# Witnessing quantum memory in non-Markovian processes

Christina Giarmatzi<EMAIL_ADDRESS>Centre for Engineered Quantum
Systems, School of Mathematics and Physics, University of Queensland, QLD 4072
Australia  Fabio Costa  Centre for Engineered Quantum Systems, School of
Mathematics and Physics, University of Queensland, QLD 4072 Australia

###### Abstract

We present a method to detect a quantum memory in a non-Markovian process. We
call a process Markovian when the environment does not provide a memory that
retains correlations across consecutive system-environment interactions. We
define two types of non-Markovian processes, depending on the required memory
being classical or quantum. We formalise this distinction using the process
matrix formalism, through which a process is represented as a multipartite
state. Within this formalism, a test for entanglement in a state can be mapped
to a test for quantum memory in the corresponding process. This allows us to
apply separability criteria and entanglement witnesses to the detection of a
quantum memory. We apply the method to a physically motivated example of a
process with quantum memory: two initially entangled qubits, representing
system and environment respectively, coupled according to the Ising model with
a transverse field. We prove Markovianity and classical memory for the special
cases of vanishing coupling and transverse field, respectively. Using the
Positive Partial Transpose criterion, we find that for almost all other
parameters the process has a quantum memory. As with entanglement witnesses,
our method of witnessing a quantum memory provides a versatile experimental
tool for open quantum systems.

#### Introduction.—

A challenge present in every experimental setup is to monitor the interaction
between some desired system and the environment at many different time steps
of an experiment. A common simplification is to assume that the environment
does not have a memory, namely that the evolution of the system during a
certain time interval is not correlated with the evolution at a different
time, or with the state of the environment before the experiment started. A
process with such a memory-less environment is generally called Markovian
Heinz-Peter and Francesco (2002) and can still have noise: for example, stray
light or temperature, as long as they do not affect different parts of the
experiment in a correlated way.

Several methods that account for noise, such as error correction for quantum
computation, typically rely on the assumption of Markovianity Terhal (2015).
However, it is often the case that system-environment interactions at
different times are correlated and/or depend on the initial state through some
memory in the environment. An important distinction is whether the memory can
or cannot be represented classically. A classical memory is typically easier
to simulate and correct for, whereas a quantum memory implies more intricate
coherences between the system and the environment. On the other hand, there
are situations where detecting a quantum memory is a desired result: one may
may regard a system of interest as ‘environment’, due to having limited access
to it. In such cases, verifying the existence of a quantum memory through a
probe certifies that the process preserves the required coherences.

Figure 1: Representation of a process with a quantum memory, where the
environment is a quantum system that interacts in a coherent way with the
system, and a classical memory where there is carrier of classical information
that interacts with the system.

Here, we give a rigorous definition of processes with classical memory and
present a technique to detect a _quantum_ memory using witnesses—a set of
measurements that verify the property without the need of full tomography. We
use the process matrix formalism Oreshkov  _et al._ (2012); Oreshkov and
Giarmatzi (2016) and in this language the process can be represented as a
multipartite state. For a specific partition of the state into subsystems,
entanglement of the state corresponds to the existence of a quantum memory.
Therefore, given a process we can employ all the known techniques that verify
entanglement, in order to verify that the process is non-Markovian with
quantum memory. We use the method that involves entanglement witnesses which
maps them to witnesses for quantum memory.

To illustrate our method, we apply it to a physically motivated example of a
process: the system and environment are qubits jointly prepared in a maximally
entangled state and later interact according to the Ising model, in between
two measurement stations $A$ and $B$ for the system. A quantum memory witness
corresponds to the operations of $A$ and $B$ to verify the property. As
separability criteria for the search of witnesses we use the positive partial
transpose (PPT) applied on the state Peres (1996) and on symmetric extensions
of the state Doherty  _et al._ (2004). We find that, in our example, the two
methods are equivalent, fully characterising the process apart from a small
set of parameters. We present our findings in a phase diagram of the
parameters of the model with areas of Markovianity, classical and quantum
memory.

#### Quantum processes.—

A typical setup can be seen as follows: in a single run of the experiment a
system goes through a number measurements stations from which we extract our
data, while in-between it may interact with the environment. We call the
process Markovian when the environment does not provide a memory to correlate
either the initial state of the system and the various interactions with the
environment, or the various interactions themselves which occur across the
experiment.

There is a considerable amount of literature to propose various definitions of
markovianity for quantum systems Piilo  _et al._ (2008); Wolf and Cirac
(2008); Breuer  _et al._ (2009); Rivas  _et al._ (2010); Hou  _et al._ (2011);
Chruściński and Maniscalco (2014); Rivas  _et al._ (2014); Breuer  _et al._
(2016); Li  _et al._ (2018), and most focus on the properties of the time-
dependent reduced state for the system, which cannot capture genuine
multipartite temporal correlations. Here we consider the notion of
markovianity originally introduced by Lindblad (1979); Accardi  _et al._
(1982), and recently reformulated within the comb formalism Pollock  _et al._
(2018). An example of a Markovian process is shown in Figure 2 for a setup
with three measurement stations, $A,B,C$, and two channels $T_{1},T_{2}$. The
channels may allow an interaction of the system with the environment but is
such that each is independent of the other and the initial state. The output
of $C$ is discarded.

Figure 2: A typical representation of a Markovian process where an initial
state is going through measurements stations $A,B,C$ and channels
$T_{1},T_{2}$ that may allow a local interaction with the environment.

The process matrix formalism Oreshkov  _et al._ (2012); Oreshkov and Giarmatzi
(2016) is a convenient way to write down quantum processes. It assumes a
number of measurement stations where arbitrary quantum operations can be
performed. An arbitrary quantum operation can be represented by a completely
positive map (CP) $\mathcal{M}^{A_{I}\rightarrow A_{O}}$ that maps the input
system $A_{I}$ to the output system $A_{O}$ of the operation. The map might
correspond to the transformation of the system resulting from the detection of
a particular measurement outcome. The collection of these maps ${\cal
J}^{A}=\\{{\cal M}^{A}\\}$, corresponding to all possible outcomes of a
measurement, represents all possible operations of a given instrument, with
${\sum}_{{\cal M}^{A}\in{\cal J}^{A}}{\cal M}^{A}$ being a CP and trace
preserving (CPTP) map. The correlations of these operations can be represented
as the joint probability of their maps given their instruments and was found
to be a linear function the maps and some matrix $W$,

$\displaystyle p({\cal M}^{A},{\cal M}^{B},\cdots|{\cal J}^{A},{\cal
J}^{B},\cdots)=$
$\displaystyle\operatorname{Tr}[W^{A_{I}A_{O}B_{I}B_{O}\cdots}(M^{A_{I}A_{O}}\otimes
M^{B_{I}B_{O}}\otimes\cdots)],$ (1)

where $M^{A_{I}A_{O}},M^{B_{I}B_{O}},\cdots$ are the Choi matrices
Jamiołkowski (1972); Choi (1975) of the maps
$M^{A_{I}A_{O}},M^{B_{I}B_{O}},\cdots$ respectively, and
$W^{A_{I}A_{O}B_{I}B_{O}\cdots}$ is the _process matrix_ that lives on the
combined Hilbert spaces of all input and output systems. The Choi matrix
$M^{A_{I}A_{O}}\in{\cal L}({\cal H}^{A_{I}}\otimes{\cal H}^{A_{I}})$,
isomorphic to a CP map ${\cal M}^{A}:{\cal L}({\cal
H}^{A_{I}})\rightarrow{\cal L}({\cal H}^{A_{O}})$ is defined as
$M^{A_{I}A_{O}}:=[{\cal I}\otimes{\cal
M}(|\mathds{1}\rangle\\!\rangle\\!\langle\\!\langle\mathds{1}|)]^{T}$, where
$\cal I$ is the identity map,
$|\mathds{1}\rangle\\!\rangle=\sum_{j=1}^{d_{A_{I}}}|jj\rangle\in{\cal
H}^{A_{I}}\otimes{\cal H}^{A_{I}}$, $\\{|j\rangle\\}^{d_{A_{I}}}_{j=1}$ is an
orthonormal basis on ${\cal H}^{A_{I}}$ and ${T}$ denotes matrix transposition
in that basis and some basis on ${\cal H}^{A_{O}}$.

