Time deformations of master equations
Sergey N. Filippov Dariusz Chruściński






Moscow Institute of Physics and Technology,
Institutskii Per. 9, Dolgoprudny, Moscow Region 141700, Russia

Institute of Physics and Technology of the Russian
Academy of Sciences, Nakhimovskii Pr. 34, Moscow 117218, Russia



Institute of Physics, Faculty of Physics, Astronomy
and Informatics 
  Nicolaus Copernicus University, Grudziadzka
5/7, 87–100 Torun, Poland


Convolutionless and convolution master equations are the two
mostly used physical descriptions of open quantum systems
dynamics. We subject these equations to time deformations: local
dilations and contractions of time scale. We prove that the
convolutionless equation remains legitimate under any time
deformation (results in a completely positive dynamical map) if
and only if the original dynamics is completely positive
divisible. Similarly, for a specific class of convolution master
equations we show that uniform time dilations preserve positivity
of the deformed map if the original map is positive divisible.
These results allow witnessing different types of non-Markovian
behavior: the absence of complete positivity for a deformed
convolutionless master equation clearly indicates that the
original dynamics is at least weakly non-Markovian; the absence of
positivity for a class of time-dilated convolution master
equations is a witness of essentially non-Markovian original
dynamics.


Time deformations of master equations
    Dariusz Chruściński
    December 30, 2023
=====================================





§ INTRODUCTION


A physical quantum system is never isolated in practice, which
leads to a concept of an open quantum system. The state of such a
system is described by a density operator ϱ on some
Hilbert space ℋ (positive semidefinite operator with
unit trace). Time evolution of the open system is governed by the
total Hamiltonian H of “system + environment” and the initial
state of the environment Ω_E. If the system and environment
are initially factorized, i.e., their state is ϱ⊗Ω_E, then the system dynamics is defined by the standard
reduction

    ϱ(t) =  Tr_E { e^-iHtϱ⊗Ω_E e^iHt}.


Formula (<ref>) defines a dynamical map Φ(t)[X] =
 Tr_E { e^-iHt X ⊗Ω_E e^iHt}, which has an important property
of being completely positive  (CP) and trace-preserving. Complete
positivity means that Φ(t) ⊗ Id_k maps any
(possibly entangled) density operator of the system +
k-dimensional ancilla into a legitimate density operator.

Physical environments usually have enormously many degrees of
freedom, which makes the dynamics ϱ(t) intractable via
formula (<ref>) unless suitable approximations are
made <cit.>.
Microscopic derivations of system evolution with the help of
projection operator techniques result in either a convolutionless
master equation <cit.>

    d ϱ(t)/d t = L(t)
    [ϱ(t)]


with a time-local generator
L(t) : ℬ(ℋ) ↦ℬ(ℋ) or
a convolution master
equation <cit.>

    dϱ(t)/dt = ∫_0^t 𝒦(t,t')[ϱ(t')]
    dt'


with a memory kernel
𝒦(t,t'):ℬ(ℋ)↦ℬ(ℋ).

Only some sufficient conditions on the time-local generator L(t)
and memory kernel 𝒦(t,t') are known, which guarantee
complete positivity and trace preservation of the corresponding
dynamical map
Φ(t) <cit.>.

Suppose that master equations (<ref>)
and (<ref>) define a legitimate quantum dynamics,
i.e., a completely positive and trace-preserving dynamical map
Φ(t). From the quantum information science perspective, the
evolution process Φ(t) can have peculiar divisibility
properties. If the dynamical map Φ(t) can be represented in
the form of concatenation Φ(t_2) = V(t_2,t_1) Φ(t_1) with
CP intermediate map V(t_2,t_1) for all t_2
> t_1 ⩾ 0, then the process Φ(t) is called CP
divisible. Analogously, if V(t_2,t_1) is positive (P) for all
t_2 > t_1 ⩾ 0, then the process Φ(t) is called P
divisible. P indivisible processes are also referred to as
essentially non-Markovian, whereas CP indivisible but P divisible
processes are sometimes called weakly
non-Markovian <cit.>. CP divisibility
and P divisibility are only two approaches to define Markovian
quantum processes <cit.>, many other
approaches include decreasing distinguishability of system
states <cit.>, monotonicity of quantum
mutual information <cit.>, decreasing capacity of quantum
channels <cit.>, independence of evolution with
respect to events preceding the causal break when the system's
state is actively reset <cit.>, and others. The
reviews of the current status in the discussion of quantum
non-Markovianity are given in the
papers <cit.>.

