# Quantum vs. classical $P$-divisibility

Fabio Benatti<EMAIL_ADDRESS>Dipartimento di Fisica, Università di
Trieste, 34151 Trieste, Italy  Istituto Nazionale di Fisica Nucleare, Sezione
di Trieste, 34151 Trieste, Italy  Dariusz Chruściński  Institute of Physics,
Faculty of Physics, Astronomy and Informatics, Nicolaus Copernicus University,
Grudziadzka 5/7, 87-100 Toruń, Poland  Giovanni Nichele  Dipartimento di
Fisica, Università di Trieste, 34151 Trieste, Italy  Istituto Nazionale di
Fisica Nucleare, Sezione di Trieste, 34151 Trieste, Italy

###### Abstract

$P$-divisibility is a central concept in both classical and quantum non-
Markovian processes; in particular, it is strictly related to the notion of
information backflow. When restricted to a fixed commutative algebra generated
by a complete set of orthogonal projections, any quantum dynamics naturally
provides a classical stochastic process. It is indeed well known that a
quantum generator gives rise to a $P$-divisible quantum dynamics if and only
if all its possible classical reductions give rise to divisible classical
stochastic processes. Yet, this property does not hold if one operates a
classical reduction of the quantum dynamical maps instead of their generators:
as an example, for a unitary dynamics, $P$-divisibility of its classical
reduction is inevitably lost, which thus exhibits information backflow.
Instead, for some important classes of purely dissipative qubit evolutions,
quantum $P$-divisibility always implies classical $P$-divisibility and thus
lack of information backflow both in the quantum and classical scenarios. On
the contrary, for a wide class of orthogonally covariant qubit dynamics, we
show that loss of classical $P$-divisibility can originate from the classical
reduction of a purely dissipative $P$-divisible quantum dynamics as in the
unitary case. Moreover, such an effect can be interpreted in terms of
information backflow, the information coming in being stored in the coherences
of the time-evolving quantum state.

##  I Introduction

In the following, we shall be concerned with $n$-level open quantum systems
interacting with their environment in a way that cannot be approximated by
means of dynamical semigroups [1, 2, 3]. The presence of memory effects is
embodied by one-parameter families of completely positive and trace-preserving
($CPTP$) maps $\Lambda_{t}$, $t\geq 0$ that act on the convex space
$\mathcal{S}(M_{n}(\mathbb{C}))$ of their positive and unit trace density
matrices $\rho\in M_{n}(\mathbb{C})$ such that
$\Lambda_{t+s}\neq\Lambda_{t}\Lambda_{s}$. We shall assume the dynamics
$\rho\mapsto\rho(t)=\Lambda_{t}[\rho]$ to consist of algebraically invertible
maps generated by time-dependent generators $\mathcal{L}_{t}$ on
$\mathcal{S}(M_{n}(\mathbb{C}))$, namely by master equations of the form

$\dot{\Lambda}_{t}=\mathcal{L}_{t}\,\Lambda_{t}\
,\quad\mathcal{L}_{t}:=\dot{\Lambda}_{t}\Lambda_{t}^{-1}\ ,$ (1)

which are solved by the time-ordered exponentials

$\displaystyle\Lambda_{t}$
$\displaystyle=\mathcal{T}_{\leftarrow}e^{\int_{0}^{t}\differential{s}\,\mathcal{L}_{s}}$
$\displaystyle=\sum_{k=0}^{\infty}\int_{0}^{t}\differential
s_{1}\int_{0}^{s_{2}}\hskip-2.84544pt\differential
s_{2}\cdots\int_{0}^{s_{k-1}}\hskip-14.22636pt\differential
s_{k}\,\mathcal{L}_{s_{1}}\cdots\mathcal{L}_{s_{k}}\ .$ (2)

Because of the invertibility of the dynamics, the intertwining maps
$\Lambda_{t,s}$ connecting time $s$ to time $t\geq s\geq 0$,
$\Lambda_{t}=\Lambda_{t,s}\Lambda_{s}$, are given by

$\Lambda_{t,s}:=\Lambda_{t}^{\phantom{.}}\,\Lambda_{s}^{-1}\ ,\quad t\geq
s\geq 0\ .$ (3)

Given $n^{2}$ Hilbert-Schmidt orthonormal matrices $F_{k}\in
M_{n}(\mathbb{C})$ such that ${\rm Tr}(F^{\dagger}_{j}F_{k})=\delta_{jk}$,
with $F_{0}=\mathds{1}/\sqrt{n}$, the generator can always be written as

$\mathcal{L}_{t}[\rho]=-i[H(t),\rho]+\mathcal{D}_{t}[\rho]\ ,$ (4)

with $H(t)$ being a time-dependent Hamiltonian and dissipator

$\mathcal{D}_{t}[\rho]=\sum_{i,j=1}^{n^{2}-1}K_{ij}(t)\left(F_{i}\rho
F^{\dagger}_{j}-\frac{1}{2}\\{F^{\dagger}_{j}F_{i},\rho\\}\right)\,,$ (5)

with Hermitian, time-dependent $(n^{2}-1)\times(n^{2}-1)$ Kossakowski matrix
$K(t)=[K_{ij}(t)]=K^{\dagger}(t)$. According to the GKSL-theorem [4, 5], for
time-independent generators, such that
$\dot{\Lambda}_{t}=\mathcal{L}\Lambda_{t}$, the dynamics becomes a semi-group,

$\Lambda_{t}={\rm e}^{t\mathcal{L}}\
,\quad\Lambda_{t+s}=\Lambda_{t}\Lambda_{s},\qquad\Lambda_{t=0}=\mathrm{id}\quad\forall
t,s\geq 0\ ,$ (6)

and the complete positivity of the maps $\Lambda_{t}$ is equivalent to the
positive semi-definiteness of the time-independent Kossakowski matrix:
$K(t)=K=[K_{jk}]\geq 0$. Instead, in the time-dependent case there are no
general constraints on $K(t)$ characterizing the complete positivity of the
generated $\Lambda_{t}$. What certainly holds is that $K(t)\geq 0$ is
equivalent to the complete positivity of all the intertwiners $\Lambda_{t,s}$
[3, 6] (and thus implies the complete positivity of $\Lambda_{t}$). In this
case, the dynamics is called $CP$-divisible. However, the complete positivity
of $\Lambda_{t}$ does not require $K(t)\geq 0$; indeed, completely positive
$\Lambda_{t}$ can very well be only $P$-divisible, namely with intertwiners
$\Lambda_{t,s}$ that are only positive and not completely positive. Even when
not semi-groups, $P$-divisible families of $CPTP$ maps may be considered to be
Markovian as they do not give rise to the phenomenon of backflow of
information from the environment to the open quantum system immersed in it
[7]. Indeed, Markovianity is nowadays identified with information being lost
from an open system to its environment, leading to increasing lack of
distinguishability of generic pairs of open system states in the course of
time.

In the following we shall study the properties inherited by the classical
dynamics that naturally emerges from $P$-divisible quantum dynamics. The
classical dynamics will be obtained by restricting both the quantum dynamics
and its tomographic reconstruction to a same (maximally) Abelian sub-algebra
of $M_{n}(\mathbb{C})$ generated by any given choice of $n$ orthogonal
rank-$1$ projections $P_{j}\in M_{n}(\mathbb{C})$ such that
$P_{j}P_{k}=\delta_{jk}$ and $\sum_{j=1}^{n}P_{j}=\mathds{1}$. In the
following, for sake of simplicity, we shall identify the commutative sub-
algebra $\mathcal{P}$ with the orthonormal, complete set
$\\{P_{i}\\}_{i=1}^{n}$ that generates it and refer to the quantum dynamics
restricted to it as its classical reduction.

Notice that one can classically reduce either the quantum generator and then
generate a classical dynamics, or the quantum dynamics itself. In the latter
case, one obtains a classical process whose generator has in general nothing
to do with the classical reduction of the quantum generator.

In Section II, we shall show that the first choice is somewhat trivial, in
that quantum $P$-divisibility is equivalent to classical divisibility, while
the second one has the peculiar effect that a $P$-divisible quantum dynamics
need not give rise to classically reduced divisible dynamics.

Thus, the classical reduction procedure is somewhat dual to the so-called
embeddability of a given classical stochastic matrix into a Markovian quantum
evolution, that may give rise to some quantum advantage [8, 9].

In Section III we focus on qubit unital dynamics, for which $P$-divisibility
of both the quantum dynamics and its classical reduction are to some extent
under control. In this context, we show that loss of $P$-divisibility for the
classical reduction is typical of unitary evolutions, but not of relevant
classes of purely dissipative evolutions ($\mathcal{L}_{t}=\mathcal{D}_{t}$),
for which the $P$-divisibility of the reduced classical stochastic processes
is equivalent to that of the quantum map that originates it.

In Section IV, we study a wide class of orthogonally covariant qubit dynamics
where necessary and sufficient conditions for $P$ and $CP$-divisibility in
terms of the generator are available. Within this class, we provide a fairly
general construction of a purely dissipative quantum dynamics whose classical
reduction is not $P$-divisible.

A backflow of information interpretation of the loss of $P$-divisibility by
classical reduction is finally provided, in Section V, in the so-called $BLP$
framework [7, 10]. In particular, we show that the backflow of information in
the classically reduced processes is due to information being stored in the
coherences of the quantum states at previous times and entirely built up by
the quantum dynamics, that generically moves the chosen commutative algebra
out of itself.

##  II Classical from quantum maps

In order to discuss the properties of the families of classical dynamical maps
induced by quantum ones we start with the case of one-parameter semigroups.

###  II.1 Classical from quantum one-parameter semi-groups

For semigroups of quantum maps as in (6), trace preservation imposes
$\Tr(\mathcal{L}[\rho])=0$ for all $\rho\in\mathcal{S}(M_{n}(\mathbb{C}))$.
Kossakowski [11] found necessary and sufficient conditions for $\mathcal{L}$
to be the generator of a positive trace-preserving ($PTP$) semigroup
$\\{\Lambda_{t}\\}_{t\geq 0}$.

######  Theorem 1.

A linear map $\mathcal{L}$ on $\mathcal{S}(M_{n}(\mathbb{C}))$ generates a
semigroup of PTP maps $\\{{\Lambda_{t}}\\}_{t\geq 0}$ iff

$\Tr(Q\mathcal{L}[P])\geq 0\ ,\quad\Tr(\mathcal{L}[\mathds{\rho}])=0$ (7)

for all pairs of projectors $P\perp Q$ and all
$\rho\in\mathcal{S}(M_{n}(\mathbb{C}))$.

######  Remark 1.

The classical analogue of a one-parameter quantum semigroup on
$\mathcal{S}(M_{n}(\mathbb{C}))$ is a one-parameter family of $n\times n$
stochastic matrices $\\{T(t)\\}_{t\geq 0}$, with entries $T_{ij}(t)$ such that
$T_{ij}(t)\geq 0$, $\sum_{i}T_{ij}(t)=1$. These matrices transform probability
vectors $\boldsymbol{p}=\\{p_{i}\\}_{i=1}^{n}$, $p_{i}\geq 0$,
$\sum_{j=1}^{n}p_{j}=1$, into themselves,
$\boldsymbol{p}(t)=T(t)\boldsymbol{p}$. Probability vectors can be represented
as density matrices $\rho_{\boldsymbol{p}}=\sum_{i=1}^{n}p_{i}\,P_{i}$ with
respect to any complete orthonormal set $\\{P_{j}\\}_{j=1}^{n}$ of rank-$1$
projections in $M_{n}(\mathbb{C})$. These projections generate a maximally
Abelian sub-algebra $\mathcal{P}$ and the classical semigroup can then be
represented by a family of linear maps
$\Lambda_{T}(t):\mathcal{P}\mapsto\mathcal{P}$ such that ${\rm
Tr}(P_{i}\Lambda_{T}(t)[P_{j}])=T_{ij}(t)$. Therefore, Theorem 1 implies that
a generic matrix $L$ generates a semigroup of stochastic matrices $T(t)={\rm
e}^{tL}$, iff the so-called Kolmogorov conditions [6, 12] are satisfied:

$L_{ij}\geq 0\qquad i\neq j,\qquad\sum_{i=1}^{n}L_{ij}=0\ ,$ (8)

so that $L_{ij}=\partial_{t}{\rm
Tr}(P_{i}\,\Lambda_{T}(t)[P_{j}])\big{|}_{t=0}\ $.

