.

# Irreversibility mitigation in unital non-Markovian quantum evolutions

Stefano Gherardini<EMAIL_ADDRESS>Department of Physics and
Astronomy & LENS, University of Florence, via G. Sansone 1, I-50019 Sesto
Fiorentino, Italy.  SISSA, via Bonomea 265, I-34136 Trieste, Italy  Stefano
Marcantoni<EMAIL_ADDRESS>Dipartimento di Fisica,
Università di Trieste & INFN, Sezione di Trieste, I-34151 Trieste, Italy.
School of Physics & Astronomy, University of Nottingham, Nottingham NG7 2RD,
UK  Centre for the Mathematics and Theoretical Physics of Quantum Non-
Equilibrium Systems, University of Nottingham, Nottingham, NG7 2RD, UK
Filippo Caruso<EMAIL_ADDRESS>Department of Physics and Astronomy &
LENS, University of Florence, via G. Sansone 1, I-50019 Sesto Fiorentino,
Italy.

###### Abstract

The relation between the thermodynamic entropy production and non-Markovian
evolutions is matter of current research. Here, we study the behavior of the
stochastic entropy production in open quantum systems undergoing unital non-
Markovian dynamics. In particular, for the family of Pauli channels we show
that in some specific time intervals both the average entropy production and
the variance can decrease, provided that the quantum dynamics fails to be
P-divisible. Although the dynamics of the system is overall irreversible, our
result may be interpreted as a transient tendency towards reversibility,
described as a delta peaked distribution of entropy production around zero.
Finally, we also provide analytical bounds on the parameters in the generator
giving rise to the quantum system dynamics, so as to ensure irreversibility
mitigation of the corresponding non-Markovian evolution.

PACS numbers

03.65.Yz, 05.70.Ln, 42.50.Lc

##  I Introduction

In out-of-equilibrium settings the entropy production is a fundamental
thermodynamic quantity allowing to measure the degree of irreversibility of
the dynamical evolution of physical systems. Classically, in the framework of
stochastic thermodynamics SeifertReview , this irreversible contribution lies
in the ratio (different from one) between the probability to observe a
specific trajectory of the system and the probability to observe its time-
reversed partner. The discrepancy between them is a consequence of an
irreversible loss of the system internal energy, usually in the form of heat
deGroot61 ; PRL2012 .

A similar framework of stochastic thermodynamics has been developed for
quantum systems too NJP2010 ; PRL2011 ; SagawaLectures2014 ; PRL2015 ;
FunoNJP2015 ; PRL2017 ; Gherardini_entropy ; ManzanoPRX ; PRL2018 ;
Irreversibility_chapter ; KimPRX2019 , where the trajectory has to be mapped
into the sequence of outcomes of measurements performed on the system. This
scheme, where just two measurements are taken into account, has been
successfully used to prove a number of fluctuation relations that hold far
from equilibrium and allow for the derivation of all the statistical moments
of thermodynamic quantities EspositoReview ; CampisiReview ; 2013Albash ;
2014Rastegin ; NJP2015 ; Mehdi1 ; Mehdi2 ; 2018GheraPRE ; 2019GomezArXiv ;
Cimini2020Arxiv . In such a framework, relevant information about the dynamics
of closed and open quantum systems can be extracted by looking at the
probability distribution of the quantum entropy production HorowitzNJP2013 ;
ManzanoPRE2015 ; Gherardini_entropy ; ManzanoPRX . In this respect, a dynamics
is said _reversible_ if the latter distribution is a Dirac-delta in zero. This
is the case if the system evolves according to a unitary dynamics and the
effect due to possible measurements is negligible. When the dynamics is non-
unitary and there are memory noise effects, one usually deals with non-
Markovian quantum dynamics, which has become a topic of extensive research in
the last decades RivasReview ; BreuerReview ; deVegaReview . Apart from the
theoretical foundational interest, this is also due to a number of
experimental platforms where non-Markovianity turns out to be necessary to
fully catch the relevant physics Exper1 ; LiuSciRep2013 ; Exper2 ;
HaasePRL2018 .

Here, we address the relation between the non-Markovianity of the evolution of
a quantum system and its thermodynamic reversibility, as described by the
first two moments of the entropy production distribution. First results on
this topic have started to appear quite recently BylickaSciRep2016 ; pezzutto
; negentroSte ; bhattacharya ; ThomasPRE2018 ; popovic ; CampbellPRA2018 ;
strasberg ; RivasThermo , only considering the average entropy production
rate. However, it is reasonable to expect that the non-Markovian character of
the quantum dynamics may display some relevant features on the whole
distribution of the entropy production, and not only on the $1$st moment. In
this respect, we show that it is possible to have time intervals where both
the $1$-th and $2$-nd cumulant of the quantum entropy distribution are
decreasing, if the dynamics display a strong form of non-Markovianity known as
essential non-Markovianity degree . It is worth noting that in this work we
specifically concentrate on the case of unital quantum dynamics, which are
customarily used to model open quantum systems subjected to white noise. This
choice is motivated by the fact that the maximally mixed state (multiple of
the identity) is a fixed point of the dynamics and represents a situation
where the experimenter can access all the possible outcomes of a certain
observable (e.g. of the energy) with equal probability. Moreover, on the
thermodynamic side, these dynamics can be thought of describing the
interaction of a quantum system with a thermal bath in the large (infinite)
temperature limit.

The importance of our result lies in the following consideration: despite
essentially non-Markovian evolutions allow for the existence of a time
interval in which the average entropy production rate is negative negentroSte
, this does not necessarily imply a mitigation of irreversibility in general.
Indeed, as it will be discussed below, there exist dynamical regimes in which,
although the mean value of the entropy distribution decreases in a given time
interval, the variance does not have the same behaviour. This implies that one
can nevertheless face with high values of the entropy production on a single
realization, occurring with low-probability but far from the corresponding
average value that is decreasing. On the other hand, we also show that,
already at the level of qubits, it is possible to have a transient reduction
of both the average entropy production and the variance, thus signalling a
tendency towards reversibility. An example and analytical bounds are provided
to corroborate our analysis.