The goal of this paper is to relate divisibility properties of
Φ(t) and the behavior of master equations under time
deformations. By time deformation of a master equation we
understand the transformation

    ϱ(t) →ϱ(τ),      dt → dτ,


where

    τ(t) = ∫_0^t α(t') dt',     dτ/dt = α(t),


and α(t) is non-negative real function
quantifying the local time stretching.

The naive interpretation of (<ref>) would be the
replacement of Φ(t) by Φ(τ(t)) but this is not the
case if the generator L(t) or memory kernel 𝒦(t,t')
is time dependent. In fact, a time deformation (<ref>)
may result in a non-legitimate master equation. Surprisingly,
non-legitimacy of a deformed master equation is closely related
with the divisibility property of the undeformed dynamics. In this
paper, we reveal this relation.

The paper is organized as follows.

In Sec. <ref>, we show that the time
deformation of the convolutionless master
equation (<ref>) results in a legitimate dynamical
maps if and only if the original dynamics is CP divisible. In
Sec. <ref>, we relate legitimacy of time
deformation of convolution master equation (<ref>)
with P divisibility of the original dynamics. In
Sec. <ref>, brief conclusions are given.




§ DEFORMATION OF CONVOLUTIONLESS MASTER EQUATIONS



Master equation (<ref>) formally defines a
dynamical map Φ(t) = T_←exp( ∫_0^t L(t')
dt' ), where T_← is the Dyson time-ordering
operator. The intermediate map V(t_2,t_1) in concatenation
Φ(t_2) = V(t_2,t_1) Φ(t_1) reads V(t_2,t_1) =
T_←exp( ∫_t_1^t_2 L(t') dt' ).

Time deformation of Eq. (<ref>) results in a
modified (inequivalent) master equation

    dϱ(τ(t))/dτ(t) = L(t)
    [ϱ(τ(t))],


where the density operator
ϱ(τ(t)) describes evolution in the deformed
time and the original generator L(t) is applied at time moments
τ(t), see Fig. <ref>(a).

In terms of the original time t Eq. (<ref>)
reads

    dϱ(t)/dt =
    dτ/dtdϱ/dτ = α(t) L(t)
    [ϱ(t)].


We will refer to Eq. (<ref>) as a time
deformation of the original time-convolutionless master equation
(<ref>).





If L is time independent, i.e. Φ(t) = e^Lt is a
semigroup, then (<ref>) results in a
deformed map Φ(t) = Φ(τ(t)). However, if
L(t) is time dependent, then Φ(t) ≠Φ(τ(t)). Moreover, Φ(t) can become not CP
even if the original map Φ(t) is legitimate (CP and trace
preserving), which can be illustrated by the following example.


 Consider a qubit map Φ(t):
ℬ(ℋ_2) ↦ℬ(ℋ_2)
given by the generator <cit.>

    L(t)[ϱ] = 1/2∑_i=1^3 γ_i(t) ( σ_i
    ϱσ_i - ϱ),


where σ_1,σ_2,σ_3 is the conventional
set of Pauli operators, γ_1(t)=γ_2(t)=1, and
γ_3(t) = -tanh(t). The map Φ_t is CP and trace
preserving for all t ⩾ 0, so it is a legitimate
dynamical map that can be realized physically, e.g., in the
deterministic collision model <cit.>.

It was shown in Ref. <cit.> that the time-deformed
master equation

    dϱ(t)/dt = α L(t) [ϱ(t)]


(obtained via constant time stretching τ = α
t) results in a CP map Φ(t) if and only if
α⩾ 1. Thus, if the original master equation is
subjected to a uniform time dilation (0<α<1), then the map
Φ_α (t) is not CP and does not correspond to
any physical evolution (of initially factorized system and
environment).

Note that Φ(t) ≠Φ(α t) because the
decoherence rates γ_k(t) are time dependent. □


Non-legitimacy of the deformed map Φ(t) in the
example above can be attributed to the fact that the master
equation (<ref>) describes so-called eternal
non-Markovian evolution, i.e. CP indivisible dynamical map
Φ(t), where V(t_2,t_1) is not CP for all t_2 >
t_1 <cit.>. On the other
hand, if the original dynamical map were CP divisible, then all
the decoherence rates would be positive. Time stretching would not
affect positiveness of decoherence rates and Φ(t)
would still be a valid dynamical map. This leads us to the
following main result.


