in the probability space.

Similarly, we can define a measure to quantify the cost of such a dynamical
evolution. This is the thermodynamic action or thermodynamic cost function
defined as crooks ; ito

$\displaystyle\mathcal{C}=\dfrac{1}{2}\int_{0}^{\tau}~{}\mathrm{d}t~{}\dfrac{\mathrm{d}s^{2}}{\mathrm{d}t^{2}}.$
(17)

In order to derive an explicit expression for the thermodynamic cost to
connect two distributions, we first compute the integrand in Eq. (17). Using
the definition of Fisher’s metric for one parameter $\lambda=t$ [see Eq.
(15)], we write

$\displaystyle\dfrac{\mathrm{d}s^{2}}{\mathrm{d}t^{2}}$
$\displaystyle=\sum_{n=1}^{N}p_{n}\bigg{(}\dfrac{\mathrm{d}\ln
p_{n}}{\mathrm{d}t}\bigg{)}^{2}$
$\displaystyle=-\sum_{n=1}^{N}p_{n}\dfrac{\mathrm{d}}{\mathrm{d}t}\bigg{(}\dfrac{1}{p_{n}}\dfrac{\mathrm{d}p_{n}}{\mathrm{d}t}\bigg{)}.$
(18)

Now, we substitute the value of terms inside the parenthesis using Eqs. (1)
and (7). Therefore, we get

$\displaystyle\dfrac{\mathrm{d}s^{2}}{\mathrm{d}t^{2}}$
$\displaystyle=-\sum_{n=1}^{N}p_{n}\dfrac{\mathrm{d}}{\mathrm{d}t}\bigg{[}\sum_{n^{\prime}\neq
n}W_{n\to n^{\prime}}(e^{-F_{n^{\prime}\to n}}-1)-\gamma_{n}+$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+\dfrac{\delta_{n,1}}{p_{n}}\sum_{m=1}^{N}\gamma_{m}p_{m}\bigg{]},$
(19)

Further, using $F_{n\to n}=0$ and $W_{n\to n}=-\sum_{n^{\prime}\neq n}W_{n\to
n^{\prime}}$ in the above equation, we rewrite

$\displaystyle\dfrac{\mathrm{d}s^{2}}{\mathrm{d}t^{2}}$
$\displaystyle=-\sum_{n=1}^{N}p_{n}\dfrac{\mathrm{d}}{\mathrm{d}t}\bigg{[}\sum_{n^{\prime}}W_{n\to
n^{\prime}}e^{-F_{n^{\prime}\to n}}-\gamma_{n}+$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+\dfrac{\delta_{n,1}}{p_{n}}\sum_{m=1}^{N}\gamma_{m}p_{m}\bigg{]}.$
(20)

Figure 2: Probability distribution, thermodynamic flux, and entropy production
with respect to time $t$. We consider a network of four states as shown in
Fig. 1(upper panel) with transitions rates $W_{2\to 1}=0.5,W_{1\to
2}=0.3~{}W_{2\to 1},W_{3\to 2}=0.1,W_{2\to 3}=0.5~{}W_{3\to 2},W_{3\to
1}=W_{1\to 3}=0.5,W_{4\to 3}=0.5,W_{3\to 4}=0.2~{}W_{4\to 3},W_{1\to
4}=0.5,W_{4\to 1}=0.5~{}W_{1\to 4}$, $\gamma_{n}=0.1$ for all $n$, and the
system is initialize as
$(p_{1}(0),p_{2}(0),p_{3}(0),p_{4}(0))=(0.3,0.05,0.1,0.55)$. In panel (a), we
show the evolution of the system, and the thermodynamic flux (7) across each
link is displayed with the observation time in panel (b). In panel (c), we
show the average entropy productions with respect to time, where the angular
brackets indicate the averaging as explained in the text. Here, we can see
that $\langle F\rangle$ (thick green curve) is the average total entropy
production [i.e., the first term on the right-hand side of Eq. (5)] which is
the sum of three contributions: 1) the average entropy production of system,
2) the average entropy production in bath, and 3) the average entropy
production due to resetting [see Eq. (5)].

In the following, we compute each term on the right-hand side of Eq. (20). Let
us first consider the first term on the right-hand side of Eq. (20):

$\displaystyle\sum_{n,n^{\prime}}p_{n}\dfrac{\mathrm{d}}{\mathrm{d}t}\bigg{[}-W_{n\to
n^{\prime}}e^{-F_{n\to n^{\prime}}}\bigg{]}$
$\displaystyle=-\sum_{n,n^{\prime}}p_{n}\bigg{[}\dfrac{\mathrm{d}W_{n\to
n^{\prime}}}{\mathrm{d}t}e^{-F_{n\to n^{\prime}}}$ $\displaystyle-W_{n\to
n^{\prime}}e^{-F_{n\to n^{\prime}}}\dfrac{\mathrm{d}F_{n\to
n^{\prime}}}{\mathrm{d}t}\bigg{]}.$ (21)

In order to proceed forward, we substitute $F_{n\to n^{\prime}}$ from Eq. (7),
and then, we employ the anti-symmetric property of both thermodynamic force
and entropy change in the bath due to transitions along bidirectional links.
Finally, we get

$\displaystyle\sum_{n,n^{\prime}}p_{n}\dfrac{\mathrm{d}}{\mathrm{d}t}\bigg{[}-W_{n\to
n^{\prime}}e^{-F_{n\to
n^{\prime}}}\bigg{]}=\bigg{\langle}\dfrac{\mathrm{d}\sigma^{\mathrm{bath}}}{\mathrm{d}t}\bigg{\rangle}-\bigg{\langle}\dfrac{\mathrm{d}F}{\mathrm{d}t}\bigg{\rangle}.$
(22)

Here the angular brackets indicate the average over all possible trajectories,
defined as $\langle\Omega\rangle=(1/2)\sum_{n,m}\Omega_{n\to m}J_{n\to m}$
Busiello_2020 . Similarly, by computing the last two terms on the right-hand
side of Eq. (20), we obtain