In conjunction with the previous observation, one derives the following
natural consequence from Theorem 1.

######  Corollary 1.

A linear map $\mathcal{L}$ on $\mathcal{S}(M_{n}(\mathbb{C}))$ such that
$\Tr(\mathcal{L}[\mathds{\rho}])=0$, generates a semigroup of $PTP$ maps
$\\{{\Lambda_{t}}\\}_{t\geq 0}$ iff the $n\times n$ matrix $L$ with entries

$L_{ij}=\Tr(P_{i}\mathcal{L}[P_{j}])$ (9)

generates a semigroup of stochastic $n\times n$ matrices, for any orthonormal,
complete set $\\{P_{i}\\}_{i=1}^{n}$ of rank-1 projectors.

Given an orthonormal complete set of orthogonal rank-1 projections
$\mathcal{P}=\\{P_{i}\\}_{i=1}^{n}$, any $PTP$ quantum map $\Phi$ on
$\mathcal{S}(M_{n}(\mathbb{C}))$ defines a stochastic matrix through the
entries

$T_{ij}:=\Tr(P_{i}\Phi[P_{j}]).$ (10)

Indeed, the positivity of $\Phi$ together with the completeness of the
projections and trace-preservation yields $T_{ij}\geq 0$ and
$\sum_{i=1}^{n}T_{ij}=1$. The converse, though, is not true: even if the
matrices $T$ defined by $\Phi$ through (10) are stochastic for all maximally
Abelian $\mathcal{P}$, $\Phi$ need not be positive. The reason is that the
positivity of $\Phi$ amounts to

$\Tr(P\Phi[Q])\geq 0$ (11)

for all one-dimensional, not necessarily orthogonal projections $P$ and $Q$,
while (10) restricts to orthogonal ones.

######  Example 1.

Let $\Phi$ be a Hermiticity and trace-preserving, unital qubit map on
$\mathcal{S}(M_{2}(\mathbb{C}))$, namely $\Phi[\sigma_{0}]=\sigma_{0}$. Here
$\sigma_{0}=\mathds{1}$ denotes the identity $2\times 2$ matrix which,
together with the other Pauli matrices, $\sigma_{1,2,3}$, after normalization,
constitutes a Hilbert-Schmidt basis: $\displaystyle{\rm
Tr}\left(\frac{\sigma_{\alpha}}{\sqrt{2}}\,\frac{\sigma_{\beta}}{\sqrt{2}}\right)=\delta_{\alpha\beta}$.
They provide a matrix representation of $\Phi$ through the entries

$\Phi_{\alpha\beta}:=\frac{1}{2}\Tr\left({\sigma_{\alpha}}\,\Phi\left[{\sigma_{\beta}}\right]\right)\
,\quad\alpha=0,\dots,3\ .$ (12)

where $\Phi_{00}=1$ and $\Phi_{0i}=0$ for $i=1,2,3$ follows from trace
preservation. Also, from unitality it follows that $\Phi_{i0}=0$; therefore,
one concentrates upon the real matrix $\widetilde{\Phi}$ with entries

$\widetilde{\Phi}_{ij}:=\frac{1}{2}\Tr(\sigma_{i}\Phi[\sigma_{j}]),\quad
i=1,2,3\,,$ (13)

so that $\Phi[\sigma_{i}]=\sum_{j=1}^{3}\widetilde{\Phi}_{ji}\,\sigma_{j},$.
Let $P_{0}$, $P_{1}=\mathds{1}-P_{0}$ arbitrary orthonormal projectors in
$M_{2}(\mathbb{C})$ and define the $2\times 2$ matrix with entries
$T_{ij}=\Tr(P_{i}\Phi[P_{j}])$. Since $\Phi$ is unital,

$T=\begin{pmatrix}T_{00}&1-T_{00}\\\ 1-T_{00}&T_{00}\end{pmatrix},$ (14)

is a (bi-)stochastic matrix iff $0\leq T_{00}\leq 1$. In the Bloch
representation, $P_{0}$ is identified by means of a unit real vector
$\boldsymbol{n}\in\mathbb{R}^{3}$,
$P_{\boldsymbol{n}}=(\mathds{1}+\boldsymbol{n}\cdot\boldsymbol{\sigma})/2$,
where $\boldsymbol{\sigma}=(\sigma_{1},\sigma_{2},\sigma_{3})$. Then, the
bistochasticity conditions read

$\displaystyle 1\geq
T_{00}=\Tr(P_{0}\Phi[P_{0}])=\frac{1}{2}+\frac{1}{2}\matrixelement{\boldsymbol{n}}{\widetilde{\Phi}}{\boldsymbol{n}}\geq
0,$

for all unit $\boldsymbol{n}\in\mathbb{R}^{3}$. Hence, $T$ is stochastic for
every choice of orthogonal projectors $P_{0}$ and $\mathds{1}-P_{0}$ iff

$\ell(\widetilde{\Phi}):=\sup_{\norm{\boldsymbol{n}}=1}|\matrixelement{\boldsymbol{n}}{\widetilde{\Phi}}{\boldsymbol{n}}|\leq
1.$ (15)

Notice that, if $\widetilde{\Phi}$ is non-symmetric,
$\widetilde{\Phi}\neq\widetilde{\Phi}^{T}$, where T denotes transposition,
$\ell(\widetilde{\Phi})$ is in general smaller than the norm

$\|\widetilde{\Phi}\|=\sup_{\begin{subarray}{c}\boldsymbol{n},\boldsymbol{m}\\\
\norm{\boldsymbol{n}}=\norm{\boldsymbol{m}}=1\end{subarray}}|\matrixelement{\boldsymbol{n}}{\widetilde{\Phi}}{\boldsymbol{m}}|\
.$ (16)

On the other hand, the map is positive iff

$\Tr(P_{\pm\boldsymbol{n}}\Phi[P_{\mathbf{m}}])=\frac{1}{2}\pm\frac{1}{2}\matrixelement{\boldsymbol{n}}{\widetilde{\Phi}}{\boldsymbol{m}}\geq
0\,,$ (17)

for all unit vectors $\boldsymbol{n},\boldsymbol{m}\in\mathbb{R}^{3}$; namely,
iff $|\matrixelement{\boldsymbol{n}}{\widetilde{\Phi}}{\boldsymbol{m}}|\leq
1$. Then, $\|\widetilde{\Phi}\|\leq 1$, since $\widetilde{\Phi}$ is real. If
$\widetilde{\Phi}=\widetilde{\Phi}^{T}$, then
$\|{\widetilde{\Phi}}\|=\ell(\widetilde{\Phi})$ and $T$ is stochastic for all
${P}_{0}$ iff the map $\Phi$ is positive.

Instead, let $\alpha\in(0,\frac{\pi}{2})$,
$\lambda>\eta\geq\lambda\cos(\alpha)>0$; then,

$\widetilde{\Phi}=\begin{pmatrix}\lambda\cos(\alpha)&-\lambda\sin(\alpha)&0\\\
\lambda\sin(\alpha)&\lambda\cos(\alpha)&0\\\ 0&0&\eta\\\
\end{pmatrix}\neq\widetilde{\Phi}^{T}$ (18)

has singular values $\lambda$ and $\eta$. Therefore,
$\|{\widetilde{\Phi}}\|=\lambda$, while
$|\matrixelement{\boldsymbol{n}}{\widetilde{\Phi}}{\boldsymbol{n}}|=\lambda\cos(\alpha)+n_{3}^{2}\,(\eta-\lambda\cos(\alpha))$.
Then, choosing $\lambda=1+\varepsilon$ and $\eta=\lambda\cos(\alpha)$, one has

$\displaystyle\ell(\widetilde{\Phi})=\sup_{\norm{\boldsymbol{n}}=1}|\matrixelement{\boldsymbol{n}}{\widetilde{\Phi}}{\boldsymbol{n}}|=(1+\varepsilon)\cos(\alpha)\leq
1<\|\widetilde{\Phi}\|.$

for $\varepsilon$ sufficiently small. With such choice of parameters, the map
${\Phi}$ is not positive, while $T_{ij}=\Tr(P_{i}\Phi[P_{j}])$ are the entries
of a well defined stochastic matrix, for all choices of orthonormal projectors
$P_{0}$ and $P_{1}$.

###  II.2 Time-dependent case

We now consider the case of time-dependent generators as in (1) and positive
intertwiners as in (3), namely the case of $P$-divisible families of $CPTP$
maps. Theorem 1 generalizes as follows [10, 6].

######  Theorem 2.

A linear map $\mathcal{L}_{t}$ that annihilates the trace generates a family
$\\{\Lambda_{t}\\}_{t\geq 0}$ of $P$-divisible, trace-preserving maps iff
$\Tr(Q\mathcal{L}_{t}[P])\geq 0$, for all $t\geq 0$, and for any pair of
rank-1, mutually orthogonal projectors $P\perp Q$.

###### Proof.

The “only if” part follows straightforwardly from the positivity of
$\Lambda_{t,s}$, since for all $s\geq 0$

$\Tr(Q\mathcal{L}_{s}[P])=\lim_{\epsilon\to
0}\frac{1}{\epsilon}{\Tr(Q\Lambda_{s+\epsilon,s}[P])}\geq 0.$ (19)

For the “if” part, rewrite
$\mathcal{T}_{\leftarrow}e^{\int_{s}^{t}\differential{s}\mathcal{L}_{s}}$ by
means of the time-splitting formula [3]

$\displaystyle\Lambda_{t,s}=\lim_{\max\limits_{0\leq k\leq
n-1}\absolutevalue{t_{k+1}-t_{k}}\to
0}\prod_{k=n-1}^{0}e^{(t_{k+1}-t_{k})\,\mathcal{L}_{t_{k}}}\,,$ (20)

with $t\equiv t_{n}\geq t_{n-1}\geq\dots\geq t_{0}\equiv s$. Fix
$k\in\mathbb{N}$, $0\leq k\leq n-1$. Since $\Tr(Q\mathcal{L}_{t_{k}}[P])\geq
0$ for all $P\perp Q$, $t\mapsto e^{({t-t_{k}})\mathcal{L}_{t_{k}}}$ is a
positive map for all $t\geq t_{k}$, by Theorem 1. Since the composition of
positive maps is positive, one concludes that $\Lambda_{t,s}$ is positive. ∎

The classical analogue of the family of $P$-divisible $CPTP$ maps is a one-
parameter family of stochastic matrices such that

$T(t)=T(t,s)T(s),$ (21)

for all $t\geq s\geq 0$, where $T(t,s)$ is a stochastic matrix,
$T_{ij}(t,s)\geq 0$, $\sum_{i}T_{ij}(t,s)=1$. The classical time-local master
equation has the form

$\dot{T}(t)=L(t)T(t).$ (22)

The quantum rendering of classical processes within a quantum setting outlined
in Remark 1 leads to the following generalizations of the Kolmogorov
conditions (8) and of Corollary 1.

######  Proposition 1.

A matrix $L(t)$ generates a classical $P$-divisible dynamics $T(t)$ iff

$L_{ij}(t)\geq 0\quad\forall\,i\neq j,\qquad\sum_{i}L_{ij}(t)=0.$ (23)

If $T(t)$ is an invertible matrix, writing $L(t)=\dot{T}(t)T(t)^{-1}$ and
$T(t,s)=T(t)T(s)^{-1}$ provides a time dependent version of Corollary 1.

######  Corollary 2.

A map $\mathcal{L}_{t}$ such that $\Tr(\mathcal{L}_{t}[\mathds{\rho}])=0$
generates a $P$-divisible family $\\{{\Lambda_{t}}\\}_{t\geq 0}$ iff the
$n\times n$ matrix defined by $L_{ij}(t)=\Tr(P_{i}\mathcal{L}_{t}[P_{j}])$
generates a classical $P$-divisible dynamics for any orthonormal set of rank-1
projectors $\\{P_{i}\\}_{i}$.

Therefore, any quantum dynamics $\\{{\Lambda_{t}}\\}_{t\geq 0}$ is
$P$-divisible iff its generator defines classically $P$-divisible processes
for any choice of orthonormal basis.