 Master equation (<ref>) with
non-singular generator L(t) describes CP divisible dynamics if
and only if the deformed map remains CP under any time
deformation (<ref>).



Necessity. Suppose the process Φ(t) is CP divisible and
L(t) is not singular; then the generator L(t) has the
time-dependent Gorini-Kossakowski-Sudarshan-Lindblad
form <cit.>

    L(t)[ϱ] = -i [H(t),ϱ] 
           + ∑_k γ_k(t) ( A_k(t) ϱ A_k^†(t) -
    1/2{A_k^†(t) A_k(t) , ϱ}),


where all the rates γ_k(t) ⩾ 0.
Multiplication of the Hamiltonian H(t) by α(t) preserves
its Hermiticity, and α(t) γ_k(t) ⩾ 0, so
α(t)L(t) is still a valid generator of the dynamical map
(see, e.g., <cit.>).

Sufficiency. Let α(t) = {[                0 if  0 ⩽ t < t_1,;                1     if  t ⩾ t_1,;                  ]. then the deformed map Φ(t) =
T_←exp( ∫_0^t α(t') L(t') dt' ) =
{[              Id, if  0 ⩽ t < t_1,;         V(t,t_1)     if  t ⩾ t_1.;                  ]. Therefore, if the deformed map
Φ(t) remains CP under any deformation, then
V(t,t_1) is CP too for all t>t_1, i.e. the original map
Φ(t) is CP divisible.


Therefore, CP divisible dynamics preserves the property of being
CP divisible (and consequently CP) under any time deformation; see
Fig. <ref>(b). More importantly, if the original dynamical
map is not CP divisible, then this fact can be revealed by a
suitable time deformation under which the deformed map becomes
nonlegitimate.

 Non-singularity of generator L(t) is needed to
guarantee invertibility of Φ(t). If Φ(t) is not
invertible, then CP divisibility of Φ(t) does not require
positivity of rates γ_k(t); see
Refs. <cit.>.
However, the generator is not uniquely defined by the dynamical
map Φ(t) in this case. In particular, if the process is CP
divisible, then there exists a corresponding (possibly singular)
time-local generator with non-negative rates.
Theorem <ref> holds true for such generators too.




§ DEFORMATION OF CONVOLUTION MASTER EQUATIONS
 

In this section, we consider time deformations of the convolution
master equation (<ref>) and make implications on P
divisibility of the dynamical map Φ(t).

Continuing the same line of reasoning as before, let us assume
that the same kernel 𝒦(t,t') is applied at deformed
time moments τ(t) and τ(t'); see Fig. <ref>. As
a result, we obtain a time deformation of
Eq. (<ref>) of the form

    dϱ(τ(t))/dτ(t) = ∫_0^τ(t)𝒦(t,t')[ϱ(τ(t'))] d τ(t'),


which in terms of the original time t reads

    dϱ(t)/dt = ∫_0^t
    α(t) α(t') 𝒦 (t,t') [ϱ(t')] dt'.







If 𝒦(t,t') = δ(t-t')L(t'), then
(<ref>) reduces to dϱ(t)/dt
= α^2(t) L(t)[ϱ(t)], i.e., to the time deformation of
the convolutionless master equation considered before.

We assume that the open system dynamics does not depend on the
particular choice of time moment t=0, when the system starts
interacting with environment. Due to this time invariance
𝒦(t,t') = K(t-t') <cit.>.
In local time deformations (<ref>), the modified
kernel α(t) α(t') K (t-t') exhibits time invariance
only if α(t) is time independent. For this reason, we
consider only uniform time deformations τ(t) = α t,
α =  const.

Denoting (A ∗ B) (t) = ∫_0^t A(t-t')B(t')dt', master
equation (<ref>) takes the form d/dtΦ(t) = (K ∗Φ) (t). Using the Laplace transform Φ_s
= ∫_0^∞Φ(t) e^-st dt, the latter equation reduces
to

    Φ_s = (s   Id - K_s)^-1.