In this language, it was found Costa and Shrapnel (2016); Giarmatzi and Costa
(2018) that the process matrix of a Markovian process must have the following
form

$\displaystyle
W^{AB\cdots}_{\textup{M}}=\rho^{A_{I}}\otimes{T}^{A_{O}B_{I}}\otimes\cdots,$
(2)

where ${T}^{A_{O}B_{I}}$ is the Choi matrix, defined as above but without the
transposition, of the channel $\cal T^{A\rightarrow B}$ describing the
evolution from station $A$ to station $B$, while $\rho$ represents the state
before the first measurement.

Now let us see what happens when the process is non-Markovian, that is, when
it cannot be written in the above form. One specific case is when the
different time evolutions between stations are classically correlated, and
possibly correlated with the initial state. Concretely, we can imagine that,
in each run of the experiment, the first station receives a state $\rho_{j}$
from a set. The evolution from the first to the second step can depend
probabilistically on $j$. Furthermore, during the evolution some classical
information might go back into the environment, and later get correlated with
the future time evolution. This case is generally represented by letting the
evolution between time steps be represented by a CP map, with the various CP
maps correlated with each other and with the initial state, and satisfying the
condition that the entire process takes place with probability 1.

Such a process can be written as

$\displaystyle W^{AB\cdots}_{\textup{Cl}}=\sum_{j}q_{j}\rho^{A_{I}}_{j}\otimes
T^{A_{O}B_{I}}_{j}\otimes\cdots,$ (3)

where $\rho_{j}$, $T_{j}$ are positive semidefinite matrices and $W$ is a
‘comb’, i.e. it satisfies a set of linear constraints that ensure
normalization of probabilities Chiribella  _et al._ (2009). This process can
be simulated with a classical memory that performs the switching between the
different Markovian ones; hence we define a _process with classical memory_ to
be one that has the above form.

If however the process cannot be written in either of the above forms, we call
it _process with quantum memory_. For example, a beam splitter that behaves
differently on the polarisation of the incoming photon will entangle the path
degree of freedom with polarisation. If we look at the evolution of the path
degree of freedom, this can only be simulated with a quantum memory that
carries the polarisation degree of freedom and interacts coherently with the
path. A more general situation would be that the external system—the quantum
memory—can be jointly prepared with the initial state and can later interact
with the system in a coherent way, as shown in Figure 3.

Figure 3: Example of a non-Markovian process with quantum memory: the system
and environment are prepared in a maximally entangled state
$|\Phi^{+}\rangle^{A_{I}E_{I}}$ and later undergo a unitary evolution $U$ that
acts coherently on both systems.

#### Detecting a quantum memory.—

From Equation 2 we observe that a Markovian process is a tensor product of a
state with Choi matrices of the channels. Choi’s theorem Choi (1975) ensures
that these matrices are positive semidefinite; hence the process matrix can be
viewed as a product state, up to normalization. In what follows we focus on
scenarios with two measurement stations, $A$ before $B$ for which the output
of $B$ can be ignored since the only contribution to non-Markovianity stems
from the initial system-environment correlations and their later interaction.
Therefore, a process with classical memory
$W_{\textup{Cl}}=\sum_{j}q_{j}\rho^{A_{I}}_{j}\otimes T^{A_{O}B_{I}}_{j}$ can
be written as a bipartite separable state $\rho_{sep}^{AB}$, where $A=A_{I}$,
$B=A_{O}B_{I}$. We can now see that detecting entanglement of the state
translates to detecting a quantum memory for the process; this is our main
result.

One approach to detecting entanglement is through entanglement witnesses. An
entanglement witness is a hermitian operator $Z$ whose expectation value is
positive for all separable states, $\langle
Z\rangle=\operatorname{Tr}(Z\rho_{sep})\geq 0.$ A negative expectation value
guarantees entanglement of the state, whereas a positive value provides no
guarantee for separability. In the case of processes, we define a _quantum
memory witness_ to be a hermitian operator $Z$ whose expectation value is
positive for all process matrices with classical memory

$\displaystyle\langle Z\rangle=\operatorname{Tr}(ZW_{\textup{Cl}})\geq 0.$ (4)

We note that an entanglement witness is also a quantum memory witness for the
associated process, but the reverse is not necessarily true. This is because
the space of process matrices is smaller than the space of states. Process
matrices obey further linear constraints $W=L(W)$, where $L$ is a projector on
the subspace of valid process matrices as defined in Araújo  _et al._ (2015),
hence
$\operatorname{Tr}(ZW)=\operatorname{Tr}(ZL(W))=\operatorname{Tr}(L(Z)W)$— $L$
is self adjoint—and so $L(Z)$ is also a quantum memory witness, although not
necessarily an entanglement witness.

A quantum memory witness is a hermitian matrix, hence it can always be
decomposed into combinations of CP maps Araújo  _et al._ (2015). From Equation
1 we see that the witness corresponds to the different maps of the measurement
stations; for example for a bipartite process

$\displaystyle Z=\sum_{i,j}\alpha_{i,j}M^{A_{I}A_{O}}_{i}\otimes
M^{B_{I}}_{j}.$ (5)

#### Separability criteria—

An entanglement witness is obtained through various separability criteria: a
property that is proved to hold for all separable states. These criteria
provide necessary (but not sufficient) conditions for a state to be separable.
The most notable criterion is based on partial transposition: if a state is
separable, it must have a positive partial transpose (PPT) Peres (1996) with
respect to any subsystem. For example, for a bipartite state $\rho^{AB}$ it is
separable if (but not only if) $\rho^{T_{A}}>0$, where $\rho^{T_{A}}$ denotes
partial transposition with respect to subsystem $A$ (transposition with
respect to $B$ provides an equivalent condition). An entanglement witness is
obtained through the eigenvector corresponding to the negative eigenvalue of
$\rho^{T_{A}}$. If $\rho^{T_{A}}<0$, there exist $|\psi\rangle$ such that
$\langle\psi|\rho^{T_{A}}|\psi\rangle<0$. Then
$Z=|\psi\rangle\\!\langle\psi|^{T_{A}}$ is an entanglement witness as its
expectation value is negative for the state $\rho$ and positive for any
separable state Peres (1996)

$\displaystyle\operatorname{Tr}(Z\rho^{A}\otimes\rho^{B})=\operatorname{Tr}(|\psi\rangle\\!\langle\psi|(\rho^{A})^{T_{A}}\otimes\rho^{B})\geq
0.$ (6)

A different family of separability criteria was introduced by Doherty _et al._
Doherty  _et al._ (2004). These criteria are based on the PPT criterion
applied on symmetric extensions of the state. For example, for $\rho^{AB}$,
one can apply the PPT criterion to the extended state $\tilde{\rho}^{ABA}$ and
obtain the following necessary conditions for the state to be separable:

$\displaystyle\tilde{\rho}>0,\ \ \tilde{\rho}^{T_{A}}>0,\ \
\tilde{\rho}^{T_{B}}>0,$ (7)

One can keep extending the state (i.e. $\rho^{ABAA}$) and obtain new families
of conditions, each at least as strong as the previous ones. This creates a
hierarchy of families of separability criteria which was proven to be
complete—for an entangled state it is guaranteed that some level of the
hierarchy will prove non-separability. The first step of the hierarchy is the
PPT criterion, the first symmetric extension is the second step, and so on. As
we discussed earlier, these results directly extend to define a complete
family of criteria for classical memory.