##  II Essential Non-Markovianity

Many different approaches to quantum non-Markovianity can be found in the
literature RivasReview ; BreuerReview ; deVegaReview . In the following, we
adopt the point of view first presented in Ref. RHP2010 , associating the
concept of quantum Markovianity to the divisibility of the dynamical
evolution. In particular, given a quantum evolution described by a one-
parameter family of completely positive (CP) and trace preserving (TP) maps
$\Lambda_{t}$, one says that the dynamics is CP-divisible if for any $t,s$
such that $t\geq s\geq 0$ one has $\Lambda_{t}=V_{t,s}\Lambda_{s},$ with
$V_{t,s}$ CP map. A dynamics is Markovian if it is CP-divisible, it is non-
Markovian otherwise. Actually, one can go a step further and distinguish
between different degrees of non-Markovianity, as suggested in degree ,
depending on whether the intertwining map $V_{t,s}$ is $k$-positive, namely
whether the map $V_{t,s}\otimes\mathrm{id}_{k}$ is positive ($\mathrm{id}_{k}$
is the identity map on $\mathbb{C}^{k}$) or not. Given a Hilbert space of
dimension $n$, dynamical maps corresponding to $n$-positive $V_{t,s}$ are CP-
divisible, those corresponding to $V_{t,s}$ that are only $1$-positive are
called P-divisible, while if a dynamics is not even P-divisible we call it as
_essentially non-Markovian_. Resorting to a recently proved inequality PosMaps
, essential non-Markovianity is necessary condition to find negative entropy
production rates negentroSte , even though it may not be sufficient popovic .
In this paper we consider unital dynamics, namely those evolutions that
preserve the identity operator, fixed point of the map. Among them we focus on
Pauli channels, whose Markovianity degree has been studied in detail in these
recent papers PLA2013 ; PRA2015 ; PRA2016 ; BenattiPRA2017 .

##  III Non-Markovian Pauli channels

In the following, we consider a two-level quantum system described by a
density matrix $\varrho_{t}$ evolving in time through a unital dynamics.
Notice that any unital qubit dynamics can be always described by a random
unitary map unital2d ; Caruso_RMP  (this is not true in higher dimension) that
belongs to the family of Pauli channels. They are defined through the
following Kraus representation:

$\Lambda_{t}(\varrho)=\sum_{\alpha=0}^{3}p_{\alpha}(t)\sigma_{\alpha}\varrho\,\sigma_{\alpha},$
(1)

where
$\\{\sigma_{\alpha}\\}_{0}^{3}=\\{\mathbbm{1},\sigma_{x},\sigma_{y},\sigma_{z}\\}$
is the set of Pauli matrices plus the identity, while the coefficients
$p_{\alpha}$ obey the relation $\sum_{\alpha}p_{\alpha}(t)=1$, $\forall t$
(trace preservation). The initial condition $\Lambda_{0}=\mathrm{id}$ implies
that $p_{0}(0)=1$ and $p_{\alpha}(0)=0$ for $\alpha\neq 0$. The map
$\Lambda_{t}(\varrho)$ is CP if $p_{\alpha}(t)\geq 0$, $\forall t,\alpha$.

The conditions for CP-divisibility of the system dynamics are usually provided
by introducing the generator $\mathcal{L}_{t}$ associated to the quantum map
(1). The generator satisfies the differential equation
$\partial_{t}\Lambda_{t}=\mathcal{L}_{t}\Lambda_{t}$. Thus, under the
hypothesis that the inverse $\Lambda_{t}^{-1}$ exists, the generator is
defined as $\mathcal{L}_{t}=\partial_{t}\Lambda_{t}\circ\Lambda_{t}^{-1}$
PLA2013  with $\circ$ denoting the composition of quantum maps. In this
respect, one finds that the invertibility of the quantum map is ensured if
$p_{1}+p_{2}$, $p_{2}+p_{3}$ and $p_{1}+p_{3}$ are different from $1/2$ for
any finite time $t$. Moreover, being $p_{1}(0)=p_{2}(0)=p_{3}(0)=0$, the
invertibility of the map together with the continuity in time of the functions
$p_{\alpha}$ implies $p_{1}+p_{2}<1/2$, $p_{2}+p_{3}<1/2$ and
$p_{1}+p_{3}<1/2$, $\forall t<\infty$. This working hypothesis will be assumed
in the following. Note however that we may relax the invertibility assumption
for asymptotically long times in order to produce a unique asymptotic state
$\Lambda_{\infty}(\varrho)=\mathbbm{1}$, for any state $\varrho$.

As discussed in Appendix A, the Pauli channels generator can be written in the
following general form PLA2013 ; degree ; PRA2015

$\mathcal{L}_{t}(\varrho)=\sum_{\alpha=1}^{3}\gamma_{\alpha}(t)\left(\sigma_{\alpha}\varrho\,\sigma_{\alpha}-\varrho\right),$
(2)

where $\gamma_{\alpha}$ are the so-called Lindblad coefficients. Therefore,
the dynamics originated from $\Lambda_{t}(\varrho)$ is CP-divisible if and
only if $\gamma_{\alpha}\geq 0$ $\forall t,\alpha$, while necessary and
sufficient condition for P-divisibily is
$\gamma_{\alpha}(t)+\gamma_{\beta}(t)\geq 0$ with $\alpha,\beta=1,2,3$ and
$\alpha\neq\beta$. Finally, one also finds that the relations linking together
all the Lindblad coefficients and the parameters $p_{\alpha}$ of the map are
given by the following equation, with $\alpha,\beta=1,2,3$ and
$\alpha\neq\beta$ (see also Appendix A):

$\exp\left(-2\int_{0}^{t}\gamma_{\alpha}(s)+\gamma_{\beta}(s)\,ds\right)=1-2(p_{\alpha}(t)+p_{\beta}(t)).$
(3)