The uniformly deformed map Φ(t) governed by
Eq. (<ref>) with α(t)=α
satisfies

    Φ_s = (s   Id - α^2 K_s)^-1.


A straightforward algebra yields the following Laplace
transform of the derivative d/dtΦ(t):

    ( dΦ/dt)_s    =   α^2
    ( dΦ/dt)_s/ Id - (α^2
    - 1)( dΦ/dt)_s
       =   α^2 ( dΦ/dt)_s∑_n=0^∞ (α^2 - 1)^n [ (
    dΦ/dt)_s] ^n,


where the second line represents a valid expansion if
the norm (α^2-1) ( dΦ/dt)_s_1 → 1 < 1. In the time domain one finds

    dΦ/dt   =   α^2 dΦ/dt +
    α^2 (α^2 - 1)
    dΦ/dt∗dΦ/dt  + …
           + α^2 (α^2 - 1)^n dΦ/dt∗…∗dΦ/dt_n+1 times + …


Let us restrict ourselves to the commutative case, i.e maps
Φ(t) satisfying Φ(t_1)Φ(t_2) = Φ(t_2)Φ(t_1) for
all t_1,t_2 ⩾ 0. Commutative maps have time-independent
eigenoperators, so the spectrum of dΦ/dt is merely
the derivative of the spectrum of Φ(t). Denote eigenvalues of
Φ(t) by λ_k(t), then for P divisible Φ(t) one
has d|λ_k(t)|/dt⩽ 0
<cit.>. If Φ(t) is
Hermitian, i.e., Φ(t) coincides with its dual map
Φ^†(t) in the Heisenberg picture, then λ_k(t)
are real. Therefore, for commutative Hermitian P divisible maps
Φ(t) we have dλ_k(t)/dt⩽ 0. On the
other hand, if dλ_k(t)/dt⩽ 0, then
(<ref>) implies
dλ_k(t)/dt⩽ 0 provided 0 <
α < 1. This way one arrives at the following result.


 Suppose the commutative Hermitian
dynamical map Φ(t) is governed by a memory kernel K(t). If the
uniform time dilation K(t) →α^2 K(t) with 0 <
α < 1 and (1-α^2) ( dΦ/dt)_s_1 → 1 < 1 results in a map
Φ(t) such that dΦ/dt has
at least one positive eigenvalue at some time t, then the
original map Φ(t) is not P divisible.


The class of commutative Hermitian dynamical maps comprises
conventional Pauli qubit maps Φ(t)[ϱ] = 1/2(  tr[ϱ] I + ∑_k=1^3 λ_k(t) 
tr[σ_k ϱ] σ_k ) as well as generalized
Pauli
channels <cit.>.
For Pauli qubit maps one can find a simpler implication of
Proposition <ref>.


 Suppose the Pauli map Φ(t) is governed by
a memory kernel K(t). If the uniform time dilation K(t)
→α^2 K(t) with 0 < α < 1 and (1-α^2)
(1 - s ∫_0^∞λ_k(t) e^- s t dt) < 1 results in
a map Φ(t), which is not positive, then the
original map Φ(t) is not P divisible.



Condition (1-α^2) (1 - s ∫_0^∞λ_k(t) e^- s
t dt) < 1 guarantees the validity of
expansion (<ref>). Let Φ(t)
be non-positive. Since the Pauli map Φ(t) is
positive if and only if -1 ⩽λ_k(t)
⩽ 1, then either λ_k(t)
> 1 or λ_k(t) < - 1 for some time t.
Note that at the initial moment λ_k(0)=1.

Suppose λ_k(t)
> 1; then there exists a time moment t_0 ∈ (0,t) such that
dλ_k(t)/dt (t_0) > 0. By
Proposition <ref>, Φ(t) is not P
divisible.

Suppose λ_k(t) < - 1; let us show that
λ_k(t) ⩾̸0. Using expansion

    Φ(t)    =   Φ(t) + (α^2 - 1)
    (dΦdt∗Φ) (t) + …
           + (α^2 - 1)^n (dΦdt∗…∗dΦdt_n times∗Φ ) (t) + …
    ,


one finds that if λ_k(t) ⩾ 0 and
dλ_k/dt⩽ 0, then a time deformation with
0 < α < 1 guarantees λ_k(t) ⩾
0. As we consider the case λ_k(t) < - 1, this
contradiction proves that λ_k(t) ⩾̸0. As a
result, the original Pauli map Φ(t) is not P divisible.


