# CP conditions for GKSL-like master equations

Akane Watanabe  Department of Physics, Waseda University, Tokyo 169-8555,
Japan<EMAIL_ADDRESS>Takayuki Suzuki  National Institute of
Information and Communications Technology, Nukui-Kitamachi 4-2-1, Koganei,
Tokyo 184-8795, Japan  Makoto Unoki  Department of Physics, Waseda University,
Tokyo 169-8555, Japan  Hiromichi Nakazato  Department of Physics, Waseda
University, Tokyo 169-8555, Japan

###### Abstract

The complete positivity (CP) of a quantum dynamical map (QDM) is, in general,
difficult to show when its master equation (ME) does not conform to the
Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) form. The GKSL ME describes the
Markovian dynamics, comprising a unitary component with time-independent
Hermitian operators and a non-unitary component with time-independent Lindblad
operators and positive time-independent damping rates. Recently, the non-
Markovian dynamics has received growing attention, and the various types of
GKSL-like MEs with time-dependent operators are widely discussed; however,
rigorous discussions on their CP conditions remain limited. This paper
presents conditions for QDMs to be CP, whose MEs take the GKSL-like form with
arbitrary time dependence. One case considered is where its ME takes the time-
local integro-differential GKSL-like form, which includes CP-divisible cases.
Another case considered is where the ME is time-non-local but can be
approximated to be time-local in the weak-coupling regime. As a special case
of the time-non-local case, the same discussion holds for the time-convoluted
GKSL-like form, which should be compared to previous studies.

##  1 Introduction

The theory of open quantum systems is the essential framework for describing
microscopic systems influenced by their surrounding environments [1, 2, 3, 4].
The dynamics of these systems is described by either a quantum dynamical map
(QDM), which maps an arbitrary initial state to another state at an arbitrary
time, or its differential equation, known as a master equation (ME).

This framework ensures the probabilistic interpretation of quantum mechanics,
guaranteed by two fundamental characteristics of a QDM. One characteristic is
the complete positivity (CP), which ensures the positivity of probabilities
when the system of interest is correlated with other systems. If a QDM has the
Kraus representation [5, 6], the QDM is CP, but the CP is generally difficult
to show in the form of MEs. The other characteristic is the trace-preserving
(TP), which ensures that all probabilities always sum to unity. This property
can be easily demonstrated by taking the trace of MEs.

The Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) ME is one of the most
significant MEs in the theory of open quantum systems [7, 8]. This represents
the most general form of a CPTP quantum dynamical semigroup, which is used to
describe the Markovian dynamics. The GKSL ME is time-local and consists of two
parts: a unitary part with time-independent Hermitian operators and a non-
unitary part with time-independent Lindblad operators and their coefficients,
which are time-independent and positive damping rates. The widely accepted
derivation of the GKSL ME resorts to the Born-Markov approximation, which is
valid for initially uncorrelated systems in the weak coupling regime [2, 3].
Since the standard derivation also involves the rotating wave approximation,
finding alternative methods for deriving the GKSL ME remains an important
problem [9, 10, 11, 12]. The GKSL ME has successfully described the Markovian
dynamics, not only theoretically but also phenomenologically [13].

The non-Markovian effect has recently attracted researchers with increasing
attention [14, 15, 16, 17, 4, 18, 19, 20, 21, 22, 23, 24, 25, 26]. Due to the
non-Markovianity, the QDM does not have the semigroup property and becomes
time inhomogeneous [18]. Since the non-Markovian systems do not follow the
GKSL ME, various alternative MEs have been proposed [27, 28, 29, 30, 31, 32,
33, 34]. Some yield a time-local GKSL-like ME characterized by time-dependent
Lindblad operators [27, 28, 29, 30, 31, 32].

The CP property has been extensively studied for the Markovian dynamics, e.g.,
a recent research investigates the relation between the CP and relaxation
rates [35, 36]. In contrast, rigorous discussions of the CP property for the
non-Markovian dynamics remain limited, particularly to the time-local cases
[37, 38, 32, 39, 40, 41]. For the time-local cases, CP divisibility, i.e.
whether a QDM can be divided into CP maps, is often studied [42, 43, 44, 45,
46]. For the time-non-local cases, positivity is discussed for the GKSL-like
ME [47, 48]. The general discussion of the CP condition is challenging, as
exemplified by the introduction of an additional mathematical condition on the
kernel [49, 50]. In the time-convoluted case, the CP property has been proven
perturbatively for the general form [18] and for the GKSL-like form [15, 16,
17].

This paper aims to rigorously describe the conditions for QDMs to be CP when
their MEs take the GKSL-like form. Specifically, the integro-differential
local and non-local GKSL-like MEs with arbitrary time dependence on a
Hermitian operator of the unitary part and Lindblad operators of the non-
unitary part are considered. The significance lies in evaluating the CP by
obtaining explicit forms of QDMs as the solutions of the MEs.

##  2 Setup and goals

###  2.1 Quantum dynamical maps (QDMs) and complete positivity (CP)

For a finite $d$-dimensional open quantum system, a quantum state is described
by a $d\times d$ positive semidefinite and unit-trace density matrix. The
dynamics is described by a QDM, which is a CPTP superoperator $\Lambda_{t}$
mapping an arbitrary initial state $\rho$ to a state at a later time $t$:
$\rho(t)=\Lambda_{t}\rho,\ \Lambda_{0}={\mathbbm{1}}$, where ${\mathbbm{1}}$
denotes the $d$-dimensional identity superoperator.

A QDM $\Lambda_{t}$ is TP iff a state at a later time
$\rho(t)=\Lambda_{t}\rho$ is also unit-trace, or its ME is traceless. A QDM
$\Lambda_{t}$ is CP iff a superoperator $\Lambda_{t}\otimes{\mathbbm{1}}$ is a
positive semidefinite superoperator. Here, the superoperator
$\Lambda_{t}\otimes{\mathbbm{1}}$ is positive semidefinite iff
$(\Lambda_{t}\otimes{\mathbbm{1}})|\Phi\rangle\langle\Phi|$ is positive
semidefinite for an arbitrary positive state $|\Phi\rangle\langle\Phi|$ of a
$2d$-dimensional system. Also, $\Lambda_{t}$ is CP iff $\Lambda_{t}$ has a
Kraus representation: $\Lambda_{t}\rho=\sum_{\alpha}K_{t}^{\alpha}\rho
K_{t}^{\alpha\dagger},\
\sum_{\alpha}K_{t}^{\alpha\dagger}K_{t}^{\alpha}={\mathbbm{1}}$. In this
paper, the CP of a map $\Lambda_{t}$ is evaluated by the positive
semidefiniteness of a measure ${\cal
M}(\Lambda_{t}):=\langle\Psi|\big{(}\Lambda_{t}\otimes\mathbbm{1}_{d}\big{)}(|\Phi\rangle\langle\Phi|)|\Psi\rangle$,
where $|\Psi\rangle$ is an arbitrary $2d$-dimensitonal vector.

###  2.2 GKSL-like master equations

Two types of integro-differential equations are discussed in this paper. The
first is the time-local case for $\Lambda^{\rm l}_{t}$ and the second is the
time-non-local case for $\Lambda^{{\rm nl}}_{t}$:

$\frac{d}{dt}\Lambda^{{\rm l}}_{t}\rho\equiv\int^{t}_{0}dt^{\prime}{\cal
L}_{t,t^{\prime}}^{\rm GKSL}(\Lambda^{{\rm l}}_{t}\rho),\ \ \
\frac{d}{dt}\Lambda^{{\rm n}{\rm l}}_{t}\rho\equiv\int^{t}_{0}dt^{\prime}{\cal
L}_{t,t^{\prime}}^{\rm GKSL}(\Lambda^{{\rm n}{\rm l}}_{t^{\prime}}\rho).$
(2.1)

For both types, the GKSL-like kernel ${\cal L}^{\rm GKSL}_{t,t^{\prime}}$ is
considered to have an arbitrary time dependence on both $t$ and the integral
variable $t^{\prime}$. Note that the only difference between them appears in
the time-dependence of its argument of the kernel. More specifically, in this
paper, we consider the GKSL-like kernel ${\cal L}^{\rm GKSL}_{t,t^{\prime}}$

$\displaystyle{\cal L}^{\rm
GKSL}_{t,t^{\prime}}\rho=-i\Big{[}H^{\prime}_{t,t^{\prime}},\rho\Big{]}+\sum_{i}\Bigg{(}L^{i}_{t,t^{\prime}}\rho
L^{i\ \dagger}_{t,t^{\prime}}-\frac{1}{2}\Big{\\{}L^{i\
\dagger}_{t,t^{\prime}}L^{i}_{t,t^{\prime}},\rho\Big{\\}}\Bigg{)},$ (2.2)

where $H^{\prime}_{t,t^{\prime}}$ and $\\{L^{i}_{t,t^{\prime}}\\}$ denote a
Hermitian operator and Lindblad operators, respectively. Notice that despite
the time dependence of the Lindblad operators, the overall sign of the non-
unitary part is positive. Previous studies showed that the discussion becomes
simple when the GKSL-like kernel ${\cal L}^{\rm GKSL}_{t,t^{\prime}}$ is
divided into two superoperators ${\cal B}_{t,t^{\prime}},$ and ${\cal
Z}_{t,t^{\prime}}$ with another operator $W_{t,t^{\prime}}$[15, 16, 17]:

$\displaystyle{\cal L}^{\rm GKSL}_{t,t^{\prime}}\rho$
$\displaystyle=\Big{(}{\cal B}_{t,t^{\prime}}-{\cal
Z}_{t,t^{\prime}}\Big{)}\rho,$ (2.3) $\displaystyle{\cal
B}_{t,t^{\prime}}\rho$ $\displaystyle=\sum_{i}L^{i}_{t,t^{\prime}}\ \rho\
L^{i\ \dagger}_{t,t^{\prime}},$ (2.4) $\displaystyle{\cal
Z}_{t,t^{\prime}}\rho$ $\displaystyle=W_{t,t^{\prime}}\ \rho+\rho\
W_{t,t^{\prime}}^{\dagger},\
W_{t,t^{\prime}}=iH^{\prime}_{t,t^{\prime}}+\frac{1}{2}\sum_{i}L^{i\
\dagger}_{t,t^{\prime}}L^{i}_{t,t^{\prime}}.$ (2.5)

We may introduce two other maps $\Lambda^{\cal B}_{t},\Lambda^{\cal Z}_{t}$ in
association with ${\cal B}_{t,t^{\prime}}$ and ${\cal Z}_{t,t^{\prime}}$ for
the local/non-local cases:

$\displaystyle\frac{d}{dt}\Lambda^{{\rm l/nl},{\cal
B}}_{t}\rho\equiv\int^{t}_{0}dt^{\prime}{\cal B}_{t,t^{\prime}}(\Lambda^{{\rm
l/nl},{\cal B}}_{t/t^{\prime}}\rho),\ \ \ \frac{d}{dt}\Lambda^{{\rm
l/nl},{\cal Z}}_{t}\rho\equiv\int^{t}_{0}dt^{\prime}\Big{(}-{\cal
Z}_{t,t^{\prime}}(\Lambda^{{\rm l/nl},{\cal Z}}_{t/t^{\prime}}\rho)\Big{)}.$
(2.6)

To solve the time-non-local GKSL-like ME, one usually considers the weak-
coupling regime. The idea comes from the standard derivation of the GKSL ME,
which starts from the Redfield equation [51]. In the standard derivation of
the Redfield equation, the coupling strength $g$ between the system of
interest and the environment is assumed to be small, and only terms up to the
second order $O(g^{2})$ are kept. We note that the time-local and time-non-
local Redfield equations describe the same dynamics up to the second order
$O(g^{2})$, which is shown in App. A.

###  2.3 Overview of the results

This subsection provides a concise overview of the main achievements of this
work. The time-local case is discussed in detail in Sec. 3, while the time-
non-local case is examined in Sec. 4.

