# Evidence of the entanglement constraint on wave-particle duality using the
IBM Q quantum computer

Nicolas Schwaller<EMAIL_ADDRESS>Institute of Physics, École
Polytechnique Fédérale de Lausanne (EPFL), Lausanne, CH-1015, Switzerland
Marc-André Dupertuis<EMAIL_ADDRESS>Institute of Physics, École
Polytechnique Fédérale de Lausanne (EPFL), Lausanne, CH-1015, Switzerland
Clément Javerzac-Galy<EMAIL_ADDRESS>Institute of Physics, École
Polytechnique Fédérale de Lausanne (EPFL), Lausanne, CH-1015, Switzerland
Miraex, EPFL Innovation Park, Bâtiment C, Lausanne, CH-1015, Switzerland

###### Abstract

We experimentally verify the link existing between entanglement and the amount
of wave-particle duality in a bipartite quantum system, with superconducting
qubits in the IBM Q quantum computer. We analyze both pure and mixed states,
and study the limitations of the state purity on the complementarity
“triality” relation. This work confirms the quantitative completion of local
Bohr’s complementarity principle by the nonlocal quantum entanglement typical
of a bipartite quantum system.

##  I Introduction

In 1924, physicist Louis De Broglie developed the theory of electron waves
debroglie:tel-00006807 , coming up with the idea that particles behave like
waves. This discovery is with no doubt one of the most stunning ideas in
physics. Indeed, four years later, Niels Bohr formulated his principle of
complementarity Bohr:1928:QPRb  dealing with this non intuitive property of
Nature. It is possible to detect particle and wave characteristics of a single
quantum object, but it never behaves fully like a wave and a particle at the
same time. This idea was democratized by Richard Feynman in 1965, who
underlined the strangeness of the so called _wave-particle duality_ : “a
phenomenon which is impossible, _absolutely_ impossible, to explain in any
classical way, and which has in it the heart of quantum mechanics. In reality,
it contains the _only_ mystery” feynman1965flp . Experiments were in
particular conducted with neutrons and photons (see e.g. RevModPhys.60.1067 ),
typically with double-slit setups, where a single quantum object has two
possible paths, before to reach a screen where its position is measured.
Knowing which path the object took indicates the object is a point-like
particle, whereas observing an interference pattern on the screen, formed by
the detected positions of the particles when the experiment is repeated, is
the manifestation of the wave characteristic of the object, which apparently
passes through both slits at once.

Wootters and Zurek initiated in 1979 a quantitative approach to wave-particle
duality in a double slit experiment PhysRevD.19.473 , applicable to
intermediate cases where the wave character would be incompletely revealed,
and their surprising results prompted in 1988 a new and simpler mathematical
description of Bohr’s principle by Greenberger and Yasin GREENBERGER1988391 ,
namely

$V^{2}+P^{2}\leq 1\,,$ (1)

where $V,P\in[0,1]$ are respectively the _a priori_ fringe _visibility_ and
the which-way _a priory distinguishability_ (also called “predictability”).
$V$ is commonly associated to the waviness and $P$ to the particleness of a
single quantum object. Equality holds for pure states, or “coherent” beams.
Such type of inequality was later investigated and extended by a few authors.
Jaeger, Shimony, and Vaidman PhysRevA.51.54  proved a similar relationship in
_bipartite systems_ (two-particle interferometer) by relating the visibility
$V$ of one-particle interference fringes to the _visibility of two-particle
fringes_ $V_{12}\in[0,1]$ , i.e. $V^{2}+V_{12}^{2}\leq 1$. Englert
PhysRevLett.77.2154  also obtained a similar result

$V^{2}+D^{2}\leq 1\,,$ (2)

where $D\in[0,1]$ is this time the _a posteriori distinguishability_ (after
detection, therefore also intrinsically bipartite). Equality holds when the
which-path detector is similarly in a pure state. The analysis of the
experiments for which relations of type (1) or (2) hold can thus be
significantly different and easily induce misleading representations. Englert
introduced a more prudent definition of the notion of (wave-particle)
_duality_ , i.e. “the observations of an interference pattern and the
acquisition of which-way information are mutually exclusive” and emphasized
also that duality might not be enforced only by the use of position-momentum
uncertainty relations as in the historical Bohr-Einstein debates (opening
another debate, a recent account on this subtle aspect of complementarity can
be found in Xiao et al.Xiaoeaav9547 ). The second duality relation (2) was
first experimentally tested by Dürr et al.PhysRevLett.81.5705  with an atom
interferometer. For the first duality relation (1), earlier experiments in
neutron interferometers implicitely tested it PhysRevA.36.4447 ;
englert_bergou_2000 . For subsequent work Englert and Bergou PhysRevA.36.4447
; englert_bergou_2000  include a short review of experimental and theoretical
work on this topic (as of 2000), and put also on record a new erasure
inequality quite similar to (2) but with different quantities outside our
scope here. Very recently Norrman et al. Norrman2020  also derived interesting
vector-light complementarity relations (1) and (2) in the case of double
pinhole vectorial interference, the relevant visibility becomes then the
Stokes visibility for polarization modulation.

Another important appreciation of wave-particle duality came with the
realization of _delayed-choice experiments_ , first proposed by Wheeler
RevModPhys.88.015005  as gedanken experiments. In such case the choice of the
type of measurement is delayed at a later stage, which allows to challenge the
idea that the measurement configuration could dictate _a priori_ the waviness
or particleness of the quantum system, as if the system would adapt to the
choice of measurement. Delayed choice experiments were carried out for single
particle by Jacques et al. in 2008 PhysRevLett.100.220402  and for two
particles by Ma et al. in 2009 PhysRevA.79.042101 . Since then a rich set of
proposals and various experiments have been carried out: Wheeler’s delayed-
choice duality Marlow1978-MARMFO ; Jacques966 , delayed-choice quantum erasure
PhysRevA.25.2208 ; PhysRevLett.84.1 ; Mandel:91  and delayed-choice
entanglement swapping doi:10.1080/09500340008244032  to mention a few.

Despite such important and extensive progress it is quite obvious that duality
relations envisaged so far, (1) and (2), are incomplete because they are
inequalities, which can only bound duality. Indeed, if for example $V=1$ is
measured then $P=0$, or vice versa, but if $V=0$ nothing can be deduced about
the range of $P$. This highlights in a striking way the _incompleteness_ of
duality relations (1) and (2). Things changed when people started to study
quantitative complementarity occuring in composite systems (see jakob2003 ;
PhysRevA.76.052107  and references therein). The simplest case is a bipartite
composite quantum system composed of two qubits. Jakob and Bergou jakob2003 ;
ref:bergou2010  have found in this case the single missing quantity which
turned out to be the entanglement with the second qubit (defined by the
concurrence $\mathscr{C}$), so that

$V^{2}+D^{2}+\mathscr{C}^{2}=1\,.$ (3)

is a “triality” relation and an equality which holds for any pure state of the
two qubits. Furthermore this equation can be interpreted as a new
complementarity relation between (wave-particle) duality of any of the two
single qubit on one hand (the first two terms), and quantum entanglement with
the second qubit on the other hand (the remaining term). So the amount of
duality/complementarity in any of the two local subsystems determines the
amount of bipartite non-local entanglement, the latter being also understood
as a property which can exclude any of the two single-partite realities if
$\mathscr{C}=1$. The beauty of this relationship resides in the fact that it
relates the two most counter intuitive phenomenon of quantum physics, namely
wave-particle duality and _quantum entanglement_ , in a single relation!
Delayed-choice entanglement swapping experiments have now illustrated in a
particularly bright manner such _entanglement-separability duality_ for
bipartite (and multipartite) systems RevModPhys.88.015005 .

More recently Qian et al. Qian2018  derived a “triality” relation which looks
totally similar to the Jakob-Bergou relation (3). In fact the underlying
mathematics is identical, thereby explaining the same identity, however it
should be kept in mind that the physical content is completely different: at
the outset two completely classical beams are considered, including the
polarization degree of freedom for both beams. The analogy between the vector
description of two classical polarized beams and a two-qubit quantum system is
well known Spreeuw1998 , and obtained at the price of the introduction of a
so-called “position cebit” together with the “polarization cebit” (standard
Jones vector of one of the beams). The notion of “classical entanglement” that
naturally ensues is still highly debated Karimi2015 ; ForbesEtAl2019 , and is
sometimes referred to as “entanglement of degrees of freedom”, “single-
particle entanglement”, or “self-entanglement”, but it shows that quantum and
classical physics do cross-fertilize again (see e.g.Spreeuw2001 ; Konrad2019
and references therein). While it lead some to argue that the quantum-
classical boundary was shifting Qian2015 , we stress that in fact only the
domain of application of Bell-like inequalities changes.

Qian et al. also carried out a follow-up experiment in the quantum limit to
verify a similar “triality” relation for single photons qian2019quantum . In
this regard it is necessary to point out that single photons can only test the
very same classical structure of the field degrees of freedom: in a lossless
linear optical system the transformation of single photon creation operators
is the same as for the classical beam amplitudes (which also tells us that for
more than one input photon other effects appear). So Ref. Qian2018  probes in
fact the same triality relation as the classical experiment Qian2018 . This
type of point of view was already exposed by Spreeuw Spreeuw1998 , we quote
his conclusion “ The term classical entanglement seems justified even though a
single particle is, strictly speaking, a quantum system. Single-photon
entanglement is what remained when we took the low-intensity limit of a
classical electromagnetic wave”.

In the present paper we offer the first experimental check of the Jakob-Bergou
“triality” relation (3) for a genuine bipartite quantum system of two qubits,
namely the superconducting qubits of the IBM Q quantum computer ibm ,
harnessed by the current fascinating progress in widely accessible quantum
technologies.

##  II Quantum waviness, particleness and entanglement

Consider a general pure state of two qubits,

$\ket{\psi}=\alpha\ket{00}+\beta\ket{01}+\gamma\ket{10}+\delta\ket{11},$ (4)

with $\alpha,\beta,\gamma,\delta\in\mathbb{C}$ satisfying the normalization

$|\alpha|^{2}+|\beta|^{2}+|\gamma|^{2}+|\delta|^{2}=1.$ (5)

The state (4) can be characterized by its density matrix,

$\rho=\begin{pmatrix}\alpha\alpha^{*}&\alpha\beta^{*}&\alpha\gamma^{*}&\alpha\delta^{*}\\\
\beta\alpha^{*}&\beta\beta^{*}&\beta\gamma^{*}&\beta\delta^{*}\\\
\gamma\alpha^{*}&\gamma\beta^{*}&\gamma\gamma^{*}&\gamma\delta^{*}\\\
\delta\alpha^{*}&\delta\beta^{*}&\delta\gamma^{*}&\delta\delta^{*}\end{pmatrix}.$
(6)

By convention, the first and second qubits will be respectively called qubit
$A$ and qubit $B$. The corresponding reduced density matrices of subsystems
$A$ and $B$ are

$\rho_{A}=Tr_{B}(\rho)=\begin{pmatrix}\alpha\alpha^{*}+\beta\beta^{*}&\alpha\gamma^{*}+\beta\delta^{*}\\\
\gamma\alpha^{*}+\delta\beta^{*}&\gamma\gamma^{*}+\delta\delta^{*}\end{pmatrix}$
(7)

and

$\rho_{B}=Tr_{A}(\rho)=\begin{pmatrix}\alpha\alpha^{*}+\gamma\gamma^{*}&\alpha\beta^{*}+\gamma\delta^{*}\\\
\beta\alpha^{*}+\delta\gamma^{*}&\beta\beta^{*}+\delta\delta^{*}\end{pmatrix}.$
(8)

Three central quantities jakob2003 ; ref:bergou2010  can then be derived.