#### Example.—

We use the above separability criteria to detect a quantum memory on the
process of Figure 3. System and environment are two qubits prepared in the
state $|\phi^{+}\rangle=1/\sqrt{2}(|00\rangle+|11\rangle)$. The system
interacts with the environment according to the Ising model for nearest
neighbor interaction between two dipoles Ernst (1925), in-between two
measurement stations, $A$ and $B$. We consider the following Hamiltonian

$\displaystyle
H=-J\sigma_{x}\sigma_{x}-h(\sigma_{z}\mathds{1}+\mathds{1}\sigma_{z}),$ (8)

where $J$ is the coupling strength with which the dipoles are aligned along
the $x$ direction and $h$ is the strength of the external magnetic field along
the $z$ direction. The operator that describes the evolution of system and
environment is $U(J,h,t)=e^{-iH(J,h)t}$ acting on $A_{O}E_{I}$. The Choi
matrix of the interaction is

$\displaystyle[[U(J,h,t)]]=[[e^{-iH(J,h)t}]]^{A_{O}E_{I}B_{I}E_{O}},$ (9)

where, for a linear operator $x$, we define $[[x]]=\mathds{1}\otimes
x|\mathds{1}\rangle\\!\rangle\\!\langle\\!\langle\mathds{1}|\mathds{1}\otimes
x^{\dagger}$. The process matrix, as we obtain in the Appendix, is

$\displaystyle
W^{A_{I}A_{O}B_{I}}(J,h,t)=\operatorname{Tr}_{E_{O}}[[U(J,h,t)]].$ (10)

To detect a quantum memory for this process, we start with the PPT criterion
Peres (1996). As stated above, we can simply check if the partial transpose of
the state is negative, $\rho^{T_{A}}=W^{T_{A_{I}}}<0$. If so, the state is
entangled and an entanglement witness can be constructed by taking the
projector of the negative eigenvector of $W^{T_{A_{I}}}$. We apply the
criterion to our process for various values of the parameters of the
Hamiltonian. We detect a quantum memory for any combination of non-zero
parameters $(J,h)$, fixing $t=1$ and present our results in the next section.
Note that although the problem is a simple check of negativity of the partial
transpose $W^{T_{A_{I}}}$, it can be cast as the following SDP

maximize $\displaystyle-\operatorname{Tr}(ZW^{T_{A_{I}}})$ subject to
$\displaystyle\operatorname{Tr}Z==1.$ (11)

For a positive value of the maximizing quantity (over the hermitian
semidefinite $Z$), the quantum memory witness is $Z_{t}=Z^{T_{A}}$. Although
this way of finding a witness is slower that to simply check if
$W^{T_{A_{I}}}<0$, the advantage of the SDP is that we can add restrictions to
the witness we want to find, in the form of linear constraints. For example,
one might wish to obtain a witness of the form
$Z=\sum_{i,j}\psi^{A_{I}}\otimes\psi^{A_{O}}\otimes E_{j}^{B_{I}}$, where
Alison only has to perform projective measurements, where the re-prepared
state coincides with the measured one, while Ben is measuring with some
projector $E_{j}$. In this case, the linear constraint is $PZP=Z$, where $P$
is a swap operator that swaps the systems $A_{I}$ and $A_{O}$ in the witness
$Z$.

The second step of the hierarchy developed by Doherty _et. al._ Doherty  _et
al._ (2004) provides stronger criteria for separability. The main idea is that
the PPT criterion is applied on a symmetric extension of the state, i.e. on
$W^{A_{I}A_{O}B_{I}A_{I}}$. We present the associated SDPs in the Appendix.

#### Results.—

We run our code using the PPT criterion for various combinations of the
parameters $(J,h,t)$. As $t$ is the time of interaction between the system and
the memory, for time $t=0$ the process is trivially Markovian for every
nonzero $(J,h)$. This is also the case for $J=0$ for every nonzero $(t,h)$
because the $h-$ term is affecting both systems separately. In the Appendix,
we find all the other parameter values for which the process is Markovian. For
$h=0$ and every nonzero $(J,t)$ the process is with classical memory (or
Markovian), as we prove in the Appendix. For a fixed time $t=1$ we run the
code for $(J,h)\in[0,10]$ and present our findings in Figure 4. The value
$\operatorname{Tr}(ZW)$ becomes negative when the PPT criterion has detected a
quantum memory while in all other areas, a value close to zero is in general
inconclusive. For this reason, we run the SDP testing the second step of the
hierarchy. We find the same results, suggesting that the areas of not
detecting a quantum memory were probably areas with classical memory. However,
as we present in the Appendix, the points in the Figure where the lines are
crossing correspond to Markovian processes. Additionally, we verify that these
are the only Markovian points by calculating the norm distance $||W-W_{M}||$
of each process with an associated Markovian one, as constructed in Giarmatzi
and Costa (2018), which becomes zero when the process is Markovian.

Figure 4: Heatmap of the value $\operatorname{Tr}(ZW)$ across the parameter
space $(J,h)\in[0,10]$ with steps of $1/15$, for $t=1$, where $Z$ is the
witness obtained through the PPT criterion and $W$ is the process matrix of
our example

Our method expands considerably the investigation of open quantum systems. Our
definition of non-Markovianity with classical memory, which leads to mapping
entanglement to a quantum memory, provides an experimentally accessible tool
and paves the way for further research. It can be used in experimental
situations to detect that an external quantum system (environment) retains
quantum correlations over time—whether this is desired or not. Furthermore,
entanglement leads to many properties for states, which can be mapped to
interesting properties for processes, for example multipartite entanglement
could correspond to novel types of quantum non-Markovianity in multi-time-step
processes. Hence, investigating other properties of entanglement could unveil
more information about quantum memory in open quantum systems.

#### Acknowledgements.—

We thank Andrew Doherty for significant clarification on their paper, on which
our SDPs are based. We thank Gerard Milburn, Kavan Modi, Felix Pollok, and
Philip Taranto for discussions. This work was supported by the Australian
Research Council (ARC) Centre of Excellence for Quantum Engineered Systems
grant (CE 110001013). Furthermore, this publication was made possible through
the support of a grant from the John Templeton Foundation. The opinions
expressed in this publication are those of the authors and do not necessarily
reflect the views of the John Templeton Foundation. Finally, we acknowledge
the traditional owners of the land on which the University of Queensland is
situated, the Turrbal and Jagera people.

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##  I Appendix

###  I.1 higher-order criteria

For the second level of the hierarchy, the problem is no longer a simple
computational task—we need to find if there exist a symmetric extension
$\tilde{\rho}^{ABA}$ of a state $\rho^{AB}$ (for a process matrix $W^{AB}$),
such the state and all of its partial transposes with respect to $A$ and $B$
are positive. The latter conditions are linear constraints on
$\tilde{\rho}^{ABA}$ and so the problem can be cast as an SemiDefinite Program
which can be solved efficiently Doherty  _et al._ (2004). We develop this
method as it is a stronger test of non-separability, although in our example
it turns out that the first level completely characterizes the type of non-
Markovinity.

The SDP has a primal and a dual form. The primal form is whether it is
possible to find an extension of the state such that the PPT criterion is
satisfied. This is called the feasibility problem and is formulated as follows

minimize $\displaystyle 0$ subject to
$\displaystyle\tilde{\rho}\oplus\tilde{\rho}^{T_{A}}\oplus\tilde{\rho}^{T_{B}}>0,$
(12)

where the variable is $\tilde{\rho}$, defined to be a semidefinite matrix.
Note that a block diagonal matrix is positive if and only if each of its block
matrices is positive.