##  IV Stochastic quantum entropy production

The distribution of the quantum entropy production, originated by a generic
quantum dynamics, can be obtained by realizing two distinct experimental
procedures, i.e., a forward and a backward protocol that are appropriately
chosen Gherardini_entropy ; ManzanoPRX ; Irreversibility_chapter ; KimPRX2019
. Both protocols are interspersed by the application of two projective
measurements at the initial and final time instants, according to the the
well-known two-point measurement (TPM) scheme CampisiReview . The two
measurements are defined as projections on the eigenstates of the arbitrary
observables $\mathcal{O}_{\rm in}$ and $\mathcal{O}_{\rm fin}$. By using the
spectral decomposition theorem, the observables $\mathcal{O}_{\rm in}$ and
$\mathcal{O}_{\rm fin}$ can be generally written as $\mathcal{O}_{\rm
in}=\sum_{k}a^{\rm in}_{k}\Pi^{\rm in}_{k}$ and $\mathcal{O}_{\rm
fin}=\sum_{m}a^{\rm fin}_{m}\Pi^{\rm fin}_{m}$, where $\\{\Pi\\}$ is the set
of projectors associated to the set of observable eigenvalues $\\{a\\}$
(measurement outcomes). The stochastic quantum entropy production
$\Delta\sigma$ is then defined as Gherardini_entropy :

$\Delta\sigma(a^{\textrm{fin}}_{m},a^{\textrm{in}}_{k})\equiv\ln\frac{p_{F}(a^{\textrm{fin}}_{m},a^{\textrm{in}}_{k})}{p_{B}(a^{\textrm{in}}_{k},a^{\textrm{ref}}_{m})},$
(4)

where $p_{F}(a^{\textrm{fin}}_{m},a^{\textrm{in}}_{k})$ and
$p_{B}(a^{\textrm{in}}_{k},a^{\textrm{ref}}_{m})$ are the joint probability to
simultaneously measure the outcomes $\\{a\\}$ in a single realization of the
forward and backward processes, respectively footnote1 . In Eq. (4)
$a^{\textrm{ref}}_{m}$ is obtained from the state after the $1$st measurement
of the backward process, which is generally named as reference state.
Explicitly, the joint probabilities read

$p_{F}(a^{\textrm{fin}}_{m},a^{\textrm{in}}_{k})={\rm Tr}[\Pi^{\rm
fin}_{m}\Lambda^{F}_{t_{\rm fin}}(\Pi^{\rm in}_{k})]p(a^{\rm in}_{k})$ (5)

and

$p_{B}(a^{\textrm{in}}_{k},a^{\textrm{ref}}_{m})={\rm Tr}[\Pi^{\rm
in}_{k}\Lambda^{B}_{t_{\rm fin}}(\Pi^{\rm fin}_{m})]p(a^{\rm ref}_{m}).$ (6)

If the CPTP map $\Lambda_{t}$ governing the system dynamics is unital, then it
is customary to consider the backward dynamics $\Lambda_{t}^{B}$ as the dual
of the forward one (because it is itself a proper quantum dynamics). As a
result, the stochastic quantum entropy production $\Delta\sigma$ becomes
$\Delta\sigma(a^{\textrm{fin}}_{m},a^{\textrm{in}}_{k})=\ln\left(p(a^{\textrm{in}}_{k})/p(a^{\textrm{ref}}_{m})\right)$,
with $p(a^{\textrm{ref}}_{m})$ denoting the probability to get the measurement
outcome $a_{m}^{\rm ref}$. It is reasonable to choose the reference state
equal to the final density operator after the $2$nd measurement of the forward
process. This means that for our purposes the stochastic quantum entropy
production is equal to

$\Delta\sigma(a^{\textrm{fin}}_{m},a^{\textrm{in}}_{k})=\ln
p(a^{\textrm{in}}_{k})-\ln p(a^{\textrm{fin}}_{m}),$ (7)

where $p(a^{\textrm{fin}}_{m})$ denotes the probability to measure the $m-$th
outcome at the final time instant $t_{\rm fin}$.

The statistics of the stochastic quantum entropy production $\Delta\sigma$ can
be computed by evaluating the corresponding probability distribution
$\textrm{Prob}(\Delta\sigma)$. Each time we repeat the TPM scheme, one has a
different realization for $\Delta\sigma$ within a set of discrete values,
whereby $\textrm{Prob}(\Delta\sigma)$ is fully determined by the knowledge of
the measurement outcomes and the respective probabilities:

$\textrm{Prob}(\Delta\sigma)=\sum_{k,m}\delta\left[\Delta\sigma-\Delta\sigma(a^{\textrm{fin}}_{m},a^{\textrm{in}}_{k})\right]p(a_{k}^{\rm
in},a_{m}^{\rm fin}),$ (8)

where $\delta[\cdot]$ denotes the Dirac delta and $p(a_{k}^{\rm in},a_{m}^{\rm
fin})={\rm Tr}[\Pi^{\rm fin}_{m}\Lambda_{t_{\rm fin}}(\Pi^{\rm
in}_{k})]p(a^{\rm in}_{k})$ with $p(a^{\rm in}_{k})={\rm
Tr}[\varrho_{0}\Pi^{\rm in}_{k}]$.

All the statistical moments of $\Delta\sigma$ can be obtained by using the
characteristic function
$G_{\Delta\sigma}(u)\equiv\int\textrm{Prob}(\Delta\sigma)e^{iu\Delta\sigma}{\rm
d}\Delta\sigma$ with $u\in\mathbb{C}$. As formally shown in Appendix B, there
exists a closed-form expression for each quantum entropy statistical moment,
provided that a TPM scheme is used to derive the entropy fluctuations. As a
consequence, one can determine the $1$st and $2$nd moments of $\Delta\sigma$.
The former equals to

$\displaystyle\langle\Delta\sigma\rangle$ $\displaystyle=$ $\displaystyle-{\rm
Tr}[\ln\varrho_{\tau}\Lambda_{t_{\rm fin}}(\varrho_{\rm in})]+{\rm
Tr}[\varrho_{\rm in}\ln\varrho_{\rm in}]$ (9) $\displaystyle=$
$\displaystyle\Delta S+S(\varrho_{\rm fin}\|\varrho_{\tau})$

with $S(\varrho\|\sigma)$ denoting the quantum relative entropy of $\varrho$
with respect to $\sigma$ and $\Delta S\equiv S(\varrho_{\rm
fin})-S(\varrho_{\rm in})$ the difference of the von-Neumann entropies of
$\varrho_{\rm in}$ and $\varrho_{\rm fin}$. In Eq. (9),