First, the _concurrence_ , defined in the bipartite pure case by

$\mathscr{C}(\psi)=2|\alpha\delta-\beta\gamma|.$ (9)

The concurrence is a measurement of the amount of entanglement between two
quantum systems PhysRevLett.78.5022 ; wooters:eofandconcurrence  as it is a
monotone of the _entanglement of formation_ , $E_{f}$, which is a measure of
entanglement based on the separability criterion: $E_{f}=0$ if and only if the
density matrix can be written as a mixture of product states. Both
$\mathscr{C}$ and $E_{f}$ take the value one for maximally entangled states.

Second, the _coherence_ $\mathscr{V}_{k}$ between the two orthogonal states
$\ket{0}$ and $\ket{1}$ of the qubit $k$, which is therefore a quantity
related to a single qubit. It is directly proportional to the norm of the off-
diagonal elements of its density matrix, and reads

$\mathscr{V}_{k}=2|\rho_{k_{12}}|,\ \ \ k=A,B.$ (10)

Note that the counterpart of coherence in an interference experiment is the
visibility.

Third, the _predictability_ $\mathscr{P}_{k}$, which quantifies the knowledge
of “which proportion” of the system $k$ is in the state $\ket{0}$ or
$\ket{1}$. It is defined by

$\mathscr{P}_{k}=|\rho_{k_{22}}-\rho_{k_{11}}|,\ \ \ k=A,B.$ (11)

The predictability is analogous to the which-path information in an
interference experiment.

By replacing the definitions (6 \- 8) in eqs. (9 \- 11) it is easy to show
that

$\mathscr{V}_{k}^{2}+\mathscr{P}_{k}^{2}+\mathscr{C}^{2}=(|\alpha|^{2}+|\beta|^{2}+|\gamma|^{2}+|\delta|^{2})^{2}.$
(12)

One notices that the right-hand side of (12) is nothing else than the norm of
the state (4) raised to the power 4. Thus, one can conclude that for a pure
state jakob2003 ; ref:bergou2010 ,

$\mathscr{V}_{k}^{2}+\mathscr{P}_{k}^{2}+\mathscr{C}^{2}=1.$ (13)

Note that (13) remarkably claims that for a pure state of two qubits, the
amount of entanglement strictly pilots the amount of duality of any qubit of
the pair, namely $\mathscr{V}_{k}^{2}+\mathscr{P}_{k}^{2}$, $k=A,B$, which has
the same value for both qubits. Conversely Eq. (13) also nicely reflect the
well-known fact that local unitary transformations on any of the qubits cannot
change the amount of mutual entanglement.

##  III Quantum experiments on IBM Q

In order to create a tunable state in term of its $\mathscr{V}_{k}$,
$\mathscr{P}_{k}$ and $\mathscr{C}$, we propose the simple circuit shown in
Fig. 1.

State preparation$\ket{0}$$\ket{0}$Ry($\alpha$)CU3($\theta$, 0, 0)TomographyAA

Figure 1: Quantum circuit composed of two gates to prepare the state as a
function of two parameters; perform tomography; and measure.

Our circuit prepares the state (see Appendix A)

$\ket{\psi}=\cos\frac{\alpha}{2}\ket{00}+\cos\frac{\theta}{2}\sin\frac{\alpha}{2}\ket{10}+\sin\frac{\theta}{2}\sin\frac{\alpha}{2}\ket{11}$
(14)

and then performs tomography allowing to retrieve the density matrix of the
quantum state PhysRevA.64.052312 , from which one computes $\mathscr{P}_{k}$,
$\mathscr{V}_{k}$ and $\mathscr{C}$. Note that using two additional ancilla
qubits it is possible to perform quantum nondemolition measurements of these
quantities PhysRevLett.98.250501 , which could be useful when using the
bipartite state in further quantum information protocols.

Using this circuit, the IBM qasm simulator allows to check the equality (13)
for a pure state of two qubits of the form (14), as reported (for qubit $A$)
in Fig. 2.

Figure 2: Simulation for qubit $A$. Selection of 13 couples of values
$\alpha$, $\theta$ emphasizing relation (13): the values of $\mathscr{V}_{A}$,
$\mathscr{P}_{A}$, $\mathscr{C}$ lie on a sphere of radius 1.

With the aim of performing the experiment on the real quantum computer, a
_noisy intermediate-scale quantum_ (NISQ) computer
Preskill2018quantumcomputingin , formulas need to be extended to mixed states.
For a mixed state with density matrix

$\rho=\sum_{j}p_{j}\ket{\phi_{j}}\bra{\phi_{j}},$ (15)

where $\ket{\phi_{j}}$ are pure states composing the complete state with
probability $p_{j}$, it is possible to measure the concurrence
PhysRevLett.80.2245  by defining the _spin flip_ matrix

$\Sigma=\sigma_{y}\otimes\sigma_{y}$ (16)

and the matrix

$R(\rho)=\rho\Sigma\rho^{*}\Sigma.$ (17)

The concurrence is given by

$\mathscr{C}=max(0,\sqrt{r_{1}}-\sqrt{r_{2}}-\sqrt{r_{3}}-\sqrt{r_{4}}).$ (18)

where $r_{1}\geq r_{2}\geq r_{3}\geq r_{4}$ are the eigenvalues of $R(\rho)$.
Using expressions (16) to (18) allows to compute the concurrence of the pair
of qubits from the tomography step.

The coherence of the qubit $k$ in the mixed bipartite case Tessier2005  is
given by

$\mathscr{V}(\rho_{k})=2\left|Tr(\rho_{k}\sigma_{+}^{(k)})\right|,$ (19)

where $\sigma_{+}^{(k)}=\begin{pmatrix}0&1\\\ 0&0\end{pmatrix}$ is the raising
operator acting on qubit $k$.
It can be written as

$\mathscr{V}_{k}=\sum_{i\neq j}\left|\rho_{k_{ij}}\right|,$ (20)

which, for a pure state, is equivalent to (10) thanks to the hermiticity of
the density matrix. Similarly, the predictability (of the state) of a qubit
Tessier2005  is given by (11) in the case of a two-qubits mixed state.

Given the possibility to compute the quantities for a mixed state, we can now
perform the experiment with the real qubits. For this, the backend ibmqx2 is
used date_meas . The measurement is repeated 10 times for each of the thirteen
states in order to calculate confidence bounds, and the number of shots is set
to 1000 for each measurement. We are expecting to observe

$\mathscr{V}_{k}^{2}+\mathscr{P}_{k}^{2}+\mathscr{C}^{2}\leq 1$ (21)

because of the mixedness of the state jakob2003 ; ref:bergou2010 , the
experimental values being limited by decoherence and noise. Nevertheless, the
measurements on the real backend can be simulated using a noise model,
directly constructed from the real backend properties thanks to _Qiskit Ignis_
Qiskit . The IBM qasm simulator is thus used to simulate the noise and
decoherence taking place in the ibmqx2 backend used for the experiment (see
Appendix C), thus allowing to normalize the measured data with respect to the
noise simulation. More precisely, the mean value of each point along axes
$\mathscr{V}_{A}$, $\mathscr{P}_{A}$ and $\mathscr{C}$ is normalized with
respect to the corresponding value in the noise simulation. Confidence bounds
are taken as 4-$\sigma$ along each axis, from the measured data distribution
for each state, considered to be normal in every direction. This results in
confidence ellipsoids visible in Fig. 3.

Figure 3: Measurements on ibmqx2, normalized with respect to the backend noise
model, with 4-$\sigma$ confidence bounds.

From this result, one sees that the normalization allows to verify (13). The
noise model could accurately simulate errors due to decoherence and noise
accounting for the mixedness of the prepared states, the tomography step as
well as the eventual readout errors.

In contrast, looking at the raw data (see Appendix C), one can see that the
measurement of the three quantities for the thirteen states give lower values
than expected for a pure state (Fig. 2). This drop is particularly consequent
for $\mathscr{C}$. It seems that it is the most delicate and sensitive to
decoherence among the three quantities. In fact, it is possible to quantify
the drop of concurrence which is due to the mixedness of the state. Indeed,
such an upper bound on concurrence can be computed PhysRevA.64.012316 .
Following the notation of Wooters for the two qubit case
wooters:eofandconcurrence , this upper bound on concurrence, quantifying the
maximal entanglement reachable with a given state purity, is given by

$\mathscr{C}_{max}(\rho)=max(0,\lambda_{1}-\lambda_{3}-2\sqrt{\lambda_{2}\lambda_{4}}),$
(22)

with $\lambda_{1}\geq\lambda_{2}\geq\lambda_{3}\geq\lambda_{4}$ the
eigenvalues of $\rho$. In other words, $\mathscr{C}_{max}$ is the maximal
concurrence a state can reach, knowing the eigenvalues of its density matrix.

The drop of concurrence in our experiment is especially visible using the
ibmqx4 backend: Fig. 4 shows that the raw measurements date_meas2  of
$\mathscr{C}$ are constrained and reach their highest value
($\mathscr{P}_{k}=\mathscr{V}_{k}=0$) close to the maximal concurrence bound
$\mathscr{C}_{max}$. Therefore, we identify it as the principal cause of the
flattening of the sphere along the $\mathscr{C}$ axis observed in the raw
data, and this cannot be improved by any mean except increasing the state
purity.

Figure 4: Concurrence $\mathscr{C}$ in the plane $\mathscr{V}_{A}=0$ in
comparison with $\mathscr{C}_{max}$ for the ibmqx4 backend, as well as for the
simulator.

Finally, the ratio
$\frac{\mathscr{C}_{\mathrm{ibmqx4}}}{\mathscr{C}_{\mathrm{simulator}}}$
appears to stay constant as $\mathscr{P}_{A}$ varies. More precisely, the
comparison of the value of the real machine with respect to the simulator
indicates that $\mathscr{C}_{\mathrm{ibmqx4}}\cong
0.6\cdot\mathscr{C}_{\mathrm{simulator}}$. One could say that the efficiency
in the preparation of an entangled state is constant, while the preparation of
an entangled state decreases the purity, in its turn limiting entanglement.

##  IV Conclusion

Our work is, to the best of our knowledge, the first experimental explicit
verification of the beautiful Jakob-Bergou “triality” relation for bipartite
pure quantum states of two qubits jakob2003 ; ref:bergou2010 ;
PhysRevA.76.052107 . This relation really represents the full quantitative
completion of local Bohr’s complementarity principle by quantum entanglement
(concurrence) for this case. The measurements on the two real superconducting
qubits of the IBM Q quantum computer, together with simulations using the
backend noise model of IBM Q, have shown that the duality of each qubit can
indeed be turned off completely, or set to any desired amount by controlling
the degree of entanglement between the qubits. Clearly, the Jakob-Bergou
relation can be separated into mutually exclusive local and nonlocal parts as

$\mathscr{S}_{k}^{2}+\mathscr{C}^{2}=1$ (23)

where $\mathscr{S}_{k}^{2}=\mathscr{P}_{k}^{2}+\mathscr{V}_{k}^{2}$ is the
amount of locality since the predictability $\mathscr{P}_{k}$ and the
visibility $\mathscr{V}_{k}$ are local with respect to subsystem $k$. Maximal
entanglement of the bipartite system ($\mathscr{C}=1$) imply that the local
realities must totally disappear ($\mathscr{S}_{k}=0$), synonym of maximal
amount of nonclassical nonlocal phenomena such as violations of Bell
inequalities.

