# On the use of total state decompositions for the study of reduced dynamics

Andrea Smirne<EMAIL_ADDRESS>Dipartimento di Fisica “Aldo
Pontremoli”, Università degli Studi di Milano, via Celoria 16, 20133 Milan,
Italy  Istituto Nazionale di Fisica Nucleare, Sezione di Milano, via Celoria
16, 20133 Milan, Italy  Nina Megier<EMAIL_ADDRESS>Dipartimento di
Fisica “Aldo Pontremoli”, Università degli Studi di Milano, via Celoria 16,
20133 Milan, Italy  Istituto Nazionale di Fisica Nucleare, Sezione di Milano,
via Celoria 16, 20133 Milan, Italy  International Centre for Theory of Quantum
Technologies (ICTQT), University of Gdańsk, 80-308 Gdańsk, Poland  Bassano
Vacchini<EMAIL_ADDRESS>Dipartimento di Fisica “Aldo
Pontremoli”, Università degli Studi di Milano, via Celoria 16, 20133 Milan,
Italy  Istituto Nazionale di Fisica Nucleare, Sezione di Milano, via Celoria
16, 20133 Milan, Italy

###### Abstract

The description of the dynamics of an open quantum system in the presence of
initial correlations with the environment needs different mathematical tools
than the standard approach to reduced dynamics, which is based on the use of a
time-dependent completely positive trace preserving (CPTP) map. Here, we take
into account an approach that is based on a decomposition of any possibly
correlated bipartite state as a conical combination involving statistical
operators on the environment and general linear operators on the system, which
allows one to fix the reduced-system evolution via a finite set of time-
dependent CPTP maps. In particular, we show that such a decomposition always
exists, also for infinite dimensional Hilbert spaces, and that the number of
resulting CPTP maps is bounded by the Schmidt rank of the initial global
state. We further investigate the case where the CPTP maps are semigroups with
generators in the Gorini-Kossakowski-Lindblad-Sudarshan form; for two simple
qubit models, we identify the positivity domain defined by the initial states
that are mapped into proper states at any time of the evolution fixed by the
CPTP semigroups.

##  I Introduction

The dynamics of open quantum systems in the presence of initial correlations
with the environment has been attracting a renewed interest, driven by both
fundamental questions and the need for a realistic description of concrete
physical systems [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12]. Relaxing the
assumption that the open system and the environment are uncorrelated at the
initial time, the very existence of a reduced dynamics defined in terms of a
time-dependent map acting on the whole set of open-system states is no longer
guaranteed. Most of the recent theoretical literature on the subject has thus
been focused on how to extend the notion of dynamical maps to the scenario
with initial system-environment correlations, resorting to the identification
of a restricted domain of allowed initial states, and possibly preserving the
notion of complete positivity in such a scenario [13, 14, 15, 16, 17, 18, 19].

Very recently [20], a different approach has been introduced, which, relying
on the frame decomposition of the bipartite initial state, leads to the
definition of a set of completely positive trace preserving (CPTP) time-
dependent maps that describe the evolution of a precisely identified set of
initial states. More specifically, the exploited decomposition relies on the
introduction of a positive frame on the set of the open-system Hilbert-Schmidt
operators and, provided that such a positive frame exists, it allows one to
deal with fully general initial conditions. The number of involved CPTP maps
depends on the chosen initial state and it is anyway bounded by the square of
the dimensionality of the open system. This approach has been used to devise
an efficient control of general open-system evolutions [21] and to
characterize multipartite photonic systems [22]; furthermore, it has been
combined with perturbative projection-operator techniques [23] to derive an
uncoupled system of homogeneous master equations that can be applied under
general initial conditions [24].

In this paper, we introduce two different positive decompositions, one
starting from the generalized Pauli matrices for a generic finite dimension,
and one directly built on any orthonormal basis of a possibly infinite
dimensional Hilbert space. The latter, in particular, explicitly shows that
the description of the open-system evolution in the presence of initial
correlations via a set of time-dependent CPTP maps can be defined in full
generality, for any initial global state and for any dimension of the open-
system Hilbert space. Moreover, in the finite dimensional case, we prove that
the number of needed CPTP maps always coincides with the Schmidt rank of the
initial global state, both for pure and mixed states. This clarifies in a
quantitative way the enhanced complexity needed to describe open-system
dynamics when moving from an initial product state, where one CPTP map is
enough, to initially correlated states. In the latter case, CPTP maps can
still be used, but at the price of increasing their number according to the
Schmidt rank of the initial global state, which will be indeed strictly larger
than one in the presence of correlations. Finally, in the last part of the
paper, we have considered the case in which the CPTP maps are semigroups with
generators in the Gorini-Kossakowski-Lindblad-Sudarshan (GKLS) form. For the
case of qubit Pauli channels, we have derived and investigated an explicit
condition that defines the set of initial open-system states that are mapped
into positive states at any subsequent time of the evolution, when each of the
semigroups is applied to a different term of the conical decomposition
yielding the initial reduced state.

The rest of the paper is organized as follows. In Sec.II, we briefly recall
the framework of open-system dynamics in the presence of initial system-
environment correlations and how it can be described through a family of time-
dependent CPTP maps via the approach introduced in [20], and we further recall
the elements of the theory of frames that we will exploit in the following. In
Sec.III, we show that this approach can be extended to infinite-dimensional
Hilbert spaces of the open-system, and we prove that in the finite dimensional
case the number of required CPTP maps can be linked in full generality to the
Schmidt rank of the initial global state. In Sec.IV, we specialize our
analysis to the case were the CPTP maps are semigroups and we give a detailed
characterization of initial reduced states compatible with such a choice, in
the case of qubit Pauli channels. Finally, in Sec.V, the conclusions of our
work are presented.

##  II One-sided positive decomposition

###  II.1 Open quantum system dynamics via completely positive maps

We consider the standard framework to describe the dynamics of an open quantum
system [23]. We have a bipartite global system, consisting of the open system
$S$, which is associated with the Hilbert space $\mathcal{H}_{S}$, and the
environment $E$, associated with $\mathcal{H}_{E}$, so that the global system
is associated with the tensor product Hilbert space
$\mathcal{H}_{S}\otimes\mathcal{H}_{E}$. Moreover, the global system is
assumed to evolve unitarily, i.e., the joint system-environment evolution is
fixed by the unitary operators $U(t)$ (here and in the following, $t_{0}=0$ is
the initial time). Let $\mathcal{S}(\mathcal{H})$ denote the set of
statistical operators, i.e. positive linear operators with unit trace, on
$\mathcal{H}$. The reduced open-system state $\rho_{S}(t)$ can be written at
any time $t$ as a map from
$\mathcal{S}\left(\mathcal{H}_{S}\otimes\mathcal{H}_{E}\right)$ to
$\mathcal{S}(\mathcal{H}_{S})$,

$\rho_{S}(t)=\operatorname{Tr}_{E}\left[U(t)\rho_{SE}U^{\dagger}(t)\right],$
(1)

where $\mbox{Tr}_{E}$ is the partial trace on the environmental degrees of
freedom. The map defined in Eq.(1) is CPTP, but its domain involves the whole
$\mathcal{S}\left(\mathcal{H}_{S}\otimes\mathcal{H}_{E}\right)$, while when
dealing with the evolution of an open quantum system, one would like to
describe the dynamics via maps defined on $\mathcal{S}(\mathcal{H}_{S})$ only.

This goal is achieved in [20], relying on the decomposition of the bipartite
statistical operator
$\rho_{SE}\in\mathcal{S}\left(\mathcal{H}_{S}\otimes\mathcal{H}_{E}\right)$ as

$\rho_{SE}=\sum_{\alpha=1}^{\mathfrak{N}}\omega_{\alpha}D_{\alpha}\otimes\rho_{\alpha},$
(2)

where $\rho_{\alpha}\in\mathcal{S}(\mathcal{H}_{E})$ and $\omega_{\alpha}\geq
0$, while the $D_{\alpha}$ are operators within the set
$\mathcal{L}_{2}(\mathcal{H}_{S})$ of Hilbert-Schmidt operators on
$\mathcal{H}_{S}$, i.e., the trace of the square of their absolute value is
finite; note that these operators are indeed not necessarily positive. We will
call the representation of $\rho_{SE}$ in Eq.(2) one-sided positive
decomposition (OPD), to stress that positive operators on the environmental
side are needed; we will further call cost of the OPD the minimum number
$\mathfrak{N}$ of terms with which the sum in Eq.(2) can be expressed. If the
$D_{\alpha}$ are positive operators, $\rho_{SE}$ in Eq.(2) is a separable
state [25]; if, in addition, the $D_{\alpha}$ or the $\rho_{\alpha}$ or both
are given by a family of orthogonal projections, $\rho_{SE}$ is a zero discord
state [26, 27, 8]. On the other hand, let us stress once more that general
bipartite states, including any kind of classical or quantum correlations, can
be decomposed via (infinitely many) OPDs.