The primal SDP detects entanglement when its output is that the problem is
infeasible. The dual SDP finds an explicit extension that yields one of the
block matrices negative; hence it finds a _witness_. A witness is a Hermitian
operator that yields a negative eigenvalue when applied on the initial state,
i.e. $\operatorname{Tr}(Z\rho)<0$. The form of the dual SDP is asking to
minimize the value $\operatorname{Tr}(Z\tilde{\rho})$. The actual
implementation of the SDP is asking to minimize a slightly different value but
the concept remains the same. To explicitly write the dual form we need some
definitions. Due to symmetries on the extension $\tilde{\rho}$, it can be
written as follows

$\displaystyle\tilde{\rho}=G(x)=G_{0}+\sum_{J}x_{J}G_{J}$ (13)

where $G_{0}$ is a fixed part that depends only on the given $\rho$ and
$x_{J}$ is the coefficients for the rest of the basis elements $G_{J}$. The
dual form of the SDP writes

maximize $\displaystyle-\operatorname{Tr}(F_{0}Z)$ subject to $\displaystyle
Z>0$ (14) $\displaystyle\operatorname{Tr}(F_{i}Z)=0,$

where $F_{0}=G_{0}\oplus G_{0}^{T_{A}}\oplus G_{0}^{T_{B}}$,
$G_{0}=\Lambda(\rho)$, and $\Lambda$ is a linear map from
$\cal{H}^{A}\otimes\cal{H}^{B}$ to
$\cal{H}^{A}\otimes\cal{H}^{B}\otimes\cal{H}^{A}$. Each $F_{i}$ is a block
diagonal matrix composed of $G_{J},G_{J}^{T_{A}},G_{J}^{T_{B}}$. The SDP is
maximizes the above value over the variable $Z$, defined to be hermitian
semidefinite, living on $\cal{H}^{A}\otimes\cal{H}^{B}\otimes\cal{H}^{A}$. The
witness is obtained by $\tilde{Z}=\Lambda^{*}(Z)$, where $\Lambda^{*}$ is the
adjoint map of $\Lambda$.

###  I.2 Derivation of the process matrix

To construct the process matrix for our example, we can use the rules to
construct combs by sequential composition of processes (Chiribella  _et al._ ,
2009). For the bipartite scenario considered here, the rules are as follows:
we start from an initial state $\rho^{A_{I}E_{I}}$, while the joint system-
environment evolution is given, in Choi form, by a matrix
$T^{A_{O}E_{I}B_{I}E_{O}}$. The comb for $A_{I}$, $A_{O}$, $B_{I}$, and
$E_{O}$ is obtained combining $\rho$ and $T$ through the _link product_ :

$\rho*T=\operatorname{tr}_{E_{I}}\left[\left(\rho^{T_{E_{I}}}\otimes\mathds{1}^{A_{O}B_{I}E_{O}}\right)\left(\mathds{1}^{A_{I}}\otimes
T\right)\right],$ (15)

with the partial transpose acting on the common subsystem that is traced over.
In our example, $T=\left[\left[U\right]\right]$ and, omitting the tensor
product symbols,

$\left[\left[U\right]\right]=\sum_{jkj^{\prime}k^{\prime}}|jk\rangle\\!\langle
j^{\prime}k^{\prime}|^{A_{O}E_{I}}\left(U|jk\rangle\\!\langle
j^{\prime}k^{\prime}|U^{{\dagger}}\right)^{B_{I}E_{O}},$ (16)

while the initial state is $\rho=|\phi^{+}\rangle\\!\langle\phi^{+}|$, so that
$\rho^{T_{E_{I}}}=\frac{1}{2}\sum_{il}|i\rangle\\!\langle
l|^{A_{I}}|l\rangle\\!\langle i|^{E_{I}}$. The link product then gives

$|\phi^{+}\rangle\\!\langle\phi^{+}|*[[U]]\\\
=\frac{1}{2}\sum_{ijk}|k\rangle\\!\langle
k^{\prime}|^{A_{I}}|j\rangle\\!\langle
j^{\prime}|^{A_{O}}\left(U|jk\rangle\\!\langle
j^{\prime}k^{\prime}|U^{{\dagger}}\right)^{B_{I}E_{O}},$ (17)

which is, up to normalisation, the same expression as (16), with the first two
factors swapped and $E_{I}$ substituted with $A_{I}$. The process matrix
describing the first qubit is obtained by tracing out $E_{O}$

###  I.3 Classical memory in the $h=0$ case

In this case the Hamiltonian is

$H(J)=-J\sigma_{x}\otimes\sigma_{x},$ (18)

so it has product eigenstates, built from the Pauli matrix’s eigenstates
$\sigma_{x}|\mu\rangle=\mu|\mu\rangle$, $\mu=\pm 1$:

$\displaystyle H(J)$
$\displaystyle|\mu\rangle|\nu\rangle=-J\mu\nu|\mu\rangle|\nu\rangle,$ (19)
$\displaystyle U(J,t)$ $\displaystyle|\mu\rangle|\nu\rangle=e^{-J\mu\nu
t}|\mu\rangle|\nu\rangle,$ (20) $\displaystyle\mu,\nu$ $\displaystyle=\pm 1.$

Using the eigenstates of $H(J)$ in the definition of the Choi isomorphism, and
taking the partial trace over $E_{O}$ of expression (17), we obtain for the
process matrix

$\displaystyle W$ ${}^{A_{I}A_{O}B_{I}}(J,t)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\sum_{\begin{subarray}{c}\mu_{1}\mu_{2}\\\ \nu_{1}\nu
2\end{subarray}}|\nu_{1}\mu_{1}\rangle\\!\langle\nu_{2}\mu_{2}|^{A_{I}A_{O}}\operatorname{tr}_{E_{O}}\left(U|\mu_{1}\nu_{1}\rangle\\!\langle\mu_{2}\nu_{2}|U^{{\dagger}}\right)^{B_{I}E_{O}}$
$\displaystyle=$
$\displaystyle\frac{1}{2}\sum_{\mu_{1}\mu_{2}\nu}e^{iJ(\mu_{1}-\mu_{2})\nu
t}|\nu\rangle\\!\langle\nu|^{A_{I}}|\mu_{1}\rangle\\!\langle\mu_{2}|^{A_{O}}|\mu_{1}\rangle\\!\langle\mu_{2}|^{B_{I}}.$

We see that this matrix is an equal mixture of two product matrices:

$\displaystyle W$ ${}^{A_{I}A_{O}B_{I}}(J,t)=\frac{1}{2}\sum_{\nu=\pm
1}|\nu\rangle\\!\langle\nu|^{A_{I}}[[e^{i\nu J\sigma_{x}t}]]^{A_{O}B_{I}}.$

Therefore, the process we are describing is the equal mixture of two Markovian
processes: for $\nu=\pm 1$ respectively, these correspond to processes where
the system is initially in state $|\nu\rangle$, while from $A$ to $B$ it
evolves according to the (single-qubit) Hamiltonian $H_{\nu}=-\nu
J\sigma_{x}$. Thus, even though in our model the initial system-environment
state is entangled, the process for the system alone can be reproduced by
substituting the environment with a classical variable, which samples with
equal probabilities the values $\nu=\pm 1$, and determines both the initial
state and the evolution from first to second time step.

###  I.4 Markovian points

Here we find the parameters $J$, $h$, for which the process is Markovian. A
sufficient condition is that the unitary matrix that represents the system-
environment evolution factorises: $U(J,h,t)=e^{-iH(J,h)t}=U_{s}\otimes U_{e}$.
As $U(J,h,t)=U(Jt,ht,1)$, it is sufficient to consider the $t=1$ case.

We find that the unitary factorises for the parameter values

$\displaystyle J$ $\displaystyle=\pi k_{1},\quad k_{1}\in\mathbb{Z},$
$\displaystyle h$
$\displaystyle=\pm\frac{\pi}{2}\sqrt{k_{2}^{2}-k_{1}^{2}},\quad
k_{2}\in\mathbb{Z},\quad k_{2}^{2}\geq k_{1}^{2},$

plus an additional point $h=0,J=\pi/2$, for which
$U(J,h,1)=i\sigma_{x}\otimes\sigma_{x}$.

In principle, there could be parameter values for which the evolution is not
trivial but the process is Markovian. We verified that this is not the case
numerically: for a range of parameters, we calculated the norm distance of $W$
from the corresponding Markovian process $W_{M}=\rho^{A_{I}}\otimes
T^{A_{O}B_{I}}$, where $\rho^{A_{I}}$ and $T^{A_{O}B_{I}}$ are obtained as
partial traces of $W$ (this is the Markovianity test used in Ref. Giarmatzi
and Costa (2018)). We found that the norm distance vanishes (and hence the
process is Markovian) only for the parameter values found above. The values of
the norm distance are shown in Figure 5.