Finally, such experiments with the superconducting qubits of the IBM Q quantum
computer could be extended in different fundamental directions. First it would
be interesting to test a generalization of the Jakob-Bergou relation derived
for non-orthogonal alternatives in a similar interference and which-state
information experiments for two qubits:

$V^{2}+D^{2}+U^{2}+\mathscr{C}^{2}=1\,.$ (24)

Here $\mathscr{C}$ remains the only purely bipartite quantity as before, but
$V,D$ would become the non-orthogonal counterparts of visibility and
distinguishability, and $U$ would be a new single qubit quantity involving the
overlap of non-orthogonal markers. Second, we are also interested in the
dynamical evolution of these relations under decoherence, which is inevitable
in a quantum computer. For the resulting mixed states it is well-known that
Eqs. (3) and (24) become undersaturated inequalities (LHS $<1$), but more
interestingly the evolution of $\mathscr{C}$ can be surprising, leading for
example to entanglement sudden death Almeida2007 , and one may wonder about
the comparative evolution of each term. Third, one expects that multipartite
quantum states which are presently realized on IBM Q quantum computers
doi:10.1002/qute.201970031 , and which are essential for applications in
quantum information, also possess rich entanglement-separability duality
relations of their own, which are of fundamental interest.

##  V Acknowledgements

###### Acknowledgements.

We thank Dr. James R. Wootton for useful discussions and comments. We
acknowledge use of the IBM Quantum Experience for this work. The views
expressed are those of the authors and do not reflect the official policy or
position of IBM or the IBM Quantum Experience team.

##  VI Data availability statement

The code that supports this study is openly available in GitHub at
https://github.com/NicoSchwaller/Duality-and-Entanglement-of-two-Qubits.

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##  Appendix A Appendix A: quantum circuit for state preparation

The left-hand side of the circuit in Fig. 1 prepares the state
$\ket{\psi}=\cos\frac{\alpha}{2}\ket{00}+\cos\frac{\theta}{2}\sin\frac{\alpha}{2}\ket{10}+\sin\frac{\theta}{2}\sin\frac{\alpha}{2}\ket{11},$
by applying the unitary transformation

$\displaystyle(CU3_{A\rightarrow B})(R_{y}\otimes\mathbb{I})=$
$\displaystyle\begin{pmatrix}1&0&0&0\\\ 0&1&0&0\\\
0&0&\cos\frac{\theta}{2}&-\sin\frac{\theta}{2}\\\
0&0&\sin\frac{\theta}{2}&\cos\frac{\theta}{2}\\\
\end{pmatrix}\left[\begin{pmatrix}\cos\frac{\alpha}{2}&-\sin\frac{\alpha}{2}\\\
\sin\frac{\alpha}{2}&\cos\frac{\alpha}{2}\\\
\end{pmatrix}\otimes\begin{pmatrix}1&0\\\ 0&1\\\ \end{pmatrix}\right].$

According to equations (9 \- 11), such a unitary operation acting on the state
$\ket{00}$ allows the five quantities $\mathscr{V}_{k}$, $\mathscr{P}_{k}$
(with $k=A,B$) and $\mathscr{C}$ to reach their extremal values, i.e. 0 and 1.
To verify it, the parameters of the circuit are set to random angles
$\alpha,\theta\in[0,\pi]$. The outcomes of the simulations are matching the
analytical expressions for $\mathscr{V}_{k}(\alpha,\theta)$,
$\mathscr{P}_{k}(\alpha,\theta)$ and $\mathscr{C}(\alpha,\theta)$. This is
shown in Fig. 5 in the case of qubit $A$.

Figure 5: Simulation of the coherence, predictability and concurrence for
qubit $A$ using the circuit of Fig. 1, with $\alpha,\theta$ randomly chosen in
$[0,\pi]$.

##  Appendix B Appendix B: maximum entanglement versus purity

It is interesting to note that eq. (22) spoils any concurrence in a state for
which Tr($\rho^{2}$)$<\frac{1}{3}$. Fig. 6 accordingly highlights the
existence of a threshold value of the state purity under which concurrence
cannot exist. These measurements were computed with the simulator, varying the
amount of noise (readout errors). Accordingly, we note that we occasionally
observed sufficient drops of fidelity in IBM Q machines to prevent any attempt
to conduct the experiment due to zero concurrence.

In accordance with Fig. 6, we expect $\mathscr{C}$ to equal
$\mathscr{C}_{max}$ for a maximally entangled state, a necessary condition to
refer to $\mathscr{C}_{max}$ as an upper bound to entanglement.

Figure 6: Evolution of concurrence $\mathscr{C}$ and concurrence maximal bound
$\mathscr{C}_{max}$ for the maximally entangled Bell state $\ket{\Phi^{+}}$
($\alpha=\frac{\pi}{2},\theta=\pi$) as purity varies.

##  Appendix C Appendix C: measured data and noise simulation

Raw data and the corresponding noise simulation used to normalize the measured
data are plotted in Fig. 7. One can see that the noise simulation effectively
simulates the noise present on the real quantum computer, which manifests
itself as a drop of the three studied quantities, and especially affects
concurrence.

(a)

(b)

Figure 7: Noisy simulation (a) and raw data (b) from ibmqx2. The experiment is
repeated 10 times for each state.


_Swift/XRT_ , _Swift/BAT_ , _ASCA_ , _Chandra_ , and _Suzaku_ by Ricci et al.
(2017a). The red circles represent the upper limit of the error bar and the
blue stars the lower limit. The dotted black line is x=y.  Figure 9: Relation
between the spectral index, $\Gamma$, and the accretion rate,
$\log(\lambda_{Edd})$. The blue points (top panel) represents the values
obtained by (Ricci et al. 2017a) using PEXRAV model from the DR1. The middle
panel (red points) represents the data points obtained in this work using
PEXMON reflection model. In the lower panel (green points) are the values
obtained in this work using borus02 reflection model. In all the panels, the
dotted black lines are the best fit model.

##  Appendix D Tables

Table 5: Observational Details.  Name | Instrument | ObsID | Date | R | Exp. Time
---|---|---|---|---|---
|  |  |  | (”) | (ks)
(1) | (2) | (3) | (4) | (5) | (6)
NGC 3998 | _XMM-Newton_ | 0790840101 | 2016-10-26 | 49 | 25
| _NuSTAR_ | 60201050002 | 2016-10-25 | 49 | 103
NGC 3718 | _XMM-Newton_ | 0795730101 | 2017-10-24 | 49 | 38
| _NuSTAR_ | 60301031002 | 2017-10-24 | 49 | 24
NGC 4258* | _XMM-Newton_ | 0400560301 | 2006-11-17 | 49 | 64
| _NuSTAR_ | 60101046004 | 2016-01-10 | 49 | 103
ESO 253-G003* | _XMM-Newton_ | 0762920501 | 2015-08-19 | 49 | 27
| _NuSTAR_ | 60101014002 | 2015-08-21 | 49 | 22
NGC 1052 | _XMM-Newton_ | 0790980101 | 2017-01-17 | 49 | 70
| _NuSTAR_ | 60201056002 | 2017-01-17 | 49 | 59
NGC 2655 | _XMM-Newton_ | 0301650301 | 2016-11-10 | 49 | 11
| _NuSTAR_ | 60160341004 | 2016-11-10 | 49 | 15
NGC 3147* | _XMM-Newton_ | 0405020601 | 2006-10-06 | 49 | 17
| _NuSTAR_ | 60101032002 | 2015-12-27 | 49 | 49
NGC 2110* | _XMM-Newton_ | 0145670101 | 2003-03-05 | 49 | 59
| _NuSTAR_ | 60061061002 | 2012-10-05 | 49 | 15
|  | 60061061004 | 2013-02-14 | 49 | 12
LEDA 96373* | _XMM-Newton_ | 0674940101 | 2012-04-09 | 49 | 56
| _NuSTAR_ | 60061073002 | 2014-07-31 | 49 | 22
NGC 2992 | _XMM-Newton_ | 0840920201 | 2019-05-07 | 49 | 134
|  | 0840920301 | 2019-05-09 | 49 | 134
| _NuSTAR_ | 90501623002 | 2019-05-10 | 49 | 57
M 51 | _XMM-Newton_ | 0852030101 | 2019-07-11 | 49 | 77
| _NuSTAR_ | 60501023002 | 2019-07-10 | 49 | 169
Table 6: Cont: Observational Details

Name | Instrument | ObsID | Date | R | Exptime
---|---|---|---|---|---
|  |  |  | (”) | (ks)
(1) | (2) | (3) | (4) | (5) | (6)
NGC 2273* | _XMM-Newton_ | 0140951001 | 2003-09-05 | 49 | 13
| _NuSTAR_ | 60001064002 | 2014-03-23 | 49 | 23
HE 1136-2304 | _XMM-Newton_ | 0741260101 | 2014-07-02 | 49 | 110
| _NuSTAR_ | 80002031003 | 2014-07-02 | 49 | 63
|  | 80002031002 | 2014-07-02 | 49 | 23
IC 4518A | _XMM-Newton_ | 0401790901 | 2006-08-07 | 11 | 134
|  | 0406410101 | 2006-08-15 | 49 | 25
| _NuSTAR_ | 60061260002 | 2013-08-02 | 49 | 7
NGC 7674* | _XMM-Newton_ | 0200660101 | 2004-06-02 | 49 | 10
| _NuSTAR_ | 60001151002 | 2014-09-30 | 49 | 51
IGRJ 11366-6002 | _NuSTAR_ | 60061213002 | 2014-10-29 | 49 | 21
|  _Swift_ /XRT | 00080058001 | 2014-10-29 | 25 | 1.6
NGC 5033 | _XMM-Newton_ | 0871020101 | 2020-12-10 | 49 | 21
| _NuSTAR_ | 60601023002 | 2020-12-08 | 49 | 104
|  | 60601023004 | 2020-12-12 | 49 | 53
111111 Notes: (Col. 1) Name, (Col. 2) instrument, (Col. 3) obsID, (Col. 4)
date, (Col. 5) radii of the extraction region and (Col. 5) exposure time (ks).
Objects with * symbols are the galaxies with not simultaneous observations.

Table 7: Spectral variability of the not simultaneous data between 3.0-10.0
keV.