From the point of view of the description of open-system dynamics, the key
advantage of the OPD is that, starting from it, the reduced state at time $t$
can always be obtained by means of a family of time-dependent CPTP maps
defined on operators acting on $\mathcal{H}_{S}$ only. Replacing Eq.(2) into
Eq.(1), we have in fact

$\displaystyle\rho_{S}(t)$
$\displaystyle=\sum_{\alpha=1}^{\mathfrak{N}}\omega_{\alpha}\Phi_{\alpha}(t)[D_{\alpha}],$
(3)

where

$\displaystyle\Phi_{\alpha}(t):\mathcal{L}_{2}(\mathcal{H}_{S})$
$\displaystyle\to\mathcal{L}_{2}(\mathcal{H}_{S})$ (4) $\displaystyle A$
$\displaystyle\mapsto\Phi_{\alpha}(t)[A]=\operatorname{Tr}_{E}[U(t)A\otimes\rho_{\alpha}U(t)^{\dagger}].$

The initial reduced state

$\rho_{S}=\sum_{\alpha=1}^{\mathfrak{N}}\omega_{\alpha}D_{\alpha}$ (5)

is mapped into the reduced state at time $t$ by the $\mathfrak{N}$ CPTP maps
$\left\\{\Phi_{\alpha}(t)\right\\}_{1,\ldots,\mathfrak{N}}$ on
$\mathcal{L}_{2}(\mathcal{H}_{S})$. An initial product state corresponds to
the case $\mathfrak{N}=1$, which directly reduces to the usual description in
terms of a single CPTP map [23, 28]. The presence of initial correlations
implies that generally $\mathfrak{N}>1$ CPTP maps are needed; on the other
hand, the same set $\left\\{\Phi_{\alpha}(t)\right\\}_{1,\ldots,\mathfrak{N}}$
of maps can be used for all the states connected by any local operation on $S$
[20].

###  II.2 Theory of frames: reconstruction formula and positive frames

Here, we briefly present the main features of the theory of frames that will
be relevant for our purposes, directly referring to the set
$\mathcal{L}_{2}(\mathcal{H}_{S})$; we will further restrict to frames
consisting of a countable set of operators, but, indeed, extensions to
uncountable sets are possible. For a more general treatment of frame theory,
the interested reader is referred to [29, 30].

Consider the Hilbert space of Hilbert-Schmidt operators
$\mathcal{L}_{2}(\mathcal{H}_{S})$ equipped with the scalar product

$\left(A,B\right)=\mbox{Tr}_{S}\left[A^{\dagger}B\right],\quad
A,B\in\mathcal{L}_{2}(\mathcal{H}_{S}),$ (6)

where $\mbox{Tr}_{S}$ denotes the trace over $\mathcal{H}_{S}$, and a family
of operators $\left\\{F_{\alpha}\right\\}_{\alpha\in I}$, where each
$F_{\alpha}\in\mathcal{L}_{2}(\mathcal{H}_{S})$ and the index $\alpha$ takes
values in the countable set $I$, such that

$\sum_{\alpha\in I}|\left(F_{\alpha},A\right)|^{2}<\infty$ (7)

for every $A\in\mathcal{L}_{2}(\mathcal{H}_{S})$. The associated map on
$\mathcal{L}_{2}(\mathcal{H}_{S})$

$\displaystyle\Xi:\mathcal{L}_{2}(\mathcal{H}_{S})$
$\displaystyle\to\mathcal{L}_{2}(\mathcal{H}_{S})$ (8) $\displaystyle A$
$\displaystyle\mapsto\Xi[A]=\sum_{\alpha\in
I}\left(F_{\alpha},A\right)F_{\alpha}$

is called _frame map_ ; the operators $F_{\alpha}$ are indeed not necessarily
orthogonal, nor linearly independent. Now, one says that the family of
operators $\left\\{F_{\alpha}\right\\}_{\alpha\in I}$ is a _frame_ of
$\mathcal{L}_{2}(\mathcal{H}_{S})$ whenever Eq.(7) holds and the corresponding
frame map $\Xi$ satisfies the following lower and upper bound:

$m\|A\|_{2}\leq(A,\Xi[A])\leq M\|A\|_{2},\quad\forall
A\in\mathcal{L}_{2}(\mathcal{H}_{S}),$ (9)

for some $0<m\leq M<\infty$, where
$\|A\|_{2}=\mbox{Tr}_{S}\left[A^{\dagger}A\right]$ is the Hilbert-Schmidt norm
induced by the scalar product in Eq.(6); in particular, the lower bound with
$m$ strictly larger than zero implies that
$\left\\{F_{\alpha}\right\\}_{\alpha\in I}$ spans
$\mathcal{L}_{2}(\mathcal{H}_{S})$. On the whole, the definition of frame is
equivalent to the requirement that $\Xi$ is bounded and invertible, with
bounded inverse $\Xi^{-1}$. Using such properties, so that we can write

$A=\Xi^{-1}\left[\Xi[A]\right]=\sum_{\alpha\in
I}\left(F_{\alpha},A\right)\Xi^{-1}\left[F_{\alpha}\right],$

we decompose any Hilbert-Schmidt operator according to

$A=\sum_{\alpha\in I}(F_{\alpha},A)\widetilde{F}_{\alpha},\quad
A\in\mathcal{L}_{2}(\mathcal{H}_{S}),$ (10)

where we defined

$\widetilde{F}_{\alpha}=\Xi^{-1}[F_{\alpha}].$ (11)

Importantly, Eq.(10) allows us to reconstruct any operator in
$\mathcal{L}_{2}(\mathcal{H}_{S})$ in terms of the elements of the chosen
frame. Indeed, this is a generalization of the reconstruction formula of any
vector of a Hilbert space by means of an orthonormal basis. In fact,
orthonormal bases are a special case of frames, for which $\Xi$ corresponds to
the identity map and thus Eq.(9) holds with $m=M=1$111Whenever Eq.(9) holds
with $m=M=1$, the frame $\left\\{F_{\alpha}\right\\}_{\alpha\in I}$ is called
a Parseval frame, as Eq.(10) reduces to $A=\sum_{\alpha\in
I}(F_{\alpha},A)F_{\alpha}$; interestingly, there are Parseval frames that are
_not_ orthonormal bases..

It is easy to see that the family
$\left\\{\widetilde{F}_{\alpha}\right\\}_{\alpha\in I}$ is itself a frame,
whose frame map coincides with $\Xi^{-1}$, so that the corresponding
reconstruction formula reads

$A=\sum_{\alpha\in I}(\widetilde{F}_{\alpha},A)F_{\alpha},\quad
A\in\mathcal{L}_{2}(\mathcal{H}_{S});$

$\left\\{\widetilde{F}_{\alpha}\right\\}_{\alpha\in I}$ is called the
_canonical dual frame_ of $\left\\{F_{\alpha}\right\\}_{\alpha\in I}$. More in
general, a dual frame of $\left\\{F_{\alpha}\right\\}_{\alpha\in I}$ is a
frame $\left\\{D_{\alpha}\right\\}_{\alpha\in I}$, such that for any operator
$A$ the following reconstruction formula holds:

$A=\sum_{\alpha\in I}(F_{\alpha},A)D_{\alpha}=\sum_{\alpha\in
I}(D_{\alpha},A)F_{\alpha},\quad A\in\mathcal{L}_{2}(\mathcal{H}_{S});$ (12)

as seen, every frame has at least one dual frame, which is the canonical dual
frame.

We can now move back to the bipartite setting that is of interest for us, so
that, besides the open system $S$, we also consider the environment $E$ and we
deal with Hilbert-Schmidt operators on the tensor product
$\mathcal{H}_{S}\otimes\mathcal{H}_{E}$, i.e.,
$O_{SE}\in\mathcal{L}_{2}(\mathcal{H}_{S}\otimes\mathcal{H}_{E})$. Given a
frame of operators referred to the open system,
$\left\\{F_{\alpha}\right\\}_{\alpha\in I}$, and a dual frame
$\left\\{D_{\alpha}\right\\}_{\alpha\in I}$, with
$F_{\alpha},D_{\alpha}\in\mathcal{L}_{2}(\mathcal{H}_{S})$, together with a
frame of environmental operators, $\left\\{X_{\beta}\right\\}_{\beta\in J}$,
along with a dual frame $\left\\{Y_{\beta}\right\\}_{\beta\in J}$, with
$X_{\beta},Y_{\beta}\in\mathcal{L}_{2}(\mathcal{H}_{E})$, one can readily see
that $\left\\{F_{\alpha}\otimes X_{\beta}\right\\}_{\alpha\in I,\beta\in J}$
and $\left\\{D_{\alpha}\otimes Y_{\beta}\right\\}_{\alpha\in I,\beta\in J}$
provide us with a frame of
$\mathcal{L}_{2}(\mathcal{H}_{S}\otimes\mathcal{H}_{E})$ and its dual. We can
thus apply the reconstruction formula to any system-environment Hilbert-
Schmidt operator,