For the time-local case, $\Lambda^{{\rm l},{\cal B}}_{t}$ is CP as shown in
Subsec. 3.1, $\Lambda^{{\rm l},{\cal Z}}_{t}$ is CP as shown in Subsec. 3.2,
and therefore $\Lambda_{t}^{{\rm l}}$ is CP as shown in Subsec. 3.3.
Specifically, if the time dependence of the kernel only appears through
c-number functions, $\Lambda^{{\rm l}}_{t}$ is CP divisible, as shown in
Subsec. 3.4. This result is a simple extension of the previous studies [42,
43, 44, 46].

For the time-non-local case, $\Lambda^{{\rm nl},{\cal B}}_{t}$ is CP as shown
in Subsec. 4.1. Other than this, the following discussions only give the
sufficient conditions for the maps to be CP. For $\Lambda^{{\rm nl},{\cal
Z}}_{t}$, the sufficient condition is strict, as shown in Subsec. 4.2. With
the perturbative approximation in the weak-coupling regime, $\Lambda^{{\rm
nl},{\cal Z}}_{t}$ becomes CP as shown in Subsec. 4.3. Therefore, in the weak-
coupling regime, $\Lambda_{t}^{{\rm n}{\rm l}}$ is CP as shown in Subsec. 4.4.

Finally, the sufficient conditions for $\Lambda^{{\rm n}{\rm l}}_{t}$ are
discussed under an additional assumption that $\Lambda^{{\rm nl},{\cal
Z}}_{t}$ is CP. With the general time dependence of the kernel, the sufficient
condition comes down to specific conditions on the Kraus representation of
$\Lambda^{{\rm nl},{\cal Z}}_{t}$, as shown in Subsec. 4.5. This is also the
case when the kernel is convoluted, as shown in Subsec. 4.6. This subsection
also discusses the relations to the previous studies [15, 16, 17].

##  3 Time-local GKSL-like form

###  3.1 $\Lambda_{t}^{{{\rm l}},{\cal B}}$ is CP

Consider the first-order, linear integro-differential equation

$\displaystyle\frac{d}{dt}\Lambda^{{{\rm l}},{\cal
B}}_{t}\rho=\int^{t}_{0}dt^{\prime}{\cal B}_{t,t^{\prime}}(\Lambda^{{{\rm
l}},{\cal
B}}_{t}\rho)=\int^{t}_{0}dt^{\prime}\Big{(}\sum_{i}L^{i}_{t,t^{\prime}}(\Lambda^{{{\rm
l}},{\cal B}}_{t}\rho)L^{i\ \dagger}_{t,t^{\prime}}\Big{)},\ \
\Lambda_{0}^{{{\rm l}},{\cal B}}={\mathbbm{1}}.$ (3.1)

Integration of Eq. (3.1) between $[0,t]$ explicitly shows the structure of
$\Lambda^{{{\rm l}},{\cal B}}_{t}$

$\displaystyle\Lambda_{t}^{{{\rm l}},{\cal B}}\rho=\rho$
$\displaystyle+\int^{t}_{0}dt_{1}\int^{t_{1}}_{0}dt_{2}\sum_{i_{1}}L^{i_{1}}_{t_{1},t_{2}}(\Lambda^{{{\rm
l}},{\cal B}}_{t_{1}}\rho)L^{i_{1}\ \dagger}_{t_{1},t_{2}}.$ (3.2)

Note that the time locality in Eq. (3.1) yields the $t_{1}$-dependence of
$\Lambda^{{{\rm l}},{\cal B}}_{t_{1}}$ on the RHS of Eq. (3.2). Its formal
solution is iteratively obtained

$\displaystyle\Lambda_{t}^{{{\rm l}},{\cal
B}}\rho=\rho+\sum_{n=1}^{\infty}\int^{t}_{0}dt_{1}\int^{t_{1}}_{0}dt_{2}\int^{t_{1}}_{0}dt_{3}\int^{t_{3}}_{0}$
$\displaystyle
dt_{4}\cdots\int^{t_{2n-3}}_{0}dt_{2n-1}\int^{t_{2n-1}}_{0}dt_{2n}$
$\displaystyle\times\sum_{i_{1},\cdots i_{n}}L^{i_{1}}_{t_{1},t_{2}}\cdots
L^{i_{n}}_{t_{2n-1},t_{2n}}\ \rho\ L^{i_{n}\ \dagger}_{t_{2n-1},t_{2n}}\cdots
L^{i_{1}\ \dagger}_{t_{1},t_{2}}.$ (3.3)

The solution might appear complicated at first sight, but a simple example in
App. B helps to see the essence.

The CP of $\Lambda_{t}^{{{\rm l}},{\cal B}}$ is shown by the positive
semidefiniteness of ${\cal M}(\Lambda_{t}^{{{\rm l}},{\cal B}})$

$\displaystyle{\cal M}(\Lambda_{t}^{{{\rm l}},{\cal
B}})=|\langle\Psi|\Phi\rangle|^{2}+\sum_{n=1}^{\infty}\int^{t}_{0}dt_{1}\cdots\int^{t_{2n-1}}_{0}dt_{2n}\sum_{i_{1},\cdots
i_{n}}|\langle\Psi|\Big{(}L^{i_{1}}_{t_{1},t_{2}}\cdots
L^{i_{n}}_{t_{2n-1},t_{2n}}\Big{)}|\Phi\rangle|^{2}\geq 0.$ (3.4)

###  3.2 $\Lambda_{t}^{{{\rm l}},{\cal Z}}$ is CP

Consider the first-order, linear integro-differential equation

$\displaystyle\frac{d}{dt}\Lambda_{t}^{{{\rm l}},{\cal
Z}}\rho=\int^{t}_{0}dt^{\prime}\Big{(}-{\cal Z}_{t,t^{\prime}}(\Lambda^{{{\rm
l}},{\cal
Z}}_{t}\rho)\Big{)}=\int^{t}_{0}dt^{\prime}\Big{(}-W_{t,t^{\prime}}(\Lambda_{t}^{{{\rm
l}},{\cal Z}}\rho)-(\Lambda_{t}^{{{\rm l}},{\cal
Z}}\rho)W_{t,t^{\prime}}^{\dagger}\Big{)},\ \ \Lambda_{0}^{{{\rm l}},{\cal
Z}}={\mathbbm{1}}.$ (3.5)

Note that the equation is local in time. Therefore, the formal solution is
obtained by using the chronological and anti-chronological exponential
functions

$\displaystyle\Lambda_{t}^{{{\rm l}},{\cal Z}}\rho=\overleftarrow{\cal
T}e^{-\int^{t}_{0}ds\int^{s}_{0}ds^{\prime}W_{s,s^{\prime}}}\ \rho\
\overrightarrow{\cal
T}e^{-\int^{t}_{0}ds\int^{s}_{0}ds^{\prime}W_{s,s^{\prime}}^{\dagger}}\equiv
V_{t}\rho V_{t}^{\dagger}.$ (3.6)

The explicit forms of $V_{t}$ and $V_{t}^{\dagger}$ are the following

$\displaystyle V_{t}$
$\displaystyle=1+\sum_{n=1}^{\infty}\frac{(-1)^{n}}{n!}\int^{t}_{0}ds_{1}\int^{s_{1}}_{0}ds_{2}\cdots\int^{t}_{0}ds_{2n-1}\int^{s_{2n-1}}_{0}ds_{2n}\overleftarrow{\cal
T}\big{(}W_{s_{1},s_{2}}\cdots W_{s_{2n-1},s_{2n}}\big{)},$ (3.7)
$\displaystyle V_{t}^{\dagger}$
$\displaystyle=1+\sum_{n=1}^{\infty}\frac{(-1)^{n}}{n!}\int^{t}_{0}ds_{1}\int^{s_{1}}_{0}ds_{2}\cdots\int^{t}_{0}ds_{2n-1}\int^{s_{2n-1}}_{0}ds_{2n}\overrightarrow{\cal
T}\big{(}W_{s_{1},s_{2}}^{\dagger}\cdots
W_{s_{2n-1},s_{2n}}^{\dagger}\big{)}.$ (3.8)

Note that the chronological and anti-chronological orderings are applied to
the odd number of the integral variables.

The CP $\Lambda_{t}^{{{\rm l}},{\cal Z}}$ is guaranteed by the positive
semidefiniteness of the measure ${\cal M}(\Lambda_{t}^{{{\rm l}},{\cal Z}})$

$\displaystyle{\cal M}(\Lambda_{t}^{{{\rm l}},{\cal
Z}})=|\langle\Psi|V_{t}|\Phi\rangle|^{2}\geq 0.$ (3.9)

###  3.3 $\Lambda_{t}^{{\rm l}}$ is CP

Consider the first-order, linear integro-differential, time-local GKSL-like
master equation

$\displaystyle\frac{d}{dt}\Lambda^{{{\rm
l}}}_{t}\rho=\int^{t}_{0}dt^{\prime}{\cal L}_{t,t^{\prime}}^{\rm
GKSL}(\Lambda^{{{\rm
l}}}_{t}\rho)=\int^{t}_{0}dt^{\prime}\Big{(}\sum_{i}L^{i}_{t,t^{\prime}}(\Lambda^{{{\rm
l}}}_{t}\rho)L^{i\ \dagger}_{t,t^{\prime}}-W_{t,t^{\prime}}(\Lambda_{t}^{{{\rm
l}}}\rho)-(\Lambda_{t}^{{{\rm
l}}}\rho)W_{t,t^{\prime}}^{\dagger}\Big{)},\Lambda_{0}^{{{\rm
l}}}={\mathbbm{1}}.$ (3.10)

To simplify Eq. (3.10), take the time derivative of the map $\Lambda_{t}^{{\rm
l}}$ under the similarity transformation $\Lambda_{t}^{{\rm l}}\equiv
V_{t}\hat{\Lambda}_{t}^{{\rm l}}V_{t}^{\dagger}$

$\displaystyle\frac{d}{dt}\Lambda_{t}^{{\rm
l}}\rho=V_{t}\Big{(}\frac{d}{dt}\hat{\Lambda}_{t}^{{\rm
l}}\rho\Big{)}V_{t}^{\dagger}+\int^{t}_{0}dt^{\prime}\Big{(}-W_{t,t^{\prime}}V_{t}(\hat{\Lambda}_{t}^{{{\rm
l}}}\rho)V_{t}^{\dagger}-V_{t}(\hat{\Lambda}_{t}^{{{\rm
l}}}\rho)V_{t}^{\dagger}W_{t,t^{\prime}}^{\dagger}\Big{)}.$ (3.11)

Comparing Eq. (3.10) and Eq. (3.11), the first-order, linear integro-
differential equation for $\hat{\Lambda}_{t}^{{\rm l}}$ is obtained

$\displaystyle V_{t}\Big{(}\frac{d}{dt}\hat{\Lambda}_{t}^{{\rm
l}}\rho\Big{)}V_{t}^{\dagger}=\int^{t}_{0}dt^{\prime}\Big{(}\sum_{i}L^{i}_{t,t^{\prime}}V_{t}(\hat{\Lambda}^{{{\rm
l}}}_{t}\rho)V_{t}^{\dagger}L^{i\ \dagger}_{t,t^{\prime}}\Big{)}.$ (3.12)

Define $\hat{L}^{i}_{s,s^{\prime}}\equiv V_{s}^{-1}L^{i}_{s,s^{\prime}}V_{s}$.
See the explicit form of $V_{t}^{-1}$ in App. C. Then, Eq. (3.12) appears the
same as Eq. (3.1)

$\displaystyle\frac{d}{dt}\hat{\Lambda}_{t}^{{\rm
l}}\rho=\int^{t}_{0}dt^{\prime}\Big{(}\sum_{i}\hat{L}^{i}_{t,t^{\prime}}(\hat{\Lambda}^{{{\rm
l}}}_{t}\rho)\hat{L}^{i\ \dagger}_{t,t^{\prime}}\Big{)}.$ (3.13)

Using the same arguments as in Subsec. 3.1, Eq. (3.10) is solved to give the
formal solution

$\displaystyle\Lambda_{t}^{{\rm
l}}\rho=V_{t}\Bigg{(}\rho+\sum_{n=1}^{\infty}\int^{t}_{0}dt_{1}\cdots\int^{t_{2n-1}}_{0}dt_{2n}\sum_{i_{1},\cdots
i_{n}}\hat{L}^{i_{1}}_{t_{1},t_{2}}\cdots\hat{L}^{i_{n}}_{t_{2n-1},t_{2n}}\
\rho\ \hat{L}^{i_{n}\ \dagger}_{t_{2n-1},t_{2n}}\cdots\hat{L}^{i_{1}\
\dagger}_{t-1,t_{2}}\Bigg{)}V_{t}^{\dagger}.$ (3.14)

The CP of $\Lambda_{t}^{{\rm l}}$ is guaranteed by the positive
semidefiniteness of the measure ${\cal M}(\Lambda_{t}^{{\rm l}})$

$\displaystyle{\cal M}(\Lambda_{t}^{{\rm
l}})=|\langle\Psi|V_{t}|\Phi\rangle|^{2}+\sum_{n=1}^{\infty}\int^{t}_{0}dt_{1}\cdots\int^{t_{2n-1}}_{0}dt_{2n}\sum_{i_{1},\cdots
i_{n}}|\langle\Psi|V_{t}\hat{L}^{i_{1}}_{t_{1},t_{2}}\cdots\hat{L}^{i_{n}}_{t_{2n-1},t_{2n}}|\Phi\rangle|^{2}\geq
0.$ (3.15)

Therefore, the QDM $\Lambda_{t}^{{\rm l}}$ for the time-local GKSL-like ME is
CP.