Name | Instrument | ObsID | Model | $\Gamma$ | NH | Norm PL | Norm G | Flux |  $\chi^{2}$/d.o.f | $\chi^{2}_{\nu}$ | Notes
---|---|---|---|---|---|---|---|---|---|---|---
|  |  |  |  | ($10^{22}$ cm-2) | ($10^{-3}$) | ($10^{-3}$) | ($10^{-12}$ erg cm-2 s-1) |  |  |
(1) | (2) | (3) | (4) | (5) | (6) | (7) | (8) | (9) | (10) | (11) | (12)
NGC 4258* | _XMM-Newton_ | 0400560301 | PL | $\\#$ | $\\#$ | 1.48${}_{1.42}^{1.92}$ | - | 3.84${}_{1.14}^{5.35}$ | - | - | We select
| _NuSTAR_ | 60101046004A |  | 1.61${}_{1.18}^{1.77}$ | 7.67${}_{6.90}^{8.66}$ | 0.84${}_{0.81}^{1.11}$ | - | 2.18${}_{0.65}^{3.06}$ | 443.2/387 | 1.145 | PL norm
|  | 60101046004B |  | $\\#$ | $\\#$ | $\\#$ | - | - | - | - | variable
ESO 253-G003* | _XMM-Newton_ | 0762920501 | PL | 1.76${}_{1.11}^{1.92}$ | $\\#$ | $\\#$ | - | 0.45${}_{**}^{0.51}$ | - | - |  $\Gamma$ variable,
| _NuSTAR_ | 60101014002A |  | 1.23${}_{**}^{1.33}$ | 2.29${}_{**}^{5.13}$ | 0.19${}_{0.11}^{0.30}$ | - | 1.07${}_{**}^{1.24}$ | 119.4/86 | 1.389 | We select only
|  | 60101014002B |  | $\\#$ | $\\#$ | $\\#$ | - | - | - | - |  _NuSTAR_ observations
NGC 3147* | _XMM-Newton_ | 0405020601 | PL | $\\#$ | $\\#$ | 0.53${}_{0.38}^{0.78}$ | - | 1.14${}_{1.05}^{1.19}$ | - | - | We select
| _NuSTAR_ | 60101032002A |  | 1.79${}_{1.69}^{2.02}$ | 2.98${}_{1.47}^{4.80}$ | 1.05${}_{0.98}^{1.56}$ | - | 2.19${}_{2.03}^{2.25}$ | 243.5/217 | 1.122 | PL norm
|  | 60101032002B |  | $\\#$ | $\\#$ | $\\#$ | - | - | - | - | variable
NGC 2110* | _XMM-Newton_ | 0145670101 | PL+G | $\\#$ | $\\#$ | 6.78${}_{6.34}^{7.46}$ | $\\#$ | 25.00${}_{21.28}^{30.10}$ | - | - | We select
| _NuSTAR_ | 60061061004A |  | 1.53${}_{1.50}^{1.61}$ | 3.84${}_{3.57}^{4.14}$ | 37.08${}_{34.72}^{40.83}$ | 0.049${}_{0.045}^{0.53}$ | 134.85${}_{112.7}^{163.9}$ | 527.3/453 | 1.164 | PL norm
|  | 60061061004B |  | $\\#$ | $\\#$ | $\\#$ | $\\#$ | - | - | - | variable
LEDA 96373* | _XMM-Newton_ | 0674940101 | PL+G | $\\#$ | $\\#$ | 0.13${}_{0.09}^{0.23}$ | $\\#$ | 1.14${}_{0.16}^{1.67}$ | - | - | We select PL norm tied, they
| _NuSTAR_ | 60061073002A |  | 0.93${}_{0.70}^{1.13}$ | 13.59${}_{10.32}^{17.14}$ | 0.17${}_{0.11}^{0.31}$ | 0.022${}_{0.20}^{0.24}$ | 1.40${}_{0.16}^{2.19}$ | 245.2/141 | 1.739 | are consistent. The model
|  | 60061073002B |  | $\\#$ | $\\#$ | $\\#$ | $\\#$ | - | - | - | is not enough to fit the data
NGC 2273* | _XMM-Newton_ | 0140951001 | PL+G | $\\#$ | $\\#$ | 0.49${}_{0.04}^{1.40}$ | $\\#$ | 0.93${}_{0.19}^{1.48}$ | - | - | We select
| _NuSTAR_ | 60001064002A |  | 1.56${}_{0.79}^{1.74}$ | 30.04${}_{21.21}^{44.99}$ | 0.53${}_{0.31}^{1.75}$ | 0.037${}_{0.031}^{0.047}$ | 0.98${}_{0.19}^{1.47}$ | 109.7/102 | 1.075 | PL norm tied, they
|  | 60001064002B |  | $\\#$ | $\\#$ | $\\#$ | $\\#$ | - | - | - | are consistent

121212(Col. 1) name, (Col. 2) instrument, (Col. 3) obsID, (Col. 4) model,
where ”PL” represents the power-law and ”G” a Gaussian component at 6.4 keV
and $\sigma$=0.01 keV. (Col. 5) photon index, (Col. 6) column density, (Col.
7) Power-law normalization, (Col. 8) Normalization of the Gaussian component,
(Col. 10 and 11) $\chi^{2}$/d.o.f., and $\chi^{2}_{\nu}$, (Col. 12) Notes
about selection of the model. The case where the Normalization of the power-
law (Norm PL) is a free parameter between the _XMM-Newton_ and _NuSTAR_ data
is marked in bold face. The parameter in this table marked with $\\#$ means
that is tied to FPMA/_NuSTAR_ . The symbol ”$-$” denotes that these components
were not require to improve the fit.

Table 8: Spectral variability not simultaneous data between 3.0-10.0 keV. Name | Instrument | ObsID | Model | $\Gamma$ | NH | Norm PL | Norm G | Flux |  $\chi^{2}$/d.o.f | $\chi^{2}_{\nu}$ | Notes
---|---|---|---|---|---|---|---|---|---|---|---
|  |  |  |  | ($10^{22}$ cm-2) | ($10^{-3}$) | ($10^{-3}$) | ($10^{-12}$ erg cm-2 s-1) |  |  |
(1) | (2) | (3) | (4) | (5) | (6) | (7) | (8) | (9) | (10) | (11) | (12)
NGC 7674* | _XMM-Newton_ | 0200660101 | PL+G | $\\#$ | $\\#$ | 0.20${}_{**}^{0.50}$ | $\\#$ | 0.052${}_{0.029}^{1.243}$ | - | - | Is a changing look, we
| _NuSTAR_ | 60001151002A |  | 1.71${}_{1.34}^{2.17}$ | 18.28${}_{13.05}^{26.89}$ | 0.49${}_{0.24}^{1.33}$ | 0.0059${}_{0.0037}^{0.0079}$ | 0.79${}_{0.03}^{1.43}$ | 12.4/76 | 1.479 | did not combine the data.
|  | 60001151002B |  | $\\#$ | $\\#$ | $\\#$ | $\\#$ | - | - |  |
IC 451A* | _XMM-Newton_ | 0406410101 | PL+G | $\\#$ | $\\#$ | 0.97${}_{0.56}^{1.06}$ | $\\#$ | 1.64${}_{0.09}^{3.46}$ | - | - | We select
| _NuSTAR_ | 60061260002A |  | 1.70${}_{1.36}^{1.83}$ | 21.57${}_{18.30}^{24.35}$ | 2.64${}_{1.44}^{2.89}$ | 0.015${}_{0.012}^{0.017}$ | 4.29${}_{0.09}^{9.22}$ | 143.4/123 | 1.166 | PL norm
|  | 60061260002B |  | $\\#$ | $\\#$ | $\\#$ | $\\#$ | - | - | - | variable

Table 9:  Final compilation of the coronal parameters of the best-fit models
for the sample.

| PEXMON | borus02 | XILLVER |
---|---|---|---|---
Name | $\Gamma$ |  $E_{\rm cut}$ (keV) |  $\chi^{2}$/d.o.f | $\chi^{2}_{\nu}$ | $\Gamma$ |  $E_{\rm cut}$ (keV) |  $\chi^{2}$/d.o.f | $\chi^{2}_{\nu}$ | $\Gamma$ |  $E_{\rm cut}$ (keV) |  $\chi^{2}$/d.o.f | $\chi^{2}_{\nu}$ | BM
NGC 3998 | 1.80${}_{1.77}^{1.83}$ | 81.78${}_{61.07}^{123.99}$ | 840.7/896 | 0.938 | 1.83${}_{1.80}^{1.85}$ | 119.02${}_{92.00}^{173.06}$ | 837.6/895 | 0.936 | 1.79${}_{1.76}^{1.83}$ | 568.75${}_{108.96}^{**}$ | 838.0/895 | 0.936 | T/D
NGC 3718 | 1.88${}_{1.79}^{1.96}$ | 111.57${}_{55.39}^{994.30}$ | 408.2/371 | 1.100 | 1.90${}_{1.86}^{1.97}$ | 252.99${}_{80.04}^{**}$ | 405.9/369 | 1.100 | 1.86${}_{1.77}^{1.95}$ | 142.89${}_{64.63}^{1140.6}$ | 404.8/369 | 1.097 | T/D
NGC 4258* | 1.83${}_{1.76}^{1.90}$ | 142.15${}_{93.54}^{396.60}$ | 740.0/666 | 1.111 | 1.80${}_{1.72}^{1.87}$ | 371.07${}_{120.95}^{**}$ | 731.5/665 | 1.100 | 1.86${}_{1.78}^{1.93}$ | 208.36${}_{91.48}^{**}$ | 739.3/668 | 1.187 | Torus
ESO 253-G003* | 1.30${}_{**}^{1.65}$ | 79.45${}_{**}^{2000.0}$ | 107.1/121 | 0.885 | 1.40${}_{**}^{1.52}$ | 90.08${}_{38.99}^{187.57}$ | 108.6/120 | 0.905 | 1.41${}_{**}^{1.62}$ | 228.04${}_{89.35}^{**}$ | 108.37/120 | 0.903 | T/D
NGC 1052 | 1.76${}_{1.73}^{1.81}$ | 134.55${}_{95.96}^{223.48}$ | 956.7/872 | 1.097 | 1.86${}_{1.85}^{1.87}$ | 2000.0${}_{374.77}^{**}$ | 949.9/872 | 1.089 | 1.45${}_{1.40}^{1.57}$ | 40.39${}_{38.09}^{42.05}$ | 1286.0/872 | 1.4747 | Torus
NGC 2655 | 1.85${}_{1.53}^{2.11}$ | 209.70${}_{59.54}^{**}$ | 171.8/168 | 1.022 | 1.85${}_{1.56}^{2.10}$ | 221.28${}_{**}^{2000.0}$ | 171.2/167 | 1.025 | 1.90${}_{1.78}^{2.23}$ | 230.86${}_{79.97}^{1183.5}$ | 178.1/169 | 1.054 | T/D
NGC 3147* | 1.71${}_{1.66}^{1.75}$ | 65.65${}_{**}^{138.94}$ | 394.3/378 | 1.043 | 1.76${}_{1.74}^{1.79}$ | 2000${}_{177.53}^{**}$ | 379.3/377 | 1.006 | 1.72${}_{1.64}^{1.80}$ | 71.68${}_{44.69}^{191.61}$ | 384.7/377 | 1.020 | T/D
NGC 2110* | 1.87${}_{1.85}^{1.88}$ | 3000.0${}_{1419.73}^{**}$ | 1600.4/1261 | 1.269 | 1.70${}_{1.68}^{1.72}$ | 354.86${}_{303.67}^{465.42}$ | 1293.2/1260 | 1.026 | 1.82${}_{1.81}^{1.83}$ | 1870.56${}_{764.82}^{**}$ | 1564.3/1263 | 1.239 | Torus
LEDA 96373* | 1.20${}_{**}^{1.30}$ | 38.02${}_{33.40}^{44.08}$ | 403.1/276 | 1.460 | 1.87${}_{1.77}^{1.93}$ | 2000.0${}_{596.45}^{**}$ | 309.9/275 | 1.127 | 1.77${}_{1.69}^{1.84}$ | 114.47${}_{53.53}^{184.62}$ | 461.0/276 | 1.670 | Torus
NGC 2992 | 1.60${}_{1.58}^{1.62}$ | 55.28${}_{51.78}^{58.83}$ | 2092.1/1619 | 1.2922 | 1.69${}_{1.68}^{1.71}$ | 259.51${}_{183.62}^{376.14}$ | 1738.7/1618 | 1.075 | 1.65${}_{1.60}^{1.68}$ | 64.80${}_{61.66}^{69.13}$ | 2140.5/1620 | 1.3213 | Torus
M 51 | 1.90${}_{1.76}^{2.06}$ | 3000${}_{381.74}^{**}$ | 482.9/341 | 1.416 | 1.56${}_{**}^{1.76}$ | 2000${}_{70.89}^{**}$ | 470.5/343 | 1.372 | 1.90${}_{1.72}^{2.24}$ | 3000${}_{395.18}^{**}$ | 493.8/341 | 1.448 | Torus
NGC 2273* | 1.80${}_{1.51}^{2.10}$ | 57.86${}_{**}^{313.43}$ | 226.8/197 | 1.151 | 1.44${}_{**}^{2.00}$ | 60.41${}_{36.33}^{**}$ | 173.0/196 | 0.882 | 2.14${}_{1.99}^{2.27}$ | 575.64${}_{298.54}^{**}$ | 246.3/196 | 1.257 | D
HE 1136-2304 | 1.74${}_{1.72}^{1.76}$ | 97.20${}_{77.12}^{136.22}$ | 1066.0/987 | 1.080 | 1.74${}_{1.72}^{1.76}$ | 394.08${}_{172.4}^{**}$ | 1054.5/986 | 1.069 | 1.567${}_{1.553}^{1.598}$ | 1014.66${}_{554.79}^{**}$ | 1126.3/986 | 1.142 | Torus
IGRJ 11366-6002 | 2.10${}_{2.03}^{2.16}$ | 2999.36${}_{375.28}^{**}$ | 261.9/245 | 1.069 | 2.06${}_{1.91}^{2.16}$ | 206.76${}_{100.57}^{**}$ | 252.3/244 | 1.034 | 1.92${}_{1.85}^{2.00}$ | 2000${}_{395.73}^{**}$ | 250.49/244 | 1.026 | T/D
IC 4518A* | 1.62${}_{1.54}^{1.89}$ | 50.0${}_{**}^{85.95}$ | 245.7/215 | 1.143 | 1.57${}_{**}^{1.83}$ | 34.19${}_{**}^{75.18}$ | 222.6/217 | 1.026 | 1.76${}_{1.66}^{1.92}$ | 73.03${}_{44.59}^{173.61}$ | 250.4/217 | 1.154 | Torus
NGC 7674* | 1.82${}_{1.45}^{2.02}$ | 180.88${}_{107.29}^{**}$ | 189.0/169 | 1.118 | 1.47${}_{**}^{1.58}$ | 30${}_{**}^{40.22}$ | 190.7/168 | 1.135 | 1.42${}_{**}^{1.84}$ | 95.68${}_{67.58}^{116.13}$ | 179.4/168 | 1.068 | D
NGC 5033 | 1.70${}_{1.69}^{1.72}$ | 50${}_{**}^{55.13}$ | 1215.4/1059 | 1.148 | 1.74${}_{1.72}^{1.76}$ | 142.95${}_{102.90}^{200.67}$ | 1177.4/1058 | 1.113 | 1.77${}_{1.72}^{1.81}$ | 61.35${}_{48.78}^{74.93}$ | 1193.8/1054 | 1.133 | Torus