$\displaystyle\sum_{n=1}^{N}p_{n}\bigg{[}\dfrac{\mathrm{d}\gamma_{n}}{\mathrm{d}t}-\dfrac{\mathrm{d}}{\mathrm{d}t}$
$\displaystyle\bigg{\\{}\dfrac{\delta_{n,1}}{p_{n}}\sum_{m=1}^{N}\gamma_{m}p_{m}\bigg{\\}}\bigg{]}=$
$\displaystyle=\sum_{n=1}^{N}\bigg{[}\dfrac{\gamma_{n}p_{n}}{p_{1}}\dfrac{\mathrm{d}p_{1}}{\mathrm{d}t}-\gamma_{n}\dfrac{\mathrm{d}p_{n}}{\mathrm{d}t}\bigg{]}$
$\displaystyle=\sum_{n=1}^{N}\gamma_{n}p_{n}\dfrac{\mathrm{d}}{\mathrm{d}t}\bigg{(}\ln\dfrac{p_{1}}{p_{n}}\bigg{)}$
$\displaystyle=\bigg{\langle}\dfrac{\mathrm{d}\sigma^{\mathrm{reset}}}{\mathrm{d}t}\bigg{\rangle}_{\gamma}.$
(23)

Here the angular brackets with the subscript $\gamma$ indicate the average
over all possible resetting steps in a trajectory, which is defined as
$\langle\omega\rangle_{\gamma}=\sum_{n}\omega_{n}J_{n}^{\mathrm{reset}}$. The
term $1/2$ is missing since in this case each transition rate is counted once
by construction.

Therefore, using Eqs. (22) and (23), we finally obtain

$\displaystyle\dfrac{\mathrm{d}s^{2}}{\mathrm{d}t^{2}}$
$\displaystyle=\bigg{\langle}\dfrac{\mathrm{d}\sigma^{\mathrm{bath}}}{\mathrm{d}t}\bigg{\rangle}-\bigg{\langle}\dfrac{\mathrm{d}F}{\mathrm{d}t}\bigg{\rangle}+\bigg{\langle}\dfrac{\mathrm{d}\sigma^{\mathrm{reset}}}{\mathrm{d}t}\bigg{\rangle}_{\gamma}.$
(24)

Similarly, the above result can also be generalized for a system with multiple
unidirectional system (see Appendix B). Substituting the above equation in
Eqs. (16) and (17), we obtain, respectively, the expression for the
statistical length $\mathcal{L}$ and the thermodynamic cost function
$\mathcal{C}$, in terms of the entropic contributions in the stochastic
resetting system.

In Fig. 2(a), we numerically evolve the dynamical equations (1) for the
network shown in Fig. 1(upper panel) and show the probability to be in each
state with respect to time. Herein, we choose all transitions rates to be
time-independent. Consequently, the system reaches a non-equilibrium steady
state [see Fig. 2(b)] as time progresses and the system entropy production (3)
vanishes in the non-equilibrium steady state [see thin solid blue curve in
Fig. 2(c)]. Moreover, we can see that $\langle
F\rangle\to\langle\sigma^{\mathrm{bath}}\rangle+\langle\sigma^{\mathrm{reset}}\rangle_{\gamma}$
as expected, in the large time limit.

In the next section, we build a relation between the thermodynamic cost
function (17) and the statistical length (16).

##  V The role of unidirectional transitions

Here, we first show that the relation between $\mathcal{L}$ and $\mathcal{C}$
can be shaped in the form of a thermodynamic bound for the speed at which the
system can go from an initial to a final distribution. In order to do so, we
employ the Cauchy-Schwartz inequality for two real functions $f(t)$ and
$g(t)$:

$\displaystyle\bigg{[}\int_{0}^{\tau}\mathrm{d}t~{}f(t)~{}g(t)\bigg{]}^{2}\leq\int_{0}^{\tau}\mathrm{d}t_{1}~{}f^{2}(t_{1})\int_{0}^{\tau}\mathrm{d}t_{2}~{}g^{2}(t_{2}).$
(25)

Now we substitute $g(t)=\phi$ (an arbitrary constant $\in\mathbb{R}$) and
$f(t)=\frac{\mathrm{d}s}{\mathrm{d}t}$ in Eq. (25), and we obtain [see Eqs.
(16) and (17)]:

$\displaystyle\mathcal{C}_{\tau}\geq\dfrac{\mathcal{L}^{2}}{2\tau^{2}},$ (26)

where $\mathcal{C}_{\tau}=\frac{\mathcal{C}}{\tau}$ is a scaled quantity, and
the term on the right-hand side can be interpreted as the square of the speed
at which the probability vector traces a path to connect initial and the final
states.

Rearranging the above inequality (26), the bound on the speed limit $\tau$ can
be expressed as:

$\displaystyle\tau\geq\dfrac{\mathcal{L}}{\sqrt{2\mathcal{C}_{\tau}}}.$ (27)

We stress that although the form of the thermodynamic bound remains same as in
Ref. ito  even for the case of system with unidirectional transition, the
difference lies in entropic contributions in both $\mathcal{L}$ and
$\mathcal{C}$.

In order to show how the presence of unidirectional transitions modifies the
bound on the speed limit, let us start considering a network without
unidirectional links. It will satisfy the inequality in Eq. (27), with:

$\frac{{\rm d}s^{2}}{{\rm d}t^{2}}=\left\langle\frac{{\rm d}\sigma^{\rm
bath}}{{\rm d}t}\right\rangle-\left\langle\frac{{\rm d}F}{{\rm
d}t}\right\rangle=\frac{{\rm d}s^{2}}{{\rm d}t^{2}}\bigg{|}_{0}.$ (28)

We name the bound on the speed limit for this network, i.e,. the right-hand
side of Eq. (27), $\tau_{0}^{\rm bound}$.