###  3.4 If the time dependence appears only through c-number functions,
$\Lambda^{{{\rm l}}}_{t}$ is CP divisible

Consider a special case of the time-local GKSL-like ME where the time
dependence of the Lindblad operators is limited to c-number functions
$h^{i}(t,t^{\prime})\in{\mathbbm{C}}$, ${\rm i.e.}$,
$L^{i}_{t,t^{\prime}}=h^{i}(t,t^{\prime})L^{i}$,

$\displaystyle\frac{d}{dt}\Lambda^{{{\rm
l}}}_{t}\rho=\int^{t}_{0}dt^{\prime}\Bigg{(}-i\big{[}H^{\prime}_{t,t^{\prime}},\Lambda^{{{\rm
l}}}_{t}\rho\big{]}+\sum_{i}|h^{i}(t,t^{\prime})|^{2}\Big{(}L^{i}(\Lambda^{{{\rm
l}}}_{t}\rho)L^{i\dagger}-\frac{1}{2}\big{\\{}L^{i\dagger}L^{i},\Lambda_{t}^{{{\rm
l}}}\rho\big{\\}}\Big{)}\Bigg{)},\ \Lambda_{0}^{{{\rm l}}}={\mathbbm{1}}.$
(3.16)

The time-dependent damping rates
$\int^{t}_{0}dt^{\prime}|h^{i}(t,t^{\prime})|^{2}$ are clearly positive
semidefinite. This supports the well-known condition for the QDM to be CP-
divisible [42, 43, 44, 46].

##  4 Time-non-local GKSL-like form

###  4.1 $\Lambda_{t}^{{\rm n}{\rm l},{\cal B}}$ is CP

Consider the first-order, linear integro-differential equation

$\displaystyle\frac{d}{dt}\Lambda^{{\rm n}{\rm l},{\cal
B}}_{t}\rho=\int^{t}_{0}dt^{\prime}{\cal B}_{t,t^{\prime}}(\Lambda^{{\rm
n}{\rm l},{\cal
B}}_{t^{\prime}}\rho)=\int^{t}_{0}dt^{\prime}\Big{(}\sum_{i}L^{i}_{t,t^{\prime}}(\Lambda^{{\rm
n}{\rm l},{\cal B}}_{t^{\prime}}\rho)L^{i\ \dagger}_{t,t^{\prime}}\Big{)},\ \
\Lambda_{0}^{{\rm n}{\rm l},{\cal B}}={\mathbbm{1}}.$ (4.1)

Integration of Eq. (4.1) between $[0,t]$ explicitly shows the structure of
$\Lambda^{{\rm n}{\rm l},{\cal B}}_{t}$

$\displaystyle\Lambda_{t}^{{\rm n}{\rm l},{\cal B}}\rho=\rho$
$\displaystyle+\int^{t}_{0}dt_{1}\int^{t_{1}}_{0}dt_{2}\sum_{i_{1}}L^{i_{1}}_{t_{1},t_{2}}(\Lambda^{{\rm
n}{\rm l},{\cal B}}_{t_{2}}\rho)L^{i_{1}\ \dagger}_{t_{1},t_{2}}.$ (4.2)

Note that the time non-locality is reflected on the $t_{2}$-dependence of
$\Lambda^{{\rm n}{\rm l},{\cal B}}_{t_{2}}$ on the RHS of Eq. (4.2). Its
formal solution is obtained

$\displaystyle\Lambda_{t}^{{\rm n}{\rm l},{\cal
B}}\rho=\rho+\sum_{n=1}^{\infty}\int^{t}_{0}dt_{1}\int^{t_{1}}_{0}dt_{2}\int^{t_{2}}_{0}$
$\displaystyle
dt_{3}\int^{t_{3}}_{0}dt_{4}\cdots\int^{t_{2n-2}}_{0}dt_{2n-1}\int^{t_{2n-1}}_{0}dt_{2n}$
$\displaystyle\times\sum_{i_{1},\cdots i_{n}}L^{i_{1}}_{t_{1},t_{2}}\cdots
L^{i_{n}}_{t_{2n-1},t_{2n}}\ \rho\ L^{i_{n}\ \dagger}_{t_{2n-1},t_{2n}}\cdots
L^{i_{1}\ \dagger}_{t_{1},t_{2}}.$ (4.3)

Note that the intervals of integrations are naturally time-ordered and are
different from those in Eq. (3.3).

The CP of $\Lambda_{t}^{{\rm n}{\rm l},{\cal B}}$ is shown by the positive
semidefiniteness of ${\cal M}(\Lambda_{t}^{{\rm n}{\rm l},{\cal B}})$

$\displaystyle{\cal M}(\Lambda_{t}^{{\rm n}{\rm l},{\cal
B}})=|\langle\Psi|\Phi\rangle|^{2}+\sum_{n=1}^{\infty}\int^{t}_{0}dt_{1}\cdots\int^{t_{2n-1}}_{0}dt_{2n}\sum_{i_{1},\cdots
i_{n}}|\langle\Psi|\Big{(}L^{i_{1}}_{t_{1},t_{2}}\cdots
L^{i_{n}}_{t_{2n-1},t_{2n}}\Big{)}|\Phi\rangle|^{2}\geq 0.$ (4.4)

###  4.2  $\Lambda_{t}^{{\rm n}{\rm l},{\cal Z}}$ is CP only if
$W_{t,t^{\prime}}$ satisfies a strict condition

Consider the first-order, linear integro-differential equation

$\displaystyle\frac{d}{dt}\Lambda_{t}^{{\rm n}{\rm l},{\cal
Z}}\rho=\int^{t}_{0}dt^{\prime}\Big{(}-{\cal Z}_{t,t^{\prime}}(\Lambda^{{\rm
n}{\rm l},{\cal
Z}}_{t^{\prime}}\rho)\Big{)}=\int^{t}_{0}dt^{\prime}\Big{(}-W_{t,t^{\prime}}(\Lambda_{t^{\prime}}^{{\rm
n}{\rm l},{\cal Z}}\rho)-(\Lambda_{t^{\prime}}^{{\rm n}{\rm l},{\cal
Z}}\rho)W_{t,t^{\prime}}^{\dagger}\Big{)},\ \ \Lambda_{0}^{{\rm n}{\rm
l},{\cal Z}}={\mathbbm{1}}.$ (4.5)

In this subsection, the conditions for the solution $\Lambda_{t}^{{\rm n}{\rm
l},{\cal Z}}$ to be CP are discussed. By integrating Eq. (4.5) between
$[0,t]$, the structure of $\Lambda_{t}^{{\rm n}{\rm l},{\cal Z}}$ is
explicitly shown

$\displaystyle\Lambda_{t}^{{\rm n}{\rm l},{\cal
Z}}\rho=\rho+\int^{t}_{0}dt_{1}\int^{t_{1}}_{0}dt_{2}\Big{(}-{\cal
Z}_{t_{1},t_{2}}(\Lambda^{{{\rm l}},{\cal Z}}_{t_{2}}\rho)\Big{)}.$ (4.6)

Iterating multiple times yields the formal solution as an infinite series
expansion

$\displaystyle\Lambda_{t}^{{\rm n}{\rm l},{\cal
Z}}\rho=\rho+\int^{t}_{0}dt_{1}\int^{t_{1}}_{0}dt_{2}\Big{(}-{\cal
Z}_{t_{1},t_{2}}\rho\Big{)}+\int^{t}_{0}dt_{1}\cdots\int^{t_{3}}_{0}dt_{4}\Big{(}{\cal
Z}_{t_{1},t_{2}}{\cal Z}_{t_{3},t_{4}}\rho\Big{)}+\cdots.$ (4.7)

The condition for $\Lambda_{t}^{{\rm n}{\rm l},{\cal Z}}$ to be CP depends on
the sign of ${\cal M}(\Lambda_{t}^{{\rm n}{\rm l},{\cal Z}})$:

$\displaystyle{\cal M}(\Lambda_{t}^{{\rm n}{\rm l},{\cal
Z}})=|\langle\Psi|\Phi\rangle|^{2}$
$\displaystyle+\int^{t}_{0}dt_{1}\int^{t_{1}}_{0}dt_{2}\langle\Psi|\Big{(}-{\cal
Z}_{t_{1},t_{2}}|\Phi\rangle\langle\Phi|\Big{)}|\Psi\rangle$
$\displaystyle+\int^{t}_{0}dt_{1}\cdots\int^{t_{3}}_{0}dt_{4}\langle\Psi|\Big{(}{\cal
Z}_{t_{1},t_{2}}{\cal
Z}_{t_{3},t_{4}}|\Phi\rangle\langle\Phi|\Big{)}|\Psi\rangle+\cdots.$ (4.8)

Showing the positive semidefiniteness of the measure ${\cal
M}(\Lambda_{t}^{{\rm n}{\rm l},{\cal Z}})$ seems in general difficult. Since
Eq. (4.8) is expressed as the power series expansion by the kernel $-{\cal
Z}_{t,t^{\prime}}$, the positive semidefiniteness of the second term of Eq.
(4.8) is sufficient to guarantee $0\leq{\cal M}(\Lambda_{t}^{{\rm n}{\rm
l},{\cal Z}})$.

It is shown, however, that the condition for the second term of Eq. (4.8) to
be positive semidefinite imposes a strong condition on the operational
structure of $W_{t,t^{\prime}}$. Express
$|\Psi\rangle=\sum_{m}\psi_{m}|m\rangle$, and an arbitrary initial state
$\rho=\sum_{n}\rho_{n}|n\rangle\langle n|$ in terms of the complete
orthonormal basis $\\{|n\rangle\\}$. The condition is then expressed as

$\displaystyle
0\leq\int^{t}_{0}dt_{1}\int^{t_{1}}_{0}dt_{2}\sum_{m,n,m^{\prime}}\psi_{m}^{*}\rho_{n}\psi_{m^{\prime}}\big{(}-\langle
m|W_{t_{1},t_{2}}|n\rangle\langle n|m^{\prime}\rangle-\langle
m|n\rangle\langle n|W_{t_{1},t_{2}}^{\dagger}|m^{\prime}\rangle\big{)}.$ (4.9)

This condition is indeed satisfied for arbitrary
$\psi_{m}^{*}\rho_{n}\psi_{m^{\prime}}$, by a trivial diagonal operator
$W_{t_{1},t_{2}}$ with some function $f(t_{1},t_{2})\in{\mathbbm{C}}$,

$\displaystyle\langle
m|W_{t_{1},t_{2}}|n\rangle\propto-f(t_{1},t_{2})\delta_{n,m},\ \ {\rm
Re}\big{[}f(t_{1},t_{2})\big{]}<0.$ (4.10)

Actually, Eq. (4.9) is inevitably violated if the operator $W_{t_{1},t_{2}}$
has a non-vanishing off-diagonal matrix element. Consider a normalized vector
$|\Psi\rangle=\psi_{n}|n\rangle+\psi_{l}|l\rangle$, and suppose $n\neq l$,
$\langle l|W_{t_{1},t_{2}}|n\rangle\neq 0$. Eq. (4.9) then becomes