131313Notes: photon index ($\Gamma$), the high energy cut-off (Ecut) in keV,
$\chi^{2}$/d.o.f and $\chi^{2}_{\nu}$ for PEXMON , borus02 (B) and XILLVER
(X). BM shows the best model according with Akaike criterion. The symbol T/D
represents that the models are indistinguishable. Objects with * symbols are
the galaxies with not simultaneous observations. The quantities in this table
marked with ** are unconstrained errors.

Table 10: Final compilation of the reflector parameters of the best-fit models
for the sample.

| PEXMON | borus02 | XILLVER |
---|---|---|---|---
Name | Rf | Incl |  $\chi^{2}$/d.o.f. | $\chi^{2}_{\nu}$ | $\log(\rm{N_{H-refl}})$ | CF | Cos($\theta_{\rm incl}$) |  $\chi^{2}$/d.o.f. | $\chi^{2}_{\nu}$ | $\log\xi$ | Rf | Incl (deg) |  $\chi^{2}$/d.o.f. | $\chi^{2}_{\nu}$ | BM
NGC 3998 | 0.07${}_{**}^{0.15}$ | 0.01${}_{**}^{90.0}$ | 840.7/896 | 0.938 | 22.50${}_{22.28}^{23.11}$ | 1.00${}_{0.20}^{1.00}$ | 0.18${}_{0.05}^{**}$ | 837.6/895 | 0.936 | 4.07${}^{**}_{3.99}$ | 4.1${}_{0.26}^{**}$ | 18.33${}_{0.94}^{58.68}$ | 837.95/895 | 0.936 | T/D
NGC 3718 | 0.01${}_{**}^{0.24}$ | 0.02${}_{**}^{90}$ | 408.2/371 | 1.100 | 22.78${}_{22.38}^{23.98}$ | 0.99${}_{0.05}^{1.0}$ | 0.30${}_{**}^{0.95}$ | 405.9/369 | 1.100 | 3.12${}_{1.35}^{3.61}$ | 4.91${}_{0.86}^{**}$ | 88.74${}_{81.30}^{89.93}$ | 404.8/369 | 1.097 | T/D
NGC 4258* | 0.78${}_{0.20}^{3.44}$ | 80.14${}_{**}^{87.67}$ | 740.0/666 | 1.111 | 23.29${}_{22.77}^{24.39}$ | 0.38${}_{0.05}^{0.81}$ | 0.95${}_{0.27}^{**}$ | 731.5/665 | 1.100 | 0.48${}_{**}^{2.06}$ | 5.39${}_{1.54}^{**}$ | 87.97${}_{67.03}^{89.32}$ | 739.3/668 | 1.187 | Torus
ESO 253-G003 | 10.00${}_{1.59}^{**}$ | 86.60${}_{69.93}^{89.03}$ | 107.1/121 | 0.885 | 24.04${}_{23.93}^{24.30}$ | 0.73${}_{0.05}^{1.00}$ | 0.76${}_{**}^{0.94}$ | 108.6/120 | 0.905 | 2.68${}_{1.34}^{2.84}$ | 2.08${}_{0.85}^{**}$ | 65.99${}_{**}^{83.64}$ | 108.3/120 | 0.903 | T/D
NGC 1052 | 1.26${}_{0.79}^{3.05}$ | 74.71${}_{61.54}^{84.16}$ | 956.7/872 | 1.097 | 24.35${}_{24.19}^{24.51}$ | 0.50${}_{0.27}^{0.54}$ | 0.55${}_{0.45}^{0.73}$ | 949.9/872 | 1.089 | 1.88${}_{1.78}^{2.00}$ | 10.00${}_{9.18}^{**}$ | 88.12${}_{86.06}^{88.29}$ | 1286.0/872 | 1.4747 | Torus
NGC 2655 | 0.07${}_{**}^{0.51}$ | 0.01${}_{**}^{80.10}$ | 171.8/168 | 1.022 | 23.24${}_{22.48}^{25.19}$ | 0.83${}_{0.11}^{1.00}$ | 0.85${}_{0.05}^{**}$ | 159.4/167 | 0.954 | 1.85${}_{**}^{3.55}$ | 9.84${}_{**}^{10}$ | 89.09${}_{24.98}^{89.51}$ | 178.1/169 | 1.054 | T/D
NGC 3147* | 3.99${}_{0.97}^{6.41}$ | 87.11${}_{**}^{88.19}$ | 394.3/378 | 1.043 | 23.27${}_{22.98}^{23.94}$ | 0.89${}_{0.26}^{1.00}$ | 0.90${}_{**}^{0.05}$ | 379.3/377 | 1.006 | 0.79${}_{0.06}^{1.15}$ | 9.48${}_{0.91}^{**}$ | 87.61${}_{62.87}^{88.14}$ | 384.7/377 | 1.020 | T/D
NGC 2110* | 1.01${}_{0.91}^{1.12}$ | 79.58${}_{79.27}^{80.68}$ | 1600.4/1261 | 1.2691 | 23.49${}_{23.03}^{23.76}$ | 0.13${}_{0.11}^{0.30}$ | 0.45${}_{0.23}^{0.73}$ | 1293.2/1260 | 1.026 | 1.99${}_{1.82}^{2.01}$ | 10.00${}_{7.80}^{**}$ | 88.94${}_{88.47}^{89.05}$ | 1564.3/1263 | 1.236 | Torus
LEDA 96373* | 10.00${}_{9.38}^{**}$ | 65.49${}_{50.08}^{69.11}$ | 403.1/276 | 1.460 | 23.40${}_{23.33}^{23.69}$ | 0.59${}_{0.41}^{0.76}$ | 0.95${}_{0.59}^{**}$ | 309.9/275 | 1.127 | 2.08${}_{2.03}^{2.17}$ | 10.00${}_{9.37}^{**}$ | 19.10${}_{1.51}^{43.32}$ | 461.0/276 | 1.670 | Torus
NGC 2992 | 0.37${}_{0.33}^{0.39}$ | 0${}_{**}^{11.76}$ | 2092.1/1619 | 1.292 | 23.17${}_{23.13}^{23.29}$ | 0.75${}_{0.05}^{1.00}$ | 0.85${}_{0.63}^{**}$ | 1738.7/1618 | 1.075 | 0.006${}_{**}^{0.08}$ | 10.00${}_{8.80}^{**}$ | 88.41${}_{88.11}^{88.55}$ | 2140.5/1620 | 1.3213 | Torus
M 51 | 10.00${}_{7.86}^{**}$ | 0.06${}_{**}^{52.70}$ | 482.9/341 | 1.416 | 25.40${}_{24.68}^{**}$ | 0.34${}_{0.05}^{0.57}$ | 0.95${}_{0.80}^{**}$ | 470.5/343 | 1.372 | 0.05${}_{**}^{1.26}$ | 10.00${}_{8.96}^{**}$ | 18.00${}_{**}^{47.80}$ | 493.8/341 | 1.448 | Torus
NGC 2273* | 10.00${}_{9.02}^{**}$ | 0.09${}_{**}^{52.65}$ | 226.8/197 | 1.151 | 23.75${}_{23.56}^{24.19}$ | 0.15${}_{0.05}^{0.59}$ | 0.94${}_{0.72}^{**}$ | 173.5/196 | 0.882 | 2.14${}_{2.02}^{2.31}$ | 9.99${}_{8.80}^{**}$ | 8.90${}_{**}^{45.34}$ | 246.3/196 | 1.257 | D
HE 1136-2304 | 3.99${}_{2.85}^{5.10}$ | 87.77${}_{86.75}^{88.41}$ | 1066.0/987 | 1.080 | 23.52${}_{23.38}^{23.64}$ | 0.59${}_{0.50}^{0.68}$ | 0.05${}_{**}^{0.39}$ | 1054.5/986 | 1.069 | 3.78${}_{3.77}^{3.79}$ | 9.93${}_{1.45}^{**}$ | 0.05${}_{**}^{26.84}$ | 1126.3/986 | 1.142 | Torus
IGRJ 11366-6002 | 0.68${}_{0.42}^{2.23}$ | 18.37${}_{**}^{77.64}$ | 261.9/245 | 1.069 | 24.75${}_{24.48}^{**}$ | 0.93${}_{0.77}^{0.97}$ | 0.94${}_{0.67}^{0.95}$ | 252.3/244 | 1.034 | 3.19${}_{3.08}^{3.40}$ | 3.68${}_{1.29}^{**}$ | 63.99${}_{37.40}^{85.73}$ | 250.49/244 | 1.026 | T/D
IC 4518A* | 3.41${}_{1.10}^{6.17}$ | 75.58${}_{**}^{83.72}$ | 245.7/215 | 1.143 | 24.59${}_{24.11}^{25.16}$ | 0.56${}_{0.39}^{1.00}$ | 0.55${}_{0.30}^{0.72}$ | 222.6/217 | 1.026 | 1.35${}_{**}^{1.73}$ | 9.99${}_{2.94}^{**}$ | 82.47${}_{59.56}^{84.84}$ | 250.4/217 | 1.154 | Torus
NGC 7674* | 3.22${}_{2.21}^{**}$ | 18.17${}_{**}^{80.86}$ | 189.0/169 | 1.118 | 25.40${}_{24.43}^{**}$ | 0.70${}_{0.52}^{0.90}$ | 0.95${}_{0.84}^{**}$ | 190.7/168 | 1.135 | 2.82${}_{2.77}^{3.02}$ | 3.25${}_{2.26}^{6.07}$ | 0.0001${}_{**}^{71.42}$ | 179.4/168 | 1.068 | D
NGC 5033 | 1.35${}_{1.00}^{**}$ | 78.47${}_{56.20}^{87.66}$ | 1232.9/1058 | 1.165 | 23.49${}_{23.34}^{23.69}$ | 0.77${}_{0.53}^{0.92}$ | 0.95${}_{0.76}^{**}$ | 1177.4/1058 | 1.113 | 0.45${}_{**}^{1.00}$ | 10.00${}_{4.88}^{**}$ | 87.41${}_{84.68}^{87.83}$ | 1193.8/1054 | 1.133 | Torus

141414 Notes: Reflection fraction (Rf), the inclination (Incl) and the
ionization degree (in log) of PEXMON and XILLVER models. The column density of
the torus like reflector ($\log(\rm{N_{H,refl}})$), the covering factor (CF)
and the inclination (Cos($\theta_{\rm incl}$)) for borus02 . BM shows the best
model according with Akaike criterion. The symbol T/D represents that the
models are indistinguishable. Objects with * symbols are the galaxies with not
simultaneous observations. The quantities in this table marked with ** are
unconstrained errors except in the case of the CF, where we report the cases
the error is limited by the model.