Consider, now, the same network with an additional time-independent
unidirectional transition rate, $\gamma$, from the node $n_{1}$ to the node
$n_{2}$. The bound maintains the same form, but the square of the intrinsic
speed becomes equal to the one reported in Eq. (24). In order to study what
happens in the limit of $\gamma$ much smaller than all other transition rates,
i.e., slow resetting, we expand the statistical length up to the first order:

$\displaystyle\mathcal{L}\approx\int_{0}^{\tau}dt\left(\sqrt{\frac{{\rm
d}s^{2}}{{\rm d}t^{2}}\bigg{|}_{0}}+\frac{\left\langle\frac{{\rm d}\sigma^{\rm
reset}}{{\rm d}t}\right\rangle_{\gamma}}{2\sqrt{\frac{{\rm d}s^{2}}{{\rm
d}t^{2}}\big{|}_{0}}}\right)\approx\mathcal{L}^{(0)}+\mathcal{L}^{(\gamma)}.$
(29)

The cost function can be rewritten as follows:

$\displaystyle\mathcal{C}_{\tau}=\frac{1}{2\tau}\int_{0}^{\tau}dt\left(\frac{{\rm
d}s^{2}}{{\rm d}t^{2}}\bigg{|}_{0}+\left\langle\frac{{\rm d}\sigma^{\rm
reset}}{{\rm
d}t}\right\rangle_{\gamma}\right)=\mathcal{C}_{\tau}^{(0)}+\mathcal{C}_{\tau}^{(\gamma)}.$
(30)

Computing the thermodynamic bound, Eq. (27), in this limit, we obtain:

$\tau\geq\tau_{0}^{\rm
bound}\left[1+\left(\frac{\mathcal{L}^{(\gamma)}}{\mathcal{L}^{(0)}}-\frac{\mathcal{C_{\tau}^{(\gamma)}}}{2C_{\tau}^{(0)}}\right)\right]\equiv\tau_{\gamma}^{\rm
bound},$ (31)

where $\tau_{0}^{\rm
bound}=\mathcal{L}^{(0)}/\sqrt{2\mathcal{C}_{\tau}^{(0)}}$ is the lower bound
on the speed limit for the resetting-links free network. We emphasize that a
similar relation can also be obtained even for a time-dependent unidirectional
rate $\gamma(t)$ by setting its temporal average as a small parameter.

Eq. (31) clearly shows that the thermodynamic bound on the speed limit in the
presence of even one slow unidirectional transition can be tighter than
$\tau_{0}^{\rm bound}$. Specifically, this occurs when:

$\frac{\mathcal{L}^{(\gamma)}}{\mathcal{L}^{(0)}}~{}\geq~{}\frac{\mathcal{C_{\tau}^{(\gamma)}}}{2C_{\tau}^{(0)}},$
(32)

which corresponds to the situation in which the relative statistical length
increases more than the relative thermodynamic cost associated to it, due to
the additional dissipation induced by the unidirectional link.

The advantage of performing a series expansion in powers of $\gamma$ is the
possibility to isolate the extra contribution given solely by the energy
dissipated through the unidirectional link. It is worth noting that the
generalization to the case in which multiple unidirectional links are present
is immediate, since the form of Eq. (31), and, consequently of Eq. (32), does
not change.

In Fig. (3) we study a simple four-state network with one unidirectional link
(indicated by a red one-head diagonal arrow), as shown in panel (a). In panel
(b), we show $\tau_{0}^{\rm bound}$ (light green dashed curve), the bound
including also the transition rate $\gamma$ (solid thin brown curve), and its
approximation in the slow resetting regime $\tau_{\gamma}^{\rm bound}$ (dot-
dashed blue curve). In this case we observe a tightening of the thermodynamic
bound (see green thick diagonal line).

##  VI Dynamical uncertainty relations for slow unidirectional transition
rates

This section is dedicated to the derivation of two inequalities and an
uncertainty relation (TUR-like) for the (average) resetting entropy
production, starting from the bounds derived in the previous sections.

First of all, we notice that, since the system depends only on evolution time
$t$, the statistical length and the thermodynamic cost function can be written
as

$\displaystyle\mathcal{L}$ $\displaystyle=$
$\displaystyle\int_{0}^{\tau}~{}\mathrm{d}t~{}\sqrt{\sum_{n=1}^{N}~{}\dfrac{(\partial_{t}p_{n})^{2}}{p_{n}}},$
(33) $\displaystyle\mathcal{C}$ $\displaystyle=$
$\displaystyle\int_{0}^{\tau}~{}\mathrm{d}t~{}\sum_{n=1}^{N}~{}\dfrac{(\partial_{t}p_{n})^{2}}{p_{n}},$
(34)

where we have used Eq. (15). This means that, even though $\mathcal{L}$ and
$\mathcal{C}$ can be expressed in terms of thermodynamic quantities, in order
to compute them, we just need information about the evolution of the
probability distribution.

Figure 3: Thermodynamic bound on the speed limit for a system with one (slow)
unidirectional transition. We consider the system as shown in panel a). Here,
we initialize the system as
$(p_{1}(0),p_{2}(0),p_{3}(0),p_{4}(0))=(0.7,0.025,0.15,0.125)$. The
unidirectional rate (in red) is $\gamma=0.1$, while the bidirectional one (in
black) are: $W_{1\to 2}=1$, $W_{2\to 1}=0.5$, $W_{2\to 3}=0.48,W_{3\to
2}=0.4,W_{1\to 3}=W_{3\to 1}=0.5$, $W_{3\to 4}=0.8,W_{4\to 3}=0.5$, $W_{1\to
4}=0.8$, and $W_{4\to 1}=0.4$. b) We show how the bound on the speed limit is
modified by the presence of $\gamma$ (brown solid line), and the validity of
the small-$\gamma$ approximation (blue dot-dashed line). In panels c) and d)
we show, respectively, that the inequalities in Eqs. (36) and (40) are
satisfied. We remark that the maximum time $\tau$ for which we can study the
system employing the small-$\gamma$ approximation becomes larger as $\gamma$
decreases.