$\displaystyle
0\leq\int^{t}_{0}dt_{1}\int^{t_{1}}_{0}dt_{2}\sum_{n}(-2\rho_{n})\Big{(}|\psi_{n}|^{2}|\langle
n|W_{t_{1},t_{2}}|n\rangle|+|\psi_{l}\psi_{n}||\langle
l|W_{t_{1},t_{2}}|n\rangle|\cos\big{(}\theta_{n}-\theta_{l}+\varphi_{l,n}\big{)}\Big{)},$
(4.11)

where relative phases $\varphi_{l,n}$ and $\theta_{n}-\theta_{l}$ are defined
by $\langle l|W_{t_{1},t_{2}}|n\rangle\equiv|\langle
l|W_{t_{1},t_{2}}|n\rangle|e^{i\varphi_{l,n}}$ and
$\psi_{l}^{*}\psi_{n}\equiv|\psi_{l}||\psi_{n}|e^{i(\theta_{n}-\theta_{l})}$.
Since $\rho_{n}$ of the initial state $\rho=\sum_{n}\rho_{n}|n\rangle\langle
n|$ is positive semidefinite, Eq. (4.11) is expressed as below

$\displaystyle\sum_{n}\Bigg{(}\int^{t}_{0}dt_{1}\int^{t_{1}}_{0}dt_{2}\frac{|\psi_{n}|}{\sqrt{1-|\psi_{n}|^{2}}}\frac{|\langle
n|W_{t_{1},t_{2}}|n\rangle|}{|\langle
l|W_{t_{1},t_{2}}|n\rangle|}+\cos\big{(}\theta_{n}-\theta_{l}+\varphi_{l,n}\big{)}\Bigg{)}\leq
0.$ (4.12)

Although the condition by Eq. (4.12) has to be satisfied for arbitrary
$\psi_{n}\in{\mathbbm{C}}$, this can always be violated by a certain choice of
the relative phase $\theta_{n}-\theta_{l}$. This concludes that no operator
$W_{t_{1},t_{2}}$ with non-vanishing off-diagonal elements is allowed for the
condition Eq. (4.9) to hold.

In summary, for the first-order, linear integro-differential equation (4.5),
if the operator $W_{t_{1},t_{2}}$ satisfies the strict condition Eq. (4.10),
the map $\Lambda_{t}^{{\rm n}{\rm l},{\cal Z}}$ is CP. The following
subsections, therefore, impose additional constraints on the map
$\Lambda_{t}^{{\rm n}{\rm l},{\cal Z}}$ to facilitate further analysis and
discussion.

###  4.3 $\Lambda^{{\rm n}{\rm l},{\cal Z}}_{t}$ is CP in the weak-coupling
regime

Consider the first-order, linear integro-differential equation

$\displaystyle\frac{d}{dt}\Lambda_{t}^{{\rm n}{\rm l},{\cal
Z}}\rho=\int^{t}_{0}dt^{\prime}\Big{(}-{\cal Z}_{t,t^{\prime}}(\Lambda^{{{\rm
l}},{\cal
Z}}_{t^{\prime}}\rho)\Big{)}=\int^{t}_{0}dt^{\prime}\Big{(}-W_{t,t^{\prime}}(\Lambda_{t^{\prime}}^{{{\rm
l}},{\cal Z}}\rho)-(\Lambda_{t^{\prime}}^{{{\rm l}},{\cal
Z}}\rho)W_{t,t^{\prime}}^{\dagger}\Big{)},\ \ \Lambda_{0}^{{\rm n}{\rm
l},{\cal Z}}={\mathbbm{1}}.$ (4.13)

In this subsection, the formal solution is perturbatively obtained. In the
weak-coupling regime, the system-bath coupling strength $g$ is assumed to be
sufficiently small and treated as a perturbation parameter. In the standard
derivation of the GKSL-like ME, the Hermitian operator
$H^{\prime}_{t,t^{\prime}}$ in the interaction picture and the Lindblad
operators $\\{L^{i}_{t,t^{\prime}}\\}$ are in proportion to $g^{2}$. The
operator $W_{t,t^{\prime}}$ is therefore proportional to $g^{2}$.

To further approximate Eq. (4.13), integrate Eq. (4.13) over $[0,t]$

$\displaystyle\Lambda_{t}^{{\rm n}{\rm l},{\cal
Z}}\rho=\rho+\int^{t}_{0}dt_{1}\int^{t_{1}}_{0}dt_{2}\Big{(}-W_{t_{1},t_{2}}(\Lambda_{t_{2}}^{{\rm
n}{\rm l},{\cal Z}}\rho)-(\Lambda_{t_{2}}^{{\rm n}{\rm l},{\cal
Z}}\rho)W_{t_{1},t_{2}}^{\dagger}\Big{)}.$ (4.14)

Iterating Eq. (4.14) and neglecting the terms of order $g^{3}$ and higher, a
simple differential equation is obtained

$\displaystyle\frac{d}{dt}\Lambda_{t}^{{\rm n}{\rm l},{\cal
Z}}\rho\approx\int^{t}_{0}dt^{\prime}\Big{(}-W_{t,t^{\prime}}\rho-\rho\
W_{t,t^{\prime}}^{\dagger}\Big{)}.$ (4.15)

Since $W_{t_{1},t_{2}}\propto g^{2}$, a further approximation:
$g^{2}\rho\approx g^{2}\Lambda_{t}^{{\rm n}{\rm l},{\cal Z}}\rho$ is valid up
to $g^{2}$. By this approximation, Eq. (4.15) appears as a familiar form

$\displaystyle\frac{d}{dt}\Lambda_{t}^{{\rm n}{\rm l},{\cal
Z}}\rho\approx\Big{(}-\int^{t}_{0}dt^{\prime}W_{t,t^{\prime}}\Big{)}\Lambda_{t}^{{\rm
n}{\rm l},{\cal Z}}\rho+\Lambda_{t}^{{\rm n}{\rm l},{\cal
Z}}\rho\Big{(}-\int^{t}_{0}dt^{\prime}W_{t,t^{\prime}}^{\dagger}\Big{)}.$
(4.16)

Eq. (4.16) coincides with Eq. (3.5). Thus the formal solution is the same as
the time-local case

$\displaystyle\Lambda_{t}^{{\rm n}{\rm l},{\cal Z}}\rho=\overleftarrow{\cal
T}e^{-\int^{t}_{0}ds\int^{s}_{0}ds^{\prime}W_{s,s^{\prime}}}\ \rho\
\overrightarrow{\cal
T}e^{-\int^{t}_{0}ds\int^{s}_{0}ds^{\prime}W_{s,s^{\prime}}^{\dagger}}\equiv
V_{t}\rho V_{t}^{\dagger}.$ (4.17)

The CP of $\Lambda_{t}^{{\rm n}{\rm l},{\cal Z}}$ is guaranteed by the
positive semidefiniteness of the measure ${\cal M}(\Lambda_{t}^{{\rm n}{\rm
l},{\cal Z}})$

$\displaystyle{\cal M}(\Lambda_{t}^{{\rm n}{\rm l},{\cal
Z}})=|\langle\Psi|V_{t}|\Phi\rangle|^{2}\geq 0.$ (4.18)

In the weak-coupling regime, this type of map is CP.

###  4.4 $\Lambda^{{\rm n}{\rm l}}_{t}$ is CP in the weak-coupling regime

Consider the first-order, linear integro-differential, GKSL-like master
equation

$\displaystyle\frac{d}{dt}\Lambda^{{\rm n}{\rm
l}}_{t}\rho=\int^{t}_{0}dt^{\prime}{\cal L}_{t,t^{\prime}}^{\rm
GKSL}(\Lambda^{{\rm n}{\rm l}}_{t^{\prime}}\rho)$
$\displaystyle=\int^{t}_{0}dt^{\prime}\Big{(}\sum_{i}L^{i}_{t,t^{\prime}}(\Lambda^{{\rm
n}{\rm l}}_{t^{\prime}}\rho)L^{i\
\dagger}_{t,t^{\prime}}-W_{t,t^{\prime}}(\Lambda_{t^{\prime}}^{{\rm n}{\rm
l}}\rho)-(\Lambda_{t^{\prime}}^{{\rm n}{\rm
l}}\rho)W_{t,t^{\prime}}^{\dagger}\Big{)},$ $\displaystyle\Lambda_{0}^{{\rm
n}{\rm l}}$ $\displaystyle={\mathbbm{1}}.$ (4.19)

In the weak-coupling regime, the discussion in Subsec. 4.3 holds, and Eq.
(4.19) is approximated up to the second order of the system-bath coupling
strength as

$\displaystyle\frac{d}{dt}\Lambda^{{\rm n}{\rm
l}}_{t}\rho\approx\int^{t}_{0}dt^{\prime}\Big{(}\sum_{i}L^{i}_{t,t^{\prime}}(\Lambda^{{\rm
n}{\rm l}}_{t^{\prime}}\rho)L^{i\ \dagger}_{t,t^{\prime}}$ $\displaystyle-
W_{t,t^{\prime}}(\Lambda_{t}^{{\rm n}{\rm l}}\rho)-(\Lambda_{t}^{{\rm n}{\rm
l}}\rho)W_{t,t^{\prime}}^{\dagger}\Big{)},$ $\displaystyle\Lambda_{0}^{{\rm
n}{\rm l}}$ $\displaystyle={\mathbbm{1}}.$ (4.20)

In the same approach as in Subsec. 3.3, the contribution from the operator
$W_{t,t^{\prime}}$ can be incorporated by using the operators
$V_{t},V_{t}^{-1}$. Defining $\bar{L}^{i_{j}}_{s,s^{\prime}}\equiv
V_{s}^{-1}L^{i_{j}}_{s,s^{\prime}}V_{s^{\prime}}$, Eq. (4.20) takes the same
form as Eq. (4.1)

$\displaystyle\frac{d}{dt}\Lambda_{t}^{{\rm n}{\rm
l}}\rho=\int^{t}_{0}dt^{\prime}\Big{(}\sum_{i}\bar{L}^{i}_{t,t^{\prime}}(\Lambda^{{{\rm
l}}}_{t^{\prime}}\rho)\bar{L}^{i\ \dagger}_{t,t^{\prime}}\Big{)}.$ (4.21)

The formal solution of Eq. (4.20) is therefore

$\displaystyle\Lambda_{t}^{{\rm n}{\rm
l}}\rho=V_{t}\Bigg{(}\rho+\sum_{n=1}^{\infty}\int^{t}_{0}dt_{1}\cdots\int^{t_{2n-1}}_{0}dt_{2n}\sum_{i_{1},\cdots
i_{n}}\bar{L}^{i_{1}}_{t_{1},t_{2}}\cdots\bar{L}^{i_{n}}_{t_{2n-1},t_{2n}}\
\rho\ \bar{L}^{i_{n}\ \dagger}_{t_{2n-1},t_{2n}}\cdots\bar{L}^{i_{1}\
\dagger}_{t_{1},t_{2}}\Bigg{)}V_{t}^{\dagger}.$ (4.22)

The CP of $\Lambda_{t}^{{\rm n}{\rm l}}$ is guaranteed by the positive
semidefiniteness of the measure ${\cal M}(\Lambda_{t}^{{\rm n}{\rm l}})$

$\displaystyle{\cal M}(\Lambda_{t}^{{\rm n}{\rm
l}})=|\langle\Psi|\Phi\rangle|^{2}+\sum_{n=1}^{\infty}\int^{t}_{0}dt_{1}\cdots\int^{t_{2n-1}}_{0}dt_{2n}\sum_{i_{1},\cdots
i_{n}}|\langle\Psi|V_{t}\bar{L}^{i_{1}}_{t_{1},t_{2}}\cdots\bar{L}^{i_{n}}_{t_{2n-1},t_{2n}}|\Phi\rangle|^{2}\geq
0.$ (4.23)

Therefore, in the weak-coupling regime, $\Lambda_{t}^{{\rm n}{\rm l}}$ of the
time-non-local GKSL-like ME is CP.