Table 11: Summary of other parameters of the best fit model.

| PEXMON | borus02 | XILLVER
---|---|---|---
Name | $\log(N_{{\rm H},los})$ | $\log(N_{{\rm H},ext})$ |  kT1 |  kT2 | BM | $\log(N_{{\rm H},los})$ | $\log(N_{{\rm H},ext})$ |  kT1 |  kT2 | BM | $\log(N_{{\rm H},los})$ | $\log(N_{{\rm H},ext})$ |  kT1 |  kT2 | BM
NGC 3998 |  20.54${}_{20.46}^{20.62}$ | - | - | - | - |  20.58${}_{20.53}^{20.61}$ | - | - | - | - |  20.51${}_{20.40}^{20.62}$ | - | - | - | -
NGC 3718 |  22.03${}_{21.98}^{22.07}$ | - |  0.89${}_{0.69}^{**}$ | - | ME |  22.02${}_{21.99}^{22.07}$ | - |  0.88${}_{0.67}^{**}$ | - | ME |  22.03${}_{21.98}^{22.08}$ | - |  0.89${}_{0.69}^{**}$ | - | ME
NGC 4258* |  22.94${}_{22.92}^{22.95}$ |  20.08${}_{**}^{20.52}$ |  0.58${}_{0.57}^{0.59}$ |  0.20${}_{0.19}^{0.21}$ | 2ME+PL |  22.94${}_{22.92}^{22.96}$ |  20.08${}_{**}^{2}0.63$ |  0.58${}_{0.57}^{0.59}$ |  0.20${}_{0.19}^{0.21}$ | 2ME+PL |  22.94${}_{22.92}^{22.96}$ |  20.08${}_{**}^{20.75}$ |  0.58${}_{0.57}^{0.59}$ |  0.20${}_{0.19}^{0.22}$ | 2ME+PL
ESO 253-G003* |  22.78${}_{22.15}^{23.09}$ | - | - | - | - |  23.05${}_{22.94}^{23.16}$ | - | - | - | - |  22.94${}_{22.74}^{23.12}$ | - | - | - | -
NGC 1052 |  23.16${}_{23.15}^{23.18}$ |  20.53${}_{**}^{20.70}$ |  0.65${}_{0.61}^{0.68}$ | - | ab*(ME+PL) |  23.20${}_{23.19}^{23.21}$ |  20.71${}_{20.60}^{20.82}$ |  0.63${}_{0.59}^{0.67}$ | - | ME+PL |  23.83${}_{23.76}^{23.87}$ | - |  0.86${}_{0.85}^{0.87}$ | - | ab*ME
NGC 2655 |  23.31${}_{23.19}^{23.36}$ | - |  0.65${}_{0.61}^{0.69}$ |  0.09${}_{**}^{0.43}$ | ab*2ME |  23.42${}_{23.25}^{23.51}$ |  20.32${}_{**}^{21.79}$ |  0.62${}_{0.58}^{0.70}$ |  0.15${}_{**}^{0.30}$ | 2ME+PL |  23.29${}_{23.22}^{23.37}$ | - |  0.66${}_{0.62}^{0.69}$ | - | ME*ab
NGC 3147* |  20.54${}_{**}^{20.61}$ | - | - | - | ab |  20.54${}_{**}^{20.62}$ | - | - | - | - |  22.78${}_{**}^{20.98}$ | - | - | - | -
NGC 2110* |  21.76${}_{21.72}^{21.85}$ | - |  0.65${}_{0.64}^{0.67}$ |  0.09${}_{**}^{0.39}$ | 2ME*2ab |  22.01${}_{21.96}^{22.06}$ |  21.27${}_{**}^{21.82}$ |  0.71${}_{0.69}^{0.72}$ | - | (ME+PL)*ab |  21.88${}_{21.83}^{21.93}$ | - |  0.70${}_{0.69}^{0.71}$ | - | ME*ab
LEDA 96373* |  22.67${}_{22.57}^{22.76}$ | - |  0.63${}_{0.55}^{0.76}$ |  0.22${}_{0.20}^{0.23}$ | 2ME*ab |  24.09${}_{24.06}^{24.17}$ | - |  0.59${}_{0.41}^{0.76}$ |  0.22${}_{0.19}^{0.23}$ | 2ME*ab |  23.56${}_{23.49}^{23.62}$ |  23.11${}_{23.03}^{23.18}$ |  0.79${}_{0.76}^{0.80}$ | - | (ME+PL)*ab
NGC 2992 |  21.89${}_{21.88}^{21.91}$ |  21.20${}_{21.11}^{21.26}$ |  0.81${}_{0.78}^{0.84}$ | - | (PL+ME)*ab |  21.89${}_{21.88}^{21.90}$ |  20.00${}_{**}^{21.25}$ |  0.79${}_{0.76}^{0.82}$ | - | (PL+ME)*ab |  21.90${}_{21.88}^{21.89}$ | - |  0.63${}_{0.62}^{0.65}$ |  0.19${}_{0.18}^{0.21}$ | 2ME
M 51 |  22.36${}_{**}^{22.88}$ | - | - | - | ab |  24.77${}_{24.45}^{**}$ | - | - | - | - |  21.93${}_{**}^{22.69}$ | - | - | - | ab
NGC 2273* |  23.15${}_{22.75}^{23.31}$ | - |  0.71${}_{0.52}^{0.79}$ | - | ME*ab |  24.55${}_{24.43}^{24.66}$ | - |  0.26${}_{0.22}^{0.34}$ | - | ME*ab |  23.36${}_{23.27}^{23.44}$ | - |  0.71${}_{0.50}^{0.80}$ | - | ME*ab
HE 1136-2304 |  20.97${}_{20.95}^{20.99}$ | - |  0.59${}_{0.50}^{0.67}$ | - | ME |  20.96${}_{20.94}^{20.98}$ | - |  0.59${}_{0.51}^{0.67}$ | - | ME |  20.96${}_{20.94}^{20.99}$ | - |  0.65${}_{0.54}^{0.77}$ | - | ME
IGRJ 11366-6002 |  21.81${}_{**}^{21.90}$ | - | - | - | - |  21.99${}_{**}^{22.24}$ | - | - | - | - |  21.81${}_{**}^{21.89}$ | - | - | - | -
IC 4518A* |  23.22${}_{23.17}^{23.32}$ |  20.96${}_{**}^{23.19}$ |  0.70${}_{0.67}^{0.72}$ | - | (ME+PL)*ab |  23.22${}_{23.16}^{23.30}$ | - |  0.71${}_{0.59}^{0.75}$ | - | ME*ab |  23.21${}_{23.15}^{23.28}$ | - |  0.70${}_{0.68}^{0.74}$ | - | ME*ab
NGC 7674* |  23.09${}_{22.96}^{23.22}$ | - | - | - | - |  23.06${}_{22.98}^{23.21}$ | - | - | - | - |  22.99${}_{22.75}^{23.08}$ | - | - | - | -
NGC 5033 |  20${}_{**}^{20.09}$ | - | - | - | - |  20${}_{**}^{20.12}$ | - | - | - | - |  20.99${}_{20.75}^{21.13}$ | - |  0.82${}_{0.59}^{**}$ |  0.23${}_{0.18}^{0.54}$ | 2ME

151515Name, column density in the hard energy band ($\log(N_{{\rm H},los})$),
column density in the soft energy band ($\log(N_{{\rm H},ext})$), temperature
of the thermal components (kT1 and kT2). BM means the best model added to the
reflection and the central emission to improve the fit. ME means MEKAL, PL
power-law in the scattered component and ab the warm absorption modelled with
zxipcf in xspec. - denotes that these were not require to improve the fit.
Objects with * symbols are the galaxies with not simultaneous observations.

Table 12: Summary of other parameters of the best fit models.

| PEXMON | borus02 | XILLVER
---|---|---|---
Name | NH,W | $\log(\xi_{W})$ | Norm PLXMM | Norm PLNu | NH,W | $\log(\xi_{W})$ | Norm PLXMM | Norm PLNu | NH,W | $\log(\xi_{W})$ | Norm PLXMM | Norm PLNu
| ($10^{22}$ cm-2) |  | ($10^{-3}$) | ($10^{-3}$) | ($10^{22}$ cm-2) |  | ($10^{-3}$) | ($10^{-3}$) | ($10^{22}$ cm-2) |  | ($10^{-3}$) | ($10^{-3}$)
NGC 3998 | - | - | $\\#$ | 2.33${}_{2.23}^{2.43}$ | - | - | $\\#$ | 2.35${}_{2.26}^{2.44}$ | - | - | $\\#$ | 2.05${}_{**}^{3.41}$
NGC 3718 | - | - | $\\#$ | 0.53${}_{0.47}^{0.59}$ | - | - | $\\#$ | 0.52${}_{0.47}^{0.58}$ | - | - | $\\#$ | 0.08${}_{0.05}^{0.09}$
NGC 4258* | - | - | 1.70${}_{1.50}^{1.83}$ | 1.13${}_{0.99}^{1.24}$ | - | - | 1.60${}_{1.42}^{1.74}$ | 1.04${}_{0.92}^{1.25}$ | - | - | 0.032${}_{0.028}^{0.051}$ | 0.021${}_{0.017}^{0.028}$
ESO 253-G003* | - | - | - | 0.17${}_{0.11}^{0.36}$ | - | - | - | 0.30${}_{0.26}^{0.46}$ | - | - | - | 0.02${}_{0.01}^{0.03}$
NGC 1052 | 0.05${}_{**}^{45.54}$ | 6${}_{2.37}^{**}$ | $\\#$ | 2.79${}_{2.66}^{3.03}$ | - | - | $\\#$ | 3.46${}_{**}^{3.51}$ | 19.25${}_{19.00}^{19.51}$ | 2.032${}_{2.029}^{2.043}$ | $\\#$ | 0.043${}_{0.042}^{0.044}$
NGC 2655 | 2.11${}_{1.23}^{2.81}$ | 1.32${}_{1.15}^{1.55}$ | $\\#$ | 0.91${}_{0.42}^{1.22}$ | - | - | $\\#$ | 0.80${}_{0.54}^{1.39}$ | 2.57${}_{1.71}^{3.77}$ | 1.36${}_{1.23}^{1.64}$ | $\\#$ | 0.02${}_{0.01}^{0.03}$
NGC 3147* | - | - | 1.89${}_{0.92}^{6.21}$ | 0.68${}_{0.64}^{0.72}$ | - | - | 0.34${}_{0.30}^{0.38}$ | 0.71${}_{0.66}^{0.75}$ | - | - | 0.0066${}_{0.0056}^{0.0071}$ | 0.012${}_{0.010}^{0.016}$
NGC 2110*w | 0.17${}_{0.13}^{0.24}$ | -0.55${}_{-0.83}^{-0.43}$ | 10.80${}_{9.70}^{13.10}$ | 59.40${}_{58.50}^{60.50}$ | 1.53${}_{1.39}^{1.66}$ | 4.30${}_{4.03}^{4.51}$ | 8.38${}_{7.51}^{8.79}$ | 4.44${}_{4.22}^{4.61}$ | 5.23${}_{5.07}^{5.37}$ | 1.36${}_{1.29}^{1.43}$ | 0.30${}_{0.28}^{0.59}$ | 1.54${}_{1.37}^{1.57}$
LEDA 96373* | 0.79${}_{0.67}^{0.87}$ | -0.93${}_{-1.07}^{-0.65}$ | $\\#$ | 0.12${}_{0.11}^{0.14}$ | 0.80${}_{0.68}^{0.83}$ | -0.55${}_{-1.50}^{-0.33}$ | $\\#$ | 5.93${}_{4.52}^{9.06}$ | 7.94${}_{7.28}^{8.85}$ | 2.01${}_{1.97}^{2.04}$ | $\\#$ | 0.005${}_{0.004}^{0.006}$
NGC 2992 | 0.49${}_{0.43}^{0.63}$ | 2.53${}_{2.39}^{2.69}$ | $\\#$ | 18.58${}_{18.16}^{19.06}$ | 0.64${}_{0.52}^{0.79}$ | 2.48${}_{2.36}^{2.59}$ | $\\#$ | 20.80${}_{20.30}^{21.40}$ | - | - | $\\#$ | 0.40${}_{0.39}^{0.41}$
M 51 | 499.96${}_{224.16}^{**}$ | 4.25${}_{3.36}^{4.53}$ | $\\#$ | 0.056${}_{0.036}^{0.099}$ | - | - | $\\#$ | 0.36${}_{0.22}^{1.18}$ | 488.99${}_{278.02}^{**}$ | 4.18${}_{3.88}^{4.38}$ | $\\#$ | 0.0012${}_{0.0011}^{0.0016}$
NGC 2273* | 2.95${}_{1.78}^{5.33}$ | 1.24${}_{0.26}^{1.49}$ | $\\#$ | 0.29${}_{0.08}^{0.57}$ | 0.94${}_{0.24}^{1.24}$ | -1.14${}_{-1.31}^{0.35}$ | $\\#$ | 5.14${}_{1.60}^{10.52}$ | 2.97${}_{2.43}^{5.72}$ | 1.23${}_{0.45}^{1.39}$ | $\\#$ | 0.006${}_{0.005}^{0.010}$
HE 1136-2304 | - | - | $\\#$ | 2.53${}_{2.47}^{2.59}$ | - | - | $\\#$ | 2.53${}_{2.48}^{2.59}$ | - | - | $\\#$ | 0.0042${}_{0.0041}^{0.069}$
IGRJ 11366-6002 | - | - | - | 2.42${}_{2.14}^{2.79}$ | - | - | - | 2.26${}_{1.87}^{2.50}$ | - | - | - | 1.43${}_{0.97}^{2.74}$
IC 4518A* | 3.79${}_{1.91}^{6.03}$ | 1.37${}_{1.09}^{1.50}$ | 0.77${}_{0.60}^{1.46}$ | 1.65${}_{1.41}^{1.98}$ | 4.41${}_{1.95}^{5.77}$ | 1.44${}_{1.22}^{1.64}$ | 0.65${}_{0.45}^{0.95}$ | 2.13${}_{1.59}^{3.00}$ | 3.97${}_{2.23}^{5.79}$ | 1.39${}_{1.31}^{1.57}$ | 0.015${}_{0.010}^{0.199}$ | 0.032${}_{0.026}^{0.041}$
NGC 7674* | - | - | - | 0.36${}_{0.14}^{0.47}$ | - | - | - | 0.24${}_{0.19}^{0.32}$ | - | - | - | 0.003${}_{0.002}^{0.005}$
NGC 5033 | - | - | $\\#$ | 1.93${}_{1.90}^{1.98}$ | - | - | $\\#$ | 1.90${}_{1.86}^{1.95}$ | - | - | $\\#$ | 0.032${}_{0.029}^{0.033}$