Consider again the case in which a set of slow unidirectional links are added
to a bidirectional backbone. By inverting Eq. (31):

$\displaystyle\Delta\tau\geq\dfrac{\mathcal{L}^{(\gamma)}}{\mathcal{L}^{(0)}}-\dfrac{\mathcal{C}_{\tau}^{(\gamma)}}{2\mathcal{C}_{\tau}^{(0)}},$
(35)

which results in

$\displaystyle\frac{1}{\tau}\int_{0}^{\tau}\mathrm{d}t\left\langle\frac{\mathrm{d}\sigma^{\rm
reset}}{\mathrm{d}t}\right\rangle_{\gamma}\geq
4\mathcal{C}_{\tau}^{(0)}\left(\frac{\mathcal{L}^{(\gamma)}}{\mathcal{L}^{(0)}}-\Delta\tau\right).$
(36)

where $\Delta\tau=(\tau-\tau_{0}^{\rm bound})/\tau_{0}^{\rm bound}$ is the
relative deviation from the bound without resetting. Defining, for simplicity,
$\mathbf{E}(\dot{\sigma}^{\rm reset})=2\mathcal{C}^{(\gamma)}_{\tau}$, we have
[see Eq. (36)]:

$\mathbf{E}(\dot{\sigma}^{\rm reset})\geq
4\mathcal{C}_{\tau}^{(0)}\left(\frac{\mathcal{L}}{\mathcal{L}^{(0)}}-1-\Delta\tau\right).$
(37)

This is the first inequality involving the temporal mean of $\langle{\rm
d}\sigma^{\rm reset}/{\rm d}t\rangle_{\gamma}$, $\mathbf{E}(\dot{\sigma}^{\rm
reset})$. In Fig. 3(c), we numerically check the validity of Eq. (36) in a
pedagogical system.

A different inequality can be derived by applying the following Cauchy-
Schwartz inequality:

$\left(\int_{0}^{\tau}\frac{|\langle\dot{\sigma}^{\rm
reset}\rangle_{\gamma}|}{2\sqrt{\frac{{\rm d}s^{2}}{{\rm
d}t^{2}}\big{|}_{0}}}\right)^{2}\leq\frac{\tau}{2}(1/\mathcal{C})_{\tau}^{(0)}\int_{0}^{\tau}{\rm
d}t\bigg{|}\left\langle\frac{{\rm d}\sigma^{\rm reset}}{{\rm
d}t}\right\rangle_{\gamma}\bigg{|}^{2},$ (38)

where the term inside the parenthesis on the left-hand side can be written as
$\mathcal{L}^{(\gamma)}(|\langle\dot{\sigma}^{\rm
reset}\rangle|)\equiv\mathcal{L}_{\rm abs}^{(\gamma)}$, and

$(1/\mathcal{C})_{\tau}^{(0)}=\dfrac{1}{2\tau}\int_{0}^{\tau}{\rm
d}t\left(\frac{{\rm d}s^{2}}{{\rm d}t^{2}}\bigg{|}_{0}\right)^{-1}.$ (39)

Here, the quantity $\mathcal{L}^{(\gamma)}_{\rm abs}$ represents the
perturbation to the statistical length due to $\gamma$, independently of the
fact the the resetting link is producing entropy by erasing
($\dot{\sigma}^{\rm reset}<0$) or writing ($\dot{\sigma}^{\rm reset}>0$)
information Busiello_2020 . Further, rearranging terms in Eq. (38), we obtain:

$\displaystyle\frac{1}{\tau}\int_{0}^{\tau}\mathrm{d}t\bigg{|}\left\langle\frac{\mathrm{d}\sigma^{\rm
reset}}{\mathrm{d}t}\right\rangle_{\gamma}\bigg{|}^{2}\geq\frac{2}{\tau^{2}}\left[\frac{\left(\mathcal{L}^{(\gamma)}_{\rm
abs}\right)^{2}}{(1/C)_{\tau}^{(0)}}\right].$ (40)

The above equation indicates that when there is at least one (slow)
unidirectional transition rates, the second temporal moment cannot be as small
as possible, being bounded from below. Moreover, as expected, the faster is
the process, the larger is its minimum value. In Fig. 3(d), Eq. (40) is
numerically verified.

In order to have a compact notation, we rename the term on the left-hand side
of Eq. (40) as $\mathcal{V}\left(|\langle\dot{\sigma}^{\rm
reset}\rangle|\right)\equiv\mathcal{V}_{\rm reset}$. Let us consider the
following inequality:

$\displaystyle\mathcal{L}^{(\gamma)}_{\rm
abs}\mathcal{L}^{(0)}\geq\frac{\tau^{2}}{2}\Bigg{(}\frac{1}{\tau}\int_{0}^{\tau}dt\sqrt{\bigg{|}\left\langle\frac{{\rm
d}\sigma^{\rm reset}}{{\rm
d}t}\right\rangle_{\gamma}\bigg{|}}~{}\Bigg{)}^{2}=$
$\displaystyle=\frac{\tau^{2}}{2}\mathbf{E}\left(\sqrt{|\dot{\sigma}^{\rm
reset}|}\right)^{2}\equiv\frac{\tau^{2}}{2}\mathcal{E}_{\rm reset},$ (41)

where we have used Cauchy-Schwarz inequality.

Finally, using Eqs. (38) and (41), we write an uncertainty relation for the
resetting entropy production:

$\frac{\mathcal{V}_{\rm reset}}{\mathcal{E}_{\rm
reset}^{2}}\geq\left(\frac{\tau}{\mathcal{T}_{0}^{\rm bound}}\right)^{2},$
(42)

where we have introduced a different characteristic time:

$\mathcal{T}_{0}^{\rm
bound}=\mathcal{L}^{(0)}\sqrt{2\left(1/C\right)_{\tau}^{(0)}}\geq\tau_{0}^{\rm
bound}.$ (43)

We remark that $\mathcal{T}_{0}^{\rm bound}$ is spiritually analogous to the
bound on the speed limit due to bidirectional transitions only, $\tau_{0}^{\rm
bound}=\mathcal{L}^{(0)}/\sqrt{2\mathcal{C}^{(0)}_{\tau}}$. While in the
former we consider the integral of the inverse of the squared intrinsic speed,
$({\rm d}s^{2}/{\rm d}t^{2})^{-1}$, the latter depends on the integral of the
squared intrinsic speed, ${\rm d}s^{2}/{\rm d}t^{2}$.