Figure 5: The norm distance $||W-W_{M}||$ across the parameter space
$(J,h)\in[0,10]$ for $t=1$.


since we are just bounding by the volume of some parabolic ball of radius $\frac{\mu}{\lambda}$. Combining the three previous bounds yields the estimate

\begin{eqnarray*}
C_2&\lesssim & \lambda^{-d-2-2a-2b} \int_{\lambda}^1\left(\frac{\mu}{\lambda}\right)^{-d-2-2b}\left(\frac{\mu}{\lambda}\right)^{-d-2-2a}\left(\frac{\mu}{\lambda}\right)^{d+2}\frac{d\mu}{\mu}\\
&=&\lambda^{d+2}\int_\lambda^1 \mu^{-2d-4-2a-2b} \frac{d\mu}{\mu}\lesssim \lambda^{-d-2-2a-2b}
\end{eqnarray*}

if $ d-2-2a-2b\neq 0 $. If $d-2-2a-2b= 0$ then $C_1=\mathcal{O}(1)$ when $\lambda>0$ goes to $0$ and we get
a logarithmic bound for $C_2$ of the form

C_2\lesssim \vert\hspace{-0.05cm} \log\lambda\vert.

So, for the moment, we proved that

\begin{eqnarray*}
C\big(\lambda^2(t_1-t_2),x,\lambda(x-y)\big) = \mathcal{O}\big(\lambda^{-d-2-2a-2b}\big)
\end{eqnarray*}

when $d+2+2a+2b>0$ and

\begin{eqnarray*}
\end{eqnarray*}

when $d+2+2a+2b=0$.
To conclude that the composite operator $C$ still belongs to $\Psi_P$, we need to prove that the above bounds still hold when we test $C$ against elements of the module $\mathcal{M}$ of vector fields tangent to the diagonal $\{t_1=t_2,x=y\}\subset \mathbb{R}^{1+d}\times \mathbb{R}^{1+d}$.

The stability of the bounds by testing against tangent vector fields is treated separately in Lemma <ref>.

The proofs of the composition Theorems is not done yet, we still need to show that we have the same estimates when we differentiate with vector fields $L_1,\dots, L_k$ that belong to the generators of the module $\mathcal{M}$ of vector fields tangent to $\{t_1=t_2,x=y\}\subset \mathbb{R}^{1+d}\times \mathbb{R}^{1+d}$.

[Composition and testing with tangent algebra]
Under the assumptions of Theorem <ref>, for every $L\in \mathcal{M}$ in the tangent module, the kernel $LC$ satisfies the same estimate as $C$:

\begin{eqnarray*}
\big\vert LC(t_1,t_2,x,y)\big\vert \leq  C_L \left(\vert t_2-t_1\vert +\vert x-y\vert^2 \right)^{-\frac{d+2+2a+2b}{2}}.
\end{eqnarray*}

We reduce the proof to some local computation in local coordinates involving generators of $\mathcal{M}$. We will do the detailed calculation for translations of the form $\partial_{x^i}+\partial_{y^i} $ (translation) and for general linear vector fields fixing the diagonal $\{x-y=0\}\subset U\times U$ of the form $M(x-y)\cdot\partial_x$. This covers the following important examples: $(x^i-y^i)\partial_{x^k}-(x^k-y^k)\partial_{x^i}$ (rotation), $(x^i-y^i)\partial_{x^i}$ (scaling), $(x^i-y^i)\partial_{x^k}+(x^k-y^k)\partial_{x^i} $ (boosts) in the situation of Theorem <ref> and we leave to the reader the other computations which follow the same pattern. We start with the translations

\begin{eqnarray*}
\left(\partial_{x^i}+\partial_{y^i}\right) C(t_1,t_2,x,y)&=&\left(\partial_{x^i}+\partial_{y^i}\right) \int_{\mathbb{R}^d}\int_{-L}^{\inf(t_1,t_2)} A(t_2-s,x,z) B(t_1-s,z,y) \chi(z) ds dz\\
&=&\int_{\mathbb{R}^d}\int_{-L}^{\inf(t_1,t_2)} ((\partial_{x^i}+\partial_{z^i} )A)(t_2-s,x,z)B(t_1-s,z,y)  \chi(z) ds dz\\
&+& \int_{\mathbb{R}^d}\int_{-L}^{\inf(t_1,t_2)} A(t_2-s,x,z) ((\partial_{y^i}+\partial_{z^i} )B)(t_1-s,z,y) \chi(z) ds dz\\
&+& \int_{\mathbb{R}^d}\int_{-L}^{\inf(t_1,t_2)} A(t_2-s,x,z)B(t_1-s,z,y) (\partial_{z^i}\chi)(z) ds dz.
\end{eqnarray*}

We see we can repeat the bounds of the proof of Theorem <ref> on each term using the crucial information that both $(\partial_{x^i}+\partial_{z^i} )$  and $(\partial_{y^i}+\partial_{z^i}) $ are in the tangent algebra of the respective diagonals $\{x=z\}$ and $\{y=z\}$ and the stability of the two kernels $A, B$ in $\Psi_P$ by derivation by the tangent Lie algebra. Given a matrix $M\in M_d(\mathbb{R})$, we use the short hand notation $M(x-y)\cdot\partial_x$ for the vector field $M_i^j(x-y)^i\partial_{x^j}$ where we sum over repeated indices. Differentiating at $t=0$ yields the exact identities

\begin{equation*} \begin{split}
M(x-y)\cdot\partial_x &\int_{-L}^{\inf(t_1,t_2)} A(t_2-s)\circ \chi B(t_1-s)  ds   \\
&= \int_{-L}^{t_1} \left(\underbrace{ M(x-z)\cdot\partial_x} A\right)\circ \chi B+\left(\underbrace{\left(M(z-y)\cdot\partial_x + M(z-y)\cdot\partial_y  \right)}A\right) \circ \chi B \\
&\hspace{1cm}+  A\circ \chi\left( \underbrace{ M(z-y)\cdot\partial_y}B\right) + A\circ \left(M(z-y)\cdot\partial_y\chi\right)B ds
\end{split} \end{equation*}

we decompose in different groups where we underbrace the vector fields which are tangent to the diagonal of the corresponding kernel. As usual, we decomposed $M(x-y)\cdot\partial_x \int_{-L}^{\inf(t_1,t_2)} A(t_2-s)\circ \chi B(t_1-s)  ds $ as a sum of three compositions of operators where the operators still satisfy the assumptions of Theorem <ref>, so we are done.

§ A COMMUTATOR IDENTITY ON $\MATHBB{R}^D$$\BOLDMATH{.}$

We prove in this appendix a commutator estimate for triple paraproducts on $\mathbb{R}^d$, the result we establish is originally due to Bony <cit.> but we thought it would be useful to include a complete detailed proof here since it plays a central role for the proof of Theorem <ref> and the original paper [12] is written in French.

We are given $(f,g,h)$ where $f\in C^{\alpha_1},g\in C^{\alpha_2}$ and $h\in C^\beta$ where $\alpha_1,\alpha_2>0$ and $\beta <0$. We would like to control a commutator:

\begin{eqnarray*}
f\prec\left( g\prec h \right)-(fg)\prec h.
\end{eqnarray*}

Let $f\in C^{\alpha_1}(\mathbb{R}^d),g\in C^{\alpha_2}(\mathbb{R}^d)$ and $h\in C^\beta(\mathbb{R}^d)$ where $\alpha_1,\alpha_2>0$ and $\beta <0$. Then we have

\begin{eqnarray*}
\|f\prec\left( g\prec h \right)-(fg)\prec h\|_{\alpha_1\wedge \alpha_2+\beta}\lesssim \|f\|_{\alpha_1}\|g\|_{\alpha_1}\|h\|_{\beta}.
\end{eqnarray*}

We first deal with the term $f\prec\left( g\prec h \right) $. By definition, we write:

\begin{eqnarray*}
f\prec\left( g\prec h \right)=\sum_{i=2}^\infty S_{i-2}(f) \Delta_i\left( \sum_{j=2}^\infty S_{j-2}(g)\Delta_j(h)\right).
\end{eqnarray*}

In the sequel, we shall repeatedly use the following result that can be found in <cit.>.