However, one may derive a classical continuous stochastic process not from the
quantum generator, but directly from the generated quantum maps, as for the
$PTP$ maps $\Phi$ in Example 1; namely, by associating to a maximally Abelian
sub-algebra $\mathcal{P}=\\{P_{i}\\}_{i=1}^{n}$ a one parameter family of
matrices with entries defined by:

$T_{ij}(t)=\Tr(P_{i}\,\Lambda_{t}[P_{j}])\ .$ (24)

If $\Lambda_{t}$ is a CPTP map, the entries $T_{ij}(t)$ will give rise to a
stochastic matrix.

######  Remark 2.

Notice that, from (24), when the matrices $T(t)$ are invertible, the classical
intertwiners are given by $T(t,s)=T(t)T(s)^{-1}$. In general, they have no
simple connection to the quantum intertwiners
$\Lambda_{t,s}=\Lambda_{t}\,\Lambda_{s}^{-1}$. A particular case is when the
dynamics $\Lambda_{t}$ is of the kind
$\Lambda_{t,s}\,\mathbb{D}=\mathbb{D}\,\Lambda_{t,s}\,\mathbb{D},$ where
$\mathbb{D}[\rho]=\sum_{i}P_{i}\rho P_{i}$ is a “decohering map” with respect
to some fixed maximally Abelian sub-algebra $\mathcal{P}=\\{P_{i}\\}_{i}$, so
that one can identify a stochastic propagator as
$T_{ik}(t,s)=\Tr(P_{i}\Lambda_{t,s}[P_{k}])$. Such maps are a subset of the
so-called non-coherence-generating-and-detecting dynamics considered in [13]
to characterize the classicality of a Markovian process.

In the light of the previous discussion, there emerges the following natural
question: Given a $P$-divisible dynamics $\\{{\Lambda_{t}}\\}_{t\geq 0}$, is
the classical process defined through (24) classically divisible?

Such an issue is not only mathematically interesting; indeed, as already
emphasized, from a physical perspective, the permanence of $P$-divisibility
while going from a quantum dynamics to its classical reduction means the
permanence of a one-directional flow of information from an open system to its
environment, be it quantum or classical.

As we shall see in the following section, the above question becomes
particularly relevant when one considers purely dissipative dynamics with
generators not containing commutators with Hamiltonians (see (4)), which would
provide non-divisible classical processes. Moreover, we shall mostly focus on
qubit unital dynamics for which the $P$-divisibility of both $\Lambda_{t}$ and
its classical reduction $T(t)$ can be controlled.

##  III Unital qubit dynamics

Let $\Lambda_{t}$ be a unital qubit dynamics,
$\Lambda_{t}[\mathds{1}]=\mathds{1}$, and $P_{0}$, $P_{1}$ two orthogonal
projectors generating a maximally Abelian sub-algebra $\mathcal{P}\subset
M_{2}(\mathbb{C})$. As in Example 1, the structure of the classical process
$T(t)$ obtained from $\Lambda_{t}$ by means of (45) is determined only by
$T_{00}(t)$:

$T(t)=\begin{pmatrix}T_{00}(t)&1-T_{00}(t)\\\
1-T_{00}(t)&T_{00}(t)\end{pmatrix},$ (25)

where $T_{00}(t)=\Tr(P_{0}\Lambda_{t}[P_{0}])$. If $T_{00}(t)\neq\frac{1}{2}$,
$T(t)$ is invertible and one computes the classical generator as

$L(t)=\dot{T}(t)T(t)^{-1}=\frac{\dot{T}_{00}(t)}{2T_{00}(t)-1}\begin{pmatrix}1&-1\\\
-1&1\end{pmatrix},$ (26)

Thus, from Proposition 1, $L(t)$ generates of a divisible classical process
iff

$f_{t}:=\frac{\dot{T}_{00}(t)}{2T_{00}(t)-1}\leq 0\qquad\forall t\geq 0\ .$
(27)

As in Example 1, we thus introduce the $3\times 3$ matrix representation of
$\Lambda_{t}$

$\widetilde{\Lambda}_{ij}(t)=\frac{1}{2}\Tr(\sigma_{i}\Lambda_{t}[\sigma_{j}]),$
(28)

and analogously $\widetilde{\mathcal{L}}(t)$ for $\mathcal{L}_{t}$, satisfying
the master equation
$\dot{\widetilde{\Lambda}}(t)=\widetilde{\mathcal{L}}(t)\widetilde{\Lambda}(t)$,
whose formal solution is given by a time-ordered exponentiation (see (2)):

$\widetilde{\Lambda}(t)=\mathcal{T}_{\leftarrow}e^{\int_{0}^{t}\differential{s}\widetilde{\mathcal{L}}(s)}.$
(29)

Using the Bloch representation of Example 1, one rewrites

$T_{00}(t)=\frac{1}{2}+\frac{1}{2}\matrixelement{\boldsymbol{n}}{\widetilde{\Lambda}(t)}{\boldsymbol{n}},\
\dot{T}_{00}(t)=\frac{1}{2}\matrixelement{\boldsymbol{n}}{\widetilde{\mathcal{L}}(t)\widetilde{\Lambda}(t)}{\boldsymbol{n}}\
.$

Then, (27) reads

$f_{t}=\frac{1}{2}\frac{\matrixelement{\boldsymbol{n}}{\widetilde{\mathcal{L}}(t)\widetilde{\Lambda}(t)}{\boldsymbol{n}}}{\matrixelement{\boldsymbol{n}}{\widetilde{\Lambda}(t)}{\boldsymbol{n}}}\leq
0\,,\qquad\forall t\geq 0\ .$ (30)

Similarly, $\Lambda_{t}$ is $P$-divisible iff

$\Tr(P_{0}\mathcal{L}_{t}[P_{1}])=-\frac{1}{2}\matrixelement{\boldsymbol{n}}{\widetilde{\mathcal{L}}(t)}{\boldsymbol{n}}\geq
0\qquad\forall\boldsymbol{n}\in\mathbb{R}^{3}\ ,$ (31)

namely iff the symmetric part of the generator in the Bloch representation has
negative eigenvalues

$-(\widetilde{\mathcal{L}}(t)+\widetilde{\mathcal{L}}^{T}(t))\geq 0\,.$ (32)

In the following, we consider the behaviour of $f_{t}$ under two oppositely
behaving dynamics. The first one is a purely unitary qubit rotation which
cannot give rise to a classically divisible process, while the second is that
of a purely dissipative Pauli dynamics for which $P$-divisibility of the
dynamics is equivalent to that of its classical reduction.

######  Example 2.

Consider the qubit Hamiltonian
$H=1/2\boldsymbol{\omega}\cdot\boldsymbol{\sigma}$,
$\boldsymbol{\omega}\in\mathbb{R}^{3}$, and the evolution
$\mathcal{U}_{t}[\rho]=U_{t}\,\rho\,{U}^{\dagger}_{t}$, $U_{t}=e^{-itH}$. The
classical reduction of its generator $\mathcal{L}[\cdot]=-i[H\,,\cdot\,]=0$ to
any commutative sub-algebra $\mathcal{P}\subset M_{2}(\mathbb{C})$ vanishes.
Indeed, $\Tr(PHQ)=\Tr(PQH)=0$. However, the Bloch representation of
$\mathcal{U}_{t}$ acts as $3\times 3$ rotation matrix
$\widetilde{\mathcal{U}}(t)=e^{t\widetilde{\mathcal{L}}}$ with generator

$\widetilde{\mathcal{L}}=\omega\,\begin{pmatrix}0&-r_{3}&r_{2}\\\
r_{3}&0&-r_{1}\\\ -r_{2}&r_{1}&0\\\ \end{pmatrix}\ ,$ (33)

where $\omega=\|\boldsymbol{\omega}\|$ and
$\boldsymbol{r}=(\omega_{1},\omega_{2},\omega_{3})/\omega\in\mathbb{R}^{3}$ is
a unit vector. Since
$\widetilde{\mathcal{L}}^{3}=-\widetilde{\mathcal{L}}=\widetilde{\mathcal{L}}^{T}$,
one rewrites

$\widetilde{\mathcal{U}}(t)=e^{t\widetilde{\mathcal{L}}}=\mathds{1}+\frac{\sin(\omega
t)}{\omega}\,\widetilde{\mathcal{L}}+\frac{1-\cos(\omega
t)}{\omega^{2}}\,\widetilde{\mathcal{L}}^{2}\ .$ (34)

Also,
$\matrixelement{\boldsymbol{n}}{\widetilde{\mathcal{L}}}{\boldsymbol{n}}=0$,
$\matrixelement{\boldsymbol{n}}{\widetilde{\mathcal{L}}^{2}}{\boldsymbol{n}}=-\|\widetilde{\mathcal{L}}\ket{\boldsymbol{n}}\|^{2}$.
Given a projector with Bloch vector $\boldsymbol{n}$, with $\theta$ the angle
between the latter and $\boldsymbol{r}$, one has
$\|\widetilde{\mathcal{L}}\ket{\boldsymbol{n}}\|=\omega\sin(\theta)$ and

$\displaystyle T_{00}(t)$ $\displaystyle=$
$\displaystyle\Tr\Big(P_{0}\,\mathcal{U}_{t}[P_{0}]\Big{missing})=\frac{1}{2}\Big{(}1+\matrixelement{\boldsymbol{n}}{\widetilde{\mathcal{U}_{t}}}{\boldsymbol{n}}\Big{)}$
$\displaystyle=$ $\displaystyle\frac{1}{2}(1+\cos^{2}(\theta)+\cos(\omega
t)\sin^{2}(\theta))\ ,$

so that (30) is rewritten as follows,

$f_{t}=-\frac{\omega}{2}\frac{\sin(\omega
t)\sin^{2}(\theta)}{\cos^{2}(\theta)+\cos(\omega t)\sin^{2}(\theta)}\ .$ (35)

Thus, the sign of $f_{t}$ changes for all $\theta\in(0,\pi)$, unless
$\theta=0,\pi$ (when $\ket{\boldsymbol{n}}$ is an eigenstate of $H$) yielding
$f_{t}=0$. Notice that the denominator in $f_{t}$ is the determinant of the
classical stochastic matrix $T(t)$ which is thus invertible if
$\cos(2\theta)>0$, namely, if $\theta\in(0,\pi/4)\cup(3\pi/4,\pi)$. In this
case, classical $P$-divisibility breaks when $\dot{T}_{00}(t)$ becomes
positive. In Section V, we shall interpret loss of $P$-divisibility for an
invertible classical reduction $T(t)$ in terms of information flow.

######  Example 3.

Let $\Lambda_{t}$ be the Pauli dynamics generated by

$\mathcal{L}_{t}[\rho]=\frac{1}{2}\sum_{k=1}^{3}\gamma_{k}(t)(\sigma_{k}\rho\sigma_{k}-\rho).$
(36)

Notice that ${[\mathcal{L}_{t},\mathcal{L}_{s}]=0}$, yielding the exponential
solution $\Lambda_{t}=e^{\int_{0}^{t}\differential{s}\mathcal{L}_{s}}$.
Moreover, the representation in the Pauli representation of the generator is
diagonal,

$\widetilde{\mathcal{L}}_{ij}(t)[\rho]=-\Gamma_{i}(t)\,\delta_{ij}\,,\qquad\Gamma_{i}(t)=\sum_{k\neq
i}\gamma_{k}(t)\,,$ (37)

and necessary and sufficient for $P$-divisibility follow from (32),

$\Gamma_{k}(t)\geq 0\,,\quad k=1,2,3\,.$ (38)

$\widetilde{\Lambda}(t)$ is also diagonal, with strictly positive eigenvalues

$\widetilde{\Lambda}_{ij}(t)=\lambda_{i}(t)\delta_{ij}\,,\qquad\lambda_{i}(t)=e^{-\int_{0}^{t}\differential{s}\Gamma_{i}(t)}\
.$

so that

$\displaystyle
f_{t}=\frac{1}{2}\frac{\matrixelement{\boldsymbol{n}}{\widetilde{\mathcal{L}}(t)\widetilde{\Lambda}(t)}{\boldsymbol{n}}}{\matrixelement{\boldsymbol{n}}{\widetilde{\Lambda}(t)}{\boldsymbol{n}}}=-\frac{1}{2}\frac{\sum_{i}\Gamma_{i}(t)\lambda_{i}(t)n_{i}^{2}}{\sum_{j}\lambda_{j}(t)n_{j}^{2}}\leq
0$ (39)

for all $\boldsymbol{n}$. Thus, $T(t)$ is $P$-divisible for all choices of the
reference basis iff $\Lambda_{t}$ is $P$-divisible.