###  4.5 $\Lambda_{t}^{{\rm n}{\rm l}}$ is CP if $\Lambda_{t}^{{\rm n}{\rm
l},{\cal Z}}$ is CP and its Kraus representation has a specific property

Consider the first-order, linear integro-differential, GKSL-like master
equation

$\displaystyle\frac{d}{dt}\Lambda^{{\rm n}{\rm
l}}_{t}\rho=\int^{t}_{0}dt^{\prime}{\cal L}_{t,t^{\prime}}^{\rm
GKSL}(\Lambda^{{\rm n}{\rm l}}_{t^{\prime}}\rho)$
$\displaystyle=\int^{t}_{0}dt^{\prime}\Big{(}\sum_{i}L^{i}_{t,t^{\prime}}(\Lambda^{{\rm
n}{\rm l}}_{t^{\prime}}\rho)L^{i\
\dagger}_{t,t^{\prime}}-W_{t,t^{\prime}}(\Lambda_{t^{\prime}}^{{\rm n}{\rm
l}}\rho)-(\Lambda_{t^{\prime}}^{{\rm n}{\rm
l}}\rho)W_{t,t^{\prime}}^{\dagger}\Big{)},$ $\displaystyle\Lambda_{0}^{{\rm
n}{\rm l}}$ $\displaystyle={\mathbbm{1}}.$ (4.24)

Assume $\Lambda_{t}^{{\rm n}{\rm l},{\cal Z}}$ which satisfies Eq. (4.5) is
CP. Then, it has a Kraus representation $\Lambda_{t}^{{\rm n}{\rm l},{\cal
Z}}\rho=\sum_{j}K_{t}^{j}\rho K_{t}^{j\dagger}$. Further, assume that the
Kraus operators satisfy specific conditions

$\displaystyle\big{(}K_{t}^{j}\big{)}^{-1}K_{t}^{k}={\mathbbm{1}}\delta_{j,k},\
\
K_{t}^{j\dagger}\big{(}K_{t}^{k\dagger}\big{)}^{-1}={\mathbbm{1}}\delta_{j,k}.$
(4.25)

This assumption is nothing but a technical requirement.

Consider another map $\hat{\Lambda}_{t}^{{\rm n}{\rm l}}$ which is defined by
the similarity transformation of $\Lambda_{t}^{{\rm n}{\rm l}}$

$\displaystyle\Lambda_{t}^{{\rm n}{\rm
l}}\rho\equiv\sum_{j}K_{t}^{j}\Big{(}\hat{\Lambda}_{t}^{{\rm n}{\rm
l}}\rho\Big{)}K_{t}^{j\dagger}.$ (4.26)

Comparing the time derivative of Eq. (4.26) and Eq. (4.24), a similar form to
Eq. (4.1) is obtained for $\hat{\Lambda}_{t}^{{\rm n}{\rm l}}$

$\displaystyle\sum_{j}K_{t}^{j}\Big{(}\frac{d}{dt}\hat{\Lambda}_{t}^{{\rm
n}{\rm
l}}\rho\Big{)}K_{t}^{j\dagger}=\int^{t}_{0}dt^{\prime}\Big{(}\sum_{i}L^{i}_{t,t^{\prime}}\big{(}\sum_{k}K_{t^{\prime}}^{k}(\hat{\Lambda}^{{\rm
n}{\rm l}}_{t^{\prime}}\rho)K_{t^{\prime}}^{k\dagger}\big{)}L^{i\
\dagger}_{t,t^{\prime}}\Big{)}.$ (4.27)

Using the assumed properties of the Kraus operators and defining
$\bar{L}^{i,j,k}_{s,s^{\prime}}\equiv\big{(}K_{s}^{j}\big{)}^{-1}L^{i}_{s,s^{\prime}}K_{s^{\prime}}^{k}$,
Eq. (4.27) appears the same as Eq. (4.1)

$\displaystyle\frac{d}{dt}\hat{\Lambda}_{t}^{{\rm n}{\rm
l}}\rho=\int^{t}_{0}dt^{\prime}\Big{(}\sum_{i}\sum_{j,k}\bar{L}^{i,j,k}_{t,t^{\prime}}(\hat{\Lambda}^{{\rm
n}{\rm l}}_{t^{\prime}}\rho)\bar{L}^{i,j,k\dagger}_{t,t^{\prime}}\Big{)}.$
(4.28)

According to Subsec. 4.1, Eq. (4.24) also has the formal solution

$\displaystyle\Lambda_{t}^{{\rm n}{\rm
l}}\rho=\sum_{m}K^{m}_{t}\Bigg{(}\rho+\sum_{n=1}^{\infty}\int^{t}_{0}dt_{1}$
$\displaystyle\cdots\int^{t_{2n-1}}_{0}dt_{2n}\sum_{i_{1},\cdots
i_{n}}\sum_{j_{1},\cdots j_{n}}\sum_{k_{1},\cdots k_{n}}$
$\displaystyle\times\bar{L}^{i_{1},j_{1},k_{1}}_{t_{1},t_{2}}\cdots\bar{L}^{i_{n},j_{n},k_{n}}_{t_{2n-1},t_{2n}}\
\rho\bar{L}^{i_{n},j_{n},k_{n}\
\dagger}_{t_{2n-1},t_{2n}}\cdots\bar{L}^{i_{1},j_{1},k_{1}\dagger}_{t_{1},t_{2}}\Bigg{)}K_{t}^{m\dagger}.$
(4.29)

The CP of $\Lambda_{t}^{{\rm n}{\rm l}}$ is guaranteed by the positive
semidefiniteness of the measure ${\cal M}(\Lambda_{t}^{{\rm n}{\rm l}})$

$\displaystyle{\cal M}(\Lambda_{t}^{{\rm n}{\rm
l}})=\sum_{m}\Bigg{(}|\langle\Psi|K_{t}^{m}|\Phi\rangle|^{2}+\sum_{n=1}^{\infty}\int^{t}_{0}dt_{1}$
$\displaystyle\cdots\int^{t_{2n-1}}_{0}dt_{2n}\sum_{i_{1},\cdots
i_{n}}\sum_{j_{1},\cdots j_{n}}\sum_{k_{1},\cdots k_{n}}$
$\displaystyle\times|\langle\Psi|K_{t}^{m}\bar{L}^{i_{1},j_{1},k_{1}}_{t_{1},t_{2}}\cdots\bar{L}^{i_{n},j_{n},k_{n}}_{t_{2n-1},t_{2n}}|\Phi\rangle|^{2}\Bigg{)}\geq
0.$ (4.30)

Therefore, the QDM $\Lambda_{t}^{{\rm n}{\rm l}}$ for the time-non-local GKSL-
like ME is CP if its associated map $\Lambda_{t}^{{\rm n}{\rm l},{\cal Z}}$ is
CP and its Kraus operators satisfy the specific conditions.

###  4.6 $\Lambda^{\rm conv}_{t}$ is CP if $\Lambda_{t}^{{\rm conv},{\cal Z}}$
is CP

As a special case, consider the first-order, linear integro-differential GKSL-
like master equation with convolution

$\displaystyle\frac{d}{dt}\Lambda^{\rm
conv}_{t}\rho=\int^{t}_{0}dt^{\prime}{\cal L}_{t-t^{\prime}}^{\rm
GKSL}(\Lambda^{\rm conv}_{t^{\prime}}\rho),\ \ \ \Lambda_{0}^{\rm conv}$
$\displaystyle={\mathbbm{1}}.$ (4.31)

Assume that the associated map $\Lambda^{{\rm conv},{\cal Z}}_{t}$ is CP and
its Kraus representation has a specific property: $\Lambda_{t}^{{\rm
conv},{\cal Z}}\rho=\sum_{j}K_{t}^{j}\rho
K_{t}^{j\dagger},\big{(}K_{t}^{j}\big{)}^{-1}K_{t}^{k}={\mathbbm{1}}\delta_{j,k},\
K_{t}^{j\dagger}\big{(}K_{t}^{k\dagger}\big{)}^{-1}={\mathbbm{1}}\delta_{j,k}.$
Then, the formal solution of Eq. (4.31) can be iteratively obtained and shown
to be CP.

A simpler discussion is given in the previous studies [15, 16, 17]. In these
works, the CP of $\Lambda^{\rm conv}_{t}$ is shown when the associated maps
$\Lambda^{{\rm conv},{\cal B}}_{t}$ and $\Lambda^{{\rm conv},{\cal Z}}_{t}$
are both CP. Since the proof relies on the Laplace transform, the explicit
form of $\Lambda^{{\rm conv},{\cal Z}}_{t}$ is irrelevant, and therefore, an
additional condition is not necessary.

##  5 Conclusion

This paper has examined the conditions for QDMs to be CP when their MEs take
the GKSL-like form. In the time-local case, a QDM is CP with arbitrary time
dependence in the GKSL-like kernel of the integro-differential ME, which
signifies that the GKSL-like form is both the necessary and sufficient
condition for a QDM to be CP. In the time-non-local case, an additional
requirement is necessary for QDMs to be CP. The condition has been proposed in
the form of a specific Kraus representation of a partial map or the weak
coupling regime. The difference between the two cases lies in the locality,
which arises from the Born approximation in the derivation in the weak
coupling regime.

## Acknowledgement

T.S. was supported by Japan’s MEXT Quantum Leap Flagship Program Grant No.
JPMXS0120319794.

##  Appendix A Approximation in the derivation of the Redfield equation

This section is aimed at showing that the time-local Redfield equation
describes the same dynamics as its time-non-local counterpart in the weak-
coupling regime. In the standard derivation, the microscopic Hamiltonian is
given by $H=H_{\rm S}+H_{\rm B}+gSB$, where $H_{\rm S},\ H_{\rm B},\ g,\ S,$
and $B$ denote the system free Hamiltonian, the bath free Hamiltonian, the
coupling strength, the system part of the interaction, and the bath part of
the interaction, respectively. In the weak-coupling regime, the coupling
strength $g$ is small and only terms up to the second order have been kept.
The initial state is assumed to be decoupled from the bath, and the system
state is taken as an arbitrary state $\rho(0)$, while the bath state is taken
as a thermal equilibrium $\rho_{\rm B}$.

In the standard derivation of the Redfield equation, an integro-differential
equation in the interaction picture is derived

$\displaystyle\frac{d}{dt}\rho^{\rm I}(t)=g^{2}\int^{t}_{0}d\tau\Big{(}-S^{\rm
I}(t)C(\tau)S^{\rm I}(-\tau)\rho^{\rm I}(t-\tau)+S^{\rm I}(t)\rho^{\rm
I}(t-\tau)C^{*}(\tau)S^{\rm I}(-\tau)\Big{)}+{\rm h.c.},$ (A.1)

where $C(\tau)={\rm Tr_{B}}\big{[}B^{\rm I}(t)B\rho_{\rm B}\big{]}$ denotes
the correlation function. The concrete expression of the first term of Eq.
(A.1) shows the $g$ dependence

$\displaystyle-g^{2}\int^{t}_{0}d\tau S^{\rm I}(t)C(\tau)S^{\rm
I}(-\tau)\rho^{\rm I}(t-\tau)$ (A.2) $\displaystyle=-g^{2}\int^{t}_{0}d\tau
S^{\rm I}(t)C(\tau)S^{\rm I}(-\tau)e^{iH_{\rm S}(t-\tau)}{\rm
Tr_{B}}\big{[}e^{iH(t-\tau)}(\rho(0)\otimes\rho_{\rm
B})e^{-iH(t-\tau)}\big{]}e^{-iH_{\rm S}(t-\tau)}.$ (A.3)

Since only the terms up to the second order of $g$ are kept, any $g$-dependent
parts in the partial trace can be neglected

$\displaystyle{\rm Tr_{B}}\big{[}e^{-iH(t-\tau)}(\rho(0)\otimes\rho_{\rm
B})e^{iH(t-\tau)}\big{]}={\rm Tr_{B}}\big{[}e^{iH\tau}\rho_{\rm
tot}(t)e^{-iH\tau}\big{]}$ $\displaystyle={\rm Tr_{B}}\Big{[}\big{(}1+i(H_{\rm
S}+H_{\rm B}+gSB)\tau-\cdots\big{)}\rho_{\rm tot}(t)\big{(}1-i(H_{\rm
S}+H_{\rm B}+gSB)\tau-\cdots\big{)}\Big{]}$ $\displaystyle\approx{\rm
Tr_{B}}\Big{[}e^{iH_{\rm S}\tau}\rho_{\rm tot}(t)e^{-iH_{\rm
S}\tau}\Big{]}=e^{iH_{\rm S}\tau}\rho(t)e^{iH_{\rm S}\tau}.$ (A.4)