161616Name, Column density and ionization degree of the warm absorber (NH,W
and $\log(\xi_{W})$) and the normalization of the power law emission of _XMM-
Newton_ and _NuSTAR_ observations (Norm PLXMM and Norm PLNu). The parameter in
this table marked with $\\#$ (power-law normalization of _XMM-Newton_ ) is
tied to _NuSTAR_ , ”$-$” denotes that these were not require to improve the
fit. Objects with * symbols are the galaxies with not simultaneous
observations. NGC 2110 (denoted as NGC 2110w) need two warm absorbers to
improve the pexmon fit. In addition to the absorber presented in this table,
the other correspond to: NH,W=5.38${}_{5.23}^{5.52}$ and
$\log(\xi_{W})$=1.31${}_{1.27}^{1.35}$.

Table 13: Summary of the Cross normalization constants between the
instruments.

|  | PEXMON |  |  | borus02 |  |  | XILLVER |
---|---|---|---|---|---|---|---|---|---
Name | $C_{\rm{A/B}}$ | $C_{\rm{XMM}}$ | $C_{\rm{BAT}}$ | $C_{\rm{A/B}}$ | $C_{\rm{XMM}}$ | $C_{\rm{BAT}}$ | $C_{\rm{A/B}}$ | $C_{\rm{XMM}}$ | $C_{\rm{BAT}}$
NGC 3998 | 1.04${}_{1.02}^{1.06}$ | 0.91${}_{0.88}^{0.94}$ | 0.98${}_{0.80}^{1.17}$ | 1.04${}_{1.02}^{1.06}$ | 0.91${}_{0.90}^{0.94}$ | 0.96${}_{0.79}^{1.13}$ | 1.04${}_{1.02}^{1.06}$ | 0.92${}_{0.89}^{0.94}$ | 0.96${}_{0.79}^{1.14}$
NGC 3718 | 0.99${}_{0.96}^{1.03}$ | 0.98${}_{0.94}^{1.03}$ | 4.53${}_{3.60}^{5.53}$ | 1.00${}_{0.96}^{1.02}$ | 0.99${}_{0.94}^{1.03}$ | 4.42${}_{3.52}^{5.26}$ | 0.99${}_{0.96}^{1.03}$ | 0.98${}_{0.94}^{1.03}$ | 4.46${}_{3.57}^{5.41}$
NGC 4258* | 0.99${}_{0.96}^{1.02}$ | 1.09${}_{1.03}^{**}$ | 2.64${}_{2.31}^{2.98}$ | 0.99${}_{0.96}^{1.02}$ | 1.07${}_{0.96}^{**}$ | 2.56${}_{2.24}^{2.93}$ | 0.99${}_{0.97}^{1.02}$ | 1.09${}_{**}^{1.10}$ | 2.60${}_{2.28}^{2.97}$
ESO 253-G003* | 1.14${}_{1.06}^{1.23}$ | - | 0.89${}_{0.66}^{1.14}$ | 1.14${}_{1.06}^{1.23}$ | - | 0.88${}_{0.67}^{1.13}$ | 1.14${}_{1.06}^{1.23}$ | - | 0.85${}_{0.65}^{1.08}$
NGC 1052 | 1.00${}_{0.98}^{1.03}$ | 0.89${}_{0.87}^{0.91}$ | 0.86${}_{0.76}^{0.96}$ | 1.00${}_{0.97}^{1.02}$ | 0.87${}_{0.85}^{0.89}$ | 0.83${}_{0.74}^{0.93}$ | 1.00${}_{0.99}^{1.02}$ | 0.89${}_{0.87}^{0.90}$ | 0.90${}_{0.79}^{1.00}$
NGC 2655 | 1.16${}_{1.05}^{1.05}$ | 1.09${}_{0.93}^{1.27}$ | 2.21${}_{1.63}^{2.90}$ | 1.15${}_{1.03}^{1.27}$ | 1.07${}_{0.91}^{1.25}$ | 2.22${}_{1.63}^{2.89}$ | 1.15${}_{1.04}^{1.28}$ | 1.07${}_{0.92}^{1.16}$ | 2.18${}_{1.62}^{2.78}$
NGC 3147* | 1.04${}_{0.99}^{1.09}$ | 0.95${}_{**}^{0.97}$ | 1.08${}_{0.71}^{1.47}$ | 1.04${}_{0.99}^{1.09}$ | 0.99${}_{0.90}^{**}$ | 1.00${}_{0.64}^{1.34}$ | 1.04${}_{0.90}^{1.09}$ | 0.92${}_{**}^{0.99}$ | 1.08${}_{0.70}^{1.47}$
NGC 2110* | 1.03${}_{1.02}^{1.05}$ | 1.03${}_{**}^{1.10}$ | 0.58${}_{0.57}^{0.59}$ | 1.04${}_{1.02}^{1.05}$ | 0.90${}_{**}^{0.97}$ | 0.60${}_{0.59}^{0.61}$ | 1.03${}_{1.02}^{1.05}$ | 0.97${}_{**}^{1.08}$ | 0.58${}_{0.57}^{0.59}$
LEDA 96373* | 1.12${}_{1.05}^{1.20}$ | 0.84${}_{0.79}^{0.90}$ | 0.87${}_{0.71}^{1.06}$ | 1.13${}_{1.05}^{1.21}$ | 0.87${}_{0.81}^{0.94}$ | 0.85${}_{0.70}^{1.03}$ | 1.12${}_{1.05}^{1.20}$ | 0.86${}_{0.80}^{0.92}$ | 0.88${}_{0.71}^{1.06}$
NGC 2992 | 1.027${}_{1.019}^{1.033}$ | 0.91${}_{0.90}^{0.92}$ | 0.127${}_{0.119}^{0.133}$ | 1.026${}_{1.020}^{1.033}$ | 0.909${}_{0.903}^{0.915}$ | 0.122${}_{0.110}^{0.136}$ | 1.026${}_{1.019}^{1.033}$ | 0.915${}_{0.909}^{0.920}$ | 0.13${}_{0.12}^{0.15}$
M 51 | 0.99${}_{0.93}^{1.06}$ | 1.06${}_{0.97}^{1.16}$ | 2.67${}_{1.95}^{3.46}$ | 1.00${}_{0.94}^{1.07}$ | 1.06${}_{0.98}^{1.16}$ | 2.08${}_{1.54}^{2.77}$ | 0.99${}_{0.92}^{1.06}$ | 1.07${}_{0.98}^{1.17}$ | 2.67${}_{1.91}^{3.52}$
NGC 2273* | 1.09${}_{0.99}^{1.18}$ | 1.15${}_{0.99}^{1.33}$ | 0.84${}_{0.54}^{1.17}$ | 1.08${}_{0.99}^{1.17}$ | 1.04${}_{0.90}^{1.19}$ | 0.65${}_{0.43}^{0.91}$ | 1.09${}_{1.00}^{1.18}$ | 1.09${}_{0.94}^{1.97}$ | 0.84${}_{0.55}^{1.14}$
HE 1136-2304 | 1.07${}_{1.05}^{1.08}$ | 1.03${}_{1.01}^{1.04}$ | 0.63${}_{0.50}^{0.77}$ | 1.064${}_{1.046}^{1.082}$ | 1.03${}_{1.01}^{1.04}$ | 0.58${}_{0.45}^{0.71}$ | 1.065${}_{1.047}^{1.081}$ | 1.023${}_{1.015}^{1.039}$ | 0.62${}_{0.49}^{0.75}$
IGRJ 11366-6002 | 1.15${}_{1.09}^{1.22}$ | - | 1.46${}_{1.21}^{1.81}$ | 1.15${}_{1.09}^{1.22}$ | - | 1.42${}_{1.14}^{1.71}$ | 1.15${}_{1.09}^{1.22}$ | - | 1.46${}_{1.16}^{1.78}$
IC 4518A* | 1.12${}_{1.04}^{1.20}$ | 0.97${}_{**}^{0.99}$ | 1.11${}_{0.95}^{1.25}$ | 1.11${}_{1.03}^{1.19}$ | 1.10${}_{0.90}^{**}$ | 1.09${}_{0.93}^{1.28}$ | 1.11${}_{1.04}^{1.19}$ | 0.90${}_{**}^{1.00}$ | 1.09${}_{0.94}^{1.27}$
NGC 7674* | 1.02${}_{0.95}^{1.09}$ | - | 1.58${}_{1.15}^{2.03}$ | 1.02${}_{0.96}^{1.09}$ | - | 1.60${}_{1.17}^{2.07}$ | 1.02${}_{0.95}^{1.09}$ | - | 1.57${}_{1.16}^{2.00}$
NGC 5033 | 1.05${}_{1.03}^{1.07}$ | 0.75${}_{0.73}^{0.76}$ | 0.28${}_{0.13}^{0.42}$ | 1.05${}_{1.04}^{1.07}$ | 0.76${}_{0.74}^{0.78}$ | 0.27${}_{0.14}^{0.41}$ | 1.05${}_{1.04}^{1.07}$ | 0.76${}_{0.74}^{0.78}$ | 0.28${}_{0.13}^{0.42}$

171717Name, cross normalization constant between the detector FPMA and FPMB of
_NuSTAR_ (CA/B, _NuSTAR_ and _XMM-Newton_ (CXMM) and _NuSTAR_ and
_Swift_(CBAT). In case of the not simultaneous data _XMM-Newton_ +_NuSTAR_
observations, where the normalization of the power-law is a free parameter
(objects denoted in bold face), we put the restriction of CXMM=[0.9,1.10]. For
IGRJ 11366-6002 we also obtain $C_{XRT}$=1.30${}_{1.09}^{1.50}$ (PEXMON),
$C_{XRT}$=1.13${}_{1.09}^{1.22}$ (borus02) and
$C_{XRT}$=1.32${}_{1.11}^{1.54}$ (XILLVER). The error of this parameters is in
the normalization of the power-law then we reported here only the best fit
value. The quantities in this table marked with ** are unconstrained errors.