$\varrho_{\rm in}=\sum_{k}p(a_{k}^{\rm in})\Pi_{k}^{\rm
in}\,\,\,\,\,\text{and}\,\,\,\,\,\varrho_{\tau}=\sum_{m}p(a_{m}^{\rm
fin})\Pi_{m}^{\rm fin}$

are, respectively, the ensemble average of the quantum system after the $1$st
and $2$nd measurements of the TPM scheme, while $\varrho_{\rm
fin}\equiv\Lambda_{t}(\varrho_{\rm in})$ is the density operator before the
$2$nd projective measurement. Instead, the $2$nd statistical moment of
$\Delta\sigma$ is given by the following relation:

$\displaystyle\langle\Delta\sigma^{2}\rangle={\rm
Tr}[(\ln\varrho_{\tau})^{2}\Lambda_{t_{\rm fin}}(\varrho_{\rm in})]$
$\displaystyle-2{\rm Tr}[\ln\varrho_{\tau}\Lambda_{t_{\rm fin}}(\varrho_{\rm
in}\ln\varrho_{\rm in})]+{\rm Tr}[\varrho_{\rm in}(\ln\varrho_{\rm in})^{2}].$
(10)

In this manuscript, we will mostly focus on the variance that is related to
the $2$nd moment as usual, i.e.,

${\rm
Var}(\Delta\sigma)\equiv\langle\Delta\sigma^{2}\rangle-\langle\Delta\sigma\rangle^{2}.$
(11)

##  V Pauli channels and stochastic entropy

We apply the formalism of the stochastic thermodynamics to the Pauli channel
model. For the sake of convenience, we take the observable $\mathcal{O}$,
associated to both the quantum projective measurements of the TPM, equal to
the Pauli operator $\sigma_{z}$. As a result, the projectors $\Pi^{\rm in}$
and $\Pi^{\rm fin}$ are described by the pure states
$|\ell\rangle\\!\langle\ell|$ with $\ell\in\\{0,1\\}$, whereby

$|0\rangle\\!\langle
0|=\frac{\mathbbm{1}+\sigma_{z}}{2}\,\,\,\,\,\text{and}\,\,\,\,\,|1\rangle\\!\langle
1|=\frac{\mathbbm{1}-\sigma_{z}}{2}\,.$

By initializing the system in the state $\varrho_{0}$, the $1$st measurement
of the TPM scheme makes the quantum system collapse in one of the eigenstates
of $\sigma_{z}$. Thus, the ensemble average of the system after such a
measurement is given by the mixed state $\varrho_{\rm in}$ with diagonal
elements $\frac{1+\zeta_{0}}{2}$ and $\frac{1-\zeta_{0}}{2}$, where
$\zeta_{0}\equiv 1-2\varrho_{0}^{(11)}$ and $\varrho_{0}^{(11)}\equiv\langle
1|\varrho_{0}|1\rangle$. Since

$\Lambda_{t}(|0\rangle\\!\langle
0|)=\frac{\mathbbm{1}+2(1/2-p_{1}-p_{2})\sigma_{z}}{2}$

with $p_{1}$, $p_{2}$ coefficients defining a quantum Pauli channel, also
$\varrho_{\rm fin}=\Lambda_{t}(\varrho_{\rm in})$ is a mixed state $\forall
t$, with $\frac{1+\zeta_{t}}{2}$ and $\frac{1-\zeta_{t}}{2}$ being its
diagonal elements. Here, $\zeta_{t}\equiv\lambda_{t}\zeta_{0}$ where

$\lambda_{t}\equiv
1-2(p_{1}(t)+p_{2}(t))=\mathrm{e}^{-2\int_{0}^{t}(\gamma_{1}(s)+\gamma_{2}(s))\mathrm{d}s}$
(12)

with $0\leq\lambda_{t}\leq 1$. The upper bound is saturated at time $t=0$,
while the lower bound is never achieved at finite time but it turns out that
$\lambda_{t}$ converges to $0$ for $t\to\infty$, thus implying that
$\lambda_{\infty}=0$.

Moreover, being $[\sigma_{z},\varrho_{\rm fin}]=0$, one has
$\varrho_{\tau}=\varrho_{\rm fin}$. Hence, simplifying the expressions of
$\langle\Delta\sigma\rangle$ and $\langle\Delta\sigma^{2}\rangle$, we obtain
$\langle\Delta\sigma\rangle=\Delta S$, while the $2$nd moment of
$\Delta\sigma$, in case of initial pure state with $\zeta_{0}=1$, reads

$\langle\Delta\sigma^{2}\rangle=\mathrm{Tr}\left[(\ln\varrho_{\rm
fin})^{2}\Lambda_{t}(\varrho_{\rm in})\right].$ (13)

As final remark, it is worth noting that, by fixing the initial state
($\zeta_{0}=1$) to be pure, there is an apparent asymmetry between the forward
and backward processes that could entail realizations of the stochastic
entropy production with values tending to infinite. However, the probability
that these realizations can occur is vanishing. This evidence is crucial in
explaining the convergence of our results and the admissibility of our choice.
Indeed, the formalism enabling the computation of the mean and variance of the
stochastic entropy production adopts the convention $0\ln 0:=0$, which
prevents entropy divergent behaviours.

##  VI Mitigating thermodynamic irreversibility

In this paragraph, we present our results relating non-Markovianity and
stochastic entropy production. Depending on the final time $t$ of the TPM
scheme, one has a different probability distribution for the entropy
production, with an average value that typically increases with time (this
always occurs if the dynamics is P-divisible) and a non-monotonic behaviour on
the variance. Below, we are going to show that it is possible to have a time
interval in which both the average and the width of the distribution decrease,
thus signaling an irreversibility mitigation. In particular, we will compute
explicitly the time-derivative of the first two statistical cumulants
$\langle\Delta\sigma\rangle$ and ${\rm Var}(\Delta\sigma)$, and then evaluate
their sign.