By this approximation, Eq. (A.1) can be made local in time

$\displaystyle\frac{d}{dt}\rho^{\rm I}(t)=g^{2}\int^{t}_{0}d\tau\Big{(}-S^{\rm
I}(t)C(\tau)S^{\rm I}(-\tau)\rho^{\rm I}(t)+S^{\rm I}(t)\rho^{\rm
I}(t)C^{*}(\tau)S^{\rm I}(-\tau)\Big{)}+{\rm h.c.}.$ (A.5)

##  Appendix B Simple example of proof

The basic idea to prove that $\Lambda_{t}^{{\rm l},{\cal B}}$ and
$\Lambda_{t}^{{\rm nl},{\cal B}}$ are CP can be grasped by considering a
simple example of $\Lambda_{t}^{{\rm l,ex},{\cal B}}$. The first-order, linear
differential equation

$\displaystyle\frac{d}{dt}\Lambda_{t}^{{\rm l,ex},{\cal
B}}\rho=L\Big{(}\Lambda_{t}^{{\rm l,ex},{\cal B}}\rho\Big{)}L^{\dagger}$ (B.1)

has the formal solution

$\displaystyle\Lambda_{t}^{{\rm l,ex},{\cal
B}}\rho=\rho+\sum_{n=1}^{\infty}\frac{t^{n}}{n!}L^{n}(\rho)\big{(}L^{\dagger}\big{)}^{n}.$
(B.2)

Taking the time derivative of Eq. (B.2) shows that it satisfies Eq. (B.1)

$\displaystyle\frac{d}{dt}\Lambda_{t}^{{\rm l,ex},{\cal
B}}\rho=\sum_{n=1}^{\infty}\frac{t^{(n-1)}}{(n-1)!}L^{n}(\rho)\big{(}L^{\dagger}\big{)}^{n}=L\Big{(}\rho+\sum_{n=1}^{\infty}\frac{t^{n}}{n!}L^{n}(\rho)\big{(}L^{\dagger}\big{)}^{n}\Big{)}L^{\dagger}.$
(B.3)

The CP of $\Lambda_{t}^{{\rm l,ex},{\cal B}}$ is guaranteed by the positive
semidefiniteness of ${\cal M}(\Lambda_{t}^{{\rm l,ex},{\cal B}})$

$\displaystyle{\cal M}(\Lambda_{t}^{{\rm l,ex},{\cal
B}})=|\langle\Psi|\Phi\rangle|^{2}+\sum_{n=1}^{\infty}\frac{t^{n}}{n!}|\langle\Psi|L^{n}|\Phi\rangle|^{2}\geq
0.$ (B.4)

##  Appendix C  $V_{t}$ and $V_{t}^{-1}$

In this section, the inverse operators $V_{t}^{-1},$ and
$(V_{t}^{\dagger})^{-1}$ are shown to exist for $V_{t},$ and $V_{t}^{\dagger}$
defined in Subsec. 3.2. Here, we introduce operators $V_{t,t_{0}}$ and
$V_{t,t_{0}}^{\dagger}$ and their alternative expressions, which become
$V_{t},$ and $V_{t}^{\dagger}$ at $t_{0}=0$

$\displaystyle V_{t,t_{0}}$ $\displaystyle\equiv\overleftarrow{\cal
T}e^{-\int^{t}_{t_{0}}ds\int^{s}_{t_{0}}ds^{\prime}W_{s,s^{\prime}}}=\overleftarrow{\cal
T}e^{-\int^{t}_{t_{0}}ds^{\prime}\int^{t}_{s^{\prime}}dsW_{s,s^{\prime}}},$
(C.1) $\displaystyle V_{t,t_{0}}^{\dagger}$
$\displaystyle\equiv\overrightarrow{\cal
T}e^{-\int^{t}_{t_{0}}ds\int^{s}_{t_{0}}ds^{\prime}W_{s,s^{\prime}}^{\dagger}}=\overrightarrow{\cal
T}e^{-\int^{t}_{t_{0}}ds^{\prime}\int^{t}_{s^{\prime}}dsW_{s,s^{\prime}}^{\dagger}}.$
(C.2)

Taking the time derivative of Eq. (C.1) and Eq. (C.2) with respect to time $t$
or $t_{0}$ explicitly shows the differences between the equivalent expressions

$\displaystyle\frac{d}{dt}\overleftarrow{\cal
T}e^{-\int^{t}_{t_{0}}ds\int^{s}_{t_{0}}ds^{\prime}W_{s,s^{\prime}}}$
$\displaystyle=\int^{t}_{t_{0}}ds_{2}\big{(}-W_{t,s_{2}}\big{)}\overleftarrow{\cal
T}e^{-\int^{t}_{t_{0}}ds\int^{s^{\prime}}_{t_{0}}ds^{\prime}W_{s,s^{\prime}}},$
(C.3) $\displaystyle\frac{d}{dt_{0}}\overleftarrow{\cal
T}e^{-\int^{t}_{t_{0}}ds^{\prime}\int^{t}_{s^{\prime}}dsW_{s,s^{\prime}}}$
$\displaystyle=\int^{t}_{t_{0}}ds_{2n-1}\overleftarrow{\cal
T}e^{-\int^{t}_{t_{0}}ds^{\prime}\int^{t}_{s^{\prime}}dsW_{s,s^{\prime}}}\big{(}-W_{s_{2n-1,t_{0}}}\big{)},$
(C.4) $\displaystyle\frac{d}{dt}\overrightarrow{\cal
T}e^{-\int^{t}_{t_{0}}ds\int^{s}_{t_{0}}ds^{\prime}W_{s,s^{\prime}}^{\dagger}}$
$\displaystyle=\int^{t}_{t_{0}}ds_{2}\overrightarrow{\cal
T}e^{-\int^{t}_{t_{0}}ds\int^{s^{\prime}}_{t_{0}}ds^{\prime}W_{s,s^{\prime}}^{\dagger}}\big{(}-W_{t,s_{2}}^{\dagger}\big{)},$
(C.5) $\displaystyle\frac{d}{dt_{0}}\overrightarrow{\cal
T}e^{-\int^{t}_{t_{0}}ds^{\prime}\int^{t}_{s^{\prime}}dsW_{s,s^{\prime}}^{\dagger}}$
$\displaystyle=\int^{t}_{t_{0}}ds_{2n-1}\big{(}-W_{s_{2n-1},t_{0}}^{\dagger}\big{)}\overrightarrow{\cal
T}e^{-\int^{t}_{t_{0}}ds^{\prime}\int^{t}_{s^{\prime}}dsW_{s,s^{\prime}}^{\dagger}}.$
(C.6)

Then, consider the following operators $\overrightarrow{\cal
T}e^{+\int^{t}_{t_{0}}ds\int^{s}_{t_{0}}ds^{\prime}W_{s,s^{\prime}}}=\overrightarrow{\cal
T}e^{+\int^{t}_{t_{0}}ds^{\prime}\int^{t}_{s^{\prime}}dsW_{s,s^{\prime}}}$
such that

$\displaystyle\frac{d}{dt}\overrightarrow{\cal
T}e^{\int^{t}_{t_{0}}ds\int^{s}_{t_{0}}ds^{\prime}W_{s,s^{\prime}}}=\int^{t}_{t_{0}}ds_{2}\overrightarrow{\cal
T}e^{\int^{t}_{t_{0}}ds\int^{s}_{t_{0}}ds^{\prime}W_{s,s^{\prime}}}W_{t,s_{2}},$
(C.7) $\displaystyle\frac{d}{dt_{0}}\overrightarrow{\cal
T}e^{\int^{t}_{t_{0}}ds^{\prime}\int^{t}_{s^{\prime}}dsW_{s,s^{\prime}}}=\int^{t}_{t_{0}}ds_{2n-1}W_{s_{2n-1,t_{0}}}\overrightarrow{\cal
T}e^{\int^{t}_{t_{0}}ds^{\prime}\int^{t}_{s^{\prime}}dsW_{s,s^{\prime}}},$
(C.8)

and $\overleftarrow{\cal
T}e^{+\int^{t}_{t_{0}}ds\int^{s}_{t_{0}}ds^{\prime}W_{s,s^{\prime}}^{\dagger}}=\overleftarrow{\cal
T}e^{+\int^{t}_{t_{0}}ds^{\prime}\int^{t}_{s^{\prime}}dsW_{s,s^{\prime}}^{\dagger}}$
such that

$\displaystyle\frac{d}{dt}\overleftarrow{\cal
T}e^{\int^{t}_{t_{0}}ds\int^{s}_{t_{0}}ds^{\prime}W_{s,s^{\prime}}^{\dagger}}=\int^{t}_{t_{0}}ds_{2}W_{t,s_{2}}^{\dagger}\overleftarrow{\cal
T}e^{\int^{t}_{t_{0}}ds\int^{s^{\prime}}_{t_{0}}ds^{\prime}W_{s,s^{\prime}}^{\dagger}},$
(C.9) $\displaystyle\frac{d}{dt_{0}}\overleftarrow{\cal
T}e^{\int^{t}_{t_{0}}ds^{\prime}\int^{t}_{s^{\prime}}dsW_{s,s^{\prime}}^{\dagger}}=\int^{t}_{t_{0}}ds_{2n-1}\overleftarrow{\cal
T}e^{\int^{t}_{t_{0}}ds^{\prime}\int^{t}_{s^{\prime}}dsW_{s,s^{\prime}}^{\dagger}}W_{s_{2n-1},t_{0}}^{\dagger}.$
(C.10)

Actually, they are the inverses of $V_{t,t_{0}}$ and
$(V_{t,t_{0}})^{\dagger}$. This can be shown by taking the time derivatives of
their products such as

$\displaystyle\frac{d}{dt}\Bigg{(}\overrightarrow{\cal
T}e^{+\int^{t}_{t_{0}}ds\int^{s}_{t_{0}}ds^{\prime}W_{s,s^{\prime}}}\overleftarrow{\cal
T}e^{-\int^{t}_{t_{0}}ds\int^{s}_{t_{0}}ds^{\prime}W_{s,s^{\prime}}}\Bigg{)}=0.$
(C.11) $\displaystyle\frac{d}{dt_{0}}\Bigg{(}\overleftarrow{\cal
T}e^{-\int^{t}_{t_{0}}ds^{\prime}\int^{t}_{s^{\prime}}dsW_{s,s^{\prime}}}\overrightarrow{\cal
T}e^{+\int^{t}_{t_{0}}ds^{\prime}\int^{t}_{s^{\prime}}dsW_{s,s^{\prime}}}\Bigg{)}=0.$
(C.12)

Therefore, the quantity in the parentheses is independent of $t$ and $t_{0}$,
which is nothing but unity. Taking $t_{0}=0,$ $V_{t}^{-1}$ is shown to be well
defined as the inverse of $V_{t}$. The same discussion holds for
$V_{t}^{\dagger}$.