The physical meaning of Proposition <ref> is that
positivity is a topological property of Pauli P divisible process
Φ(t), which is preserved under uniform time dilations.


 Consider a pure dephasing qubit map
Φ(t)[ϱ] = 1/2(  tr[ϱ] I +
∑_k=1^3 λ_k(t)  tr[σ_k ϱ] σ_k
) with λ_1(t) = λ_2(t) = 1 - 2 Γ t e^-
Γ t and λ_3(t) = 1. This is a valid dynamical map
if Γ > 0. Such a map is a solution of the convolution
master equation

    dϱ(t)/dt    =    ∫_0^t  ( Γδ(t-t') - Γ^2 sinΓ (t-t')
    )
    
              × [σ_z ϱ(t') σ_z - ϱ(t')]
    dt'.


Condition (1-α^2) (1 - s ∫_0^∞λ_k(t) e^- s t dt) < 1 is fulfilled automatically if 0 <
α < 1. The uniform time dilation of the memory kernel
K(t-t') →α^2 K(t-t') results in the deformed
Pauli map Φ(t) with

    λ_1(t) =
    λ_2(t) = 1 - 2 α^2 e^- α^2 Γ t sin( √(1-α^4) Γ t )
    /√(1-α^4)


and λ_3(t) = 1. When the
trigonometric function sin(·) takes negative values,
λ_1(t) = λ_2(t)
> 1, see Fig. <ref>, so the deformed map
Φ(t) is not positive. By
Proposition <ref>, it clearly indicates that the
original map Φ(t) is not P divisible.

Note that for the equivalent original convolutionless
equation, the uniform time deformation τ = α t results
in λ'_i(t) = [λ_i(t)]^α,
i=1,2,3. In this case, the deformed map Φ'(t)
remains CP and does not reveal P indivisibility of Φ(t).
□







 Let us consider a qubit evolution where the
rescaling of the memory kernel is compatible with P divisibility
of the dynamical map. Following <cit.>, let
Φ(t) be a Pauli qubit dynamical map governed by the memory
kernel

    K(t)[ϱ] = 1/2∑_k=1^3 ϰ_k(t) σ_k 
    tr[σ_k ϱ],


where the time-dependent eigenvalues ϰ_k(t)
are defined (in the Laplace transform domain) via

    (ϰ_k)_s = -s f_s/a_k - f_s .


In the above definition the positive numbers
{a_1,a_2,a_3} satisfy triangle inequality a_i^-1 + a_j^-1≥ a_k^-1 for all permutations of {i,j,k}, f(t) is a
real function satisfying f(t) ⩾ 0 and f_0 =
∫_0^∞ f(t)dt ≤ 4  (a_1^-1 + a_2^-1 + a_3^-1)^-1. The corresponding eigenvalues of Φ(t) are
given by λ_k(t) = 1 - a_k^-1∫_0^t f(t')dt'.

The dynamical map Φ(t) is known to be P divisible if
additionally f(t) satisfies the requirement <cit.>

    f_0 = ∫_0^∞ f(t)dt ≤ a_ min ,


where a_ min = min{a_1,a_2,a_3}. Suppose
condition (<ref>) is fulfilled, then f_s ⩽ a_
min for all s ⩾ 0. The deformed eigenvalue

    ( λ_k )_s = 1/s( 1 + α^2
    f_s/a_k - f_s) = 1/s( 1 - f_s/a_k) ∑_n=0^∞ (1-α^2)^n
    (f_s/a_k)^n


in time domain is a convolution of two non-negative
functions: the original eigenvalue λ_k(t) ∈ (0,1] and
the inverse Laplace transform of ∑_n=0^∞
(1-α^2)^n (f_s/a_k)^n. Hence,
λ_k(t) ⩾ 0. If 0 < α < 1, then
the latter function is less than or equal to the inverse Laplace
transform of ∑_n=0^∞(f_s/a_k)^n =
(1-f_s/a_k)^-1. Therefore,
λ_k(t) is less than or equal to the inverse
Laplace transform of function (λ_k)_s
(1-f_s/a_k)^-1 = 1/s, i.e.,
λ_k(t) ⩽ 1. Thus, the deformed map is
positive for 0 < α < 1 because the original map is P
divisible; see Eq. (<ref>).