Notice that, for $\gamma_{i}(t)=const$, the Pauli generator includes the case
of a unital purely dissipative GKSL evolution (the time-independent version of
(44)), which thus yields $P$-divisible classically reduced dynamics.

######  Example 4.

Adding a Hamiltonian to a Pauli generator $\mathcal{L}_{t}^{P}$ as in (36) may
generally lead to a positive $f_{t}$, so that the process $T(t)$ is not
divisible. Indeed, consider

$\mathcal{L}_{t}[\rho]=-i[\sigma_{z},\rho]+\mathcal{L}_{t}^{P}[\rho],$ (40)

and take, for the sake of simplicity,
$\Gamma_{1}(t)=\Gamma_{2}(t)\equiv\Gamma(t)$, so that the Hamiltonian
generator commutes with the Pauli one. Hence, the evolution will have the form

$\Lambda_{t}=\mathcal{U}_{t}\,\Lambda_{t}^{P}\implies\widetilde{\Lambda}(t)=\widetilde{\mathcal{U}}(t)\,\widetilde{\Lambda}^{P}(t)\,.$
(41)

Using (30) and (34), it follows that $f_{t}=f_{t}^{P}+f_{t}^{H}$, where
$f_{t}^{P}$ is as in (39) and always negative, while $f_{t}^{H}$ is as in (35)
and generally oscillates between positive and negative values. In particular,
$P$-divisibility is lost due to the divergence of the $f_{t}^{H}$ for some
$\boldsymbol{n}$. On the other hand, for those $\boldsymbol{n}$ that lead to
an invertible $T(t)$, $f_{t}^{H}$ is bounded, so that $f_{t}$ can stay
negative for all times for large enough Pauli rates $\Gamma_{i}(t)$.

The previous examples suggest that the presence of the commutator in the
generator is in general responsible for the loss of $P$-divisibility for the
classically reduced process defined by (24), while the classical reduction of
Pauli dynamics preserves $P$-divisibility. Therefore, in order to study which
$P$-divisible quantum dynamics $\Lambda_{t}$ give certainly rise to
classically divisible processes defined by (24), one better focus upon purely
dissipative generators.

From (4), it follows that the time-local generator of a trace and Hermiticity
preserving, qubit dynamics can be always expressed in the form [4]

$\mathcal{L}_{t}[\rho]=-i[H(t),\rho]+\mathcal{D}_{t}[\rho],$ (42)

with dissipative part

$\mathcal{D}_{t}[\rho]=\frac{1}{2}\sum_{i,j=1}^{3}K_{ij}(t)\left(\sigma_{i}\rho\sigma_{j}-\frac{1}{2}\anticommutator{\sigma_{j}\,\sigma_{i}}{\rho}\right)\
.$ (43)

For a unital qubit dynamics the following then holds.

######  Lemma 1.

A qubit dynamics generated by $\mathcal{L}_{t}$ is unital iff the Kossakowski
matrix $K(t)$ is real and symmetric, so that $\mathcal{D}_{t}$ can be written
in the diagonal form

$\mathcal{D}_{t}[\rho]=\frac{1}{2}\sum_{i}\gamma_{i}(t)\left(\sigma_{i}(t)\rho\sigma_{i}(t)-\rho\right),$
(44)

with respect to a set of matrices (generally time dependent)
$\\{\sigma_{i}(t)\\}_{i=1}^{3}$ that fulfil the Pauli algebra. Moreover, the
dynamics is purely dissipative, namely $\mathcal{L}_{t}=\mathcal{D}_{t}$, iff
the time-independent Bloch-representation of the generator

$\widetilde{\mathcal{L}}_{ij}(t)=\frac{1}{2}\Tr(\sigma_{i}\mathcal{L}_{t}[\sigma_{j}]),\quad
i,j=1,2,3\ ,$ (45)

consists of a symmetric matrix.

###### Proof.

Let us recall that $\Lambda_{t}$ is unital if and only if
$\mathcal{L}_{t}[\mathds{1}]=0$ for all $t\geq 0$. Indeed, if $\Lambda_{t}$ is
unital,
$0=\dot{\Lambda}_{t}[\mathds{1}]=\mathcal{L}_{t}\Lambda_{t}[\mathds{1}]=\mathcal{L}_{t}[\mathds{1}]$;
conversely, if the generator kills the identity at all times, unitality
follows from the Dyson series (2). Hence, in the qubit case, plugging the
identity into (43) $\Lambda_{t}$ is unital iff

$\sum_{i,j=1}^{3}K_{ij}(t)[\sigma_{i},\sigma_{j}]=0,$ (46)

which is satisfied iff $K_{ij}(t)=K_{ji}(t)={K_{ij}^{*}}(t)$. Then, the real
symmetric Kossakowski matrix $K(t)$ can be diagonalized by means of an
orthogonal transformation,

$K(t)=O(t)\gamma(t)O(t)^{T}\
,\quad\gamma(t)=\textrm{diag}\\{\gamma_{1}(t),\gamma_{2}(t),\gamma_{3}(t)\\}\
.$

Therefore, $O(t)O^{T}(t)=O^{T}(t)O(t)=\mathds{1}$ guarantees that the matrices
$\sigma_{i}(t)$ satisfy the Pauli algebra. Moreover, the Bloch representation
(45) of the dissipative part $\mathcal{D}_{t}$ of the generator reads

$\displaystyle\widetilde{\mathcal{D}}_{ij}(t)$
$\displaystyle=\frac{1}{2}\sum_{k}\gamma_{k}(t)\left[\Tr\left(\sigma_{i}\sigma_{k}(t)\sigma_{j}\sigma_{k}(t)\right)-2\delta_{ij}\right]$
$\displaystyle=\widetilde{\mathcal{D}}_{ji}(t).$ (47)

Thus, an anti-symmetric contribution to the matrix representation of the full
generator $\mathcal{L}(t)$ can only come from the commutator with the
Hamiltonian. It follows that the unital dynamics is purely dissipative iff
${\widetilde{\mathcal{L}}(t)=\widetilde{\mathcal{L}}^{T}(t)}$. ∎

######  Remark 3.

The dual, $\Lambda^{\ddagger}$, of a state-transformation $\Lambda$ is defined
as

$\Tr(X\Lambda[\rho])=\Tr(\Lambda^{\ddagger}[X]\rho),$

for all system states $\rho$ and system operators $X$. Since $\Lambda$ is
trace-preserving $\Lambda^{\ddagger}$ is unital
$\Lambda^{\ddagger}[\mathds{1}]=\mathds{1}$. A state-transformation $\Lambda$
is self-dual iff $\Lambda=\Lambda^{\ddagger}$ so that it is necessarily
unital, while the reverse is obviously not true. When $\Lambda_{t}$ is a qubit
dynamical map, going to the Bloch representation, one then has that self-
duality is equivalent to $\widetilde{\Lambda}(t)=\widetilde{\Lambda}^{T}(t)$.
On the other hand, if the generator $\mathcal{L}_{t}$ of the qubit dynamics
$\Lambda_{t}$ is self-dual,
$\mathcal{L}_{t}^{\phantom{\ddagger}}=\mathcal{L}_{t}^{\ddagger}$, it must be
purely dissipative, $\mathcal{L}_{t}=\mathcal{D}_{t}$, and $\Lambda_{t}$
unital. Indeed,
$\mathcal{L}_{t}[\mathds{1}]=\mathcal{L}_{t}^{\ddagger}[\mathds{1}]=0$ yields
a symmetric Kossakowski matrix which in turn implies the absence of the
commutator with a Hamiltonian. The unitality of $\Lambda_{t}$ then follows
from its Dyson expansion. Moreover, in the qubit case, Lemma 1 implies that a
purely dissipative generator of a unital dynamics is necessarily self-dual. In
the following Section IV, we shall provide a concrete qubit construction for a
non self-dual unital dynamics arising from a self-dual generator. In
particular, this cannot occur if the solution is of the form
$\Lambda_{t}=e^{\int_{0}^{t}\differential{s}\mathcal{L}_{s}}$.

It follows that if $\Lambda_{t}$ is unital and purely dissipative,
$\widetilde{\mathcal{L}}(t)$ is real symmetric and then, from (32), the
$P$-divisibility of $\Lambda_{t}$ becomes equivalent to
$-\widetilde{\mathcal{L}}(t)\geq 0$. The result for Pauli dynamics can be then
generalized to self-dual qubit dynamics with self-dual generators, for which
$P$-divisibility is preserved by their classical reduction.

######  Proposition 2.

Let $\Lambda_{t}^{\phantom{\ddagger}}=\Lambda_{t}^{\ddagger}$ be a self-dual,
purely dissipative, invertible qubit dynamics. Then, the associated classical
stochastic process $T(t)$ is divisible if and only if $\Lambda_{t}$ is
$P$-divisible.

###### Proof.

One has to check when (30) holds. Since
$\Lambda_{t}^{\phantom{\ddagger}}=\Lambda_{t}^{\ddagger}$, this implies that
their Bloch representations satisfy
$\widetilde{\Lambda}(t)=\widetilde{\Lambda}^{T}(t)$. Also, the assumed pure
dissipativeness of $\Lambda_{t}$ entails that the generator itself is self-
dual, $\mathcal{L}_{t}=\mathcal{L}_{t}^{\ddagger}$, so that
$\widetilde{\mathcal{L}}^{T}(t)=\widetilde{\mathcal{L}}(t)$ (see Lemma 1).
Then, taking the time derivative of both members of the inequality, one gets

$\dot{\widetilde{\Lambda}}(t)=\widetilde{\mathcal{L}}(t)\widetilde{\Lambda}(t)=\widetilde{\Lambda}(t)\widetilde{\mathcal{L}}(t)\
,$ (48)

or, equivalently,
$[{\widetilde{\Lambda}(t)},{\dot{\widetilde{\Lambda}}(t)}]=0$. The
invertibility of $\Lambda_{t}$ means that none of the real eigenvalues of
$\widetilde{\Lambda}(t)=\widetilde{\Lambda}^{T}(t)$ can change sign with
varying $t$. Since, at $t=0$, $\widetilde{\Lambda}_{t}=\mathds{1}$ all of them
must remain positive. Therefore, one can consistently express the generator as
a logarithmic derivative:

$\widetilde{\mathcal{L}}(t)=\dot{\widetilde{\Lambda}}(t)\widetilde{\Lambda}(t)^{-1}=\widetilde{\Lambda}(t)^{-1}\dot{\widetilde{\Lambda}}(t)=\derivative{t}\log\widetilde{\Lambda}(t)\
.$ (49)

Thus,
$\widetilde{\Lambda}(t)=e^{\int_{0}^{t}\differential{s}\widetilde{\mathcal{L}}(s)}$
without time-ordering; furthermore,

$f_{t}=-\frac{1}{2}\frac{\matrixelement{\boldsymbol{n}}{\sqrt{\widetilde{\Lambda}(t)}(-\widetilde{\mathcal{L}}(t))\sqrt{\widetilde{\Lambda}(t)}}{\boldsymbol{n}}}{\matrixelement{\boldsymbol{n}}{\widetilde{\Lambda}(t)}{\boldsymbol{n}}}\leq
0\,.$ (50)

The proof is completed by observing that, because of (32), the dynamics
$\Lambda_{t}$ is $P$-divisible iff $-\widetilde{\mathcal{L}}(t)\geq 0$. ∎

######  Remark 4.

The assumptions in Proposition 2 lead to a proper exponential solution
$\widetilde{\Lambda}(t)=e^{\int_{0}^{t}\differential{s}\widetilde{\mathcal{L}}(s)}$
without asking for commuting generators at different times as in Example 3.
Indeed, a sufficient condition for the exponential to solve
$\dot{\widetilde{\Lambda}}(t)=\widetilde{\mathcal{L}}(t)\widetilde{\Lambda}(t)$
is that $\Big{[}\widetilde{\mathcal{L}}(t)\,,\,\int_{0}^{t}{\rm
d}s\,\widetilde{\mathcal{L}}(s)\Big{]}=0$ for all $t\geq 0$ (notice that, from
the power series of the exponential, (48) also follows). As an example,
consider a generator of a unital dynamics of the form

$\widetilde{\mathcal{L}}(t)=\left\\{\begin{matrix}\begin{pmatrix}-1&\cos
t&0\cr\cos t&-1&0\cr 0&0&-2\cr\end{pmatrix}&\quad 0\leq
t\leq\pi\cr-\begin{pmatrix}1&0&0\cr 0&2&0\cr 0&0&1\end{pmatrix}&\quad\pi\leq
t\end{matrix}\right.\ .$

$\widetilde{\mathcal{L}}(t)$ and $\widetilde{\mathcal{L}}(s)$ do not commute
when $0\leq s\leq\pi$ and $t>\pi$, while they do when either $0\leq
s,t\leq\pi$ or $s,t>\pi$; on the other hand, since

$\int_{0}^{t}{\rm d}s\,\widetilde{\mathcal{L}}(s)=\begin{pmatrix}-t&\sin
t&0\cr\sin t&-t&0\cr 0&0&-2t\end{pmatrix}$

when $0\leq t\leq\pi$ and, when $t\geq\pi$,

$\int_{0}^{t}{\rm d}s\,\widetilde{\mathcal{L}}(s)=-\pi\begin{pmatrix}1&0&0\cr
0&1&0\cr 0&0&2\end{pmatrix}-(t-\pi)\begin{pmatrix}1&0&0\cr 0&2&0\cr
0&0&1\end{pmatrix}$

the condition $\Big{[}\widetilde{\mathcal{L}}(t),\int_{0}^{t}{\rm
d}s\,\widetilde{\mathcal{L}}(s)\Big{]}=0$, for all $t\geq 0$, that avoids
time-ordering is fulfilled. Moreover, $\widetilde{\mathcal{L}}(t)$ is
symmetric (self-dual generator), so that the dynamics is purely dissipative.
The exponential solution then entails the self-duality of the map.
$-\widetilde{\mathcal{L}}(t)\geq 0$ finally assures $P$-divisibility of the
generated dynamics, so that all the hypothesis of Proposition 2 are matched.