$\displaystyle O_{SE}$ $\displaystyle=\sum_{\alpha\in I,\beta\in
J}(F_{\alpha}\otimes X_{\beta},O_{SE})D_{\alpha}\otimes Y_{\beta}$
$\displaystyle=\sum_{\alpha\in I,\beta\in
J}\mbox{Tr}_{SE}\left[(F^{\dagger}_{\alpha}\otimes
X^{\dagger}_{\beta})O_{SE}\right]D_{\alpha}\otimes Y_{\beta}$
$\displaystyle=\sum_{\alpha\in I}D_{\alpha}\otimes\left\\{\sum_{\beta\in
J}\mbox{Tr}_{E}\left[\mbox{Tr}_{S}\left[(F^{\dagger}_{\alpha}\otimes\mathbb{1}_{E})O_{SE}\right]X^{\dagger}_{\beta}\right]Y_{\beta}\right\\},$

where in the second identity we used the definition of the Hilbert-Schmidt
scalar product of $\mathcal{L}_{2}(\mathcal{H}_{S}\otimes\mathcal{H}_{E})$, as
in Eq.(6) but where the trace is now taken over all the global $S-E$ degrees
of freedom, while in the third identity we used that
$\mbox{Tr}_{SE}\left[(F^{\dagger}_{\alpha}\otimes
X^{\dagger}_{\beta})O_{SE}\right]=\mbox{Tr}_{E}\left[\mbox{Tr}_{S}\left[(F^{\dagger}_{\alpha}\otimes\mathbb{1}_{E})O_{SE}\right]X^{\dagger}_{\beta}\right]$,
with $\mathbb{1}_{E}$ the identity operator on $\mathcal{H}_{E}$. Hence,
noting that the expression in the curly brackets in the last line is simply
the reconstruction formula for the environmental operator
$\operatorname{Tr}_{S}\left[(F^{\dagger}_{\alpha}\otimes\mathbb{1}_{E})O_{SE}\right]$
on the frame $\left\\{X_{\beta}\right\\}_{\beta\in J}$ and its dual
$\left\\{Y_{\beta}\right\\}_{\beta\in J}$, we conclude that any global $S-E$
Hilbert-Schmidt operator
$O_{SE}\in\mathcal{L}_{2}(\mathcal{H}_{S}\otimes\mathcal{H}_{E})$ can be
decomposed as

$O_{SE}=\sum_{\alpha\in
I}D_{\alpha}\otimes\operatorname{Tr}_{S}\left[(F^{\dagger}_{\alpha}\otimes\mathbbm{1}_{E})O_{SE}\right];$
(13)

in other terms, a frame of open-system operators naturally induces a
decomposition formula for the global operators.

Comparing Eq.(13) with Eq.(2), we see that in order to define an OPD we have
to guarantee that when we focus on system-environment states,
$\rho_{SE}\in\mathcal{S}(\mathcal{H}_{S}\otimes\mathcal{H}_{E})\subset\mathcal{L}_{2}(\mathcal{H}_{S}\otimes\mathcal{H}_{E})$,
the decomposition can be expressed in terms of environmental statistical
operators and positive coefficients. Indeed, this is the case if we have a
_positive frame_ , i.e., a frame $\left\\{F_{\alpha}\right\\}_{\alpha\in I}$
made up of positive operators, $F_{\alpha}\geq 0$ for any $\alpha$, which
implies that
$\mbox{Tr}_{S}\left[(F^{\dagger}_{\alpha}\otimes\mathbb{1}_{E})\rho_{SE}\right]=\mbox{Tr}_{S}\left[(F_{\alpha}\otimes\mathbb{1}_{E})\rho_{SE}\right]\geq
0$. In fact, introducing the trace of these operators

$\omega_{\alpha}=\mbox{Tr}_{E}\left[\mbox{Tr}_{S}\left[(F^{\dagger}_{\alpha}\otimes\mathbb{1}_{E})\rho_{SE}\right]\right]\geq
0$ (14)

and defining the environmental statistical operators
$\rho_{\alpha}\in\mathcal{S}(\mathcal{H}_{E})$ via the assignment

$\omega_{\alpha}\rho_{\alpha}=\mbox{Tr}_{S}\left[(F^{\dagger}_{\alpha}\otimes\mathbb{1}_{E})\rho_{SE}\right]$
(15)

(so that $\rho_{\alpha}$ is arbitrary when
$\mbox{Tr}_{S}\left[(F^{\dagger}_{\alpha}\otimes\mathbb{1}_{E})\rho_{SE}\right]=0$),
we end up with an OPD. Furthermore, we note that the coefficients
$\omega_{\alpha}$ only depend on the reduced system state
$\rho_{S}=\mbox{Tr}_{E}\left[\rho_{SE}\right]$, since they are equivalently
expressed by

$\omega_{\alpha}=\mbox{Tr}_{S}\left[F^{\dagger}_{\alpha}\rho_{S}\right].$ (16)

In the next section, we present two different explicit choices of positive
operators allowing us to express every initial global state with an OPD, but
before that let us note the following. Once we have assigned a certain initial
global state $\rho_{SE}$, there is a (slightly) weaker sufficient condition
than using a positive frame to define an OPD. We could allow some elements of
the frame not to be positive, as long as the corresponding environmental
operator
$\mbox{Tr}_{S}\left[(F^{\dagger}_{\alpha}\otimes\mathbb{1}_{E})\rho_{SE}\right]$
is equal to the null operator, and then also the corresponding
$\omega_{\alpha}$ as in Eq.(14) is equal to zero. More in general, it might be
useful to consider an OPD where the frame is chosen according to the specific
initial state (or set of initial states) at hand, and possibly even according
to the global unitary evolution, so as to simplify the dynamical
representation in Eq.(3) as much as possible. We leave the study of how to
optimize the choice of the frame in the sense now indicated for future
investigation.

##  III Existence and cost of the decomposition

###  III.1 Positive decomposition in finite and infinite-dimensional systems

We introduce now two different constructions of an OPD valid for any global
state $\rho_{SE}$; the first one is referred to a finite dimensional Hilbert
space $\mathcal{H}_{S}$ and relies on the definition of a positive frame,
which generalizes straightforwardly the frame for qubit systems introduced in
[20]. The second OPD, instead, includes fully general Hilbert spaces
$\mathcal{H}_{S}$ and is defined in terms of a family of positive operators
that, in the infinite dimensional case, do not constitute a frame but still
allow us to exploit a reconstruction formula as in Eq.(13).

Let us then first consider a $d$-dimensional $\mathcal{H}_{S}$, with
$d<\infty$, and the $d^{2}$ generalized Pauli matrices (also known as
generalized Gell-Mann matrices) [32]

$\displaystyle\mathfrak{f}_{kj}^{(d)}$
$\displaystyle=\frac{1}{\sqrt{2}}\begin{cases}\ket{k}\bra{j}+\ket{j}\bra{k};\text{
for }k>j\\\ -i\Big{(}\ket{j}\bra{k}-\ket{k}\bra{j}\Big{)};\text{ for
}k<j\end{cases}$ $\displaystyle\mathfrak{h}_{k}^{(d)}$
$\displaystyle=\begin{cases}\frac{1}{\sqrt{d}}\mathbb{1}_{d};\text{ for
}k=1\\\ \mathfrak{h}_{k}^{(d-1)}\oplus 0;\text{ for }k=2,\dots,d-1\\\
\frac{1}{\sqrt{d(d-1)}}\big{[}\mathbb{1}_{(d-1)}\oplus(1-d)\big{]};\text{ for
}k=d\end{cases},$ (17)

where $\left\\{\ket{k}\right\\}_{k=1,\ldots d}$ is an orthonormal basis of
$\mathcal{H}_{S}$ and $\oplus$ denotes the matrix direct sum; these matrices
are Hermitian and traceless, apart from $\mathfrak{h}_{1}^{(d)}$ with trace
$\sqrt{d}$, and they correspond to the Pauli and Gell-Mann matrices for $d=2$
and $d=3$, respectively. A positive frame is thus defined by setting (with a
mapping of the index $\alpha$ to the couple $(k,j)$)

$\displaystyle F_{11}^{(d)}$ $\displaystyle=\mathfrak{h}_{1}^{(d)};$
$\displaystyle F_{kk}^{(d)}$
$\displaystyle=\sqrt{d}\sqrt{\frac{k-1}{k}}\mathfrak{h}_{1}^{(d)}+\mathfrak{h}_{k}^{(d)};\text{
for }k=2,...,d$ $\displaystyle F_{kj}^{(d)}$
$\displaystyle={\frac{1}{\sqrt{2}}}\mathbb{1}_{d}+\mathfrak{f}_{kj}^{(d)};\text{
for }k\neq j=1,...,d.$ (18)

The positivity of the frame elements in Eq.(III.1) directly follows from the
fact that the minimum eigenvalue of $\mathfrak{h}_{k}^{(d)}$ is
$-\sqrt{(k-1)/k}$, while $\sqrt{d}\mathfrak{h}_{1}^{(d)}$ is equal to the
identity matrix, and the minimum eigenvalue of $\mathfrak{f}_{kj}^{(d)}$ is
$-1/\sqrt{2}$. The inverse of the frame map provides us with the canonical
dual frame, see Eq.(11),

$\displaystyle D_{11}^{(d)}$
$\displaystyle=\mathfrak{h}_{1}^{(d)}-\sqrt{d}\sum_{k=2}^{d}\sqrt{\frac{k-1}{k}}\mathfrak{h}_{k}^{(d)}-\sqrt{\frac{d}{2}}\sum_{j\neq
k}\mathfrak{f}_{kj}^{(d)};$ $\displaystyle D_{kk}^{(d)}$
$\displaystyle=\mathfrak{h}_{k}^{(d)};\text{ for }k=2,...,d$ $\displaystyle
D_{kj}^{(d)}$ $\displaystyle=\mathfrak{f}_{kj}^{(d)};\text{ for }k\neq
j=1,...,d.$ (19)

We will then call _Pauli-OPD_ the decomposition in Eq.(2) fixed by the
positive frame in Eq.(III.1) and its dual in Eq.(III.1), see also Eqs.(14) and
(15). Importantly, we note that since $D_{11}^{(d)}$ is the only element of
the dual frame with non-vanishing trace, Eq.(2) implies that
$\rho_{E}=\operatorname{Tr}_{S}[\rho_{SE}]$ coincides with the first
environmental statistical operator appearing in the Pauli-OPD,
$\rho_{E}=\rho_{1}$.