Interestingly, the map Φ(t) being positive and
trace-preserving is in general not completely positive and hence
the kernel deformation K(t) →α^2 K(t) does not
lead to the legitimate dynamical map. In fact, consider the
behavior of λ_k(t) when t →∞. By the final value theorem

    lim_t →∞λ_k(t) = lim_s
    → 0 s(λ_k)_s = 1/1 + α^2
    f_0/a_k - f_0.


Suppose a_1 ⩽ a_2 ⩽ a_3. The deformed map is CP
if and only if the condition λ_i +
λ_j ⩽ 1 + λ_k is
fulfilled for permutations of indices {i,j,k}. In the limit t
→∞ this condition reduces to inequality

    f_0 ( 1/a_2-f_0 + 1/a_3-f_0 -
    1/a_1-f_0) + 2 α^2 f_0^2/(a_2 -
    f_0)(a_3 - f_0)
           + α^4 f_0^3/(a_1 - f_0)(a_2 - f_0)(a_3 -
    f_0)⩾ 0.


The obtained inequality is fulfilled for all 0 < α
< 1 if and only if (a_2-f_0)^-1 + (a_3-f_0)^-1⩾
(a_1-f_0)^-1, which is surprisingly equivalent to CP
divisibility of the original map Φ(t); cf.
Ref. <cit.>. Thus the dynamical map governed by the
memory kernel (<ref>) is CP divisible only if the
deformed map is CP for all 0 < α < 1. □






 
Consider CP indivisible Pauli dynamical map Φ(t) as in
Example <ref> but now in terms of the convolution
equation dΦ/dt = K ∗Φ. The explicit form of
the kernel K(t) is given in Ref. <cit.>. The uniform
time deformation K(t) →α^2 K(t) leads to the
deformed eigenvalues λ_1(t) =
λ_2(t) = (1+α^2)^-1 [ 1 + α^2
e^-(1+α^2)t ] and λ_3(t) = e^- 2
α^2 t. The deformed map Φ(t) is never CP
for t>0 and 0 < α < 1 since the corresponding set of
eigenvalues violates the Fujiwara-Algoet conditions for complete
positivity <cit.> (cf. Fig. <ref>).
□


Considered examples allow us to make a conjecture that a
general Pauli dynamical map Φ(t), defined by a convolution
master equation, is CP divisible if and only if the deformed map
Φ(t) is CP for all 0 < α < 1.  It is
tempting to pose a similar conjecture for general dynamical maps
governed by memory kernel master equations, namely, the map is CP
divisible iff the corresponding rescaled kernel α^2 K(t) is
physically legitimate for  0 < α < 1. This, however,
requires further analysis.



§ CONCLUSIONS
 

We have analyzed different forms of non-Markovianity in terms of
the time deformations of governing master equations. If some
deformation of the time-local equation results in a map, which is
not CP, then the original map is not CP divisible (it is at least
weakly non-Markovian). Analogously, if a deformation of the proper
time-convolution equation results is a map, which is not P, then
the original map is not P divisible (it is essentially
non-Markovian).

As the analysis of convolution master equations is particularly
complicated, we have managed to obtain only a necessary condition
for P divisible Hermitian commutative dynamical maps
(Proposition <ref>). We have illustrated
implications of this condition for Pauli dynamical qubit maps
(Proposition <ref> and Example <ref>). We
have also considered Examples <ref> and <ref>
of Pauli dynamical maps defined via a convolution master equation,
for which CP divisibility is equivalent to CP property of the
deformed map for all uniform time dilations.

In addition to witnessing non-Markovianity, the achieved results
clarify legitimate forms of dissipators and memory kernels, which
naturally emerge due to relativistic and gravitational time
dilation <cit.> as well as acceleration of quantum
systems <cit.>.



S.N.F. thanks the Russian Science Foundation for support under
project No. 16-11-00084. D.C. was partially supported by the
National Science Centre project 2015/17/B/ST2/02026.