[Decomposition Hölder]
Let $p$ be a real number, $p\notin \mathbb{N}$ and $0<a<b$ be given.
If we are given a sequence $a_j$ of smooth functions such that
$\Vert a_j \Vert_{L^\infty}=\mathcal{O}(2^{-jp})$
and each $a_j$ is Fourier supported in coronas $\big\{ a2^j\leq  \vert \xi\vert \leq  b2^j\big\}$, then the series
$ \sum_{j=0}^\infty a_j$
converges in the Hölder space $C^p=B^p_{\infty,\infty}$.

The first crucial observation, since $\Delta_i$ localizes in Fourier space on the corona $\{ 2^{i-1}\leq  \vert \xi\vert\leq  2^{i+1} \}$ and that each $S_{j-2}(g)\Delta_j(h)$ is supported in the corona $2^{j-2}\leq  \vert \xi\vert\leq   2^{j+2}$, necessarily the double sum over both $i,j$ localizes on the diagonal $ \vert i-j \vert\leq  3$.
So we rewrite the previous term as a double sum

\begin{eqnarray*} \label{eq:diaglocalization}
f\prec\left( g\prec h \right)=\sum_{\vert i-j\vert\leq  3, i,j\geqslant 2} S_{i-2}(f) \Delta_i\big( S_{j-2}(g)\Delta_j(h)\big).
\end{eqnarray*}

The second observation is that if we fix $j$, then the sum of the five terms:

\begin{eqnarray} \label{eq:asumgivesidentity}
\sum_{i=j-3}^{j+3}\Delta_{i}\big( S_{j-2}(g)\Delta_j(h)\big) = S_{j-2}(g)\Delta_j(h)
\end{eqnarray}

this is because
$ \psi(2^{-j+3}\xi)+\dots + \psi(2^{-j-3}\xi)=1$ on the corona
$2^{j-2}\leq  \vert \xi\vert\leq   2^{j+2}$ by construction of the Littlewood-Paley-Stein partition of unity.
Therefore at fixed $j$, we can add and subtract as follows

\begin{eqnarray*}
&&\sum_{i=j-3}^{j+3} S_{i-2}(f) \Delta_i\big( S_{j-2}(g)\Delta_j(h)\big)   \\
&=&\sum_{i=j-3}^{j+3} S_{j-5}(f) \Delta_i\big( S_{j-2}(g)\Delta_j(h)\big) - \sum_{i=j-3}^{j+3} (S_{i-2}-S_{j-5})(f) \Delta_i\big( S_{j-2}(g)\Delta_j(h)\big)   \\
&=&S_{j-5}(f)  S_{j-2}(g)\Delta_j(h)
-\sum_{i=j-3}^{j+3} (S_{[j-5,i-2]})(f) \Delta_i\big( S_{j-2}(g)\Delta_j(h)\big)
\end{eqnarray*}

then in the second line, we used the second miracle equation (<ref>). We define $(S_{[j-5,i-2]})$ as the difference $(S_{i-2}-S_{j-5})$ and we observe that $(S_{[j-5,i-2]})(f)$ is Fourier supported on some corona contained in the shell $ \vert \xi\vert\leq  2^{i-2}$. Since $\Delta_i\left( S_{j-2}(g)\Delta_j(h)\right)$ is supported in the shell $2^{i-1}\leq  \vert \xi\vert\leq  2^{i+1}$ because of the localizing property of $\Delta_i$, the discrepancy $i-2,i$ makes the support of the product $(S_{[j-5,i-2]})(f) \Delta_i\left( S_{j-2}(g)\Delta_j(h)\right)$ a corona around $\vert\xi\vert\simeq 2^i$. From the Hölder regularities assumptions on the functions $(f,g,h)$, we get the bound

\begin{eqnarray*}
\Big\Vert \sum_{i=j-3}^{j+3}  (S_{[j-5,i-2]})(f) \Delta_i\left( S_{j-2}(g)\Delta_j(h)\right) \Big\Vert_{L^\infty} \lesssim \sum_{i=j-3}^{j+3} \big\Vert (S_{[j-5,i-2]})(f)\big\Vert_{L^\infty}\Vert \Delta_j(h)\Vert_{L^\infty}\\ \lesssim 2^{-(j-5)\alpha_1}2^{-j\beta}\lesssim 2^{-j(\alpha_1+\beta)}
\end{eqnarray*}

where we use the fact that $S_{j-2}(g)$ is bounded uniformly in the index $j$ since $g\in C^{\alpha_2}$ for $\alpha_2>0$.
So the series

\begin{eqnarray*}
\sum_j \left( \sum_{i=j-3}^{j+3}  (S_{[j-5,i-2]})(f) \Delta_i\left( S_{j-2}(g)\Delta_j(h)\right)\right)
\end{eqnarray*}

is a series of functions
supported in coronas $ a2^j\leq  \vert \xi\vert \leq  b2^j$ for $0<a<b$ and thus converges absolutely in $C^{\alpha_1+\beta}$. This tells us that in the equality

\begin{equation*} \begin{split}
\sum_{i=j-3}^{j+3} S_{i-2}(f) \Delta_i&\big( S_{j-2}(g)\Delta_j(h)\big)   \\
&= S_{j-5}(f)  S_{j-2}(g)\Delta_j(h) - \sum_{i=j-3}^{j+3} (S_{[j-5,i-2]})(f) \Delta_i\big( S_{j-2}(g)\Delta_j(h)\big)
\end{split} \end{equation*}

the sum $\sum_{i=j-3}^{j+3} (S_{[j-5,i-2]})(f) \Delta_i\big( S_{j-2}(g)\Delta_j(h)\big)$ is a good term absorbed in a good remainder and we should only keep $S_{j-5}(f)  S_{j-2}(g)\Delta_j(h)$.

So for the moment, we just proved that

\begin{eqnarray*} \label{eq:simplificationhuge}
f\prec\left( g\prec h \right)=\sum_{j\geqslant 5} S_{j-5}(f)  S_{j-2}(g)\Delta_j(h)+C^{\alpha_1+\beta}.
\end{eqnarray*}

Now we would like to compare this quantity with

\begin{eqnarray*}
\left(fg\right)\prec h=\sum_{j\geqslant 2} S_{j-2}(fg)\Delta_j(h)
\end{eqnarray*}

so the difference $f\prec\left( g\prec h \right)-\left(fg\right)\prec h$ reads

\begin{eqnarray*}
\sum_{j\geqslant 5} \big(S_{j-5}(f)  S_{j-2}(g)-S_{j-2}(fg)\big) \Delta_j(h)+C^{\alpha_1+\beta}.
\end{eqnarray*}

Everything boils down to studying the difference $S_{j-5}(f)  S_{j-2}(g)-S_{j-2}(fg)$ which we treat as follows. First decompose $fg=S_{j-5}(f)S_{j-5}(g)+R$ where the remainder contains at least either one of the two terms $ \sum_{i\geqslant j-5} \Delta_i(f)$ or  $\sum_{i\geqslant j-5} \Delta_i(g) $ in factor. Observe that

\begin{eqnarray*}
\sum_{i\geqslant j-5} \Delta_i(f)=\mathcal{O}_{C^{\alpha_1}}(2^{-j\alpha_1})\\
\sum_{i\geqslant j-5} \Delta_i(g)=\mathcal{O}_{C^{\alpha_2}}(2^{-j\alpha_2})
\end{eqnarray*}

this is almost by construction of these objects and by definition of the Hölder norms. Therefore using the continuity of
$S_{j-2}:C^\bullet\mapsto C^\bullet$ acting on Hölder spaces where this is bounded uniformly in $j$, we deduce that
and $\sum_j S_{j-2}(R)\Delta_j(h)$ is a series of functions each term
supported in coronas $ a2^j\leq  \vert \xi\vert \leq  b2^j$ for $0<a<b$, $ \Vert S_{j-2}(R)\Delta_j(h)\Vert_{L^\infty}=\mathcal{O}(2^{-j(\alpha_1\wedge\alpha_2+\beta)}) $  and thus
converges absolutely in $C^{\alpha_1\wedge\alpha_2+\beta}$.
Using the magic identity