Eq. (42) poses a limit on how much the resetting entropy production has to
fluctuate, if we aim at erasing ($\langle\dot{\sigma}^{\rm reset}\rangle<0$)
or writing ($\langle\dot{\sigma}^{\rm reset}\rangle>0$) information at a given
rate, exploiting unidirectional transitions Busiello_2020 . We highlight that
the quantities appearing in this TUR-like inequality are related to the ones
usually involved in analogous relationships (variance and mean), even though
they are not the same. Notably, in the limit of slow transitions, the right-
hand side of Eq. (42) depends solely on zeroth order quantities (without
resetting).

An additional remark is that the time $\tau$ for which a system can be studied
under the small-$\gamma$ approximation gets larger when the value of $\gamma$
decreases. In the system under investigation in Fig. (3) the value of the
resetting link is not particularly small, even if all inequalities still
remain valid, as evidenced by simulations.

A further refinement to the TUR-like inequality in Eq. (42) can be derived by
considering the case of small $\tau$. In this limit, in fact, we have that
$\mathcal{T}_{0}^{\rm bound}\to\tau_{0}^{\rm bound}$, which, in turns,
implies:

$\frac{\mathcal{V}_{\rm reset}}{\mathcal{E}_{\rm
reset}^{2}}\gtrsim\left(\frac{\tau}{\tau_{0}^{\rm bound}}\right)^{2}.$ (44)

In Fig. 4 we numerically show the validity of Eqs. (42) and (44) for the four-
state system depicted in Fig. 3(a). Numerical simulations evidences that this
approximated version of the inequality provides a tighter bound to the
precision-to-cost ratio even out of its range of validity, i.e., for not-small
times. We leave for future investigations a more in-depth study of this
empirical observation.

Figure 4: Thermodynamic uncertainty relation for the resetting entropy
production [see Eqs. (42) and (44)]. The system under investigation is the
same as in Fig. 3(a). The green dot-dashed line is the left-hand side term of
Eq. (42), indicating the precision-to-cost ratio for resetting, divided by a
factor $\tau^{2}$. The blue solid line represents $1/(\mathcal{T}_{0}^{\rm
bound})^{2}$, which is the bound derived in the text. The red dashed line is
$1/(\tau_{0}^{\rm bound})^{2}$, indicating the bound in the small $\tau$
regime, Eq. (44). In this particular case, the latter term provides a tighter
bound for the TUR-like inequality even for not-small times.

##  VII Thermodynamic bound on the speed limit

In this section, we obtain a new bound on the speed at which the system goes
from an initial to a final distributions in terms of thermodynamic cost
function and statistical length. To this aim, we use the Milne’s inequality
milne-1 ; milne-2 ; milne-3 :

$\displaystyle\bigg{[}\int_{0}^{\tau}\mathrm{d}t~{}f(t)~{}g(t)\bigg{]}^{2}\leq\int_{0}^{\tau}\mathrm{d}t_{1}~{}[f^{2}(t_{1})+g^{2}(t_{1})]$
$\displaystyle\times\int_{0}^{\tau}\mathrm{d}t_{2}~{}\dfrac{f^{2}(t_{2})~{}g^{2}(t_{2})}{f^{2}(t_{2})+g^{2}(t_{2})}\leq\int_{0}^{\tau}\mathrm{d}t_{1}~{}f^{2}(t_{1})\int_{0}^{\tau}\mathrm{d}t_{2}~{}g^{2}(t_{2}),$
(45)

to construct a relation between $\mathcal{L}$ and $\mathcal{C}$. Notice that,
in principle, Milne’s inequality may constitute a refinement of the Cauchy-
Schwartz inequality. Proceeding as before, and substituting $g(t)=\phi$ (an
arbitrary constant $\in\mathbb{R}$) and $f(t)=\frac{\mathrm{d}s}{\mathrm{d}t}$
in the above expression (45), we get [see Eqs. (16) and (17)]

$\displaystyle\mathcal{L}^{2}\leq(\phi^{2}\tau+2\mathcal{C})\bigg{[}\tau$
$\displaystyle-\overbrace{\phi^{2}\int_{0}^{\tau}~{}\mathrm{d}t~{}\bigg{\\{}\phi^{2}+\frac{\mathrm{d}s^{2}}{\mathrm{d}t^{2}}\bigg{\\}}^{-1}}^{\mathcal{M}(\tau,\phi)}\bigg{]}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\leq
2\tau\mathcal{C},$ (46)

where the equality is achieved when the square of the intrinsic speed
$\frac{\mathrm{d}s^{2}}{\mathrm{d}t^{2}}$ is independent of time. Notice that
both integrands of $\mathcal{C}$ and $\mathcal{M}(\tau,\phi)$ contain the same
term, which is the square of the intrinsic speed,
$\frac{\mathrm{d}s^{2}}{\mathrm{d}t^{2}}$. The latter is related to
thermodynamic quantities as displayed in Eq. (24).

Figure 5: Thermodynamic bound on the speed limit for constant resetting rate.
We consider the system as detailed in Fig. 2. In panel (a), we show each
component on the right-hand side of Eq. (24). Since bidirectional transitions
are independent of time,
$\big{\langle}\frac{\mathrm{d}\sigma^{\mathrm{bath}}}{\mathrm{d}t}\big{\rangle}$
is equal to zero for all time. Moreover, we show the verification of the Eq.
(24), where the left-hand side is computed using Eq. (18). In panel (b), we
plot the inequality relating the statistical length and the shortest path
between two states separated by time $\tau$: $\Lambda\leq\mathcal{L}$. Panel
(c) demonstrates the inequality given in Eq. (47), where the blue circles are
obtained using the central term for random values of time $\tau\in[0,10]$ and
$\phi\in[0,5]$ drawn from a uniform distribution. Clearly, the inequality
obtained from first and second term could be tighter than the one derived
using Cauchy-Schwartz inequality, presented in ito , and involving first and
third term of Eq. (47). In panels (d), we show that the speed limit $\tau$
obeys a new inequality given in Eq. (51). The numerical result hints at the
possibility to find a tighter bound. Herein, blue circles are obtained from
the second term given in Eq. (51) using random values of $\tau$ and $\phi$
[similar to panel (c)]. Figure 6: Thermodynamic bound on the speed limit for
periodic resetting. We consider the system as shown in Fig. 1(upper-panel).
Here, we initialize the system as
$(p_{1}(0),p_{2}(0),p_{3}(0),p_{4}(0))=(0.7,0.025,0.15,0.125)$. The
unidirectional and bidirectional rates are
$\gamma(t)=\gamma_{n}(t)=2\cos^{2}(t),W_{2\to 1}(t)=\tanh(0.5t),W_{1\to
2}(t)=W_{2\to 1}(0.3t),W_{3\to 2}(t)=\tanh(0.1t),W_{2\to 3}(t)=W_{3\to
2}(0.5t),W_{3\to 1}(t)=W_{1\to 3}(t)=\tanh(0.5t),W_{4\to
3}=\tanh(0.5t),W_{3\to 4}(t)=W_{4\to 3}(0.2t),W_{1\to
4}(t)=\tanh(0.5t),W_{4\to 1}(t)=W_{1\to 4}(0.5t)$. In analogy to Fig. 5, in
panel (a), we show each component of Eq. (24); in panel (b), we plot the
inequality $\mathcal{L}\geq\Lambda$; finally, in panels (c) and (d), we verify
the inequalities in Eqs. (49) and (51).