As first step, we start from the computation of the time-derivative of
$\langle\Delta\sigma\rangle$. The latter, for the class of models we consider,
equals to

$\displaystyle\displaystyle{\Delta S\equiv S(\varrho_{\rm fin})-S(\varrho_{\rm
in})=}$
$\displaystyle\displaystyle{=\frac{1+\zeta_{0}}{2}\ln\left(\frac{1+\zeta_{0}}{2}\right)+\frac{1-\zeta_{0}}{2}\ln\left(\frac{1-\zeta_{0}}{2}\right)}$
$\displaystyle\displaystyle{-\frac{1+\zeta_{t}}{2}\ln\left(\frac{1+\zeta_{t}}{2}\right)-\frac{1-\zeta_{t}}{2}\ln\left(\frac{1-\zeta_{t}}{2}\right)}$
(14)

where $\zeta_{0}$ and $\zeta_{t}$ have been defined in the previous section.
For simplicity, let us assume $\zeta_{0}=1$. As we will show below, this
simple choice of the initial state is just sufficient to find an evidence of
the mechanism ruling irreversibility mitigation. Notice that for a non-P
divisible unital quantum map the time-derivative of
$\langle\Delta\sigma\rangle$ is not necessarily positive. Indeed,
$\partial_{t}\langle\Delta\sigma\rangle$ explicitly reads

$\partial_{t}\langle\Delta\sigma\rangle=\lambda_{t}\ln\left(\frac{1+\lambda_{t}}{1-\lambda_{t}}\right)(\gamma_{1}(t)+\gamma_{2}(t))$
(15)

with $\lambda_{t}\ln(\frac{1+\lambda_{t}}{1-\lambda_{t}})$ always non
negative, so that $\partial_{t}\langle\Delta\sigma\rangle$ is negative
whenever the sum $\gamma_{1}(t)+\gamma_{2}(t)$ becomes negative. This happens
when the dynamics fails to be P-divisible. Let us observe that, while the
time-derivative of the average entropy production is divergent for
$\lambda_{t}=1$ (occurring only at $t=0$), $\langle\Delta\sigma\rangle$ always
takes finite values, again due to the properties of the function $x\ln x$.

As a second step, in order to see some effects on the system reversibility due
to non-Markovianity, we look at the time-derivative of the variance ${\rm
Var}(\Delta\sigma)$ and study its sign. By substituting the expressions of
$\varrho_{\rm in}$ and $\varrho_{\rm fin}$ (depending on $\lambda_{t}$) in the
formula for the second moment
$\langle\Delta\sigma^{2}\rangle=\mathrm{Tr}\left[(\ln\varrho_{\rm
fin})^{2}\Lambda_{t}(\varrho_{\rm in})\right]$, one finds that

$\partial_{t}\langle\Delta\sigma^{2}\rangle=-2\left(1+\frac{1}{2}\ln\Big{(}\frac{1-\lambda_{t}^{2}}{4}\Big{)}\right)\partial_{t}\langle\Delta\sigma\rangle\,.$
(16)

As a consequence, the time-derivative of the stochastic entropy variance reads

$\partial_{t}{\rm
Var}(\Delta\sigma)=2f_{t}\,\partial_{t}\langle\Delta\sigma\rangle\,,$ (17)

where the function $f_{t}$ is defined as follows

$\displaystyle f_{t}$
$\displaystyle\equiv-\left(\langle\Delta\sigma\rangle+1+\frac{1}{2}\ln\Big{(}\frac{1-\lambda_{t}^{2}}{4}\Big{)}\right)$
$\displaystyle=\frac{\lambda_{t}}{2}\ln{\left(\frac{1+\lambda_{t}}{1-\lambda_{t}}\right)}-1\,.$
(18)

Thus, given the sign of $\partial_{t}\langle\Delta\sigma\rangle$, one can also
determine the sign of the variance by looking at the function $f_{t}$. On the
one hand, the function $f_{t}$ is known to be always greater or equal than
$-1$ and is such that $\lim_{t\to 0}f_{t}=+\infty$ and
$\lim_{t\to\infty}f_{t}=-1$. On the other hand, by computing
$\partial_{t}f_{t}$ , namely

$\partial_{t}f_{t}=\left(\frac{1}{2}\ln{\left(\frac{1+\lambda_{t}}{1-\lambda_{t}}\right)}+\frac{\lambda_{t}}{1-\lambda_{t}^{2}}\right)\partial_{t}{\lambda_{t}}\,,$

one observes that $f_{t}$ is increasing or decreasing depending of
$\partial_{t}{\lambda}_{t}$. In particular, it is increasing in the region
where $\gamma_{1}(t)+\gamma_{2}(t)$ is negative that means when P-divisibility
is broken. Therefore, assuming $\gamma_{1}+\gamma_{2}\leq 0$ in a single
interval $[t_{1},t_{2}]$, the function $f_{t}$ is decreasing up to time
$t_{1}$, then increases from $t_{1}$ to $t_{2}$ and finally decreases for
$t>t_{2}$. As a result, one can have three different cases for the sign of
$f_{t}$ :

$\displaystyle{\rm(I)}\,\,\,f_{t_{1}}\geq 0\,\,\,\text{and}\,\,\,f_{t_{2}}\geq
0\,;\,\,\,\,\,\,\,{\rm(II)}\,\,\,f_{t_{1}}<0\,\,\,\text{and}\,\,\,f_{t_{2}}\geq
0\,;$
$\displaystyle{\rm(III)}\,\,\,f_{t_{1}}<0\,\,\,\text{and}\,\,\,f_{t_{2}}<0\,.$
(19)

Figure 1: Pictorial representation of the three cases (I), (II) and (III),
respectively green solid, red dashed and blue dotted lines, concerning the
possible behaviour of $f_{t}$ as a function of time $t$.