##  IV A class of orthogonally covariant dynamics

In the following, we study whether the $P$-divisibility of a quantum qubit
dynamics $\Lambda_{t}$ is inherited by its classical reductions $T(t)$ for non
self-dual, purely dissipative dynamics. We focus upon the following family of
maps,

$\displaystyle\Phi^{({A,\lambda,\mu})}[\rho]=$
$\displaystyle\sum_{i,j=0}^{1}A_{ij}E_{ij}\rho E_{ji}+\lambda E_{00}\rho
E_{11}+\overline{\lambda}\,E_{11}\rho E_{00}$
$\displaystyle+\mu\,E_{11}\rho^{T}E_{00}+\overline{\mu}\,E_{00}\rho^{T}E_{11}\,,$
(51)

where $E_{ij}=\outerproduct{i}{j}$ are the matrix units associated with the
basis of eigenvectors of $\sigma_{3}$,
$\sigma_{3}\ket{0}=\ket{0},\sigma_{3}\ket{1}=-\ket{1}$. The maps are not in
general of Pauli type and depend parametrically on a triplet
$(A,\lambda,\mu)$, given by a $2\times 2$ matrix ${A=[A_{ij}]}$ and
coefficients $\lambda,\mu\in\mathbb{C}$.

They were studied in the $d$-dimensional case in [14]. In particular, they
satisfy the following group composition law,

$\Phi^{({A,\lambda,\mu})}\Phi^{({A^{\prime},\lambda^{\prime},\mu^{\prime}})}=\Phi^{({AA^{\prime},\,\lambda\lambda^{\prime}+\mu\overline{\mu^{\prime}},\,\lambda\mu+\mu\overline{\lambda^{\prime}}})}\
.$ (52)

Moreover, if $\mu=0$, the maps $\Phi^{(A,\lambda,0)}$ satisfy the diagonal
unitary-covariance property

$\Phi^{(A,\lambda,0)}[UXU^{\dagger}]=U\Phi^{(A,\lambda,0)}[X]U^{\dagger}\
,\quad U=\sum_{i}e^{i\theta_{i}}E_{ii}\ .$

Conversely, if $\lambda=0$, the maps $\Phi^{(A,0,\mu)}$ are conjugate diagonal
unitary-covariant, namely

$\Phi^{(A,0,\mu)}[UXU^{\dagger}]=\overline{U}\Phi^{(A,0,\mu)}[X]U^{T}\ .$

Lastly, if both $\lambda,\mu$ are different from zero, the only symmetry left
is with respect to rotations around the $z$ axis, corresponding to the
diagonal orthogonal covariance

$\Phi^{(A,\lambda,\mu)}[OXO^{T}]=O\Phi^{(A,\lambda,\mu)}[X]O^{T}\ ,$

where $O=\sum_{i}o_{i}E_{ii}$, $o_{i}\in\\{-1,1\\}$.

Consider a time-dependent family of hermiticity and trace-preserving maps
within the above class, $\Lambda_{t}=\Phi^{(A(t),\lambda_{t},\mu_{t})}$. Then,
$A_{t}$ is to be taken real and of the form $A_{ij}(t)\in\mathbb{R}$ and
$\sum_{i}A_{ij}(t)=1$, so that

$A(t)=\begin{pmatrix}a_{t}&1-b_{t}\\\ 1-a_{t}&b_{t}\end{pmatrix}\ ,\quad
a_{t},b_{t}\in\mathbb{R}\ .$ (53)

Asking that $\Lambda_{t=0}=\mathrm{id}_{2}$ yields $\lambda_{0}=1,\mu_{0}=0$
and $a_{0}=1$; whereas positivity can be proved to be equivalent to [14]

$\displaystyle A_{ij}(t)\geq 0\,,$
$\displaystyle\absolutevalue{\lambda_{t}}+\absolutevalue{\mu_{t}}\leq\sqrt{a_{t}b_{t}}+\sqrt{(1-a_{t})(1-b_{t})}\,.$
(54)

so that $A(t)$ has to be a stochastic matrix. On the other hand, the
conditions for complete positivity can be obtained by imposing positivity of
the associated Choi matrix,

$\Lambda_{t}\otimes\mathrm{id}_{2}\bigg{[}\sum_{ij}E_{ij}\otimes
E_{ij}\bigg{]}\geq 0\,,$ (55)

leading to the stronger necessary and sufficient conditions

$\absolutevalue{\lambda_{t}}\leq\sqrt{a_{t}b_{t}}\,,\quad\absolutevalue{\mu_{t}}\leq\sqrt{(1-a_{t})(1-b_{t})}\,.$
(56)

Recently, Schwartz-positivity for maps within this class has also been fully
characterized [15]. Sufficient conditions for complete positivity in terms of
the generator for this class of maps were also investigated in [16]. We shall
now characterize the divisibility properties of the dynamics within class
(IV). Due to the composition property (52), the generator
${\mathcal{L}_{t}=\dot{\Lambda}^{\phantom{.}}_{t}\Lambda_{t}^{-1}}$ will
itself belong to the class (IV),
${\mathcal{L}_{t}=\Phi^{(B(t),\ell_{t},m_{t})}}$, with $B(t)$ of the form

$B(t)=\dot{A}(t)A(t)^{-1}=\begin{pmatrix}-\gamma_{-}(t)&\gamma_{+}(t)\\\
\gamma_{-}(t)&-\gamma_{+}(t)\end{pmatrix}$ (57)

where $\gamma_{-}(t)$, $\gamma_{+}(t)$ are related to $a_{t}$, $b_{t}$ via

$\gamma_{-}(t)=-\frac{\dot{a}_{t}(1-b_{t})+\dot{b}_{t}a_{t}}{a_{t}+b_{t}-1},\quad\gamma_{+}(t)=-\frac{\dot{a}_{t}b_{t}+\dot{b}_{t}(1-a_{t})}{a_{t}+b_{t}-1},$

and $l_{t}$, $m_{t}$ are related to $\lambda_{t}$, $\mu_{t}$ via

$\displaystyle l_{t}$
$\displaystyle=\frac{\dot{\lambda}_{t}\overline{\lambda_{t}}-\dot{\mu}_{t}\overline{\mu_{t}}}{\absolutevalue{\lambda_{t}}^{2}-\absolutevalue{\mu_{t}}^{2}}\,,\qquad
m_{t}=\frac{\dot{\mu}_{t}\lambda_{t}-\dot{\lambda}_{t}{\mu}_{t}}{\absolutevalue{\lambda_{t}}^{2}-\absolutevalue{\mu_{t}}^{2}}\,.$
(58)

It is convenient to introduce time-dependent “transversal” and “longitudinal”
rates,

$\Gamma_{T}(t):=-\real(l_{t})\,,\qquad\Gamma_{L}(t):={\gamma_{+}(t)+\gamma_{-}(t)}\,.$
(59)

and let also $\omega(t):=-\imaginary(l_{t})$. Notice that $\Gamma_{T}(t)$ only
depends on the absolute values of $\lambda_{t}$ and $\mu_{t}$,

$\Gamma_{T}(t)=-\frac{1}{2}\frac{\partial_{t}\absolutevalue{\lambda_{t}}^{2}-\partial_{t}\absolutevalue{\mu_{t}}^{2}}{|\lambda_{t}|^{2}-|\mu_{t}|^{2}}\,.$
(60)

Necessary and sufficient conditions for $P$ and $CP$-divisibility of the
dynamics can be then determined from the generator.

######  Proposition 3.

Let ${\mathcal{L}_{t}=\Phi^{(B(t),l_{t},m_{t})}}$ be the generator of an
evolution $\Lambda_{t}=\Phi^{(A(t),\lambda_{t},\mu_{t})}$. Then,

  * •

$\Lambda_{t}$ is $P$-divisible iff

$\displaystyle\gamma_{\pm}(t)\geq 0\,,$ (61)
$\displaystyle\Gamma_{T}(t)-\frac{\Gamma_{L}(t)}{2}+\sqrt{\gamma_{+}(t)\gamma_{-}(t)}\geq\absolutevalue{m_{t}}\,;$
(62)

  * •

$\Lambda_{t}$ is $CP$-divisible iff

$\displaystyle\gamma_{\pm}(t)\geq
0\,,\,\qquad{\gamma_{+}(t)\gamma_{-}(t)}\geq\absolutevalue{m_{t}}^{2}\,,$ (63)
$\displaystyle\Gamma_{T}(t)\geq\frac{\Gamma_{L}(t)}{2}\ .$ (64)

Though in a different form, this result has already been obtained in [17]; in
Appendix A we report the proof since it is somewhat simpler that in the quoted
reference. Notice that condition (64) is the time-dependent version of the
celebrated constraint between relaxation rates already discussed in seminal
paper [4], which has been proved to be universal for qubit completely positive
semigroups [18], and recently conjectured to be universal for all completely
positive dynamical semigroups [19].