In what follows, for convenience, we will not write the time dependence in
$\mathcal{M}(\tau,\phi)$. Rearranging the inequality (46), we obtain

$\displaystyle\mathcal{C}_{\tau}\geq\dfrac{1}{2}[1-\mathcal{M}_{\tau}(\phi)](\phi^{2}+2\mathcal{C}_{\tau})\geq\dfrac{\mathcal{L}^{2}}{2\tau^{2}},$
(47)

Notice that $\mathcal{M}_{\tau}(\phi)$ is a non-null quantity.

Consider now the expressions of the statistical length $\mathcal{L}$ and the
thermodynamic cost $\mathcal{C}$ in terms of the probability distribution,
respectively Eqs. (33) and (34). These two functions can be simultaneously
optimised with the constraint of the normalization of the probability
distribution, by using the method of Lagrange multipliers. Following the steps
presented in PRX , the shortest path connecting two distributions in a finite
time $\tau$, at the minimum cost, is

$\displaystyle\Lambda=2\arccos\bigg{(}\sum_{n=1}^{N}\sqrt{p_{n}(0)p_{n}(\tau)}\bigg{)}.$
(48)

Therefore, the statistical length of any other path is always greater or equal
to the shortest path at all times, i.e., $\mathcal{L}\geq\Lambda$. We remark
that the optimal distribution $p^{*}_{n}(t)$, derived in PRX , simultaneously
minimizes $\mathcal{L}$ and $\mathcal{C}$, and let Eq. (46) holds as an
equality. Indeed, when evaluated for $p_{n}^{*}(t)$,
$\frac{\mathrm{d}s^{2}}{\mathrm{d}t^{2}}=\Lambda^{2}$, being constant in time,
that is the condition for the equality to hold in Eq. (46).

Using the definition of $\Lambda$, the relation (47) becomes

$\displaystyle\mathcal{C}_{\tau}\geq\dfrac{1}{2}[1-\mathcal{M}_{\tau}(\phi)](\phi^{2}+2\mathcal{C}_{\tau})\geq\dfrac{\mathcal{L}^{2}}{2\tau^{2}}\geq\dfrac{\Lambda^{2}}{2\tau^{2}},$
(49)

where the fourth term on the right-hand side is obtained using the shortest
length $\Lambda$. Clearly, from first and last term, one can conclude that the
(scaled) cost function is bounded from below by the net kinetic energy PRX
associated to the motion along the shortest path in the probability space in a
time $\tau$, independently on how the path is traversed. Furthermore, using
second, third and fourth terms of the inequality (49), we find a new
uncertainty relation for $\mathcal{M}_{\tau}(\phi)$ and $\mathcal{C}_{\tau}$:

$\displaystyle[1-\mathcal{M}_{\tau}(\phi)](\phi^{2}+2\mathcal{C}_{\tau})\geq\dfrac{\mathcal{L}^{2}}{\tau^{2}}\geq\dfrac{\Lambda^{2}}{\tau^{2}}.$
(50)

Unfortunately, the physical interpretation of $\mathcal{M}_{\tau}(\phi)$ is
not clear to us. However, as we showed earlier, this quantity contains the
same term as in $\mathcal{C}_{\tau}$ [see Eq. (46)]. Nonetheless, from the
above relation (50), we can say that for any physical system connecting two
states in a finite time $\tau$, the product of two functions on the left-hand
side will be always greater than or equal to the square of the speed of the
system over the shortest path. We stress that the bound shown in Eq. (49) does
not require that a system has both unidirectional and bidirectional links, but
is also applicable to those with only bidirectional transitions.

Using Eq. (50), we obtain the thermodynamic bound on the speed limit:

$\displaystyle\tau$
$\displaystyle\geq\dfrac{\mathcal{L}}{\sqrt{[1-\mathcal{M}_{\tau}(\phi)](\phi^{2}+2\mathcal{C}_{\tau})}}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\geq\dfrac{\Lambda}{\sqrt{[1-\mathcal{M}_{\tau}(\phi)](\phi^{2}+2\mathcal{C}_{\tau})}}.$
(51)

Clearly, in the limit $\phi\to 0$, we reduce to the bound
$\tau\geq\mathcal{L}/\sqrt{2\mathcal{C}_{\tau}}$, derived in the Ref. ito .