99

alicki-1987
R. Alicki and K. Lendi, Quantum Dynamical Semigroups and
Applications (Springer, Berlin, 1987).

breuer-petruccione-2002
H.-P. Breuer and F. Petruccione, The Theory of Open Quantum
Systems (Oxford University Press, Oxford, 2002).

gardiner-2004
C. Gardiner, P. Zoller, Quantum Noise: A Handbook of
Markovian and Non-Markovian Quantum Stochastic Methods with
Applications to Quantum Optics (Springer, Berlin, 2004).

carmichael-2009
H. Carmichael, An Open Systems Approach to Quantum Optics,
(Springer, Berlin, 2009).

li-2018
L. Li, M. J. W. Hall, and H. M. Wiseman, Concepts of quantum
non-Markovianity: a hierarchy, Physics Reports 759, 1
(2018).

nakajima-1958
S. Nakajima, On Quantum Theory of Transport Phenomena: Steady
Diffusion, Prog. Theor. Phys. 20, 948 (1958).

zwanzig-1960
R. Zwanzig, Ensemble Method in the Theory of Irreversibility, J.
Chem. Phys. 33, 1338 (1960).

haake-1973
F. Haake, Statistical Treatment of Open Systems by
Generalized Master Equations, Springer Tracts in Modern Physics
Vol. 66 (Springer, Berlin, 1973).

hall-2008
M. J. W. Hall, Complete positivity for time-dependent qubit master
equations, J. Phys. A: Math. Theor. 41, 205302 (2008).

barnett-2001
S. M. Barnett and S. Stenholm, Hazards of reservoir memory, Phys.
Rev. A 64, 033808 (2001).

budini-2004
A. A. Budini, Stochastic representation of a class of
non-Markovian completely positive evolutions, Phys. Rev. A 69, 042107 (2004).

shabani-2005
A. Shabani and D. A. Lidar, Completely positive post-Markovian
master equation via a measurement approach, Phys. Rev. A 71,
020101(R) (2005).

maniscalco-2005
S. Maniscalco, Limits in the characteristic-function description
of non-Lindblad-type open quantum systems, Phys. Rev. A 72,
024103 (2005).

maniscalco-pertuccione-2006
S. Maniscalco and F. Petruccione, Non-Markovian dynamics of a
qubit, Phys. Rev. A 73, 012111 (2006); Phys. Rev. A 73, 029902(E) (2006); Phys. Rev. A 75, 059905(E) (2007).

breuer-vacchini-2008
H.-P. Breuer and B. Vacchini, Quantum Semi-Markov Processes, Phys.
Rev. Lett. 101, 140402 (2008).

nalezyty-2015
F. A. Wudarski, P. Należyty, G. Sarbicki, and D.
Chruściński, Admissible memory kernels for random unitary
qubit evolution, Phys. Rev. A 91, 042105 (2015).

chruscinski-kossakowski-2016
D. Chruściński and A. Kossakowski, Sufficient conditions
for a memory-kernel master equation, Phys. Rev. A 94,
020103(R) (2016).

vacchini-2016
B. Vacchini, Generalized Master Equations Leading to Completely
Positive Dynamics, Phys. Rev. Lett. 117, 230401 (2016).

chruscinski-kossakowski-2017
D. Chruściński and A. Kossakowski, Generalized semi-Markov
quantum evolution, Phys. Rev. A 95, 042131 (2017).

zanardi-2017
J. Marshall, L. C. Venuti, and P. Zanardi, Noise suppression via
generalized-Markovian processes, Phys. Rev. A 96, 052113
(2017).

chruscinski-maniscalco-2014
D. Chruściński and S. Maniscalco, Degree of
non-Markovianity of quantum evolution, Phys. Rev. Lett. 112,
120404 (2014).

wolf-prl-2008
M. M. Wolf, J. Eisert, T. S. Cubitt, and J. I. Cirac, Assessing
Non-Markovian Quantum Dynamics, Phys. Rev. Lett. 101, 150402
(2008).

rivas-2010
Á. Rivas, S. F. Huelga, and M. B. Plenio, Entanglement and
Non-Markovianity of Quantum Evolutions, Phys. Rev. Lett. 105, 050403 (2010).

breuer-2009
H.-P. Breuer, E.-M. Laine, and J. Piilo, Measure for the degree of
non-Markovian behavior of quantum processes in open systems, Phys.
Rev. Lett. 103, 210401 (2009).

laine-2010
E.-M. Laine, J. Piilo, and H.-P. Breuer, Measure for the non-
Markovianity of quantum processes, Phys. Rev. A 81, 062115
(2010).