In Fig. 1 we report a sketch of the possible behaviour of $f_{t}$ as a
function of time in the three different cases (I), (II) and (III),
corresponding to three different situations for the sign of $\partial_{t}{\rm
Var}(\Delta\sigma)$ in the interval $[t_{1},t_{2}]$, where the time-derivative
of the average is also negative. In particular, one has:

$\displaystyle{\rm(I)}\,\,\,\partial_{t}{\rm Var}(\Delta\sigma)\leq
0\,\,\,\text{in}\,\,\,[t_{1},t_{2}]$
$\displaystyle{\rm(II)}\,\,\,\partial_{t}{\rm Var}(\Delta\sigma)\leq
0\,\,\,\text{in}\,\,\,[t_{3},t_{2}]\,\,\,\text{with}\,\,\,t_{1}<t_{3}<t_{2}$
$\displaystyle{\rm(III)}\,\,\,\partial_{t}{\rm Var}(\Delta\sigma)\geq
0\,\,\,\text{in}\,\,\,[t_{1},t_{2}].$ (20)

In cases (I) and (II) there is a time interval in which the system tends to be
more reversible, in the sense that both the average and the variance of
$\textrm{Prob}(\Delta\sigma)$ are reducing, so that the distribution becomes
sharper. As a matter of fact, the reversibility of a quantum system dynamics
is associated to a shrinking of the quantum entropy distribution
$\textrm{Prob}(\Delta\sigma)$ up to approach a Dirac delta
$\delta\left[\Delta\sigma\right]$. So, the decreasing of
$\partial_{t}\langle\Delta\sigma\rangle$ and $\partial_{t}{\rm
Var}(\Delta\sigma)$ in a given time interval represents an evident tendency
towards reversibility in the transient, induced by the presence of non-
Markovian effects.

Now, we provide analytical bounds on the coefficients $p_{\alpha}(t)$ of the
Pauli channel that are sufficient to mitigate irreversibility. Above, provided
that the dynamics of the system is not P-divisible, we have shown that the
tendency of ${\rm Var}(\Delta\sigma)$ to decrease just depends on the sign of
$f_{t}$. By introducing
$\phi_{t}\equiv\int_{0}^{t}(\gamma_{1}(s)+\gamma_{2}(s))\mathrm{d}s$, the
inequality $f_{t}\geq 0$ can be recast in the relation

$e^{-2\phi_{t}}\ln\left(\frac{1+e^{-2\phi_{t}}}{1-e^{-2\phi_{t}}}\right)\geq
2\,.$ (21)

The function $x\ln(\frac{1+x}{1-x})-2$, with $0\leq x<1$, has an unique zero
at $x^{\ast}\approx 0.8336$ and is positive for $x\geq x^{\ast}$. This implies
that the inequality (21) is verified for $x\geq x^{\ast}$, i.e.,

$0\leq\phi_{t}\leq\phi^{\ast}\equiv-\frac{1}{2}\ln(x^{\ast})\approx 0.091$
(22)

for all $t>0$. Eq. (22) clearly shows that the irreversibility mitigation can
be found only in a quite small range of dynamical parameters. This means that
essential non-Markovianity has to be usually associated to irreversibility,
except some narrow regimes whereby a transient tendency to reversibility could
be observed.

##  VII Analytical example

Here, we present an example of legitimate (namely completely positive and
trace preserving) unital dynamics for a qubit such that the evolution is not
P-divisible in a single time interval $[t_{1},t_{2}]$. As discussed before,
one has to satisfy the following constraints:

$\displaystyle{\rm(i)}\,\,\,\gamma(t)=\gamma_{1}(t)+\gamma_{2}(t)\leq
0\,\,\,\text{in}\,\,\,[t_{1},t_{2}]\,\,\,\text{(no P-divisibility)}$
$\displaystyle{\rm(ii)}\,\,\,\phi_{t}\geq 0\,\,\,\text{for any}\,\,\,t\geq
0\,\,\,\text{(CP dynamics)}.$

This in turn implies that $\lambda_{t}=\mathrm{e}^{-2\phi_{t}}\leq 1$. We
assume the following explicit form for the function $\gamma(t)$

$\gamma(t)=\beta-\mathrm{e}^{-\alpha t}\left(1-\mathrm{e}^{-\alpha t}\right),$
(23)

where $\alpha,\beta>0$ are two positive parameters. As a consequence the
function $\phi_{t}$ reads

$\phi_{t}=\beta t-\frac{\left(1-\mathrm{e}^{-\alpha
t}\right)^{2}}{2\alpha}\,.$

Then, the sign of $\gamma(t)$ can be easily studied. In particular, one finds
that two zeros exist at times $t_{1}$ and $t_{2}$ corresponding to

$t_{1}=-\frac{1}{\alpha}\ln\left(\frac{1}{2}+\frac{\sqrt{1-4\beta}}{2}\right),\quad
t_{2}=-\frac{1}{\alpha}\ln\left(\frac{1}{2}-\frac{\sqrt{1-4\beta}}{2}\right)$
(24)

provided that $\beta<\frac{1}{4}$. Moreover, it turns out that $\gamma(t)$ is
negative between $t_{1}$ and $t_{2}$ and positive otherwise, thus satisfying
condition (i). Instead, condition (ii) corresponds to the requirement
$\beta\geq\frac{\left(1-\mathrm{e}^{-\alpha t}\right)^{2}}{2\alpha t}$,
$\forall t>0$. Therefore, one has to impose a lower bound $\overline{\beta}$
to $\beta$, which is given by

$\overline{\beta}=\max_{t>0}\frac{\left(1-\mathrm{e}^{-\alpha
t}\right)^{2}}{2\alpha t}.$ (25)

Figure 2: Plot of the values of $\phi_{t_{1}}$ (green solid line) and
$\phi_{t_{2}}$ (blue dashed line) as a function of $\alpha$, with $\beta=0.23$
($0.2<\beta<0.25$). The horizontal line (red diamonds) corresponds to
$\phi^{\ast}$ and allows to distinguish the three different cases (I), (II)
and (III) for the sign of $\partial_{t}{\rm Var}(\Delta\sigma)$.

It can be easily found that the maximum of the function is implicitly defined
by the relation $\mathrm{e}^{z_{\rm max}}=1+2z_{\rm max}$, with $z_{\rm
max}\equiv\alpha t_{\rm max}$. Numerically, one obtains the value $z_{\rm
max}\approx 1.25$ that in turn implies $\overline{\beta}=2\,z_{\rm
max}\,\mathrm{e}^{-2z_{\rm max}}\approx 0.2$ for any value of $\alpha$. As a
result, conditions (i) and (ii) bound the parameter $\beta$ to be

$\overline{\beta}<\beta<\frac{1}{4}$ (26)

because $\overline{\beta}\approx 0.2$ is a nontrivial lower bound smaller than
$1/4$.