Table 14: Soft (2-10 keV) and hard (10–79 keV) intrinsic and observed
luminosities for all the sample.

| Observed | Intrinsic |
---|---|---|---
Name | $\log(L_{2.0-10.0})$ | $\log(L_{10.0-79.0})$ | $\log(L_{2.0-10.0})$ | $\log(L_{10.0-79.0})$ | Model
NGC 3998 | 41.14${}_{41.12}^{41.15}$ | 41.33${}_{41.27}^{41.34}$ | 41.15${}^{41.16}_{41.14}$ | 40.67${}^{40.71}_{40.62}$ | borus02
| 41.28${}_{41.12}^{41.40}$ | 41.46${}_{41.16}^{41.62}$ | 41.28${}_{41.12}^{41.40}$ | 41.16${}_{40.88}^{41.33}$ | XILLVER
NGC 3718 | 40.45${}_{40.39}^{40.46}$ | 40.64${}_{40.45}^{40.65}$ | 40.49${}_{40.46}^{40.52}$ | 40.64${}_{40.55}^{40.72}$ | borus02
| 40.45${}_{40.10}^{40.61}$ | 40.63${}_{40.03}^{40.71}$ | 40.49${}^{40.67}_{40.20}$ | 40.63${}^{40.79}_{40.36}$ | XILLVER
NGC 4258* | 39.93${}^{37.90}_{40.65}$ | 40.14${}_{40.07}^{40.22}$ | 40.15${}^{40.05}_{40.70}$ | 40.15${}^{40.22}_{40.07}$ | borus02
ESO 253-G003${}^{*}_{Nu}$ | 42.66${}_{42.10}^{42.69}$ | 43.56${}_{43.11}^{43.67}$ | 42.92${}_{42.05}^{43.10}$ | 43.59${}_{43.00}^{43.81}$ | borus02
| 42.70${}_{42.52}^{43.04}$ | 43.62${}_{43.15}^{43.65}$ | 42.89${}_{42.23}^{43.15}$ | 43.62${}_{43.42}^{43.76}$ | XILLVER
NGC 1052 | 41.44${}_{41.36}^{41.48}$ | 42.13${}_{41.83}^{42.18}$ | 41.76${}^{41.70}_{41.81}$ | 42.15${}^{42.23}_{42.04}$ | borus02
NGC 2655 | 40.61${}_{39.90}^{40.67}$ | 41.18${}_{40.76}^{41.21}$ | 41.03${}^{41.11}_{40.95}$ | 41.19${}^{41.26}_{41.08}$ | borus02
| 40.60${}_{40.19}^{40.74}$ | 40.19${}_{40.09}^{40.32}$ | 40.96${}_{40.66}^{41.14}$ | 41.20${}_{39.85}^{41.51}$ | XILLVER
NGC 3147* | 41.40${}_{40.08}^{41.60}$ | 42.01${}_{41.69}^{42.04}$ | 41.40${}^{41.56}_{41.16}$ | 42.01${}^{42.12}_{41.87}$ | borus02
| 40.39${}_{40.18}^{41.77}$ | 41.95${}_{41.93}^{42.07}$ | 41.39${}_{40.62}^{41.73}$ | 41.95${}_{41.86}^{42.02}$ | XILLVER
NGC 2110* | 42.44${}_{42.25}^{42.58}$ | 43.61${}_{43.60}^{43.62}$ | 42.52${}^{42.65}_{42.34}$ | 43.61${}^{43.62}_{43.60}$ | borus02
LEDA 96373* | 42.34${}_{42.10}^{42.34}$ | 43.41${}_{42.98}^{43.42}$ | 43.48${}^{43.56}_{43.38}$ | 43.80${}^{43.92}_{43.63}$ | borus02
NGC 2992 | 42.911${}_{42.907}^{42.914}$ | 43.318${}_{43.290}^{43.322}$ | 42.95${}^{42.95}_{42.94}$ | 43.32${}^{43.34}_{43.30}$ | borus02
M 51 | 38.94${}_{38.72}^{40.11}$ | 39.81${}_{37.38}^{40.41}$ | 39.68${}^{39.80}_{39.51}$ | 40.25${}^{40.75}_{40.32}$ | borus02
NGC 2273* | 40.89${}_{39.91}^{41.00}$ | 41.76${}_{39.70}^{40.95}$ | 42.33${}_{40.94}^{42.54}$ | 41.99${}_{41.88}^{42.20}$ | XILLVER
HE 1136-2304 | 43.23${}_{43.19}^{43.24}$ | 43.54${}_{43.36}^{43.55}$ | 43.23${}^{43.25}_{43.21}$ | 43.54${}^{43.61}_{43.46}$ | borus02
IGRJ 11366-6002 | 42.35${}_{42.28}^{42.35}$ | 42.59${}_{42.14}^{42.52}$ | 42.39${}^{42.59}_{42.02}$ | 42.59${}^{42.46}_{42.69}$ | borus02
| 42.33${}_{42.14}^{42.57}$ | 42.58${}_{42.10}^{42.70}$ | 42.39${}_{42.05}^{42.57}$ | 42.58${}_{42.26}^{42.76}$ | XILLVER
IC 4518A* | 41.97${}_{41.37}^{42.10}$ | 43.08${}_{42.88}^{43.13}$ | 42.38${}^{42.42}_{42.35}$ | 43.10${}^{43.16}_{43.02}$ | borus02
NGC 7674${}_{nu}^{*}$ | 42.17${}_{41.52}^{42.98}$ | 43.03${}_{42.03}^{43.58}$ | 42.42${}^{42.31}_{42.50}$ | 43.07${}^{4300}_{43.12}$ | XILLVER
NGC 5033 | 41.04${}_{40.01}^{40.05}$ | 41.43${}_{41.34}^{41.44}$ | 41.04${}_{41.02}^{41.06}$ | 41.43${}_{41.38}^{41.47}$ | borus02
181818Name, soft and hard luminosities (observed and intrinsic) and the best
fit model. Objects with * symbols are the galaxies with not simultaneous
observations. nu mark objects with only _NuSTAR_ data.

Table 15: Bolometric correction, bolometric luminosity and accretion rate for
the sample of galaxies.

Name | k2.0-10.0 | LBol | $\log(\lambda_{\rm Edd})$
---|---|---|---
(1) | (2) | (3) | (4)
NGC 3998 | 15.35${}^{15.41}_{15.29}$ | 42.33${}^{42.34}_{42.32}$ | -4.71${}^{-5.40}_{-4.02}$
NGC 3718 | 15.33${}^{15.40}_{15.27}$ | 41.68${}^{41.85}_{41.38}$ | -4.67${}^{-5.10}_{-4.17}$
NGC 4258* | 15.33${}^{15.39}_{15.27}$ | 41.08${}^{41.11}_{41.05}$ | -4.59${}^{-5.26}_{-3.92}$
NGC 5033 | 15.34${}^{15.40}_{15.28}$ | 42.02${}^{42.03}_{42.00}$ | -4.00${}^{-4.46}_{-3.09}$
ESO 253-G003* | 15.82${}^{16.11}_{15.57}$ | 44.09${}^{44.34}_{43.77}$ | -3.76${}^{-4.21}_{-3.38}$
NGC 1052 | 15.39${}^{15.45}_{15.33}$ | 42.92${}^{42.95}_{42.89}$ | -3.86${}^{-4.53}_{-3.19}$
NGC 2655 | 15.34${}^{15.40}_{15.28}$ | 42.22${}^{42.30}_{42.13}$ | -4.09${}^{-4.71}_{-3.48}$
NGC 3147* | 15.36${}^{15.43}_{15.29}$ | 42.59${}^{42.75}_{42.35}$ | -4.33${}^{-4.87}_{-3.87}$
NGC 2110* | 15.59${}^{15.70}_{15.47}$ | 43.71${}^{43.84}_{43.53}$ | -3.78${}^{-4.35}_{-3.26}$
LEDA 96373* | 16.74${}^{16.99}_{16.46}$ | 44.73${}^{44.84}_{44.60}$ | -2.59${}^{-3.18}_{-2.02}$
NGC 2992 | 15.88${}^{15.92}_{15.83}$ | 44.15${}^{44.15}_{44.14}$ | -2.29${}^{-2.99}_{-1.60}$
M 51 | 15.33${}^{15.39}_{15.27}$ | 40.87${}^{40.99}_{40.69}$ | -3.83${}^{-4.41}_{-3.31}$
NGC 2273* | 15.54${}^{15.69}_{15.35}$ | 43.60${}^{43.84}_{43.06}$ | -2.50${}^{-2.96}_{-2.34}$
HE 1136-2304 | 16.22${}^{16.27}_{16.16}$ | 44.44${}^{44.46}_{44.42}$ | -3.06${}^{-3.74}_{-2.38}$
IGRJ11366 | 15.53${}^{15.66}_{15.38}$ | 43.58${}^{43.78}_{43.20}$ | -3.09${}^{-3.59}_{-2.77}$
IC 4518A | 15.51${}^{15.61}_{15.39}$ | 43.51${}^{43.67}_{43.26}$ | -3.39${}^{-3.93}_{-2.94}$
NGC 7674* | 15.53${}^{15.72}_{15.35}$ | 43.59${}^{43.89}_{43.03}$ | -3.70${}^{-4.10}_{-3.56}$
191919(Col. 1) name, (Col. 2) bolometric correction, (Col. 3) bolometric
luminosity and (Col. 4) accretion rate. All these values were estimated using
Duras et al. (2020).

##  Appendix E Spectral models

Figure 10: Spectral modelling of NGC 3998, NGC 3718, NGC 4258* and ESO
253-G003*. The plots correspond to borus02 (left) and XILLVER (right).

Figure 11: Spectral modelling of NGC 1052, NGC 2655, NGC 3147 and NGC 2110.
The plots correspond to borus02 (left) and XILLVER (right).

Figure 12: Spectral modelling of LEDA 96373, NGC 2992, M 51 and NGC 2273. The
plots correspond to borus02 (left) and XILLVER (right).

Figure 13: Spectral modelling of HE 1136-2304, IGRJ 11366-6002, IC4518 and NGC
7674. The plots correspond to borus02 (left) and XILLVER (right).  Figure 14:
Spectral modelling of NGC 5033. The plots correspond to borus02 (left) and
XILLVER (right).