Let us now consider a possibly infinite dimensional Hilbert space
$\mathcal{H}_{S}$ and an orthonormal basis
$\left\\{\ket{k}\right\\}_{k=1,\ldots,d;k\in\mathbbm{N}}$ (where the first set
of values of $k$ refers to a finite $d$-dimensional Hilbert space and the
second to an infinite dimensional one; to simplify the notation, from now on
the range of values of $k$ will be implied). We start by defining

$\displaystyle\mathfrak{b}_{kj}$
$\displaystyle=\frac{1}{\sqrt{2}}\begin{cases}\ket{k}\bra{j}+\ket{j}\bra{k};\text{
for }k>j\\\ -i\Big{(}\ket{j}\bra{k}-\ket{k}\bra{j}\Big{)};\text{ for
}k<j\end{cases}$ (20) $\displaystyle\mathfrak{b}_{kk}$
$\displaystyle=\ket{k}\bra{k},$

that is an orthonormal basis of Hermitian operators in
$\mathcal{L}_{2}(\mathcal{H}_{S})$; note that in $d$ dimensions this basis
differs from the basis of generalized Pauli operators only in its diagonal
elements. Using a basis of
$\mathcal{L}_{2}(\mathcal{H}_{S}\otimes\mathcal{H}_{E})$ made of tensor
product operators between the elements of, respectively, the basis in Eq.(20)
and a basis of $\mathcal{L}_{2}(\mathcal{H}_{E})$ (analogously to what has
been done to derive Eq.(13)), we can decompose any
$O_{SE}\in\mathcal{L}_{2}(\mathcal{H}_{S}\otimes\mathcal{H}_{E})$ as

$O_{SE}=\sum_{kj}\mathfrak{b}_{kj}\otimes\mbox{Tr}_{S}\left[(\mathfrak{b}_{kj}\otimes\mathbb{1}_{E})O_{SE}\right].$
(21)

If we now consider 2 families of operators
$P_{kj}=\sum_{k^{\prime}j^{\prime}}M^{k^{\prime}j^{\prime}}_{kj}\mathfrak{b}_{k^{\prime}j^{\prime}}$
and
$Q_{kj}=\sum_{k^{\prime}j^{\prime}}N^{k^{\prime}j^{\prime}}_{kj}\mathfrak{b}_{k^{\prime}j^{\prime}}$
such that the corresponding real coefficients satisfy

$\sum_{k^{\prime\prime}j^{\prime\prime}}M^{kj}_{k^{\prime\prime}j^{\prime\prime}}N^{k^{\prime}j^{\prime}}_{k^{\prime\prime}j^{\prime\prime}}=\delta_{kk^{\prime}}\delta_{jj^{\prime}},$
(22)

Eq.(21) is equivalent to

$O_{SE}=\sum_{kj}Q_{kj}\otimes\mbox{Tr}_{S}\left[(P_{kj}\otimes\mathbb{1}_{E})O_{SE}\right];$
(23)

indeed, if the operators $P_{kj}$ are positive, with the same reasoning as the
one at the end of Sec.II.2, we can conclude that the previous relation
provides any
$\rho_{SE}\in\mathcal{L}_{2}(\mathcal{H}_{S}\otimes\mathcal{H}_{E})$ with a
valid OPD. Our choice for the positive operators, aimed at reproducing the
non-diagonal elements of Eq. (III.1) and using simple diagonal operators, is
the following:

$\displaystyle P_{kk}$ $\displaystyle=\ket{k}\bra{k};$ $\displaystyle P_{kj}$
$\displaystyle=\frac{1}{\sqrt{2}}\mathbb{1}+\mathfrak{b}_{kj};\text{ for
}k\neq j.$ (24)

Using Eq.(22), we can thus complete the decomposition in Eq.(23) with the
family of operators

$\displaystyle Q_{kk}$
$\displaystyle=\ket{k}\bra{k}-\frac{1}{\sqrt{2}}\sum_{k\neq
j}\mathfrak{b}_{kj};$ $\displaystyle Q_{kj}$
$\displaystyle=\mathfrak{b}_{kj};\text{ for }k\neq j.$ (25)

We will call the OPD fixed by Eqs.(23)-(III.1) basis-induced OPD, to stress
its connection with the initial choice of the basis $\left\\{\ket{k}\right\\}$
of $\mathcal{H}_{S}$. Such an OPD can be used both for finite and infinite
dimensional systems, but we stress that only in the former case the family of
operators defined in Eq.(III.1) is actually a frame. In the infinite
dimensional case, in fact, the upper bound in Eq.(9) does not hold, as, e.g.,
if we consider the operator $A=\mathfrak{b}_{kk}=\ket{k}\bra{k}$ we find the
divergent series

$\sum_{k^{\prime}j^{\prime}}|\operatorname{Tr}_{S}[\mathfrak{b}_{kk}P_{k^{\prime}j^{\prime}}]|^{2}=1+\sum_{k^{\prime}\neq
j^{\prime}}\frac{1}{2}.$

Thus, the basis-induced OPD shows that, strictly speaking, frames are not
necessary to define a proper OPD, the key point rather being the definition of
a reconstruction formula as in Eq.(13), or, equivalently, Eq.(23) via a family
of positive open-system operators ($F_{\alpha}$ or $P_{kj}$).

###  III.2 Cost of the decomposition and Schmidt rank of the initial global
state

Moving back to a finite-dimensional Hilbert space $\mathcal{H}_{S}$ with
dimension $d$, both the Pauli and the basis-induced OPD allow us to express
any global state by means of an OPD with $d^{2}$ terms. On the other hand,
assigned a certain state $\rho_{SE}$, it might well be that it is possible to
write it equivalently via an OPD with a lower number of terms. Recall that we
denote with $\mathfrak{N}$ the minimal number of terms appearing in the OPD of
a given state $\rho_{SE}$, that is the cost of the OPD of $\rho_{SE}$;
moreover, since we deal now with finite-dimensional Hilbert spaces, we will
consider OPDs induced by positive frames. Here, we show that the cost
$\mathfrak{N}$ of the OPD of a given state $\rho_{SE}$ is equal to the Schmidt
rank of $\rho_{SE}$.

As first step, we show that given a bipartite statistical operator
$\rho_{SE}$, the cost $\mathfrak{N}$ of its OPD equals the number
$\mathfrak{I}$ of linearly independent operators in the set of environmental
states $\rho_{\alpha}$ associated with non-zero coefficients
$\omega_{\alpha}$. Importantly, this number does not depend on the specific
frame chosen to perform the decomposition and therefore the cost is a property
of the operator itself. Hence, consider any positive frame
$\left\\{F_{\alpha}\right\\}_{\alpha=1,\ldots\mathfrak{D}}$ of
$\mathcal{L}_{2}(\mathcal{H}_{S})$, with $\mathfrak{D}$ elements
($\mathfrak{D}\geq d^{2}$ as the frame spans
$\mathcal{L}_{2}(\mathcal{H}_{S})$), along with a dual frame
$\left\\{D_{\alpha}\right\\}_{\alpha=1,\ldots\mathfrak{D}}$. As shown
Sec.II.2, given any bipartite statistical operator $\rho_{SE}$, we can write
its OPD with respect to this frame as

$\rho_{SE}=\sum_{\alpha=1}^{N}\omega_{\alpha}D_{\alpha}\otimes\rho_{\alpha},$

where $\rho_{\alpha}$ are environmental statistical operators defined via
Eq.(15) and we have ordered the frame elements such that the first
$N\leq\mathfrak{D}$ coefficients $\omega_{\alpha}$ defined as in Eq.(14) are
strictly positive, while the last $\mathfrak{D}-N$ are equal to zero. Now, let
us denote as $\mathfrak{I}$ the number of linearly independent $\rho_{\alpha}$
in the set $\left\\{\rho_{\alpha}\right\\}_{\alpha=1,\ldots N}$, which indeed
coincides with the number of linearly independent operators in the set
$\\{\omega_{\alpha}\rho_{\alpha}\\}_{\alpha=1,\ldots\mathfrak{D}}$; if
$N>\mathfrak{I}$, we can write
$\rho_{N}=\sum_{\alpha=1}^{N-1}c_{\alpha}\rho_{\alpha}$ for some coefficients
$c_{\alpha}\in\mathbbm{R}$, from which it follows that $\rho_{SE}$ is
equivalently represented by