Figure 3: Time-derivatives of $\langle\Delta\sigma\rangle$ and ${\rm
Var}(\Delta\sigma)$, respectively average entropy production and entropy
variance, for the analytical example of Section VII. Given the explicit form
of the function $\gamma(t)$ (Eq. (23)), we take $\beta=0.23$ as in Fig. 2 and
three values of $\alpha$, i.e. $0.31,0.38,0.45$, corresponding to the cases
(I), (II) and (III), respectively represented by green solid, red dash-dotted
and blue dotted lines. In each of them, the time-derivative of the entropy
variance has a different behaviour in the time interval $[1.43,2.28]$ where
$\partial_{t}\langle\Delta\sigma\rangle$ is negative.

As shown in Fig. 2, one can span the three different regimes (I), (II) and
(III) for the sign of the variance time-derivative, by tuning the parameter
$\alpha$. According to Eq. (22), these regimes are obtained by comparing the
values $\phi_{t_{1}}(\alpha)$ and $\phi_{t_{2}}(\alpha)$ with
$\phi^{\ast}=-\frac{1}{2}\ln(x^{*})\approx 0.091$, which is the value
corresponding to a vanishing function $f_{t}$. Finally, to corroborate the
results of our analysis, in Fig. 3 we plot the time-derivatives of
$\langle\Delta\sigma\rangle$ and ${\rm Var}(\Delta\sigma)$ as a function of
time, for $\beta=0.23$ and $\alpha\in\\{0.31,0.38,0.45\\}$ corresponding,
respectively, to the regimes (I), (II) and (III). As shown in the figure,
there exist at least one time interval $[t_{1},t_{2}]$ (in the example,
$t_{1}=1.43$ and $t_{2}=2.28$) where the time-derivative of the average
entropy production is negative for the three chosen values of $\alpha$.
Nevertheless, the time-derivative of the entropy variance ${\rm
Var}(\Delta\sigma)$ has not the same behaviour. Indeed, as predicted by our
theoretical analysis, only cases (I) and (II) allow for negative values of
$\partial_{t}{\rm Var}(\Delta\sigma)$ within $[t_{1},t_{2}]$. Once again, this
evidence shows that in a non-Markovian quantum dynamics the mitigation of the
thermodynamic irreversibility (i) occurs only for specific values of the
parameters governing the dynamics of the system, and (ii) cannot be just
ensured by the negativity of the average entropy rate.

##  VIII Conclusions

We have further investigated on the relations between entropy production and
non-Markovianity using the formalism of stochastic thermodynamics. We have
shown that it is possible to have legitimate non-Markovian dynamics, namely
1-parameter families of completely positive and trace preserving maps that
allow for both the average entropy production and its variance to be
transiently decreasing. This can happen when the dynamics is not P-divisible.
Being a dynamics reversible if the distribution of the entropy production is a
Dirac delta, we interpret our finding as a transient tendency to
reversibility. Our analysis deals with unital qubit dynamics, for which we
provide analytical bounds in the parameter space corresponding to
irreversibility mitigation. The calculation is done assuming a pure initial
state because already in this simple case we find evidence of the phenomenon
we are interested in. As a concluding remark, we also note that, provided that
the system dynamics is not P-divisible, it could be in principle possible to
find legitimate dynamics that allow for the same phenomenology both in non-
unital dynamics and higher dimensional quantum systems. This will be matter
for future investigation.

###### Acknowledgements.

S.G. and S.M. equally contributed to this work. The authors thank F. Benatti
for fruitful discussions and F. Carollo for useful comments on the manuscript.
S.G. and F.C. were financially supported by the Fondazione CR Firenze through
the projects Q-BIOSCAN and QUANTUM-AI, PATHOS EU H2020 FET-OPEN Grant No.
828946, the UNIFI Grant Q-CODYCES, and the MISTI Global Seed Funds MIT-FVG
Collaboration Grant “NVQJE”. S.M. was financially supported by the EPSRC Grant
no. EP/R04421X/1.

## Appendices

### A. Brief overview on Pauli channels

Pauli channels are quantum dynamical maps of the form

$\Lambda_{t}(\varrho)=\sum_{\alpha=0}^{3}p_{\alpha}(t)\sigma_{\alpha}\varrho\,\sigma_{\alpha},$
(27)

where
$\\{\sigma_{\alpha}\\}_{0}^{3}=\\{\mathbbm{1},\sigma_{x},\sigma_{y},\sigma_{z}\\}$
is the set of Pauli matrices plus the identity, the coefficients $p_{\alpha}$
obey the relation $\sum_{\alpha}p_{\alpha}(t)=1$, $\forall t$ (trace
preservation), and the initial condition $\Lambda_{0}={\rm id}$ enforces
$p_{0}(0)=1$. Each map $\Lambda_{t}$ is completely positive if
$p_{\alpha}(t)\geq 0\,\forall\,\alpha,t\geq 0$. One can easily check that the
Pauli matrices are the eigen-operators of the linear map $\Lambda_{t}$ and, in
particular, one has

$\displaystyle\Lambda_{t}(\mathbbm{1})=\mathbbm{1},$ (28)
$\displaystyle\Lambda_{t}(\sigma_{1})=(1-2p_{2}(t)-2p_{3}(t))\sigma_{1},$ (29)
$\displaystyle\Lambda_{t}(\sigma_{2})=(1-2p_{1}(t)-2p_{3}(t))\sigma_{2},$ (30)
$\displaystyle\Lambda_{t}(\sigma_{3})=(1-2p_{1}(t)-2p_{2}(t))\sigma_{3}.$ (31)