In Fig. 5, we consider the system as discussed in Fig. 2. We show the
evolution of each component on the right-hand side of Eq. (24) with respect to
time in Fig. 5(a). Therein, we choose all bidirectional transitions to be
independent of time. Thus, the quantity
$\big{\langle}\frac{\mathrm{d}\sigma^{\mathrm{bath}}}{\mathrm{d}t}\big{\rangle}$
remains zero at all times. We also verify Eq. (24), where the left-hand side
is computed using Eq. (18). In Fig. 5(b), we confirm the inequality
$\Lambda\leq\mathcal{L}$. Then, we show in Fig. 5(c), the numerical evidence
of the bound in Eq. (47), indicating that the inequality obtained from first
and second term could be tighter than the one derived using Cauchy-Schwartz
inequality Cover , and involving first and third term of Eq. (47). The blue
circles are obtained from the central term of the inequality (47), using
random values of time $\tau\in[0,10]$ and $\phi\in[0,5]$, drawn from a uniform
distribution. Finally, in Fig. 5(d), we plot the thermodynamic bound on the
speed limit $\tau$ given in Eq. (51), where the blue circles are obtained
using the second term of the inequality (51) in which $\tau\in[0,10]$ and
$\phi\in[0,5]$ are drawn randomly from a uniform distribution. Clearly, the
comparison between the bound on the speed limit previously obtained ito  and
the one presented in Eq. (51) suggests that the latter could be tighter,
depending on the value of $\phi$. Furthermore, we also perform the same
analysis for the case when the resetting rate is a periodic function of time:
$\gamma=\gamma_{n}=2\cos^{2}(t)$. It is illustrated in Fig. 6.

##  VIII Tighter bound

In this section, we obtain the tighter bound on the speed limit. In Figs.
5(c)-(d) and 6(c)-(d), the blue circles are obtained from central terms of Eq.
(49) and (51), where those terms are functions of $\phi$.

Clearly, one can minimize the central term in Eq. (49), obtaining the
following tighter inequality [see Fig. 7(a)]:

$\displaystyle\mathcal{C}_{\tau}\geq\min_{\phi}\bigg{\\{}\dfrac{1}{2}[1-\mathcal{M}_{\tau}(\phi)](\phi^{2}+2\mathcal{C}_{\tau})\bigg{\\}}\geq\dfrac{\mathcal{L}^{2}}{2\tau^{2}},$
(52)

Analogously, a tighter bound on the speed limit can also be derived by
maximizing the terms on the right-hand sides of Eq. (51):

$\displaystyle\tau$
$\displaystyle\geq\max_{\phi}\bigg{\\{}\dfrac{\mathcal{L}}{\sqrt{[1-\mathcal{M}_{\tau}(\phi)](\phi^{2}+2\mathcal{C}_{\tau})}}\bigg{\\}}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\geq\max_{\phi}\bigg{\\{}\dfrac{\Lambda}{\sqrt{[1-\mathcal{M}_{\tau}(\phi)](\phi^{2}+2\mathcal{C}_{\tau})}}\bigg{\\}}.$
(53)

Figure 7: Tighter thermodynamic bound on speed limit. Here, we consider the
system discussed in Fig. 2. (a): We show the inequality (52), where orange
circles are given by central term. (b): A tighter bound on the speed limit
$\tau$, where orange circles and black squares, respectively, are obtained
from the central and and the last term of Eq. (53).

In Fig. 7(b), we show that the speed limit $\tau$ is tightly bounded by the
functions in Eq. (53), and this bound is stronger than the previous one
presented in the literature ito  (see dashed and solid line).

It is worth to remark that the relations in Eqs. (36) and (40) can, in
principle, be improved by using the Milne’s inequality, and employing the
optimization scheme, instead of the Cauchy-Schwartz inequality. However, they
might result to be particularly cumbersome, so we decided to not report them
in this manuscript.

##  IX Summary

In this paper, we considered a general discrete state-space system with both
unidirectional and bidirectional transitions. It evolves from a given initial
distribution, following a Master Equation. We computed the bound on the
evolution time to reach a final distribution in terms of the statistical
length and the thermodynamic cost. These are two thermodynamic quantities that
can be written as a function of three contributions: 1) the total entropy
production, 2) the environmental entropy productions, and 3) the
unidirectional (or resetting) entropy production. In particular, the latter
contains information regarding unidirectional transitions, whose contribution
has been highlighted, in particular in the limit of slow unidirectional
transitions. Two inequalities and an uncertainty relation (TUR-like) involving
the average resetting entropy production have been also derived. Using the
Milne’s inequality milne-1 ; milne-2 ; milne-3 , we built a new thermodynamic
bound on $\tau$, which is the time to connect two distributions, parameterized
by a real number $\phi$. By means of an optimization procedure, we showed that
our bound is tighter than the one previously reported ito . Finally, we
numerically validated our findings, using simple four-state models that
stochastically reset with constant and periodic resetting rates.

Beyond refinement of the thermodynamic bound on the speed limit, and the
generalization of the formalism presented in ito , rooted on the theory of
information geometry, to the case in which unidirectional transitions are
present, here, we discuss a possible future application of this study.
Consider a discrete-state system in which some information is accessible, e.g.
the probability distributions at different times, or dissipation. It is
possible, at least in principle, to employ and/or modify the uncertainty
relations here derived to detect the presence of unidirectional transitions in
the system. The possibility to pinpoint unidirectional links, whose unnoticed
presence might irreversibly affect the system, would allow preventing
detrimental changes. The investigation is under progress, and we hope that
such results might be crucial for system suffering from catastrophic events
sandro  (e.g. social and ecological communities).

###### Acknowledgements.

D. G. is supported by “Excellence Project 2018” of the Cariparo foundation. D.
G. and D. M. B. are grateful to Amos Maritan for insightful discussions and
the careful reading of the manuscript.

##  Appendix A System with multiple unidirectional links

In this section, we consider a generalization of the system introduced in the
main text, whose dynamics is described by Eq. (1). In the same fashion, the
system has bidirectional links such that $W_{n\to m}$ ($W_{m\to n}$) is the
transition rate from a state $n$ to a state $m$ ($m$ to $n$). In addition to
that, it also has unidirectional links such that $Y_{n\to m}>0$ is the
transition rate from $n$ to $m$, but there is no conjugate transition rate:
$Y_{m\to n}=0$. Following Ref. Busiello_2020 , the master equation of this
system can be written as

$\displaystyle\dfrac{{\rm d}p_{n}}{dt}$ $\displaystyle=\sum_{m\neq
n}^{N}[W_{m\to n}p_{m}(t)-W_{n\to m}p_{n}(t)]$ $\displaystyle+\sum_{m\neq
n}^{N}[Y_{m\to n}p_{m}(t)-Y_{n\to m}p_{n}(t)].$ (54)

where implicitly either $Y_{m\to n}=0$ or $Y_{n\to m}=0$, as discussed above.
Note that, according to Eq. (54), $Y_{n\to n}=-\sum_{m\neq n}Y_{n\to m}$.
Further, we can recover the master equation (1) for a system resetting to a
particular state $m=n_{0}$ (where we set $n_{0}=1$) by setting the transition
rate $Y_{m\to n}=\gamma_{m}\delta_{n,n_{0}}$ and $Y_{n\to
m}=\gamma_{n}\delta_{m,n_{0}}$