$\rho_{SE}=\sum_{\alpha=1}^{N-1}\omega_{\alpha}\overline{D}_{\alpha}\otimes\rho_{\alpha},$

where we introduced the new frame

$\displaystyle\overline{D}_{\alpha}$
$\displaystyle=D_{\alpha}+\frac{\omega_{N}}{\omega_{\alpha}}c_{\alpha}D_{N}\text{
; for }1\leq\alpha\leq N-1$ $\displaystyle\overline{D}_{\alpha}$
$\displaystyle=D_{\alpha}\text{ ; for }N\leq\alpha\leq\mathfrak{D}.$ (26)

Correspondingly, the duality relation is preserved if we also define

$\displaystyle\overline{F}_{\alpha}$ $\displaystyle=F_{\alpha}\text{ ; for
}1\leq\alpha\leq N-1$ $\displaystyle\overline{F}_{N}$
$\displaystyle=F_{N}-\sum_{\alpha=1}^{N-1}\frac{\omega_{N}}{\omega_{\alpha}}c_{\alpha}F_{\alpha}$
$\displaystyle\overline{F}_{\alpha}$ $\displaystyle=F_{\alpha}\text{ ; for
}N+1\leq\alpha\leq\mathfrak{D}.$ (27)

This frame is not necessarily positive since $\overline{F}_{N}$ may have
negative eigenvalues, but this does not affect the resulting OPD, as

$\operatorname{Tr}_{S}\big{[}(\overline{F}_{N}\otimes\mathbb{1}_{E})\rho_{SE}\big{]}=0;$
(28)

compare with the discussion at the end of Sec.II.2. Indeed, this procedure can
be repeated if also $\\{\rho_{\alpha}\\}_{\alpha=1,\dots,N-1}$ are linearly
dependent, i.e., $N-1>\mathfrak{I}$, and so on: if there are $\mathfrak{I}$
linear independent operators in $\\{\rho_{\alpha}\\}_{\alpha=1,\dots,N}$, it
is always possible to write an OPD with only $\mathfrak{I}$ terms. Crucially,
no further reduction is possible as can be shown by reductio ad absurdum.
Suppose, in fact, that one can construct a different OPD with
$\mathfrak{I}^{\prime}<\mathfrak{I}$ terms
$\rho_{SE}=\sum_{\beta=1}^{\mathfrak{I}^{\prime}}\lambda_{\beta}Q_{\beta}\otimes\eta_{\beta}$,
starting from a possibly different frame of operators
$\\{Q_{\beta}\\}_{\beta=1,\ldots,\mathfrak{D}^{\prime}}$, dual to a positive
frame $\\{P_{\beta}\\}_{\beta=1,\ldots,\mathfrak{D}^{\prime}}$. For what seen
above, the set of statistical operators
$\\{\eta_{\beta}\\}_{\beta=1,\dots,\mathfrak{I}^{\prime}}$ can be taken
linearly independent and
$\lambda_{\beta}=\operatorname{Tr}_{SE}\left[(P_{\beta}\otimes\mathbb{1}_{E})\rho_{SE}\right]=0$
for $\beta>\mathfrak{I}^{\prime}$, see Eq.(28), without loss of generality.
But then using the decomposition of $F_{\alpha}$ on the frame
$\\{P_{\beta}\\}_{\beta=1,\ldots,\mathfrak{D}^{\prime}}$,
$F_{\alpha}=\sum_{\beta=1}^{\mathfrak{D}^{\prime}}q_{\beta\alpha}P_{\beta}$,
we obtain

$\displaystyle\omega_{\alpha}\rho_{\alpha}$
$\displaystyle=\operatorname{Tr}_{S}\big{[}(F_{\alpha}\otimes\mathbb{1}_{E})\rho_{SE}\big{]}=\sum_{\beta=1}^{\mathfrak{D}^{\prime}}q_{\beta\alpha}\lambda_{\beta}\eta_{\beta}=\sum_{\beta=1}^{\mathfrak{I}^{\prime}}q_{\beta\alpha}\lambda_{\beta}\eta_{\beta}.$
(29)

This equation states that the set of linearly independent vectors
$\\{\omega_{\alpha}\rho_{\alpha}\\}_{\alpha=1,\dots,\mathfrak{I}}$ is
generated by a family
$\\{\lambda_{\beta}\eta_{\beta}\\}_{\beta=1,\dots,\mathfrak{I}^{\prime}}$ with
$\mathfrak{I}^{\prime}<\mathfrak{I}$, which is a contradiction. We can thus
conclude that, starting from any OPD, it is always possible to reduce the
number of terms appearing in it to the number $\mathfrak{I}$ of independent
operators in the set
$\\{\omega_{\alpha}\rho_{\alpha}\\}_{\alpha=1,\ldots\mathfrak{D}}$, but not
more, i.e., $\mathfrak{N}=\mathfrak{I}$; note that the reasoning used in the
reductio ad absurdum also implies that $\mathfrak{I}$ does not depend on the
specific frame used to define the OPD of $\rho_{SE}$.

The connection between the cost $\mathfrak{N}$ of an OPD and the number
$\mathfrak{I}$ of linearly independent environmental states appearing in the
OPD directly leads us to link $\mathfrak{N}$ with the Schmidt rank. Any
bipartite state
$\rho_{SE}\in\mathcal{S}(\mathcal{H}_{S}\otimes\mathcal{H}_{E})$ can be
associated with a Schmidt decomposition, which reads:

$\rho_{SE}=\sum_{k=1}^{\mathfrak{R}}\lambda_{k}G_{k}^{S}\otimes G_{k}^{E},$
(30)

with $\lambda_{k}>0$ and $\\{G_{k}^{S,E}\\}_{k=1,\ldots\mathfrak{R}}$
orthonormal Hermitian operators on $\mathcal{H}_{S},\mathcal{H}_{E}$;
$\mathfrak{R}$ is the Schmidt rank of $\rho_{SE}$ and $\mathfrak{R}\leq
d^{2}$; importantly, the Schmidt rank is directly related with the amount of
correlations in $\rho_{SE}$ [33, 34]. Now, from Eq. (30) it follows

$\lambda_{k}G_{k}^{E}=\operatorname{Tr}_{S}\big{[}(G_{k}^{S}\otimes\mathbb{1}_{E})\rho_{SE}\big{]}$
(31)

and that the operators $\\{\lambda_{k}G_{k}^{E}\\}_{k=1,\dots,\mathfrak{R}}$
are linearly independent. For an OPD, see Eq.(2), we have the similar relation
Eq. (15) and we can always take into account a minimal OPD of $\rho_{SE}$ such
that the $\mathfrak{N}$ operators
$\\{\omega_{\alpha}\rho_{\alpha}\\}_{\alpha=1,\dots,\mathfrak{N}}$ are
linearly independent, as discussed above. In addition, we consider,
respectively, an orthonormal basis $\\{G_{k}^{S}\\}_{k=1,\ldots,d^{2}}$ of
$\mathcal{L}_{2}(\mathcal{H}_{S})$ whose first $\mathfrak{R}$ elements
coincide with the open-system Schmidt operators in Eq.(30), also implying
$\operatorname{Tr}_{S}\big{[}(G_{k}^{S}\otimes\mathbb{1}_{E})\rho_{SE}\big{]}=0$
for $k=\mathfrak{R}+1,\ldots,d^{2}$, and a complete frame
$\\{F_{\alpha}\\}_{\alpha=1,\ldots,\mathfrak{D}}$ whose first $\mathfrak{N}$
elements coincide with those defining the minimal OPD, while the others
satisfy
$\operatorname{Tr}_{S}\big{[}(F_{\alpha}\otimes\mathbb{1}_{E})\rho_{SE}\big{]}=0$
for $\alpha=\mathfrak{N}+1,\ldots,\mathfrak{D}$, see Eq.(28). Exploiting the
decompositions,
$G_{k}^{S}=\sum_{\alpha=1}^{\mathfrak{D}}g_{k\alpha}F_{\alpha}$ and
$F_{\alpha}=\sum_{k=1}^{d^{2}}f_{\alpha k}G_{k}^{S}$, we thus obtain from Eqs.
(15) and (31)

$\displaystyle\lambda_{k}G_{k}^{E}$
$\displaystyle=\sum_{\alpha=1}^{\mathfrak{N}}g_{k\alpha}\omega_{\alpha}\rho_{\alpha};$
$\displaystyle\omega_{\alpha}\rho_{\alpha}$
$\displaystyle=\sum_{k=1}^{\mathfrak{R}}f_{\alpha k}\lambda_{k}G_{k}^{E}.$
(32)

Hence, since the linearly independent sets
$\\{\lambda_{k}G_{k}^{E}\\}_{k=1,\dots,\mathfrak{R}}$ and
$\\{\omega_{\alpha}\rho_{\alpha}\\}_{\alpha=1,\dots,\mathfrak{N}}$ generate
each other, they have the same number of elements, i.e.,
$\mathfrak{N}=\mathfrak{R}$. We thus conclude that the cost $\mathfrak{N}$ of
the OPD coincides with the Schmidt rank of the global statistical operator
$\rho_{SE}$, in this way generalizing to mixed states the link between Schmidt
rank and OPD discussed in [20] for pure global states.