######  Remark 5.

The Bloch representation of $\Lambda_{t}$ as in (28) leads to the following
dynamics of the Bloch vector:

$\boldsymbol{n}_{t}=\widetilde{\Lambda}(t)\boldsymbol{n}+\boldsymbol{u}_{t}\,,\qquad\boldsymbol{u}_{t}=(0,0,a_{t}-b_{t})\,,$
(65)

with

$\widetilde{\Lambda}(t)=\begin{pmatrix}\real(\lambda_{t}+\mu_{t})&\imaginary(\lambda_{t}-\mu_{t})&0\\\
-\imaginary(\lambda_{t}+\mu_{t})&\real(\lambda_{t}-\mu_{t})&0\\\
0&0&a_{t}+b_{t}-1\\\ \end{pmatrix}.$

The block diagonal structure of $\widetilde{\Lambda}(t)$ makes evident the
orthogonal covariance of the map with respect to orthogonal transformations
diagonal in the $\sigma_{3}$ basis. The generator can also be rewritten in
GKSL Pauli form as in (42) and (43): with Hamiltonian

$H(t)=\frac{\omega(t)}{2}\sigma_{3}\ .$ (66)

Furthermore, letting $\kappa(t)=-\real(m_{t})$, $\eta(t)=-\imaginary(m_{t})$
and $\delta(t)=(\gamma_{+}(t)-\gamma_{-}(t))/2$, with Kossakowski matrix

$K(t)=\begin{pmatrix}\frac{\Gamma_{L}(t)}{2}-\kappa(t)&\eta(t)+i\delta(t)&0\\\
\eta(t)-i\delta(t)&\frac{\Gamma_{L}(t)}{2}+\kappa(t)&0\\\
0&0&\Gamma_{T}(t)-\frac{\Gamma_{L}(t)}{2}\end{pmatrix}\ .$

Notice that $CP$-divisibility conditions can be then read off asking for
$K(t)\geq 0$. If $\Lambda_{t}$ is also unital, $A(t)$ has to be bistochastic,
thus $a_{t}=b_{t}$ and

$\gamma_{+}(t)=\gamma_{-}(t)=\frac{\Gamma_{L}(t)}{2}=\frac{\dot{a}_{t}}{2a_{t}-1}\,,$

so that the $P$-divisibility conditions simplify to

$\frac{\Gamma_{L}(t)}{2}\geq 0\,,\qquad\Gamma_{T}\geq\absolutevalue{m_{t}}\,.$
(67)

Indeed, the generator $\widetilde{\mathcal{L}}(t)$ will also be Bloch
diagonal, and the conditions (67) follow from
$-(\widetilde{\mathcal{L}}(t)+\widetilde{\mathcal{L}}^{T}(t))\geq 0$. On the
other hand, the conditions for $CP$-divisibility in the unital case simplify
to

$\Gamma_{T}(t)\geq\frac{\Gamma_{L}(t)}{2}\geq\absolutevalue{m_{t}}\,.$ (68)

###  IV.1 Non-self dual dynamics from self-dual generator.

As already stressed, generators of dynamical maps in the class (IV) are also
in the same class; moreover, we shall focus upon purely dissipative generator;
for qubits, this amounts to requiring that
$\mathcal{L}_{t}^{\phantom{\ddagger}}=\mathcal{L}_{t}^{\ddagger}$ (see Lemma
1). From the general form (IV), the dual of $\Phi^{(A,\lambda,\mu)}$ acts on a
matrix $X\in M_{2}(\mathbb{C})$ as

$\displaystyle(\Phi^{(A,\lambda,\mu)})^{\ddagger}[X]=$
$\displaystyle\sum_{ij}A_{ji}E_{ij}XE_{ji}+\overline{\lambda}E_{00}XE_{11}$
$\displaystyle\hskip-28.45274pt+\lambda E_{11}XE_{00}+\mu
E_{00}X^{T}E_{11}+\overline{\mu}E_{11}X^{T}E_{00}\ .$

Then, self-duality corresponds to $A=A^{T}$ and $\lambda\in\mathbb{R}$.
Therefore, a generator $\mathcal{L}_{t}=\Phi^{(B(t),l_{t},m_{t})}$ is self-
dual iff $\gamma_{+}(t)=\gamma_{-}(t)$, implying unitality of the dynamics,
and $l_{t}$ is real:

$l_{t}=-\Gamma_{T}(t)\,,\qquad\omega(t)=0\,.$ (69)

Let $\lambda_{t}=\absolutevalue{\lambda_{t}}e^{i\varphi_{t}}$ and
$\mu_{t}=\absolutevalue{\mu_{t}}e^{i\theta_{t}}$,
$\varphi_{t},\theta_{t}\in\mathbb{R}$; then, (58) and (69) imply the following
relation between the phases and the moduli of $\lambda_{t}$ and $\mu_{t}$ of
${\Lambda}_{t}=\Phi^{(A(t),\lambda_{t},\mu_{t})}$,

$\dot{\varphi}_{t}\absolutevalue{\lambda_{t}}^{2}=\dot{\theta}_{t}\absolutevalue{\mu_{t}}^{2}\,.$
(70)

Notice that this fixes $\dot{\varphi}_{t=0}=0$. We now provide a construction
for a $P$-divisible, non self-dual dynamics arising from a self-dual, purely
dissipative generator. Let us define

$g_{t}:=\absolutevalue{\lambda_{t}}+\absolutevalue{\mu_{t}}\,,\qquad
h_{t}:=\absolutevalue{\lambda_{t}}-\absolutevalue{\mu_{t}}\,,$ (71)

with $0<h_{t}\leq g_{t}$, $g_{0}=h_{0}=1$. Then, (60) can be recast as

$\Gamma_{T}(t)=-\frac{1}{2}\frac{\partial_{t}(g_{t}h_{t})}{g_{t}h_{t}}=-\frac{\dot{g}_{t}}{2g_{t}}-\frac{\dot{h}_{t}}{2h_{t}}\,,$
(72)

which must be positive for $P$-divisibility. On the other hand, substituting
(71) and the self-duality condition (70) into the second of (58), one recasts
$\absolutevalue{m_{t}}$ as

$\displaystyle\absolutevalue{m_{t}}=\absolutevalue{\frac{\dot{g}_{t}}{2g_{t}}-\frac{\dot{h}_{t}}{2h_{t}}+i\,\dot{\theta}_{t}\,\frac{g_{t}-h_{t}}{g_{t}+h_{t}}}\,.$

Taking the square of the inequality $\Gamma_{T}(t)\geq\absolutevalue{m_{t}}$,
the following necessary and sufficient conditions for $P$-divisibility can be
finally found,

$|\dot{\theta}_{t}|^{2}\left(\frac{g_{t}-h_{t}}{g_{t}+h_{t}}\right)^{2}\leq\frac{\dot{g}_{t}}{g_{t}}\frac{\dot{h}_{t}}{h_{t}}\,.$
(73)

From (72) and (73) one deduces, in particular, that $g_{t}$, $h_{t}$ must be
monotonically decreasing.

######  Example 5.

Let $C\geq 0$ and define $\Lambda_{t}=\Phi^{(A(t),\lambda_{t},\mu_{t})}$ by

$\displaystyle\absolutevalue{\lambda_{t}}$
$\displaystyle=e^{-2t}\cosh(t)\,,\qquad\varphi_{t}=C\tanh^{3}(t)\,,$ (74)
$\displaystyle\absolutevalue{\mu_{t}}$
$\displaystyle=e^{-2t}\sinh(t)\,,\qquad\ \theta_{t}=3C\tanh(t)\,,$ (75)
$\displaystyle a_{t}$ $\displaystyle=b_{t}=e^{-t}\,\,\cosh(t)\,,$ (76)

corresponding to the positive and monotonically decreasing functions
$h_{t}=e^{-3t}$, $g_{t}=e^{-t}$. As one easily checks, (56) are satisfied and
the map is thus completely positive. On the other hand, the self-dual
generator is given by

$\displaystyle l_{t}=-\Gamma_{T}(t)$ $\displaystyle=-2\,,\qquad
m_{t}=\sqrt{1+\,r_{t}^{2}}\,e^{i(\theta_{t}+\varphi_{t})}\,,$ (77)
$\displaystyle\gamma_{+}(t)$
$\displaystyle=\gamma_{-}(t)=\frac{\Gamma_{L}(t)}{2}=1\,,$ (78)

where $r_{t}=3C(1-\tanh^{2}(t))\tanh(t)$. Notice that

$\absolutevalue{m_{t}}=\sqrt{1+r_{t}^{2}}\geq 1=\frac{\Gamma_{L}(t)}{2}\,,$

so the dynamics is $CP$-divisible iff $C=0$. On the other hand, the
$P$-divisibility condition (73) reduces to $r_{t}^{2}\leq 3$. Since $r_{t}$
reaches a maximum of $2C/\sqrt{3}$, the dynamics is $P$-divisible iff $C\leq
3/2$.

###  IV.2 Classical map from pure dissipation: loss of $P$-divisibility.

For the class $\Lambda_{t}=\Phi^{(A(t),\lambda_{t},\mu_{t})}$, clearly, there
exists a preferred basis $E_{00},E_{11}$, yielding

$T(t)=A(t)\,,$ (79)

whose $P$-divisibility is necessary to ensure that of $\Lambda_{t}$ (see
(61)). We shall now see that choosing a different basis generally breaks
$P$-divisibility of $T(t)$ even for purely dissipative generators.
Interestingly, for a suitable basis, this can occur with $T(t)$ being
invertible for all $t\geq 0$. A classical reduction of
$\Lambda_{t}=\Phi^{(A(t),\lambda_{t},\mu_{t})}$ builds upon fixing a rank-1
projector $P_{\boldsymbol{n}}=\sum_{i,j=0,1}P_{\boldsymbol{n}}^{ij}E_{ij}$ and
considering

$\displaystyle
T_{00}(t)=\Tr(P_{\boldsymbol{n}}\Lambda_{t}[P_{\boldsymbol{n}}])=\sum_{ij}A_{ij}(t)$
$\displaystyle P_{\boldsymbol{n}}^{jj}P_{\boldsymbol{n}}^{ii}$
$\displaystyle\hskip-19.91684pt+2\real(\lambda_{t})|P_{\boldsymbol{n}}^{01}|^{2}+2\real(\mu_{t}(P_{\boldsymbol{n}}^{10})^{2})\,.$
(80)

which is sufficient to construct $T(t)$ in the unital case. Letting
$\boldsymbol{n}=(\sin(\chi)\cos(\xi),\sin(\chi)\sin(\xi),\cos(\chi))$, (IV.2)
can be then recast as

$\displaystyle T_{00}(t)=$
$\displaystyle\frac{1}{2}\big{(}1+(2a_{t}-1)\cos^{2}(\chi)+\absolutevalue{\lambda_{t}}\cos(\varphi_{t})\sin^{2}(\chi)$
$\displaystyle+\absolutevalue{\mu_{t}}\sin(\chi)\cos(\theta_{t}+2\xi)\big{)}\,.$
(81)

As we already noted, for $\chi=0$, $P=E_{00}$, $T_{00}(t)=a_{t}$ yielding a
$P$-divisible process. Considering instead ${\chi={\pi}/{2}}$, one has

$\displaystyle
T_{00}(t)=\frac{1}{2}\big{(}1+\absolutevalue{\lambda_{t}}\cos(\varphi_{t})+\absolutevalue{\mu_{t}}\cos(\theta_{t}+2\xi)\big{)}\,.$
(82)

Recalling that $P$-divisibility of the classical reduction amounts to check
condition (27), let $\xi=\pi/4$ and consider the map of Example 5,

$\displaystyle 2T_{00}(t)-1=$
$\displaystyle\,e^{-2t}\cosh(t)\cos(C\tanh^{3}(t))$
$\displaystyle-e^{-2t}\sinh(t)\sin(3C\tanh(t))\,.$ (83)

Choosing $C=3/2$, (73) saturates for some $t$. Noting that
$0\leq\varphi\leq\pi/2$ and $0\leq\theta\leq 3\pi/2$, so that
$\sin(\theta_{t})<0$ $\,\forall\,t>\rm{artanh}(2\pi/9)$, one can verify that
$2T_{00}(t)-1>0,\,\forall t\geq 0$. Thus, $T(t)$ is invertible. Nevertheless,
as displayed in Fig. 1, it is not monotonically decreasing, since
$\dot{T}_{00}(t)$ can become positive. Hence, $f_{t}$ also becomes positive
implying that $T(t)$ is not $P$-divisible.

Figure 1: Plot of $T_{00}(t)$ from Example 5, with $C=3/2$ and $\xi=\pi/4$
corresponding to the Bloch vector $(\sqrt{2}/2,\sqrt{2}/2,0)$ which defines
the reference classical basis.