S_{j-2}\big(S_{j-5}(f)S_{j-5}(g)\big) = S_{j-5}(f)S_{j-5}(g),

this means that the difference can be simplified as

S_{j-5}(f)  S_{j-2}(g)-S_{j-2}(fg) = S_{j-5}(f)\big(  S_{j-2}(g)-S_{j-5}(g)\big) - S_{j-2}(R)

and combining with the fact that $\sum_j S_{j-2}(R)\Delta_j(h)\in C^{\alpha_1\wedge\alpha_2+\beta}$, the difference $f\prec\left( g\prec h \right)-\left(fg\right)\prec h$ now reads

\sum_{j\geqslant 5} S_{j-5}(f)S_{[j-5,j-2]}(g)\Delta_j(h)+C^{\alpha_1\wedge\alpha_2+\beta},

we are done using again the fact that it is a series of functions supported in annular domains and that
$\Vert S_{j-5}(f)S_{[j-5,j-2]}(g)\Delta_j(h)\Vert_{L^\infty}=\mathcal{O}(2^{-j(\alpha_2+\beta)})$.
So we get

f\prec\left( g\prec h \right)-\left(fg\right)\prec h \in C^{\alpha_1\wedge\alpha_2+\beta}

as required.

§ PARALINEARIZATION IN THE BUNDLE CASE.

Let $F\in C^\infty( \mathbb{R}^d\times M_n(\mathbb{R}), M_n(\mathbb{R}))$ be a smooth function of the two variables $(x,m)\in \mathbb{R}^d\times M_n(\mathbb{R})$ where the second variable is matrix and $F$ is matrix valued.

Assume we are given a map

$$M: x\in \mathbb{R}^d\mapsto M(y)\in M_n(\mathbb{R})$$
of Hölder regularity $C^\alpha$, we would like to define a paralinearization of the composite function
$$ x\in\mathbb{R}^d \longmapsto F(x,M(x))$$
generalizing the work of Bony to the noncommutative matrix case. This allows to extend paralinearization to composition of nonlinear bundle maps as we will later explain.

Start from the telescopic series, we put $S_jM=M_j$:

\begin{eqnarray*}
\end{eqnarray*}

the series converges uniformly to $F(.,M(.))$.

We have the nice identity

\begin{eqnarray}
F(.,M_{j+1})-F(.,M_j)= \mathbf{M}_j\Delta_jM
\end{eqnarray}

where $\mathbf{M}_j$ is the linear operator :

\begin{eqnarray*}
\mathbf{M}_j=\int_0^1 dtD_mF(.,M_j+t\Delta_jM)\in C^\infty(\mathbb{R}^d\times M^n(\mathbb{R}), End(M^n(\mathbb{R}),M^n(\mathbb{R})) ).
\end{eqnarray*}

If $M\in C^\alpha$ for $\alpha\in (0,+\infty)$ and $ \sup_{x\in \mathbb{R}^p,M\in M_n(\mathbb{R})\setminus\{0\} } \frac{\Vert D_mF(x,M)\Vert_{M_n(\mathbb{R})}}{\Vert M\Vert_{M_n(\mathbb{R})}} <+\infty $, then
there exists constants $C_\alpha$ s.t.

$$\Vert \partial^\alpha \mathbf{M}_j\Vert_\infty \leqslant C_\alpha 2^{j\vert\alpha \vert}.$$

By definition of the Besov-Hölder norm, $\Vert\Delta_jM\Vert_\infty \leqslant \Vert M\Vert_{C^\alpha}2^{-j\alpha} $ therefore
$\Vert S_jM\Vert_\infty\leqslant \sum_{p\leqslant j} \Vert \Delta_jM\Vert_\infty\leqslant \sum_{p\leqslant j} \Vert M\Vert_{C^\alpha}2^{-p\alpha} \leqslant \Vert M\Vert_{C^\alpha}$.

Therefore each projected function $S_jM$ is bounded in $L^\infty$ norm by $\Vert M\Vert_{C^\alpha}$ and Fourier supported in some ball of radius $\lesssim 2^{j+1}$.
Therefore by Bernstein's Lemma, we deduce the inequality
$\Vert\partial^\alpha S_jM\Vert_\infty \leqslant \Vert S_jM\Vert_\infty  2^{j\vert\alpha\vert}\leqslant \Vert M\Vert_{C^\alpha} 2^{j\vert\alpha\vert}$.
We need to prove that the above bounds for the sequence of smooth functions $(M_j)_j$ remain stable under composition with some given smooth function $G\in C^\infty(\mathbb{R}^d\times M_n(\mathbb{R}))$.

By the Faa-di-Bruno formula:

\begin{eqnarray*}
&&\Vert\partial^\beta G(.,M_j(.))\Vert_\infty \\ &\leqslant &\sum_{ \alpha_1+\alpha_2=\beta, \beta_1+\dots+\beta_k=\alpha_2 } \left(\begin{array}{c}
\beta\\
\alpha_1,\alpha_2
\end{array} \right) \left( \begin{array}{c}
\alpha_2\\
\beta_1\dots\beta_k
\end{array} \right)  \Vert \partial_x^{\alpha_1}\partial_m^{\alpha_2}G \left(\partial^{\beta_1}M_j, \dots, \partial^{\beta_k}M_j\right)\Vert_\infty\\
&\lesssim & \sum_{ \alpha_1+\alpha_2=\beta, \beta_1+\dots+\beta_k=\alpha_2 } \Vert \partial_x^{\alpha_1}\partial_m^{\alpha_2}G \Vert_\infty
\underset{2^{j(\vert\beta_1\vert+\dots+\vert\beta_k\vert)}}{ \underbrace{ \Vert\partial^{\beta_1}M_j\Vert_\infty \dots\Vert \partial^{\beta_k}M_j\Vert_\infty}}\\
&\lesssim &\sum_{ \alpha_1+\alpha_2=\beta, \beta_1+\dots+\beta_k=\alpha_2 } \Vert \partial_x^{\alpha_1}\partial_m^{\alpha_2}G \Vert_\infty 2^{j\vert\alpha_2\vert}\lesssim 2^{j\vert \beta\vert}
\end{eqnarray*}

where we sum over all $k$ and all multiindices $(\beta_1,\dots,\beta_k)$, , $(\alpha_1,\alpha_2)$  satisfying certain integer partitions
with constraints: $\alpha_1+\alpha_2=\beta, \beta_1+\dots+\beta_k=\alpha_2 $.

Finally, by the application of the Faa–di–Bruno formula for the composition $D_mF(.,M_j+t\Delta_jM)$ as in <cit.>, we deduce the desired bound. We use the fact that Hölder functions of positive regularity forms an algebra for the pointwise product.

Consider the linear operator $\mathcal{L}$ defined as

\begin{eqnarray*}
\mathcal{L}U=\sum_j\mathbf{M}_j\Delta_jU
\end{eqnarray*}

the operator $\mathcal{L}$ belongs to $\Psi^0_{1,1}(\mathbb{R}^d\times M_n(\mathbb{R}), End(M_n(\mathbb{R}),M_n(\mathbb{R})))$.
Indeed the symbol of this operator reads

$$ \sum_j\mathbf{M}_j(x,m)\psi(2^{-j}\xi)\in C^\infty( \mathbb{R}^d\times M_n(\mathbb{R}) , End(M_n(\mathbb{R}),M_n(\mathbb{R})) ) $$

and the estimate of the above Lemma implies that the symbol belongs to the space $S^0_{1,1}$.

Following Hörmander <cit.>, we define a pararegularized operator as :

\begin{eqnarray}
\left(D_mF(.,M)\prec M\right)_{ij}:= \sum_{(k,\ell)\in \{1,\dots,n\}^2} \frac{\partial F_{ij}}{\partial m_{k\ell}}(.,M(.)) \prec M_{k\ell}
\end{eqnarray}

where $(i,j)\in \{1,\dots,n\}^2$ and the paraproduct is taken w.r.t. the variable $x\in \mathbb{R}^d$

$$ \sum_{\ell=3}^\infty \sum_{(k,\ell)\in \{1,\dots,n\}^2} S_{\ell-2}\left(\frac{\partial F_{ij}}{\partial m_{k\ell}}(.,M(.))\right) \Delta_\ell M_{k\ell} $$

where the LP projection is taken w.r.t. the variable $x\in \mathbb{R}^d$.