##  IV Case study: combination of semigroup maps

We now provide an explicit example of an open-system evolution fixed via a
family of CPTP maps $\Phi_{\alpha}(t)$ according to Eq.(3). In Ref.[24] we
exploited the possible relevance of OPD in order to obtain a perturbative
expansion of a microscopically motivated open system dynamics in the presence
of initial correlations. We will here take a complementary phenomenological
approach, exploring whether convenient choices of the CPTP maps
$\Phi_{\alpha}(t)$ can lead to a well defined description of the system
evolution. In particular, we take the maps $\Phi_{\alpha}(t)$ to be CPTP
semigroups, i.e.,

$\Phi_{\alpha}(t)=e^{\mathcal{L}_{\alpha}t},$ (33)

where $\mathcal{L}_{\alpha}$ are generators in the GKLS form [35, 36]

$\mathcal{L}_{\alpha}[\rho]=\sum_{j=1}^{d^{2}-1}\gamma_{\alpha,j}\left(L_{\alpha,j}\rho
L^{\dagger}_{\alpha,j}-\frac{1}{2}\left\\{L^{\dagger}_{\alpha,j}L_{\alpha,j},\rho\right\\}\right),$
(34)

with $\gamma_{\alpha,j}\geq 0$ and $L_{\alpha,j}$ linear independent linear
operators on $\mathcal{H}_{S}$, and we recall that $d$ is the finite dimension
of the open-system Hilbert space $\mathcal{H}_{S}$.

The idea of combining a family of semigroup maps to go beyond the description
of Markovian dynamics has been used by Chruściński and Kossakowski in several
pioneering works [37, 38, 39], aiming at the identification of well-defined
integro-differential master equations accounting for non-Markovian dynamics.
In [40] a system of GKSL evolutions was used to take into account statistical
correlations between system and environment leading to a non-Markovian
dynamics within a projection operator approach. Here, by using Eq.(3) to
connect the initial reduced state with its value at a generic time, we explore
whether a family of semigroups can be exploited in the context of open-system
dynamics in the presence of initial correlations with the environment. Indeed,
while every $e^{\mathcal{L}_{\alpha}t}$ is CPTP, so that also a convex
combination of these maps would be CPTP, we are now applying each
$e^{\mathcal{L}_{\alpha}t}$ to a distinct element of the dual frame
$D_{\alpha}$, which needs not to be positive. As a consequence, it is a-priori
not guaranteed that even though we start from an initial positive state
$\rho_{S}$ in Eq.(5), the state at time $t$ fixed by Eq.(3) will be positive,
too. In the following, we show that it is actually possible to identify a
proper set of initial states that are mapped into states at any time $t$, for
a relevant class of two-level system dynamics. Let us stress that, as common
in phenomenological approaches, the fact that we can introduce a well-defined
evolution on open-system states does not mean by itself that the evolution can
be derived from a full microscopic model. Within the formalism exploited here,
the existence of a microscopic model would consist in the presence of initial
global states and a global unitary evolution such that the action of the maps
$\Phi_{\alpha}(t)$ could be expressed as in Eq.(4); whether this is the case
is a challenging question, which we leave for future investigation.

###  IV.1 Pauli channels

We consider a two-level open quantum system and we further exploit the
positive frame induced by the Pauli basis of operators, introduced in
Sec.III.1. In particular, the dual-frame operators $D_{\alpha}$ are as in
Eq.(III.1), which for the case $d=2$ we are dealing with simply read

$\displaystyle D_{0}$
$\displaystyle=\frac{1}{\sqrt{2}}\left(\sigma_{0}-\sum\limits_{\alpha=1}^{3}\sigma_{\alpha}\right),$
$\displaystyle D_{\alpha}$ $\displaystyle=\frac{1}{\sqrt{2}}\sigma_{\alpha},$
$\displaystyle\text{for }{\alpha}=1,2,3,$ (35)

where $\sigma_{0}$ is the identity, the $\sigma_{\alpha}$, $\alpha=1,2,3$ are
the usual Pauli matrices, and we reordered the frame indices as
$\alpha=0,\ldots,3$. Given a reduced operator of the form (compare with
Eq.(5))

$\displaystyle\rho_{S}=\frac{1}{\sqrt{2}}\sum\limits_{\alpha=0}^{3}\upsilon_{\alpha}D_{\alpha},$
(36)

where for the sake of convenience we have set
$\upsilon_{\alpha}=\sqrt{2}\omega_{\alpha}$ with $\omega_{\alpha}$ as in
Eq.(5), it has trace one if and only if

$\displaystyle\upsilon_{0}=1,$ (37)

while its positivity is ensured by the validity of

$\displaystyle\left(1-\upsilon_{1}\right)^{2}+\left(1-\upsilon_{2}\right)^{2}+\left(1-\upsilon_{3}\right)^{2}\leq
1.$ (38)

Hence, the initial positivity domain, that is the set of points
$\\{v_{1},v_{2},v_{3}\\}$ for which $\rho_{S}$ as in Eq.(36) can be taken as a
proper initial state, is a ball with radius 1 and origin in the point
$\\{1,1,1\\}$.

Our goal is then to determine whether there is a subset of the initial
positivity domain such that the state $\rho_{S}(t)$ obtained via Eq.(3) is
positive at any time $t$. In particular, we take into account the maps
corresponding to the so-called Pauli channels [41, 42], whose generators are
given by

$\mathcal{L}_{\alpha}[\rho]=\sum_{j=1}^{3}\gamma_{\alpha,j}(t)\left(\sigma_{j}\rho\sigma_{j}-\rho\right).$
(39)

When the rates $\gamma_{\alpha,j}(t)$ are non-negative and time independent
each of this generator corresponds to a GKSL master equation, otherwise the
dynamics can be highly non-Markovian also for a single $\mathcal{L}_{\alpha}$
[43]. Here, as said, we restrict to semigroups; note that while the
coefficients $\gamma_{\alpha,j}$ can be different for the different
$\mathcal{L}_{\alpha}$s, the Lindblad operators are always the same, i.e., the
Pauli matrices. The corresponding CPTP maps take the form

$\displaystyle\phi_{\alpha}(t)[\rho]=\sum\limits_{j=0}^{3}p_{\alpha,j}(t)\sigma_{j}\rho\sigma_{j},$
(40)

where the coefficients $p_{\alpha,j}(t)$ are readily expressed by introducing
the vectors $\vec{p}_{\alpha}(t)$, whose components are
$(\vec{p}_{\alpha}(t))_{j}=p_{\alpha,j}(t)$, so that one has

$\displaystyle\vec{p}_{\alpha}(t)=\frac{1}{4}\begin{pmatrix}1&1&1&1\\\
1&1&-1&-1\\\ 1&-1&1&-1\\\ 1&-1&-1&1\end{pmatrix}\vec{\lambda}_{\alpha}(t)$
(41)

where $(\vec{\lambda}_{\alpha}(t))_{j}=\lambda_{\alpha,j}(t)$ with

$\displaystyle\lambda_{\alpha,0}=1,$
$\displaystyle\lambda_{\alpha,j}=e^{-2(\gamma_{\alpha,k}+\gamma_{\alpha,m})t},$
$\displaystyle j\neq k\neq m=1,2,3,$ (42)

i.e., the exponential decays characterizing semigroup dynamics. Note that the
$\left\\{\lambda_{\alpha,j}(t)\right\\}_{j=0,\ldots,3}$ are eigenvalues of the
map $\phi_{\alpha}(t)$

$\displaystyle\phi_{\alpha}(t)[\sigma_{j}]=\lambda_{\alpha,j}(t)\sigma_{j},$
(43)

while the coefficients $\left\\{p_{\alpha,j}(t)\right\\}_{j=0,\ldots,3}$ form
a probability distribution, i.e. $p_{\alpha,j}(t)\geq 0$ and
$\sum\limits_{j=0}^{3}p_{\alpha,j}(t)=1$.