##  V Coherence-assisted backflow of information

We shall now interpret the loss of $P$-divisibility by the classical reduction
$T(t)$ in terms of information backflow. In the so-called $BLP$ approach to
quantum non-Markovianity [7], the Holevo-Helstrom distinguishability is
interpreted as “internal information” of a quantum system as embodied by two
states $\rho$ and $\sigma$ under the dynamics $\Lambda_{t}$:

$\mathcal{I}^{q}_{t}(\rho,\sigma;\mu)=\norm{\Lambda_{t}[\Delta_{\mu}(\rho,\sigma)]}_{1}\,,$
(84)

where

$\Delta_{\mu}(\rho,\sigma)=\mu\rho-(1-\mu)\sigma$ (85)

is the so-called Helstrom matrix, with $\mu\in[0,1]$ being the prior
probability in preparing $\rho$ and $\sigma$, so that (84) reduces for
$\mu=1/2$ to the standard trace distance

$D(\rho,\sigma)=\frac{1}{2}\norm{\rho-\sigma}_{1}\ .$ (86)

In a commutative setting as in Remark 1, the Helstrom matrix becomes
$\boldsymbol{\delta}_{\mu}(\boldsymbol{r},\boldsymbol{s})=\mu\boldsymbol{r}-(1-\mu)\boldsymbol{s}$,
with $\boldsymbol{r},\boldsymbol{s}$ probability vectors, and its trace norm
reduces to the so-called Kolmogorov distance

$\norm{\boldsymbol{\delta}_{\mu}(\boldsymbol{r},\boldsymbol{s})}_{\ell_{1}}=\sum_{i}|\mu
r_{i}-(1-\mu)s_{i}|\,,$ (87)

with respect to the $\ell_{1}$-norm
$\norm{\boldsymbol{x}}_{\ell_{1}}=\sum_{i}\absolutevalue{x_{i}}$. Therefore,
the internal information
$\mathcal{I}^{cl}_{t}(\boldsymbol{r},\boldsymbol{s};\mu)$ of the classical
reduction is given by

$\mathcal{I}^{cl}_{t}(\boldsymbol{r},\boldsymbol{s};\mu)=\norm{T(t)\boldsymbol{\delta}_{\mu}(\boldsymbol{r},\boldsymbol{s})}_{\ell_{1}}\
.$ (88)

For invertible quantum dynamics, $P$-divisibility is known to be equivalent
[20] to the monotonicity in time of $\mathcal{I}^{q}_{t}(\rho,\sigma;\mu)$,

$\partial_{t}\norm{\Lambda_{t}[\Delta_{\mu}(\rho,\sigma)]}_{1}\leq 0\,,$ (89)

for any choice of the initial Helstrom matrix (not true in the non invertible
case [21]). Similarly, for classical invertible dynamics, $P$-divisibility is
equivalent to

$\partial_{t}\norm{T(t)\boldsymbol{\delta}_{\mu}}_{\ell_{1}}\leq 0\,,$ (90)

for all choices of $\boldsymbol{\delta}_{\mu}\in\mathbb{R}^{d}$. We stress
that lack of $P$-divisibility straightforwardly implies a violation of (90)
only if $T(t)$ is invertible. Furthermore, in the usual open party scenario, a
system is immersed in an environment and the compound state then evolves
unitarily, $\rho_{SE}(t)=\mathcal{U}^{SE}_{t}[\rho\otimes\rho_{E}]$, so that
the distinguishability of two system-environment states is constant and one
defines accordingly an “external information” relative to two states
$\rho_{S}$ and $\sigma_{S}$ of a quantum system $S$ as

$\displaystyle\mathcal{E}^{q}_{t}(\rho_{S},\sigma_{S};\mu)$
$\displaystyle=\norm{\mathcal{U}_{t}^{SE}[\Delta_{\mu}(\rho_{S},\sigma_{S})\otimes\rho_{E}]}_{1}-\mathcal{I}^{q}_{t}(\rho_{S},\sigma_{S};\mu)$
$\displaystyle=\norm{\Delta_{\mu}(\rho_{S},\sigma_{S})\otimes\rho_{E}}_{1}-\mathcal{I}^{q}_{t}(\rho_{S},\sigma_{S};\mu)\
.$

Indeed, the trace-norm is invariant under unitary transformations, yielding
$\dot{\mathcal{E}^{q}_{t}}(\rho_{S},\sigma_{S};\mu)=-\dot{\mathcal{I}^{q}_{t}}(\rho_{S},\sigma_{S};\mu)$.
Thus, an increase of the internal information leads necessarily to a decrease
of the external one. In the $BLP$ approach, this is interpreted as backflow of
information from the environment to the system. Such an interpretation, which
involves the degrees of freedom of the environment, is also supported by the
observation [10, 22, 23] that the difference of internal quantum informations
at different times,

$\Delta\mathcal{I}^{q}_{t,s}(\rho_{S},\sigma_{S};\mu):=\mathcal{I}^{q}_{t}(\rho_{S},\sigma_{S};\mu)-\mathcal{I}^{q}_{s}(\rho_{S},\sigma_{S};\mu)\
,$

can be upper-bounded by means of trace distances (86) involving the marginal
density matrices $\rho_{S}(t)=\Tr_{E}\rho_{SE}(t)$,
$\rho_{E}(t)=\Tr_{S}\rho_{SE}(t)$. Namely,

$\displaystyle\Delta\mathcal{I}^{q}_{t,s}\leq$
$\displaystyle\,\,2\mu\,D(\rho_{SE}(s),\rho_{S}(s)\otimes\rho_{E}(s))$
$\displaystyle+2(1-\mu)D(\sigma_{SE}(s),\sigma_{S}(s)\otimes\sigma_{E}(s))\,$
(91) $\displaystyle+2\min\\{\mu,1-\mu\\}D(\rho_{E}(s),\sigma_{E}(s))\ .$

It follows that $\Delta\mathcal{I}^{q}_{t,s}(\rho_{S},\sigma_{S};\mu)>0$ can
only occur if system environment-correlations are present at time $s$ or if
the environment marginals differ at time $s$.

Let us now consider a decohering map $\mathbb{D}[X]=\sum_{i}P_{i}XP_{i}\,,$
with respect to a maximally Abelian subalgebra $\mathcal{P}=\\{P_{i}\\}_{i}$,
and a Helstrom matrix
$\Delta_{\mu}(\rho_{\boldsymbol{r}},\sigma_{\boldsymbol{s}})\in\mathcal{P}$
with $\rho_{\boldsymbol{r}}=\sum_{i}r_{i}P_{i}$ and
$\sigma_{\boldsymbol{s}}=\sum_{i}s_{i}P_{i}$.

Although the chosen density matrices commute and are incoherent classical
states with respect to $\mathcal{P}$, the dynamics $\Lambda_{t}$ takes them
out of the commutative algebra $\mathcal{P}$ and generates non-vanishing
coherences, while the classical reduction to $\mathcal{P}$ eliminates them.
Notice that the classical internal information of the classical reduction to
$\mathcal{P}$ of the quantum dynamics reads

$\mathcal{I}_{t}^{cl}(\boldsymbol{r},\boldsymbol{s};\mu)=\norm{T(t)\boldsymbol{\delta}_{\mu}(\boldsymbol{r},\boldsymbol{s})}_{\ell_{1}}=\norm{\mathbb{D}\Lambda_{t}[\Delta_{\mu}(\rho_{\boldsymbol{r}},\sigma_{\boldsymbol{s}})]}_{1}\
.$ (92)

Let us then introduce the difference between the quantum and classical
internal information relative to two classical states:

$\displaystyle\hskip-14.22636pt\mathcal{C}_{t}(\boldsymbol{r},\boldsymbol{s};\mu)=\mathcal{I}^{q}_{t}(\rho_{\boldsymbol{r}},\rho_{\boldsymbol{s}};\mu)-\mathcal{I}^{cl}_{t}(\boldsymbol{r},\boldsymbol{s};\mu)$
$\displaystyle=\norm{\Lambda_{t}[\Delta_{\mu}(\rho_{\boldsymbol{r}},\sigma_{\boldsymbol{s}})]}_{1}-\norm{\mathbb{D}\Lambda_{t}[\Delta_{\mu}(\rho_{\boldsymbol{r}},\sigma_{\boldsymbol{s}})]}_{1}\
.$ (93)

Since $\mathbb{D}$ is a contraction, the quantity
$\mathcal{C}_{t}(\boldsymbol{r},\boldsymbol{s};\mu)$ cannot be negative;
moreover, it allows to write the quantum internal information of two quantumly
evolving classical states as

$\mathcal{I}^{q}_{t}(\rho_{\boldsymbol{r}},\rho_{\boldsymbol{s}};\mu)=\mathcal{I}_{t}^{cl}(\boldsymbol{r},\boldsymbol{s};\mu)+\mathcal{C}_{t}(\boldsymbol{r},\boldsymbol{s};\mu)\
.$ (94)

It is thus appropriate to name
$\mathcal{C}_{t}(\boldsymbol{r},\boldsymbol{s};\mu)$ coherent internal
information. Then, the $P$-divisibility of $\Lambda_{t}$ is equivalent to
$\mathcal{I}^{q}_{t}(\rho_{\boldsymbol{r}},\rho_{\boldsymbol{s}};\mu)$
monotonically decreasing in time, namely to the information contained in the
system leaking towards the environment and never coming back. Hence, for
$t\geq s\geq 0$,

$\mathcal{I}_{t}^{cl}(\boldsymbol{r},\boldsymbol{s};\mu)+\mathcal{C}_{t}(\boldsymbol{r},\boldsymbol{s};\mu)\leq\mathcal{I}_{s}^{cl}(\boldsymbol{r},\boldsymbol{s};\mu)+\mathcal{C}_{s}(\boldsymbol{r},\boldsymbol{s};\mu)\
.$ (95)

Moreover, one can upper-bound the variation of the classical internal
information between times $s$ and $t\geq s$ as (see Appendix B)

$\displaystyle\Delta\mathcal{I}_{t,s}^{cl}(\boldsymbol{r},\boldsymbol{s};\mu):=\mathcal{I}_{t}^{cl}(\boldsymbol{r},\boldsymbol{s};\mu)-\mathcal{I}_{s}^{cl}(\boldsymbol{r},\boldsymbol{s};\mu)$
$\displaystyle\hskip
28.45274pt\leq\,\mathcal{C}_{s}(\boldsymbol{r},\boldsymbol{s};\mu)$
$\displaystyle\hskip
28.45274pt\leq\,\mu\,C_{\ell_{1}}(\Lambda_{s}[\rho_{\boldsymbol{r}}])\,+\,(1-\mu)\,C_{\ell_{1}}(\Lambda_{s}[\rho_{\boldsymbol{s}}])\,,$
(96)

where $C_{\ell_{1}}$ denotes the so-called $\ell_{1}$ norm of coherence of a
state [24, 25],

$C_{\ell_{1}}(\rho):=\sum_{i\neq j}\absolutevalue{\rho_{ij}}\,,$ (97)

with respect to the classical subalgebra $\mathcal{P}$. Therefore,
$C_{\ell_{1}}(\Lambda_{s}[\rho_{\boldsymbol{r}}])$ and
$C_{\ell_{1}}(\Lambda_{s}[\rho_{\boldsymbol{s}}])$ measure the amount of
coherence produced by the dynamics at time $s$ acting on the diagonal states
$\rho_{\boldsymbol{r}}$ and $\rho_{\boldsymbol{s}}$.

In analogy with (V), from (96), we can thus interpret the revival of the
classical internal information,
$\Delta\mathcal{I}_{t,s}^{cl}(\boldsymbol{r},\boldsymbol{s};\mu)>0$, between
times $s$ and $t\geq s$ as a classical backflow of information, which can
occur only if a certain degree of quantum coherence has been built up to time
$s$ in the quantumly evolving classical states. Hence, in the classical
reduction of the quantum evolution of pairs of commuting quantum states,
quantum coherences play an information storing role as the environment does in
the quantum scenario, as emerges by comparing the r.h.s of (96) and (V).