For $s\in (0,+\infty)$,
if $F\in $ and $M\in C^s$ then
the difference term

\begin{eqnarray}
F(.,M)- D_mF(.,M)\prec M\in C^{2s}
\end{eqnarray}

has higher regularity.

It suffices to show that the difference pseudodifferential operators
$\mathcal{L}-D_mF(.,M)\prec  $ belong to the space of pseudodifferential operators $\Psi^{-s}_{1,1}$. We compute its symbol

\begin{eqnarray*}
\rho(x,M;\xi)=\sum_{j=2}^\infty Q_j(x,M) \psi(2^{-j}\xi)
\end{eqnarray*}

$$Q_j(x,M)=\int_0^1 dtD_mF(.,M_j+t\Delta_jM)-S_{j-2}\left(\frac{\partial F}{\partial m}(.,M(.))\right) .$$

It suffices to show the decay estimate

\begin{eqnarray*}
\Vert \partial^\beta Q_j \Vert_\infty\leqslant C_\beta 2^{-js+j\vert \beta\vert}.
\end{eqnarray*}

We start with $\beta=0$.
The first idea is to note that $M\in C^s$ hence $\Vert M-S_jM\Vert_{\infty}\leqslant \Vert M\Vert_{C^s}2^{-js}$ hence

$$ \Vert D_mF(.,M)-D_mF(.,S_jM+t\Delta_jM)\Vert_{\infty}\lesssim 2^{-js} $$
uniformly in $t\in [0,1]$,
since the differential $D_mF(.,.)$ depends smoothly hence in a Lipschitz way in its second argument $M$.
But $D_mF(.,M(.)) $ also belongs to $C^s$ as a composition of a smooth function with a Hölder function (obvious from the definition of Hölder functions in terms of controlled increments).
Therefore we also have
an estimate of the form

$$ \Vert D_mF(.,M(.)) -S_{j-2}D_mF(.,M(.))  \Vert_\infty\lesssim 2^{-js} $$
and therefore by the triangle inequality

$$ \Vert D_mF(.,S_jM+t\Delta_jM)- S_{j-2}D_mF(.,M(.)) \Vert_\infty \lesssim 2^{-js}  $$

uniformly in $t\in [0,1]$.

Then we deal with the case where $\vert \beta\vert>s$.
In this case, we no longer control the difference term but we rather need to prove directly that

\begin{eqnarray}\label{eq:derivativeprojected}
\Vert \partial^\beta S_{j-2} D_mF(.,M(.)) \Vert_\infty\lesssim 2^{-js+j\vert \beta\vert}
\end{eqnarray}

and also

\begin{eqnarray*}\label{eq:derivativefunctionproj}
\Vert \partial^\beta D_mF(.,S_jM+t\Delta_jM) \Vert_\infty\lesssim 2^{-js+j\vert \beta\vert}.
\end{eqnarray*}

The first inequality (<ref>) is easy to prove since for all $g\in C^s$, if $\vert\beta\vert >s $ then

$$ \Vert\partial^\beta S_jg\Vert_\infty \leqslant \sum_{p\leqslant j}
\Vert\partial^\beta \Delta_pg\Vert_\infty\leqslant  \sum_{p\leqslant j}  2^{(p+1)\vert\beta\vert} \Vert \Delta_pg\Vert_\infty\lesssim  \sum_{p\leqslant j}  2^{(p+1)\vert\beta\vert} 2^{-ps}\lesssim 2^{(p+1)(\vert\beta\vert-s)} $$

where we just used Bernstein's Lemma to estimate $L^\infty$ norms of derivatives of .functions with Fourier support in corona's of radius $\simeq 2^p$. The second inequality (<ref>) set
$M_j=S_jM+t\Delta_jM$, we keep the following properties

\begin{eqnarray}\label{ineq:controlledincrement}
\Vert M_{j+1}-M_j\Vert_\infty\lesssim 2^{-js}
\end{eqnarray}

\begin{eqnarray}\label{ineq:controlledderivatives}
\Vert\partial^\beta M_j \Vert_\infty\lesssim 2^{j(\vert \beta\vert-s)}.
\end{eqnarray}

We only need to following general fact, for any sequence $(M_j)_j$ of functions in $C^\infty(\mathbb{R}^d,M_n(\mathbb{R}))$ satisfying inequalities <ref> and <ref>, then for any smooth $G(.,.)\in C^\infty(\mathbb{R}^d\times M_n(\mathbb{R}),M_n(\mathbb{R}))$, the sequence of composite functions $ G(.,M_j(.))_j$ satisfies the same type of estimates, stability of the above estimates by composition. The estimate of the form of <ref> is obviously satisfied for $G(.,M_j(.))_j$ and we focus on <ref>. The main difficulty is that when we apply Faa-di–Bruno formula for $\partial^\beta G(.,M_j(.))$, $\vert \beta\vert > s$ this involves derivatives $\partial^\alpha M_j$ of order $\vert \alpha\vert\leqslant s$ on which our assumptions give no control.

Inequality <ref> implies that

$$\Vert \partial^\beta (M_{j+1}-M_j)\Vert_\infty\lesssim 2^{-js+j\vert\beta\vert}$$

for all multiindex $\beta$ since our functions are supported in balls of radius $\simeq 2^j$ and by Bernstein's Lemma. So by triangular inequality we deduce that for multindices $\vert \beta\vert < s$, we still have:

$$ \Vert \partial^\beta M_j \Vert_\infty \leqslant \sum_{p\leqslant j-1}
\Vert \partial^\beta (M_{p+1}-M_p)\Vert_\infty\lesssim \sum_{p\leqslant j-1} 2^{-ps+p\vert\beta\vert} \lesssim  1$$

and when $\vert \beta\vert=s$ we get
a slightly worse bound of the form:

$$ \Vert \partial^\beta M_j \Vert_\infty \leqslant \sum_{p\leqslant j-1}
\Vert \partial^\beta (M_{p+1}-M_p)\Vert_\infty\lesssim \sum_{p\leqslant j-1} 1 \lesssim  j.$$

With these slightly worse bound we can still control $\Vert \partial^\beta G(.,M_j(.)) \Vert_\infty$ by using the Faa-di–Bruno formula.

A consequence of the above result is the following paralinearization Theorem for bundle maps

Let $(E_1,E_2,E_3)\mapsto M$ be a triple of smooth vector bundles, $f\in C^s(Hom(E_1,E_2))$ a bundle map of Hölder regularity $s\in (0,+\infty)$ and $F$ a smooth local map
sending sections of $E_2$ to $E_3$ in the sense that $F(x,s(x))\in E_{3,x}$ depends only on $x,s(x)\in M\times E_{2,x}$.

Then there exists a generalized paraproduct $\prec$ such that
the difference

\begin{eqnarray*}
F(.,f(.))-D_sF(.,f(.))\prec f\in C^{2s}.
\end{eqnarray*}

has Hölder regularity $2s$.

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$\bullet$ I. Bailleul – Univ Brest, CNRS UMR 6205, Laboratoire de Mathématiques de Bretagne Atlantique, France. E-mail<EMAIL_ADDRESS>
$\bullet$ N.V. Dang – Sorbonne Université and Université Paris Cité, CNRS, IMJ-PRG, F-75005 Paris, France. Institut Universitaire de France, Paris, France.
E-mail<EMAIL_ADDRESS>
$\bullet$ L. Ferdinand –
Laboratoire de Physique des 2 infinis Irène Joliot-Curie, UMR 9012, Université Paris-Saclay, Orsay, France.
E-mail<EMAIL_ADDRESS>
$\bullet$ T.D. Tô – Sorbonne Université and Université Paris Cité, CNRS, IMJ-PRG, F-75005 Paris, France. E-mail<EMAIL_ADDRESS>