For the sake of simplicity, we now restrict to two different semigroup maps,
one for the dual frame element $D_{0}$, which we denote as
$\phi(t):=\phi_{0}(t)$, and another for the other elements, which we denote as
$\tilde{\phi}(t):=\phi_{1}(t)=\phi_{2}(t)=\phi_{3}(t)$; accordingly all the
parameters referred to the first (second) map will be indicated without (with)
a tilde. The state at time $t$ obtained via Eq.(3) then reads

$\displaystyle\rho_{S}(t)=\frac{1}{2}\left(\leavevmode\hbox{\small
1\kern-3.8pt\normalsize
1}+\left(\tilde{\lambda}_{1}(t)v_{1}-\lambda_{1}(t)\right)\sigma_{1}+\left(\tilde{\lambda}_{2}(t)v_{2}-\lambda_{2}(t)\right)\sigma_{2}+\left(\tilde{\lambda}_{3}(t)v_{3}-\lambda_{3}(t)\right)\sigma_{3}\right),$
(44)

so that the positivity domain at a generic time is specified by the condition

$\displaystyle\left(\lambda_{1}(t)-\tilde{\lambda}_{1}(t)v_{1}\right)^{2}+\left(\lambda_{2}(t)-\tilde{\lambda}_{2}(t)v_{2}\right)^{2}+\left(\lambda_{3}(t)-\tilde{\lambda}_{3}(t)v_{3}\right)^{2}\leq
1,$ (45)

which defines the interior of an axis-aligned ellipsoid with semi-axes lengths
$1/\tilde{\lambda}_{1}(t)$, $1/\tilde{\lambda}_{2}(t)$ and
$1/\tilde{\lambda}_{3}(t)$, centered at the point
$\\{\lambda_{1}(t)/\tilde{\lambda}_{1}(t),\lambda_{2}(t)/\tilde{\lambda}_{2}(t),\lambda_{3}(t)/\tilde{\lambda}_{3}(t)\\}$.

###  IV.2 Results

Summarizing the previous discussion, our primary goal is thus to find the
values $\\{v_{1},v_{2},v_{3}\\}$ that satisfy both Eq. (38) and (45), where
the former inequality defines the initial positivity domain, while the latter
depends on the dynamical parameters $\lambda_{\alpha}(t)$ and
$\tilde{\lambda}_{\alpha}(t)$.

To this aim, it is advantageous to decouple the quantities referred to,
respectively, $\phi(t)$ and $\tilde{\phi}(t)$. Note that we can interpret Eq.
(45) as an equation of a ball with the axes rescaled by $\tilde{\lambda}$s:

$\displaystyle\left(\lambda_{1}(t)-\tilde{x}(t)\right)^{2}+\left(\lambda_{2}(t)-\tilde{y}(t)\right)^{2}+\left(\lambda_{3}(t)-\tilde{z}(t)\right)^{2}\leq
1.$ (46)

We call the set of points $\\{\tilde{x}(t),\tilde{y}(t),\tilde{z}(t)\\}$
satisfying this inequality an evolved positivity ball, as for $t=0$ it gives
the initial positivity domain as in Eq. (38). On the other hand, Eq. (38)
after such rescaling takes the form

$\displaystyle\frac{1}{\tilde{\lambda}_{1}^{2}(t)}\left(\tilde{\lambda}_{1}(t)-\tilde{x}(t)\right)^{2}+\frac{1}{\tilde{\lambda}_{2}^{2}(t)}\left(\tilde{\lambda}_{2}(t)-\tilde{y}(t)\right)^{2}+\frac{1}{\tilde{\lambda}_{3}^{2}(t)}\left(\tilde{\lambda}_{3}(t)-\tilde{z}(t)\right)^{2}\leq
1,$ (47)

and the set of all evolved initial points is an axis-aligned ellipsoid
centered at
$\\{\tilde{\lambda}_{1}(t),\tilde{\lambda}_{2}(t),\tilde{\lambda}_{3}(t)\\}$
with semi-axes lengths $\tilde{\lambda}_{1}(t)$, $\tilde{\lambda}_{2}(t)$ and
$\tilde{\lambda}_{3}(t)$. Accordingly, the intersection of this ellipsoid with
the evolved positivity ball with radius 1 and origin in
$\\{\lambda_{1}(t),\lambda_{2}(t),\lambda_{3}(t)\\}$ gives the wanted set of
$v$s. If the whole ellipsoid is contained within the evolved positivity ball
at a given time $t$, all initial density operators $\rho_{S}$ stay positive at
this point of time. By virtue of the rescaling introduced above, the form and
location of the ellipsoid is only governed by the $\tilde{\lambda}$s, while
the positivity ball evolves according to the $\lambda$s.

In Fig.1 and 2 we report the ellipsoid of evolved intial points defined by
Eq.(47) (in red) and the evolved positivity ball defined by Eq.(46) (in blue),
for two different choices of the parameters defining the semigroup maps.

Example I: In a generic case all the $\lambda$s and $\tilde{\lambda}$s
describe exponential decay, i.e. according to Eq.(42) at most one of the
$\gamma$s and $\tilde{\gamma}$s is zero. In this case the centers of both the
evolved positivity ball and the evolved ellipsoid converge to $\\{0,0,0\\}$.
Additionally, the ellipsoid asymptotically shrinks to a point, while the size
of the evolved positivity ball does not change. Hence, for large enough times
the ellipsoid will be completely contained into the positivity ball. For
intermediate times, a portion of the ellipsoid can leave the evolved
positivity ball, as seen in Fig. 1 for the following choice of the parameters:
$\gamma_{1}=\tilde{\gamma}_{1}=0$,
$\gamma_{2}=\gamma_{3}=2\tilde{\gamma}_{2}=2\tilde{\gamma}_{3}=\gamma$, which
correspond to

$\displaystyle\lambda_{1}(t)=e^{-4\gamma t},$
$\displaystyle\lambda_{2}(t)=e^{-2\gamma t},$ $\displaystyle\lambda_{3}(t)=$
$\displaystyle e^{-2\gamma t},$
$\displaystyle\tilde{\lambda}_{1}(t)=e^{-2\gamma t},$
$\displaystyle\tilde{\lambda}_{2}(t)=e^{-\gamma t},$
$\displaystyle\tilde{\lambda}_{3}(t)=$ $\displaystyle e^{-\gamma t}.$ (48)

The larger values of the $\gamma$s with respect to the $\tilde{\gamma}$s
result in a quicker motion of the positivity ball than of the ellipsoid.
Accordingly, some of the states in the initial positivity domain will anyhow
lead to temporarily negative evolved matrix $\rho_{S}(t)$.

Figure 1:  Example I: evolved initial points (red) and the evolved positivity
ball (blue). Left: $\gamma t=1/10$, Middle: $\gamma t=1/2$, Right: $\gamma
t=2$.

Example II: If, on the other hand, two of the $\tilde{\gamma}$s equal zero,
one can see a qualitatively different behaviour, as some of the choices of
initial points lead to eternally negative evolved matrix $\rho_{S}(t)$. For
example, for $\gamma_{1}=\tilde{\gamma}_{1}=\tilde{\gamma}_{2}=0$,
$\gamma_{2}=\gamma_{3}=\tilde{\gamma}_{3}=\gamma$, which correspond to

$\displaystyle\lambda_{1}(t)=e^{-4\gamma t},$
$\displaystyle\lambda_{2}(t)=e^{-2\gamma t},$ $\displaystyle\lambda_{3}(t)=$
$\displaystyle e^{-2\gamma t},$
$\displaystyle\tilde{\lambda}_{1}(t)=e^{-2\gamma t},$
$\displaystyle\tilde{\lambda}_{2}(t)=e^{-2\gamma t},$
$\displaystyle\tilde{\lambda}_{3}(t)=$ $\displaystyle 1,$ (49)

the $z$-coordinate of the initial points is constant: $\\{e^{-2\gamma
t}v_{1},e^{-2\gamma t}v_{2},v_{3}\\}$. Accordingly, the semi-axes length in
this direction of the ellipsoid does not change in time and its center remains
on the same x-y plane, see Fig. 2. All of the choices of points with $v_{3}>1$
will at some point of the evolution leave the evolved positivity ball forever.

Figure 2: Example II: evolved initial points (red) and the evolved positivity
ball (blue). Left: $\gamma t=1/10$, Middle: $\gamma t=1/2$, Right: $\gamma
t=2$.

##  V Conclusion

In this paper, we have considered an approach to describe the dynamics of open
quantum systems that allows one to take into account fully general initially
correlated system-environment states. The starting point is the OPD
decomposition of the initial global state, which involves positive and
normalized operators on the environmental side and fixes the evolution of the
open system through a set of time-dependent CPTP maps. Such a decomposition
can be defined via a positive frame on the open quantum system and it is in
general highly non-unique. In particular, we introduced two OPD
decompositions, one that is based on the generalized Pauli matrices and one
directly on the basis elements of the open-system Hilbert space; notably, the
latter can be defined also for infinite-dimensional Hilbert spaces, thus
showing how the OPD-based approach to open-system dynamics is not restricted
to the finite-dimensional case. In addition, we have demonstrated that the
cost of the decomposition, that is the number of terms involved in it and thus
the number of resulting CPTP maps fixing the reduced dynamics, is always
bounded by the Schmidt rank of the initial global state. This is a
particularly useful feature of the approach, since it implies that the open-
system evolution can be fixed by a number of equations, for example as in
[24], that is a direct expression of the amount of correlations between the
open system and the environment and it is ultimately bounded by the
dimensionality of the open system. Finally, we have studied the case where the
CPTP maps defining the reduced dynamics along with the OPD decomposition are
semigroups fixed by the Gorini-Kossakowski-Lindblad-Sudarshan form. In
particular, in the case of qubit Pauli channels, we have derived explicit
conditions identifying the positivity domain defined by the initial reduced
states that are mapped to proper states at any time and we have distinguished
two qualitatively different regimes; one where all the states compatible with
a given initial decomposition are eventually mapped into proper states after a
transient time interval, and one where some states lead to eternally negative
evolved matrices.