######  Remark 6.

Our previous considerations focussed on checking when $P$-divisibility of a
purely dissipative qubit dynamical map is inherited by its classical reduction
and found that in some cases it can be lost, namely the classical reduction
can become non $P$-divisible. We now like to comment that the contrary can
also occur; namely, the classical reduction of a non $P$-divisible purely
dissipative qubit dynamics can become $P$-divisible. In Fig. 2, an instance of
such behaviour is shown by the maps of Example 5 for a suitable $C>3/2$ for
which $\Lambda_{t}$ is not $P$-divisible. Interestingly, the corresponding
backflow of information can be witnessed by suitable orthogonal projections
$P_{\pm\boldsymbol{n}}$ such that

$\derivative{t}\frac{1}{2}\norm{\Lambda_{t}[P_{\boldsymbol{n}}-P_{-\boldsymbol{n}}]}_{1}=\derivative{t}\|\widetilde{\Lambda}(t)\boldsymbol{n}\|>0\,,$

for some $t>0$, as shown by the non-monotonic behaviour of
$\mathcal{I}^{q}_{t}(P_{\boldsymbol{n}},P_{-\boldsymbol{n}};1/2)=\|\widetilde{\Lambda}(t)\boldsymbol{n}\|$
in the inset of Fig. 2. Nevertheless, the stochastic process $T(t)$ defined
through the subalgebra $\mathcal{P}_{\boldsymbol{n}}$ generated by them is
$P$-divisible. This means that, unlike the quantum dynamics it originates
from, such a $T(t)$ cannot exhibit classical backflow of information. The
orthogonal projections $P_{\pm\boldsymbol{n}}$ provide classical probabilities
and, concerning information backflow, they are concrete instances of the
following behaviour. Suppose that there exist
$\rho_{\boldsymbol{r}},\rho_{\boldsymbol{s}}$ in a commutative subalgebra
$\mathcal{P}$, that is classical probability distributions, and $t>s>0$ such
that

$\mathcal{I}^{q}_{t}(\rho_{\boldsymbol{r}},\rho_{\boldsymbol{s}};\mu)-\mathcal{I}^{q}_{s}(\rho_{\boldsymbol{r}},\rho_{\boldsymbol{s}};\mu)>0\,.$

In such case, $\Lambda_{t}$ is clearly not $P$-divisible since it shows
backflow of information. Using (95), one now has

$\mathcal{I}_{t}^{cl}(\boldsymbol{r},\boldsymbol{s};\mu)+\mathcal{C}_{t}(\boldsymbol{r},\boldsymbol{s};\mu)>\mathcal{I}_{s}^{cl}(\boldsymbol{r},\boldsymbol{s};\mu)+\mathcal{C}_{s}(\boldsymbol{r},\boldsymbol{s};\mu)\,.$
(98)

In addition, suppose that the classical dynamics $T(t)$, obtained by
restricting the quantum dynamics $\Lambda_{t}$ on the same subalgebra
$\mathcal{P}$, is $P$-divisible, yielding thus

$\mathcal{C}_{t}(\boldsymbol{r},\boldsymbol{s};\mu)-\mathcal{C}_{s}(\boldsymbol{r},\boldsymbol{s};\mu)>\mathcal{I}_{s}^{cl}(\boldsymbol{r},\boldsymbol{s};\mu)-\mathcal{I}_{t}^{cl}(\boldsymbol{r},\boldsymbol{s};\mu)\geq
0\,.$ (99)

Therefore, the backflow of information from the environment to the open system
only affects the coherent contribution to $\mathcal{I}_{t}^{q}$ and cannot be
witnessed by the classical reduced map.

Figure 2: Global maximum of $f_{t}$ as defined in (27), for the classical
reduction of the map of Example 5, with $C=1.64$ and reference classical
algebra defined by the projector $P_{\boldsymbol{n}}$ with
$\boldsymbol{n}=(\cos(\xi),\sin(\xi),0)$ and $\xi=\pi/8$. Numerically, it is
checked that $f_{t}$ reaches a maximum of $-0.006$, so that $f_{t}<0$ for all
$t$ and $T(t)$ is $P$-divisible. In the inset, the non-monotonic behaviour of
$\mathcal{I}_{t}^{q}(P_{\boldsymbol{n}},P_{-\boldsymbol{n}};1/2)$ is
displayed, corresponding to a backflow of information that thus involves only
the coherences w.r.t. such basis.

##  VI Conclusions

Generic quantum dynamics restricted to commutative algebras of orthogonal
projections generate quantum coherences among them. With respect to that
commutative algebra, by restricting to the diagonal components of its time-
evolving orthogonal projections, one obtains a so-called classical reduction
of the quantum dynamics, of course depending on the chosen set of orthogonal
projections. In the manuscript, we have studied whether and how the
divisibility properties of time-dependent, non-Markovian qubit quantum
dynamics are inherited by their classical reductions. We have done it by
classically reducing the dynamics and by then studying the generator of the
classical reduction. For $P$-divisible and unital qubit dynamics we showed
that 1) the Hamiltonian contributions to the time-dependent quantum generators
generally give rise to qubit quantum dynamics with non $P$-divisible classical
reductions; 2) purely dissipative, self-dual dynamics always have
$P$-divisible classical reductions while 3) purely dissipative, non self-dual
dynamics may give rise to non $P$-divisible classical reductions. Finally, we
have argued that the lack of $P$-divisibility in classical reductions can be
ascribed to information created by the quantum dynamics which is stored in the
quantum coherences and then released back into the classical component of the
dynamics. Though this behaviour is somewhat typical of unitary quantum
dynamics, yet it can also emerge from purely dissipative quantum evolutions
due to the presence of a non-trivially time-ordered Dyson expansion.

## Acknowledgments

The authors heartily thank Roberto Floreanini for his comments on the paper.
FB and GN acknowledge financial support from the PNRR MUR project
PE0000023-NQSTI. DC was supported by the Polish National Science Center
project No. 2018/30/A/ST2/00837.

##  Appendix A Proof of Proposition 3

###### Proof.

Let $P=\outerproduct{\psi}{\psi}$ be a generic $2\times 2$ projector, with
$\ket{\psi}=(w_{1},w_{2})=(w_{1},\absolutevalue{w_{2}}e^{-i\frac{\Omega}{2}})$,
$w_{1},\Omega\in\mathbb{R}$,
$\absolutevalue{w_{1}}^{2}+\absolutevalue{w_{2}}^{2}=1$ and let $Q$ be its
orthogonal complement $Q=\mathds{1}_{2}-P$. Letting also
$m_{t}=\absolutevalue{m_{t}}e^{i\chi_{t}}$, one has

$\displaystyle\Tr(Q\mathcal{L}_{t}[P])$
$\displaystyle=\sum_{ij}B_{ij}(t)Q_{ii}P_{jj}+l_{t}P_{01}Q_{10}+\overline{l_{t}}P_{10}Q_{01}$
$\displaystyle\hskip
28.45274pt+m_{t}P_{10}Q_{10}+\overline{m_{t}}P_{01}Q_{01}$
$\displaystyle=\gamma_{-}(t)\absolutevalue{w_{1}}^{4}+\gamma_{+}(t)\absolutevalue{w_{2}}^{4}+2(\Gamma_{T}(t)-\frac{\Gamma_{L}(t)}{2})\absolutevalue{w_{1}}^{2}\absolutevalue{w_{2}}^{2}$
$\displaystyle\hskip
28.45274pt-2\absolutevalue{m_{t}}\absolutevalue{w_{1}}^{2}\absolutevalue{w_{2}}^{2}\cos(\chi_{t}-\Omega).$
(100)

For the sufficiency part, let $\gamma_{\pm}(t)\geq 0$ and

$\mathcal{G}(t):=\Gamma_{T}(t)-\frac{\Gamma_{L}(t)}{2}+\sqrt{\gamma_{+}(t)\gamma_{-}(t)}-\absolutevalue{m_{t}}\geq
0\,.$

Then,

$\displaystyle\Tr(Q\mathcal{L}_{t}[P])=\big{(}\sqrt{\gamma_{-}(t)}\absolutevalue{w_{1}}^{2}-\sqrt{\gamma_{+}(t)}\absolutevalue{w_{2}}^{2}\big{)}^{2}$
$\displaystyle+2\big{(}\Gamma_{t}(t)-\frac{\Gamma_{L}(t)}{2}+\sqrt{\gamma_{+}(t)\gamma_{-}(t)}\big{)}\absolutevalue{w_{1}}^{2}\absolutevalue{w_{2}}^{2}$
$\displaystyle-2\,\absolutevalue{m_{t}}\cos(\chi_{t}-\Omega)\big{)}\absolutevalue{w_{1}}^{2}\absolutevalue{w_{2}}^{2}$
$\displaystyle\geq\big{(}\sqrt{\gamma_{-}(t)}\absolutevalue{w_{1}}^{2}-\sqrt{\gamma_{+}(t)}\absolutevalue{w_{2}}^{2}\big{)}^{2}+2\mathcal{G}(t)\absolutevalue{w_{1}}^{2}\absolutevalue{w_{2}}^{2}$
$\displaystyle\geq 0\,,$

so that $P$-divisibility follows from Theorem 2. To prove the necessity part,
first notice that, asking for $\Tr(Q\mathcal{L}_{t}[P])\geq 0$ the choices
$w_{1}=1$, and $w_{2}=1$ imply that $\gamma_{\pm}(t)\geq 0$. Choose instead,
for fixed $t\geq 0$,

$\displaystyle\absolutevalue{w_{1}}^{2}=\frac{\sqrt{\gamma_{+}(t)}}{\sqrt{\gamma_{+}(t)}+\sqrt{\gamma_{-}(t)}}\,,\
\absolutevalue{w_{2}}^{2}=\frac{\sqrt{\gamma_{-}(t)}}{\sqrt{\gamma_{+}(t)}+\sqrt{\gamma_{-}(t)}}\,,$

and $\Omega=\chi_{t}$ so that, from (100),

$\displaystyle
0\leq\Tr(Q\mathcal{L}_{t}[P])=2\mathcal{G}(t)\absolutevalue{w_{1}}^{2}\absolutevalue{w_{2}}^{2},$

which implies (62). For $CP$-divisibility, a necessary and sufficient
condition for $\mathcal{L}_{t}$ to be in the GKSL form for all $t\geq 0$ is
[26, 27]

$Y_{t}\equiv(\mathds{1}_{4}-P_{2}^{+})\mathcal{L}_{t}\otimes\mathrm{id}_{2}[P_{2}^{+}](\mathds{1}_{4}-P_{2}^{+})\geq
0\,,$ (101)

Noting that

$Y_{t}=\begin{pmatrix}\frac{2\Gamma_{T}(t)-\Gamma_{L}(t)}{4}&0&0&-\frac{2\Gamma_{T}(t)-\Gamma_{L}(t)}{4}\\\
0&\gamma_{+}(t)&m_{t}&0\\\ 0&\overline{m_{t}}&\gamma_{-}(t)&0\\\
-\frac{2\Gamma_{T}(t)-\Gamma_{L}(t)}{4}&0&0&\frac{2\Gamma_{T}(t)-\Gamma_{L}(t)}{4}\\\
\end{pmatrix}.$

which is positive iff conditions (63) and (64) are verified. ∎

##  Appendix B Proof of inequaility (96)

From (95),

$\displaystyle\Delta\mathcal{I}_{t,s}^{cl}(\boldsymbol{r},\boldsymbol{s};\mu)$
$\displaystyle=\mathcal{I}_{t}^{cl}(\boldsymbol{r},\boldsymbol{s};\mu)-\mathcal{I}_{s}^{cl}(\boldsymbol{r},\boldsymbol{s};\mu)$
$\displaystyle\leq\mathcal{C}_{s}(\boldsymbol{r},\boldsymbol{s};\mu)-\mathcal{C}_{t}(\boldsymbol{r},\boldsymbol{s};\mu)$
$\displaystyle\leq\mathcal{C}_{s}(\boldsymbol{r},\boldsymbol{s};\mu)\,,$

since $\mathcal{C}_{s}(\boldsymbol{r},\boldsymbol{s};\mu)\geq 0$. Let then
$\mathbb{D}^{\perp}:={\rm id}-\mathbb{D}$ and apply twice the triangle
inequality, so to get

$\displaystyle\mathcal{C}_{s}(\boldsymbol{r},\boldsymbol{s};\mu)$
$\displaystyle\leq\|\mathbb{D}^{\perp}\Lambda_{s}[\Delta_{\mu}(\boldsymbol{r},\boldsymbol{s})]\|_{1}$
$\displaystyle\leq\,\mu\,\|\mathbb{D}^{\perp}\Lambda_{s}[\rho_{\boldsymbol{r}}]\|_{1}\,+\,(1-\mu)\,\|\mathbb{D}^{\perp}\Lambda_{s}[\rho_{\boldsymbol{s}}]\|_{1}\
.$ (102)

Noting that ${\norm{\outerproduct{i}{j}}_{1}=1}$, with $\ket{i},\ket{j}$
belonging to the reference orthonormal basis, one applies the triangle
inequality once again to the r.h.s. of (102) so to obtain

$\displaystyle\Delta\mathcal{I}_{t,s}^{cl}(\boldsymbol{r},\boldsymbol{s};\mu)$
$\displaystyle\leq\mu\,\|\mathbb{D}^{\perp}\Lambda_{s}[\rho_{\boldsymbol{r}}]\|_{1}\,+\,(1-\mu)\,\|\mathbb{D}^{\perp}\Lambda_{s}[\rho_{\boldsymbol{s}}]\|_{1}\
$ $\displaystyle\leq\mu\
C_{\ell_{1}}(\Lambda_{s}[\rho_{\boldsymbol{r}}])\,+\,(1-\mu)\
C_{\ell_{1}}(\Lambda_{s}[\rho_{\boldsymbol{s}}])\,.$

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