## References

  * [1] Robert Alicki and Karl Lendi  “Quantum dynamical semigroups and applications”  Springer, 2007  URL: https://link.springer.com/book/10.1007/3-540-70861-8
  * [2] Heinz-Peter Breuer and Francesco Petruccione  “The Theory of Open Quantum Systems”  Oxford University Press, 2002  URL: https://academic.oup.com/book/27757
  * [3] Angel Rivas and Susana F Huelga  “Open quantum systems: An Introduction”  Springer, 2012  URL: https://link.springer.com/book/10.1007/978-3-642-23354-8
  * [4] Jyrki Piilo Heinz-Peter Breuer and Bassano Vacchini  “Colloquium: Non-Markovian dynamics in open quantum systems”  In _Reviews of Modern Physics_ 88.2 APS, 2016, pp. 021002
  * [5] K. Kraus  “General state changes in quantum theory”  In _Annals of Physics_ 64.2, 1971, pp. 311–335  DOI: https://doi.org/10.1016/0003-4916(71)90108-4
  * [6] Man-Duen Choi  “Completely positive linear maps on complex matrices”  In _Linear Algebra and its Applications_ 10.3, 1975, pp. 285–290  DOI: https://doi.org/10.1016/0024-3795(75)90075-0
  * [7] Andrzej Kossakowski Vittorio Gorini and Ennackal Chandy George Sudarshan  “Completely positive dynamical semigroups of N-level systems”  In _Journal of Mathematical Physics_ 17.5 American Institute of Physics, 1976, pp. 821–825
  * [8] Goran Lindblad  “On the generators of quantum dynamical semigroups”  In _Communications in Mathematical Physics_ 48.2 Springer, 1976, pp. 119–130
  * [9] Bassano Vacchini  “Completely Positive Quantum Dissipation”  In _Phys. Rev. Lett._ 84 American Physical Society, 2000, pp. 1374–1377  DOI: 10.1103/PhysRevLett.84.1374
  * [10] F. Benatti, R. Floreanini and U. Marzolino  “Entangling two unequal atoms through a common bath”  In _Phys. Rev. A_ 81 American Physical Society, 2010, pp. 012105  DOI: 10.1103/PhysRevA.81.012105
  * [11] Donato Farina and Vittorio Giovannetti  “Open-quantum-system dynamics: Recovering positivity of the Redfield equation via the partial secular approximation”  In _Phys. Rev. A_ 100 American Physical Society, 2019, pp. 012107  DOI: 10.1103/PhysRevA.100.012107
  * [12] Ziwen Huang et al.  “Completely Positive Map for Noisy Driven Quantum Systems Derived by Keldysh Expansion”  In _Quantum_ 7 Verein zur Förderung des Open Access Publizierens in den Quantenwissenschaften, 2023, pp. 1158  DOI: 10.22331/q-2023-11-03-1158
  * [13] Crispin Gardiner and Peter Zoller  “Quantum noise: a handbook of Markovian and non-Markovian quantum stochastic methods with applications to quantum optics”  Springer Science & Business Media, 2004  URL: https://link.springer.com/book/9783540223016
  * [14] Heinz-Peter Breuer  “Genuine quantum trajectories for non-Markovian processes”  In _Phys. Rev. A_ 70 American Physical Society, 2004, pp. 012106  DOI: 10.1103/PhysRevA.70.012106
  * [15] Andrzej Kossakowski and Rolando Rebolledo  “On completely positive non-Markovian evolution of a d-level system”  In _Quantum Bio-Informatics II_ WORLD SCIENTIFIC, 2009  DOI: 10.1142/9789814273756_0011
  * [16] Andrzej Kossakowski and Rolando Rebolledo  “On the structure of generators for non-Markovian master equations”  In _arXiv preprint arXiv:0902.2294_ , 2009  URL: https://arxiv.org/abs/0902.2294
  * [17] Dariusz Chruściński and Sabrina Maniscalco  “Degree of Non-Markovianity of Quantum Evolution”  In _Phys. Rev. Lett._ 112 American Physical Society, 2014, pp. 120404  DOI: 10.1103/PhysRevLett.112.120404
  * [18] Dariusz Chruściński  “Time inhomogeneous quantum dynamical maps”  In _Scientific Reports_ 12.1 Nature Publishing Group, 2022, pp. 1–10
  * [19] Michael J.W. Hall Li Li and Howard M. Wiseman  “Concepts of quantum non-Markovianity: A hierarchy” Concepts of quantum non-Markovianity: A hierarchy  In _Physics Reports_ 759, 2018, pp. 1–51  DOI: https://doi.org/10.1016/j.physrep.2018.07.001
  * [20] Heinz-Peter Breuer  “Genuine quantum trajectories for non-Markovian processes”  In _Phys. Rev. A_ 70 American Physical Society, 2004, pp. 012106  DOI: 10.1103/PhysRevA.70.012106
  * [21] S. Maniscalco J. Piilo and K.-A. Suominen  “Open system dynamics with non-Markovian quantum jumps”  In _Phys. Rev. A_ 79 American Physical Society, 2009, pp. 062112  DOI: 10.1103/PhysRevA.79.062112
  * [22] H.-P. Breuer S. Maniscalco L. Mazzola and J. Piilo  “Phenomenological memory-kernel master equations and time-dependent Markovian processes”  In _Physical Review A_ 81.6 APS, 2010, pp. 062120  URL: https://journals.aps.org/pra/abstract/10.1103/PhysRevA.81.062120
  * [23] Chan U Lei and Wei-Min Zhang  “A quantum photonic dissipative transport theory”  In _Annals of Physics_ 327.5, 2012, pp. 1408–1433  DOI: https://doi.org/10.1016/j.aop.2012.02.005
  * [24] Bassano Vacchini  “Non-Markovian master equations from piecewise dynamics”  In _Physical Review A_ 87.3 APS, 2013, pp. 030101
  * [25] Li Li Michael J. W. Hall and Erika Andersson  “Canonical form of master equations and characterization of non-Markovianity”  In _Physical Review A_ 89.4 APS, 2014, pp. 042120  DOI: 10.1103/PhysRevA.89.042120
  * [26] Inés Vega and Daniel Alonso  “Dynamics of non-Markovian open quantum systems”  In _Rev. Mod. Phys._ 89 American Physical Society, 2017, pp. 015001  DOI: 10.1103/RevModPhys.89.015001
  * [27] Heinz-Peter Breuer  “Foundations and measures of quantum non-Markovianity”  In _Journal of Physics B: Atomic, Molecular and Optical Physics_ 45.15 IOP Publishing, 2012, pp. 154001  DOI: 10.1088/0953-4075/45/15/154001
  * [28] Michael J. W. Hall, James D. Cresser, Li Li and Erika Andersson  “Canonical form of master equations and characterization of non-Markovianity”  In _Phys. Rev. A_ 89 American Physical Society, 2014, pp. 042120  DOI: 10.1103/PhysRevA.89.042120
  * [29] Ling-Na Wu Tobias Becker and André Eckardt  “Lindbladian approximation beyond ultraweak coupling”  In _Phys. Rev. E_ 104 American Physical Society, 2021, pp. 014110  DOI: 10.1103/PhysRevE.104.014110
  * [30] Frederik Nathan and Mark S. Rudner  “Universal Lindblad equation for open quantum systems”  In _Phys. Rev. B_ 102 American Physical Society, 2020, pp. 115109  DOI: 10.1103/PhysRevB.102.115109
  * [31] Koki Chinzei Tatsuhiko N. Ikeda and Masahiro Sato  “Nonequilibrium steady states in the Floquet-Lindblad systems: van Vleck’s high-frequency expansion approach”  In _SciPost Phys. Core_ 4 SciPost, 2021, pp. 033  DOI: 10.21468/SciPostPhysCore.4.4.033
  * [32] Evgeny Mozgunov and Daniel Lidar  “Completely positive master equation for arbitrary driving and small level spacing”  In _Quantum_ 4 Verein zur Förderung des Open Access Publizierens in den Quantenwissenschaften, 2020, pp. 227  DOI: 10.22331/q-2020-02-06-227
  * [33] A. P. Babu K. Mølmer M. Möttönen S. Alipour and T. Ala-Nissila  “Correlation-Picture Approach to Open-Quantum-System Dynamics”  In _Phys. Rev. X_ 10 American Physical Society, 2020, pp. 041024  DOI: 10.1103/PhysRevX.10.041024
  * [34] Anton Trushechkin  “Unified Gorini-Kossakowski-Lindblad-Sudarshan quantum master equation beyond the secular approximation”  In _Physical Review A_ 103.6 APS, 2021, pp. 062226  DOI: 10.1103/PhysRevA.103.062226
  * [35] Dariusz Chruściński, Ryohei Fujii, Gen Kimura and Hiromichi Ohno  “Constraints for the spectra of generators of quantum dynamical semigroups”  In _Linear Algebra and its Applications_ 630 Elsevier, 2021, pp. 293–305  URL: https://www.sciencedirect.com/science/article/pii/S0024379521003098
  * [36] Dariusz Chruściński, Gen Kimura, Andrzej Kossakowski and Yasuhito Shishido  “Universal constraint for relaxation rates for quantum dynamical semigroup”  In _Physical Review Letters_ 127.5 APS, 2021, pp. 050401  URL: https://link.aps.org/doi/10.1103/PhysRevLett.127.050401
  * [37] Robert Alicki  “The quantum open system as a model of the heat engine”  In _Journal of Physics A: Mathematical and General_ 12.5, 1979, pp. L103  DOI: 10.1088/0305-4470/12/5/007
  * [38] Gernot Schaller and Tobias Brandes  “Preservation of positivity by dynamical coarse graining”  In _Phys. Rev. A_ 78 American Physical Society, 2008, pp. 022106  DOI: 10.1103/PhysRevA.78.022106
  * [39] Dragomir Davidović  “Completely Positive, Simple, and Possibly Highly Accurate Approximation of the Redfield Equation”  In _Quantum_ 4 Verein zur Förderung des Open Access Publizierens in den Quantenwissenschaften, 2020, pp. 326  DOI: 10.22331/q-2020-09-21-326
  * [40] Iman Sargolzahi  “Linearity conditions leading to complete positivity”  In _Phys. Rev. A_ 106 American Physical Society, 2022, pp. 032205  DOI: 10.1103/PhysRevA.106.032205
  * [41] G. Théret and D. Sugny  “Complete positivity, positivity, and long-time asymptotic behavior in a two-level open quantum system”  In _Phys. Rev. A_ 108 American Physical Society, 2023, pp. 032212  DOI: 10.1103/PhysRevA.108.032212
  * [42] Michael M Wolf and J Ignacio Cirac  “Dividing quantum channels”  In _Communications in Mathematical Physics_ 279 Springer, 2008, pp. 147–168  URL: https://link.springer.com/article/10.1007/s00220-008-0411-y
  * [43] Christian Majenz, Tameem Albash, Heinz-Peter Breuer and Daniel A. Lidar  “Coarse graining can beat the rotating-wave approximation in quantum Markovian master equations”  In _Phys. Rev. A_ 88 American Physical Society, 2013, pp. 012103  DOI: 10.1103/PhysRevA.88.012103
  * [44] Dariusz Chruściński and Paweł Należyty  “Non-Markovian Quantum Evolution: Time-Local Generators and Memory Kernels”  In _Reports on Mathematical Physics_ 77.3, 2016, pp. 399–414  DOI: https://doi.org/10.1016/S0034-4877(16)30036-2
  * [45] Gustavo Montes Cabrera, David Davalos and Thomas Gorin  “Positivity and complete positivity of differentiable quantum processes”  In _Physics Letters A_ 383.23, 2019, pp. 2719–2728  DOI: https://doi.org/10.1016/j.physleta.2019.05.049
  * [46] Antonio D’Abbruzzo, Vasco Cavina and Vittorio Giovannetti  “A time-dependent regularization of the Redfield equation”  In _SciPost Phys._ 15 SciPost, 2023, pp. 117  DOI: 10.21468/SciPostPhys.15.3.117
  * [47] Joshua Wilkie  “Positivity preserving non-Markovian master equations”  In _Phys. Rev. E_ 62 American Physical Society, 2000, pp. 8808–8810  DOI: 10.1103/PhysRevE.62.8808
  * [48] Gernot Schaller and Tobias Brandes  “Preservation of positivity by dynamical coarse graining”  In _Phys. Rev. A_ 78 American Physical Society, 2008, pp. 022106  DOI: 10.1103/PhysRevA.78.022106
  * [49] Vasily E. Tarasov  “General Non-Markovian Quantum Dynamics”  In _Entropy_ 23.8, 2021  DOI: 10.3390/e23081006
  * [50] Vasily E. Tarasov  “Non-Markovian dynamics of open quantum system with memory”  In _Annals of Physics_ 434, 2021, pp. 168667  DOI: https://doi.org/10.1016/j.aop.2021.168667
  * [51] A. G. Redfield  “On the Theory of Relaxation Processes”  In _IBM Journal of Research and Development_ 1.1, 1957, pp. 19–31  DOI: 10.1147/rd.11.0019


2022 2nd Workshop on Astronomy Beyond the Common Senses

# The universe in words: Astronomy for all through audio description within
the outreach project Astroaccesible

E. Pérez-Montero  11affiliation: Instituto de Astrofísica de Andalucía - CSIC,
Glorieta de la Astronomía s/n, 18008, Granada, Spain<EMAIL_ADDRESS>garcia@iaa.es). C. Barnés-Castaño  22affiliation: Department of Translation
and Interpreting, University of Granada, Granada, Spain (cbc@ugr.es). and E.J.
García Gómez-Caro11affiliationmark:

###### Abstract

Astroaccesible is an outreach project hosted by the Instituto de Astrofísica
de Andalucía (CSIC) and leaded by a blind astronomer. It is aimed at the
teaching and popularisation of astronomy and astrophysics among all people
regardless of their abilities using a wide range of inclusive resources. Audio
description is one of the most extended inclusive resources for making films
and artworks accessible for blind and partially sighted people. However, its
potential for the description of astronomical images has not been widely
exploited. In this contribution, we introduce The Universe in words, a series
of videos describing images of some of the most popular objects in the Messier
catalogue. These audio descriptions do not only have a prominent inclusive
aspect, but also imply a better and deeper understanding of the represented
objects for everybody. The resulting videos can also be used as a complement
to other resources for in-person activities, such as sonifications or tactile
models of the same or similar astronomical objects.