The map is invertible provided that $p_{1}(t)+p_{2}(t)\neq
1/2,\,p_{1}(t)+p_{3}(t)\neq 1/2,\,p_{2}(t)+p_{3}(t)\neq 1/2$ at any time
$t>0$. Since initially the coefficients $p_{i}$ with $i\in\\{1,2,3\\}$ are
vanishing (because $p_{0}(0)=1$) we can enforce continuity of the functions
$p_{i}(t)$ and invertibility of the map $\Lambda_{t}$ at any time if the
constraints
$p_{1}(t)+p_{2}(t)<1/2,\,p_{1}(t)+p_{3}(t)<1/2,\,p_{2}(t)+p_{3}(t)<1/2$ are
satisfied. For an invertible dynamics the time-dependent generator turns out
to be $\mathcal{L}_{t}=\partial_{t}\Lambda_{t}\circ\Lambda_{t}^{-1}$. By
comparing the following ansatz for the generator

$\mathcal{L}_{t}(\varrho)=\sum_{i=1}^{3}\gamma_{i}(t)\big{(}\sigma_{i}\varrho\sigma_{i}-\varrho\big{)},$
(32)

with the expression derived computing $\partial_{t}\Lambda_{t}$ and
$\Lambda^{-1}_{t}$ one obtains the following relation between the functions
$\gamma_{i}(t)$ and the functions $p_{i}(t)$

$\gamma_{i}(t)+\gamma_{j}(t)=\frac{\partial_{t}(p_{i}(t)+p_{j}(t))}{1-2(p_{i}(t)+p_{j}(t))}$
(33)

for any pair $i,j$ with $i\neq j$. The previous differential equations can be
easily integrated by recognizing that

$\frac{\partial_{t}(p_{i}(t)+p_{j}(t))}{1-2(p_{i}(t)+p_{j}(t))}=-\frac{1}{2}\partial_{t}\ln\big{(}1-2p_{i}(t)-2p_{j}(t)\big{)},$
(34)

so that finally one has

$p_{i}(t)+p_{j}(t)=\frac{1-\mathrm{e}^{-2\int_{0}^{t}(\gamma_{i}+\gamma_{j})\mathrm{d}s}}{2}.$
(35)

Therefore, we have two equivalent characterizations of the dynamics, one based
on the Lindblad coefficients $\gamma_{i}$ and the other one based on the
parameters $p_{i}$ of the Krauss decomposition, and we know how to connect the
two. This is important because the conditions for complete positivity (CP) are
easily given for the $p_{i}$ while conditions for CP-divisibility and
P-divisibility are given on the $\gamma_{i}$. These conditions are reported in
the main text, as first derived in Ref. PRA2015 .

### B. Closed-form expression of quantum entropy statistical moments

The statistical moments of a random variable $X$ with probability distribution
$\textrm{Prob}(X)$ can be generally computed by introducing the characteristic
function

$G_{X}(u)\equiv\int\textrm{Prob}(X)e^{iuX}{\rm d}X$ (36)

associated to $\textrm{Prob}(X)$, with $u$ complex number. In this regard, it
holds that $\langle
X^{\ell}\rangle=\left.(-i)^{\ell}\partial_{u}^{\ell}G_{X}(u)\right|_{u=0}$
namely the $\ell$-th statistical moment of $X$ is proportional to the $\ell$th
derivative of $G_{X}(u)$ with respect to $u$ and evaluated at $u=0$. This
property can be thus applied to the computation of the statistical moments of
$\Delta\sigma$ so that we are allowed to write

$\langle\Delta\sigma^{\ell}\rangle=\left.(-i)^{\ell}\partial_{u}^{\ell}G_{\Delta\sigma}(u)\right|_{u=0}.$
(37)

Provided that a TPM scheme is used to derive the fluctuations of entropy, here
we show that there exists a closed-form expression for each quantum entropy
statistical moment. In particular, the $\ell$-th statistical moment
$\langle\Delta\sigma^{\ell}\rangle$, with $\ell\geq 1$ ($\ell$ arbitrary
integer), is equal to

$\displaystyle\langle\Delta\sigma^{\ell}\rangle=$
$\displaystyle=\displaystyle{\sum_{n=0}^{\ell}(-1)^{\ell-n}\binom{\ell}{n}{\rm
Tr}\left[(\ln\varrho_{\tau})^{\ell-n}\Lambda_{t_{\rm
fin}}\left((\ln\varrho_{\rm in})^{n}\varrho_{\rm in}\right)\right]}$

where $\varrho_{\rm in}\equiv\sum_{k}p(a_{k}^{\rm in})\Pi_{k}^{\rm in}$ and
$\varrho_{\tau}\equiv\sum_{m}p(a_{m}^{\rm fin})\Pi_{m}^{\rm fin}$ are,
respectively, the ensemble average of the quantum system after the $1$st and
$2$nd measurement of the TPM scheme. The validity of (B. Closed-form
expression of quantum entropy statistical moments) can be easily shown
starting from the definition of the stochastic variable $\Delta\sigma$.
Indeed, one gets

$\displaystyle\langle\Delta\sigma^{\ell}\rangle$
$\displaystyle=\sum_{m,k}p(a_{k}^{\rm in},a_{m}^{\rm
fin})\left[\Delta\sigma(a^{\textrm{fin}}_{m},a^{\textrm{in}}_{k})\right]^{\ell}=\sum_{m,k}{\rm
Tr}\left[\Pi^{\rm fin}_{m}\Lambda_{t_{\rm fin}}(\Pi^{\rm
in}_{k})\right]p(a^{\rm in}_{k})\left[\ln p(a^{\textrm{in}}_{k})-\ln
p(a^{\textrm{fin}}_{m})\right]^{\ell}=$
$\displaystyle=\sum_{n=0}^{\ell}(-1)^{\ell-n}\binom{\ell}{n}{\rm
Tr}\left[\sum_{m}\Pi^{\rm fin}_{m}\left(\ln
p(a^{\textrm{fin}}_{m})\right)^{\ell-n}\Lambda_{t_{\rm
fin}}\left(\sum_{k}\Pi^{\rm in}_{k}p(a^{\rm in}_{k})(\ln
p(a^{\textrm{in}}_{k}))^{n}\right)\right]=$
$\displaystyle=\sum_{n=0}^{\ell}(-1)^{\ell-n}\binom{\ell}{n}{\rm
Tr}\left[(\ln\varrho_{\tau})^{\ell-n}\Lambda_{t_{\rm
fin}}\left((\ln\varrho_{\rm in})^{n}\varrho_{\rm in}\right)\right].$ (39)

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