luo-2012
S. Luo, S. Fu, and H. Song, Quantifying non-Markovianity via
correlations, Phys. Rev. A 86, 044101 (2012).

bylicka-2014
B. Bylicka, D. Chruściński, and S. Maniscalco,
Non-Markovianity and reservoir memory of quantum channels: a
quantum information theory perspective, Sci. Rep. 4, 5720
(2014).

pollock-2018
F. A. Pollock, C. Rodríguez-Rosario, T. Frauenheim, M.
Paternostro, and K. Modi, Operational Markov condition for quantum
processes, Phys. Rev. Lett. 120, 040405 (2018).

rivas-2014
Á. Rivas, S. F. Huelga, and M. B. Plenio, Quantum
non-Markovianity: characterization, quantification and detection,
Rep. Prog. Phys. 77, 094001 (2014).

breuer-2016
H.-P. Breuer, E.-M. Laine, J. Piilo, and B. Vacchini, Colloquium:
Non-Markovian dynamics in open quantum systems, Rev. Mod. Phys.
88, 021002 (2016).

de-vega-2017
I. de Vega and D. Alonso, Dynamics of non-Markovian open quantum
systems, Rev. Mod. Phys. 89, 015001 (2017).

hall-2014
M. J. W. Hall, J. D. Cresser, L. Li, and E. Andersson, Canonical
form of master equations and characterization of non-Markovianity,
Phys. Rev. A 89, 042120 (2014).

fpmz-2017
S. N. Filippov, J. Piilo, S. Maniscalco, and M. Ziman,
Divisibility of quantum dynamical maps and collision models, Phys.
Rev. A 96, 032111 (2017).

benatti-2017
F. Benatti, D. Chruściński, and S. Filippov, Tensor power
of dynamical maps and positive versus completely positive
divisibility, Phys. Rev. A 95, 012112 (2017).

megier-2016
N. Megier, D. Chruściński, J. Piilo, and W. T. Strunz,
Eternal non-Markovianity: from random unitary to Markov chain
realisations, Sci. Rep. 7, 6379 (2017).

wudarski-2015
D. Chruściński and F. A. Wudarski, Non-Markovianity degree
for random unitary evolution, Phys. Rev. A 91, 012104
(2015).

gks-1976
V. Gorini, A. Kossakowski, and E. C. G. Sudarshan, Completely
positive dynamical semigroups of N-level systems, J. Math. Phys.
17, 821 (1976).

lindblad-1976
G. Lindblad, On the Generators of Quantum Dynamical Semigroups,
Commun. Math. Phys. 48, 119 (1976).

andersson-2007
E. Andersson, J. D. Cresser, and M. J. W. Hall, Finding the Kraus
decomposition from a master equation and vice versa, J. Mod. Opt.
54, 1695 (2007).

chruscinski-rivas-stormer-2017
D. Chruściński, Á. Rivas, and E. Størmer,
Divisibility and information flow notions of quantum Markovianity
for noninvertible dynamical maps, Phys. Rev. Lett. 121,
080407 (2018).

chruscinski-macchiavello-maniscalco-2017
D. Chruściński, C. Macchiavello, and S. Maniscalco,
Detecting Non-Markovianity of Quantum Evolution via Spectra of
Dynamical Maps, Phys. Rev. Lett. 118, 080404 (2017).

nathanson-ruskai-2007
M. Nathanson and M. B. Ruskai, Pauli diagonal channels constant on
axes, J. Phys. A 40, 8171 (2007).

chruscinski-siudzinska-2016
D. Chruściński and K. Siudzińska, Generalized Pauli
channels and a class of non-Markovian quantum evolution, Phys.
Rev. A 94, 022118 (2016).

fujiwara-algoet-1999
A. Fujiwara and P. Algoet, One-to-one parametrization of quantum
channels, Phys. Rev. A 59, 3290 (1999).

pikovski-2017
I. Pikovski, M. Zych, F. Costa, and Č. Brukner, Time dilation
in quantum systems and decoherence, New J. Phys. 19, 025011
(2017).

richter-2017
B. Richter, H. Terças, Y. Omar, and I. de Vega, Collective
dynamics of accelerated atoms, Phys. Rev. A 96, 053612
(2017).