Astroaccesible es un proyecto de divulgación impulsado desde el Instituto de
Astrofísica de Andalucía (CSIC) y dirigido por un astrónomo ciego, cuyo
objetivo es el de enseñar y divulgar la astronomía y la astrofísica entre
todas las personas con independencia de sus capacidades mediante el uso de una
amplia variedad de recursos inclusivos. La audiodescripción se ha convertido
en uno de los recursos inclusivos más extendidos, sobre todo con el fin de
hacer accesibles películas y obras de arte para personas ciegas y con baja
visión. Sin embargo, su potencial para describir imágenes astronómicas aún no
ha sido desarrollado. En esta contribución presentamos El Universo en
palabras, una serie de vídeos en los que se describen imágenes de algunos de
los objetos más conocidos del catálogo de Messier. Estas audiodescripciones no
solo tienen un claro aspecto inclusivo, sino que también conllevan una mejor y
más profunda comprensión de los objetos representados para todo el público.
Los vídeos resultantes también pueden ser usados como complemento a otros
recursos, en toda clase de actividades presenciales, como sonificaciones o
maquetas de los mismos objetos astronómicos o similares.

Astronomy science education teaching material teaching practice

##  0.1 The Astroaccesible project

The primary objective of the popularization and teaching of science for
everyone can only be satisfactorily accomplished with the inclusion of all
people regardless of their abilities, overcoming those barriers that prevent
them totally accessing all the divulged contents. In the case of astronomy,
blind and partially sighted people are excluded from having whole and
satisfactory access to all its contents, given that visual media (e.g.
pictures, videos, animations and graphics) play a leading role in the
dissemination of astronomical knowledge and research results.

The project Astroaccesible 111http://astroaccesible.csic.iaa.es (Pérez-Montero
et al., 2017) was born with the aim of bringing astronomy closer to blind and
partially sighted people by means of conferences, in-person classes, and the
creation of adapted contents and resources to make the outreach of astronomy
more accessible. The project has also a social dimension, as the principal
investigator, Enrique Pérez-Montero, is blind and is member of the Spanish
National Organization for Blind People (ONCE). This helps to give visibility
to scientists with disabilities conveying an idea of social inclusion in STEM
areas (Pérez-Montero, 2019).

The main objective of in-person activities is to spark the interest for the
study of astronomy for all challenging the idea that it can only be studied
using the sense of sight (Galvez et al., 2017). This goal is achieved with
different adapted resources and materials aiming at a complete description of
several astronomical objects, their sizes, and relative distances using
examples, helping blind and partially sighted people to acquire spatial
notions using touch, complementing the verbal explanations and also providing
high-contrast images for those people with useful residual vision. This
inclusive experience is enriched with more complex models, including
hemispheres of the night sky in relief, or models of the Moon and other rocky
planets of the Solar system (Ortiz-Gil et al., 2019; Ortiz Gil, 2020).

Despite these wide variety of options, the main resource for an accessible
popularization of astronomy for blind and partially sighted people is a
complete description of all elements, which can be either voiced or provided
in paper format. This is the reason why we have focused on detailed
descriptions of the concepts and contents presented in the conferences and in
the articles available in the project webpage. This strategy is the most
adequate from the point of view of blind and partially sighted people and it
is the one that needs less additional resources.

In addition, the situation caused by the COVID-19 pandemic in the last two
years has forced many outreach events to move online, avoiding any direct
interaction with touchable resources, and fostering the use of alternative
strategies, such as oral descriptions, which can also be supported by other
resources, such as sonifications (see also the contribution by García-Benito
et al. in this volume with the project Cosmonic). In this contribution, we
describe the project based on audio descriptions of astronomical images,
called El Universo en palabras (The Universe in words), within the framework
of the different resources designed for Astroaccesible and that can be used
both for in-person and virtual activities, with great level of acceptance from
both sighted and blind people.

##  0.2 Audio description as a resource

Audio description is an access service in which images, be they dynamic (as in
films) or static (as in paintings) are rendered into words. AD is regarded as
an intersemiotic translation practice (Chica-Núñez & Jiménez-Hurtado, 2020),
as it involves the transfer of information that is conveyed by visual signs to
linguistic ones. According to the Spanish AD standard, which includes
guidelines on how to audio describe approved by the official Spanish
standardization body, the main purpose of AD should be that of compensating
for the lack of access to visual information in order that people with sight
loss can comprehend the audio described source text (e. g. a film or a
painting) in the most similar way as sighted people do. Although AD is mainly
targeted at both blind and partially sighted people, these guidelines
underscore that it has to be scripted with the information needs of
congenitally blind people in mind.

AD practice and scholarship has so far focused on AD for films and audiovisual
fiction, however it is gaining momentum in other areas, such as museums and
live events (Barnés-Castaño et al., 2021). Its application in science is still
limited and it has only been explored from the production perspective, but its
potential use in an astronomical context for both research and educational
purposes is huge.

AD is also evolving in terms of the purposes and the audiences it might serve.
For instance, it has been highlighted that AD can be successfully used in the
foreign language classroom and help people with emotion recognition
difficulties. In the context of museum AD, Eardley et al. (2017) underline
that “the standard visuocentric presentation of works of art assumes not only
that people can see but also that they know how to use that vision to ‘look’
at a piece of art or a historical artifact to be able to extract some part of
the rich cultural heritage associated with it” (17, 203). However, research
into how art expertise influences looking patterns suggests that they are
driven by previous knowledge, with experts focusing more on formal features
rather than on easily distinguishable elements (Koide et al., 2015). On the
basis of these results, Eardley et al. (2017) propose that “guided looking may
enhance the non-expert visitor’s experience”, thus directing visitors
attention to those features that can be overlooked by the non-expert gaze.
Drawing on studies from cognitive science, these authors also stress that this
guided looking experience provided by the AD could also benefit sighted people
as they would be exposed to a multisensory experience (at least visual and
aural) which would help them not only to make sense of what they are seeing,
but also to improved information processing and memorability. In a recent
study, this claim has been empirically supported, as museum AD enhances the
memorability of the museum experience also for sighted visitors as compared
with traditional audio guides (Hutchinson & Eardley, 2021). These encouraging
results make it the more worthwhile to explore the potential of AD for being a
truly inclusive resource benefiting blind and sighted people alike in all
contexts where it can be applied, especially in those in which the lack of
expert knowledge can negatively affect meaning-making processes, such as in
astronomy.

##  0.3 The Universe in words

El Universo en palabras is a series of audio described videos released in
YouTube made in collaboration with the TRACCE research group of the University
of Granada 222tracce.ugr.es.

These audio descriptions were scripted by final-year Translation and
Interpreting students as a group project for the subject Multimedia
Translation during the courses 2019/2020 and 2020/2021. The main aim of this
project is for students to put in practice all the competences they have
acquired over the course of the subject and, at the same time, to apply this
knowledge to a project which helps them delve into innovative translation
modalities. Given the specialized nature of the source texts, images of
astronomy objects, their project was conducted under the supervision of not
only the AD trainer, but also of astronomers with an expertise in science
dissemination strategies to ensure that the astronomical knowledge and
terminology were accurately conveyed for the intended audience. On the one
hand, the scripts include all the relevant visual details of the audio
described object, helping thus blind and partially sighted people to create a
mental image of the object and serving as a guided looking experience for
sighted people, as the depicted object and its elements are zoomed in and out
in the video edition as a function of the AD script. On the other hand, all
essential information to ensure understanding of astronomy concepts by non-
experts is provided, which makes these guides a science dissemination resource
both for blind and sighted people.

The program has focused so far on objects of the Messier catalogue, but it
will very soon also cover other objects with or without tactile aids along
with astronomical facilities with a view to expanding the scope of this
project by including in-person activities.

All the published videos are in Spanish, but it is foreseen to adapt them to
other languages in the near future. An example in English using a synthetic
voice is available in the presentation at the webpage of the workshop.

The list of astronomical objects published so far are:

  * •

Messier 1. The Crab nebula: a supernova remnant. This resource could be
enriched with several available sonifications of the radio emission from the
central pulsar and the emission of the surrounding gas.

  * •

Messier 13. The Hercules cluster: a globular cluster. This resource can be
supported using tactile aids.

  * •

Messier 16. The Eagle nebula: star-forming complex with very bright blue stars
and surrounding gas clouds and pillars of dust and molecular gas. There are
also sonifications of this object.

  * •

Messier 51. The Whirlpool Galaxy: an example of a face-on spiral galaxy with
multiple star-forming knots. For this object there are sonifications and
images in relief.

  * •

Messier 87. The Virgo A galaxy: a giant elliptical galaxy with a radio jet
emitting from the central supermassive black hole. This can be completed with
available touchable models and sonifications.

###### Acknowledgements.

We especially thank all the students who have participated in the elaboration
of the scripts of the audio descriptions of El Universo en palabras: A. León,
A. Pérez Domínguez, A. Gil López, A. Benito Aparicio, A. Hermán Carvajal, C.
Salmerón Borja, C. Ghidini, E. Alonso Ochoa, E. Lacanna, L. Pastor Iradier, M.
Burguillos, M. Gómez Regalado and S. Gutiérrez Gullón. We also thank the
project Estallidos de formación estelar en galaxias AYA2016-79724-C4 for its
financial support for the narration of the videos. Participation of CBC in
this project is supported by the Spanish Ministry of Science, Innovation and
Universities grant FPU17/0490

## References

  * Barnés-Castaño et al. (2021) Barnés-Castaño, C. et al. 2021, New Voices in Translation Studies, 25, 1-26.
  * Chica-Núñez & Jiménez-Hurtado (2020) Chica-Núñez, A. & Jiménez-Hurtado, 2020, Journal of Audiovisual Translation, 3(2), 264-285.
  * Eardley et al. (2017) Eardley, A. et al. 2017, Inclusion, disability and culture, 195-207, Dordrecht: Springer
  * Galvez et al. (2017) Galvez, A., Ballesteros, F., García-Frank, A., et al. 2017, European Planetary Science Congress
  * Hutchinson & Eardley (2021) Hutchinson, R. & Eardley, A. 2021, Museum Management and Curatorship, 36(4), 427-446. https://doi.org/10.1080/09647775.2021.1891563
  * Koide et al. (2015) Koide et al. 2015, PloS one, 10(2), e0117696
  * Ortiz-Gil et al. (2019) Ortiz-Gil, A., Burguet-Castell, J., Pérez-Montero, E., et al. 2019, EPSC-DPS Joint Meeting 2019
  * Ortiz Gil (2020) Ortiz Gil, A. 2020, XIV.0 Scientific Meeting (virtual) of the Spanish Astronomical Society, 265
  * Pérez-Montero (2019) Pérez-Montero, E. 2019, Nature Astronomy, 3, 114. doi:10.1038/s41550-019-0693-3
  * Pérez-Montero et al. (2017) Pérez-Montero, E.J., García Gómez-Caro, E., Sánchez Molina, Y., et al. 2017, Highlights on Spanish Astrophysics IX, 742