Using the same strategy shown in Eqs. (2)–(4), we obtain the system entropy
production:

$\displaystyle\dot{S}^{{\rm sys}}=\sum_{n,m}W_{n\to
m}p_{n}\log\dfrac{p_{n}}{p_{m}}+\sum_{n,m}Y_{n\to
m}p_{n}\log\dfrac{p_{n}}{p_{m}},$ (55)

where the dot again represents the derivative with respect to time. Rewriting
the above equation in the spirit of Eq. (5), we get

$\displaystyle\dot{S}^{{\rm sys}}$ $\displaystyle=$
$\displaystyle\overbrace{\dfrac{1}{2}\sum_{n,m}J_{n\to m}~{}F_{n\to
m}}^{\dot{S}^{{\rm tot}}}-\overbrace{\dfrac{1}{2}\sum_{n,m}J_{n\to
m}~{}\sigma_{n\to m}^{\rm bath}}^{\dot{S}^{\rm bath}}+$
$\displaystyle-\underbrace{\sum_{n,m}J^{{\rm uni}}_{n\to m}~{}\sigma_{n\to
m}^{\rm{uni}}}_{\dot{S}^{\rm uni}},$ (56)

where the quantities in the first line of the above equation are defined in
Eqs. (6)–(8), and the ones in the second line are

$\displaystyle J^{{\rm uni}}_{n\to m}$ $\displaystyle=Y_{n\to m}~{}p_{n},$
(57) $\displaystyle\sigma_{n\to m}^{\rm{uni}}$
$\displaystyle=\ln\dfrac{p_{m}}{p_{n}},$ (58)

which respectively, are the current and the entropy change due to the
unidirectional transition (indicated by the superscript ‘uni’) from $n$ to
$m$. When $m=1$ is a state where the system is allowed to reset, as for the
case in the main text, the quantities in Eqs. (57) and (58) can be easily
replaced by Eq. (10) and Eq. (9), respectively.

##  Appendix B Fisher information metric: Unidirectional links

In the following, we compute the Fisher information metric for the system
discussed in Appendix A. To do so, we recall Eq. (18):

$\displaystyle\dfrac{\mathrm{d}s^{2}}{\mathrm{d}t^{2}}=-\sum_{n=1}^{N}p_{n}\dfrac{\mathrm{d}}{\mathrm{d}t}\bigg{(}\dfrac{1}{p_{n}}\dfrac{\mathrm{d}p_{n}}{\mathrm{d}t}\bigg{)}.$
(59)

We substitute the dynamics (54) in the above equation and get:

$\displaystyle\dfrac{\mathrm{d}s^{2}}{\mathrm{d}t^{2}}$
$\displaystyle=\sum_{m,n}p_{n}\dfrac{{\rm}d}{{\rm d}t}\bigg{[}-W_{n\to
m}e^{-F_{n\to m}}\bigg{]}$ $\displaystyle-\sum_{m,n}p_{n}\dfrac{{\rm}d}{{\rm
d}t}\bigg{[}Y_{m\to n}\dfrac{p_{m}}{p_{n}}\bigg{]},$ (60)

where the first term is given by Eq. (22). Let us now evaluate the second
term:

$\displaystyle\sum_{m,n}p_{n}\dfrac{{\rm d}}{{\rm d}t}\bigg{[}Y_{m\to
n}\dfrac{p_{m}}{p_{n}}\bigg{]}$
$\displaystyle=\sum_{m,n}\bigg{[}p_{m}\dfrac{{\rm d}Y_{m\to n}}{{\rm d}t}$
$\displaystyle+p_{n}Y_{m\to n}\dfrac{{\rm d}}{{\rm
d}t}\bigg{(}\dfrac{p_{m}}{p_{n}}\bigg{)}\bigg{]}=$
$\displaystyle=\sum_{m,n}\bigg{[}p_{m}\dfrac{{\rm d}Y_{m\to n}}{{\rm d}t}$
$\displaystyle-p_{m}Y_{m\to n}\dfrac{{\rm d}}{{\rm
d}t}\bigg{(}\ln\dfrac{p_{n}}{p_{m}}\bigg{)}\bigg{]},$ (61)

where the first term on the right hand side is zero since $\sum_{n}Y_{m\to
n}=0$ (considering the diagonal part), and the second term using Eqs. (57) and
(58) can be written as

$\displaystyle\sum_{m,n}p_{n}\dfrac{{\rm d}}{{\rm d}t}\bigg{[}Y_{m\to
n}\dfrac{p_{m}}{p_{n}}\bigg{]}$ $\displaystyle=-\sum_{m,n}J^{{\rm uni}}_{m\to
n}\dfrac{{\rm d}\sigma_{m\to n}^{\rm{uni}}}{{\rm d}t}$
$\displaystyle=-\bigg{\langle}\dfrac{{\rm d}\sigma^{\rm{uni}}}{{\rm
d}t}\bigg{\rangle}_{\gamma},$ (62)

where the angular brackets represents the average over unidirectional
trajectories.

Therefore, using Eq. (22) and (62), we write the Fisher information metric as

$\displaystyle\dfrac{\mathrm{d}s^{2}}{\mathrm{d}t^{2}}$
$\displaystyle=\bigg{\langle}\dfrac{\mathrm{d}\sigma^{\mathrm{bath}}}{\mathrm{d}t}\bigg{\rangle}-\bigg{\langle}\dfrac{\mathrm{d}F}{\mathrm{d}t}\bigg{\rangle}+\bigg{\langle}\dfrac{\mathrm{d}\sigma^{\mathrm{uni}}}{\mathrm{d}t}\bigg{\rangle}_{\gamma},$
(63)

which is an expression formally identical to the one derived for the case
presented in the main text.

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