## Acknowledgments

All authors acknowledge support from UniMi, via Transition Grant H2020, PSR-2
2020 and PSR-2 2021. NM was funded by the Alexander von Humboldt Foundation in
form of a Fedor-Lynen Fellowship and project ApresSF, supported by the
National Science Centre under the QuantERA programme, funded by the European
Union’s Horizon 2020 research and innovation programme.

## References

  * Grabert  _et al._ [1988] H. Grabert, P. Schramm, and G.-L. Ingold, Quantum Brownian motion: The functional integral approach, Phys. Rep. 168, 115 (1988).
  * Pechukas [1994] P. Pechukas, Reduced dynamics need not be completely positive, Phys. Rev. Lett. 73, 1060 (1994).
  * Alicki [1995] R. Alicki, Comment on “Reduced dynamics need not be completely positive”, Phys. Rev. Lett. 75, 3020 (1995).
  * Lindblad [1996] G. Lindblad, On the existence of quantum subdynamics, J. Phys. A: Mathematical and General 29, 4197 (1996).
  * Štelmachovič and Bužek [2001] P. Štelmachovič and V. Bužek, Dynamics of open quantum systems initially entangled with environment: Beyond the Kraus representation, Phys. Rev. A 64, 062106 (2001).
  * Myöhänen  _et al._ [2008] P. Myöhänen, A. Stan, G. Stefanucci, and R. van Leeuwen, A many-body approach to quantum transport dynamics: Initial correlations and memory effects, EPL (Europhysics Letters) 84, 67001 (2008).
  * Pomyalov  _et al._ [2010] A. Pomyalov, C. Meier, and D. Tannor, The importance of initial correlations in rate dynamics: A consistent non-Markovian master equation approach, Chem. Phys. 370, 98 (2010).
  * Modi  _et al._ [2012] K. Modi, A. Brodutch, H. Cable, T. Paterek, and V. Vedral, The classical-quantum boundary for correlations: Discord and related measures, Rev. Mod. Phys. 84, 1655 (2012).
  * Vacchini and Amato [2016] B. Vacchini and G. Amato, Reduced dynamical maps in the presence of initial correlations, Sci. Rep. 6, 37328 (2016).
  * Schmid  _et al._ [2019] D. Schmid, K. Ried, and R. W. Spekkens, Why initial system-environment correlations do not imply the failure of complete positivity: A causal perspective, Phys. Rev. A 100, 022112 (2019).
  * Majeed and Chaudhry [2019] M. Majeed and A. Z. Chaudhry, Effect of initial system–environment correlations with spin environments, Eur. Phys. J. D 73, 16 (2019).
  * [12] M. Merkli, Correlation decay and Markovianity in open systems, arXiv:2107.02515 (2021).
  * Jordan  _et al._ [2004] T. F. Jordan, A. Shaji, and E. C. G. Sudarshan, Dynamics of initially entangled open quantum systems, Phys. Rev. A 70, 052110 (2004).
  * Jordan  _et al._ [2006] T. F. Jordan, A. Shaji, and E. C. G. Sudarshan, Mapping the Schrödinger picture of open quantum dynamics, Phys. Rev. A 73, 012106 (2006).
  * Rodríguez-Rosario  _et al._ [2008] C. A. Rodríguez-Rosario, K. Modi, A. meng Kuah, A. Shaji, and E. C. G. Sudarshan, Completely positive maps and classical correlations, J. Phys. A: Math. and Theor. 41, 205301 (2008).
  * Shabani and Lidar [2009] A. Shabani and D. A. Lidar, Vanishing quantum discord is necessary and sufficient for completely positive maps, Phys. Rev. Lett. 102, 100402 (2009).
  * Brodutch  _et al._ [2013] A. Brodutch, A. Datta, K. Modi, A. Rivas, and C. A. Rodríguez-Rosario, Vanishing quantum discord is not necessary for completely positive maps, Phys. Rev. A 87, 042301 (2013).
  * Buscemi [2014] F. Buscemi, Complete positivity, Markovianity, and the quantum data-processing inequality, in the presence of initial system-environment correlations, Phys. Rev. Lett. 113, 140502 (2014).
  * Dominy and Lidar [2016] J. M. Dominy and D. A. Lidar, Beyond complete positivity, Quant. Inf. Proc. 15, 1349 (2016).
  * Paz-Silva  _et al._ [2019] G. A. Paz-Silva, M. J. W. Hall, and H. M. Wiseman, Dynamics of initially correlated open quantum systems: Theory and applications, Phys. Rev. A 100, 042120 (2019).
  * Chalermpusitarak  _et al._ [2021] T. Chalermpusitarak, B. Tonekaboni, Y. Wang, L. M. Norris, L. Viola, and G. A. Paz-Silva, Frame-based filter-function formalism for quantum characterization and control, PRX Quantum 2, 030315 (2021).
  * Hamedani Raja  _et al._ [2020] S. Hamedani Raja, K. P. Athulya, A. Shaji, and J. Piilo, Photonic dephasing dynamics and the role of initial correlations, Phys. Rev. A 101, 042127 (2020).
  * Breuer and Petruccione [2002] H.-P. Breuer and F. Petruccione,  _The Theory of Open Quantum Systems_ (Oxford University Press, Oxford, 2002).
  * Trevisan  _et al._ [2021] A. Trevisan, A. Smirne, N. Megier, and B. Vacchini, Adapted projection operator technique for the treatment of initial correlations, Phys. Rev. A 104, 052215 (2021).
  * Bengtsson and Zyczkowski [2006] I. Bengtsson and K. Zyczkowski,  _Geometry of quantum states: an introduction to quantum entanglement_ (Cambridge University Press, Cambridge, 2006).
  * Ollivier and Zurek [2001] H. Ollivier and W. H. Zurek, Quantum discord: A measure of the quantumness of correlations, Phys. Rev. Lett. 88, 017901 (2001).
  * Henderson and Vedral [2001] L. Henderson and V. Vedral, Classical, quantum and total correlations, J. Phys. A: Math. and Gen. 34, 6899 (2001).
  * Rivas  _et al._ [2014] Á. Rivas, S. F. Huelga, and M. B. Plenio, Quantum non-Markovianity: characterization, quantification and detection, Rep. Progr. Phys. 77, 094001 (2014).
  * Ali  _et al._ [2000] S. T. Ali, J.-P. Antoine, and J.-P. Gazeau,  _Coherent States, Wavelets, and Their Generalizations_ , Theoretical and Mathematical Physics (Springer, New York, 2000).
  * Renes  _et al._ [2004] J. M. Renes, R. Blume-Kohout, A. J. Scott, and C. M. Caves, Symmetric informationally complete quantum measurements, J. Math. Phys. 45, 2171 (2004).
  * Note [1] Whenever Eq.(9) holds with $m=M=1$, the frame $\left\\{F_{\alpha}\right\\}_{\alpha\in I}$ is called a Parseval frame, as Eq.(10) reduces to $A=\sum@\slimits@_{\alpha\in I}(F_{\alpha},A)F_{\alpha}$; interestingly, there are Parseval frames that are _not_ orthonormal bases.
  * Kimura [2003] G. Kimura, The Bloch vector for N-level systems, Phys. Lett. A 314, 339 (2003).
  * Cariello [2014] D. Cariello, Separability for weakly irreducible matrices, Quantum Inf. Comput. 14, 1308 (2014).
  * Cariello [2015] D. Cariello, Does symmetry imply PPT property?, Quantum Inf. Comput. 15, 812 (2015).
  * Lindblad [1976] G. Lindblad, On the generators of quantum dynamical semigroups, Comm. Math. Phys. 48, 119 (1976).
  * Gorini  _et al._ [1976] V. Gorini, A. Kossakowski, and E. C. G. Sudarshan, Completely positive dynamical semigroups of $N$-level systems, J. Math. Phys. 17, 821 (1976).
  * Chruściński and Kossakowski [2010] D. Chruściński and A. Kossakowski, Non-Markovian quantum dynamics: Local versus nonlocal, Phys. Rev. Lett. 104, 070406 (2010).
  * Chruściński and Kossakowski [2011] D. Chruściński and A. Kossakowski, Local approach to the non-Markovian evolution of quantum systems, Int. J. Quant. Inf. 09, 129 (2011).
  * Chruściński and Kossakowski [2012] D. Chruściński and A. Kossakowski, From Markovian semigroup to non-Markovian quantum evolution, EPL (Europhysics Letters) 97, 20005 (2012).
  * Breuer [2007] H.-P. Breuer, Non-Markovian generalization of the Lindblad theory of open quantum systems, Phys. Rev. A 75, 022103 (2007).
  * Vacchini [2012] B. Vacchini, A classical appraisal of quantum definitions of non-markovian dynamics, Journal of Physics B: Atomic, Molecular and Optical Physics 45, 154007 (2012).
  * Chruściński and Wudarski [2013] D. Chruściński and F. A. Wudarski, Non-Markovian random unitary qubit dynamics, Phys. Lett. A 377, 1425 (2013).
  * Megier  _et al._ [2017] 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).