# Squashed entanglement in one-dimensional quantum matter

Alfonso Maiellaro  Dipartimento di Fisica ”E.R. Caianiello”, Università di
Salerno, Via Giovanni Paolo II, 132, I-84084 Fisciano (SA), Italy  Francesco
Romeo  Dipartimento di Fisica ”E.R. Caianiello”, Università di Salerno, Via
Giovanni Paolo II, 132, I-84084 Fisciano (SA), Italy  INFN, Sezione di Napoli,
Gruppo collegato di Salerno,Italy  Roberta Citro  Dipartimento di Fisica ”E.R.
Caianiello”, Università di Salerno, Via Giovanni Paolo II, 132, I-84084
Fisciano (SA), Italy  INFN, Sezione di Napoli, Gruppo collegato di Salerno,
Italy  Fabrizio Illuminati<EMAIL_ADDRESS>Dipartimento di Ingegneria
Industriale, Università di Salerno, Via Giovanni Paolo II, 132, I-84084
Fisciano (SA), Italy  INFN, Sezione di Napoli, Gruppo collegato di
Salerno,Italy

(March 6, 2023)

###### Abstract

Squashed entanglement and its universal upper bound, the quantum conditional
mutual information, are faithful measures of bipartite quantum correlations
defined in terms of multipartitions. As such, they are sensitive to the fine-
grain structure of quantum systems. Building on this observation, we introduce
the concept of quantum conditional mutual information between the edges of
quantum many-body systems. We show that this quantity characterizes
unambiguously one-dimensional topological insulators and superconductors,
being equal to Bell-state entanglement in the former and to half Bell-state
entanglement in the latter, mirroring the different statistics of the edge
modes in the two systems. The edge-to-edge quantum conditional mutual
information is robust in the presence of disorder or local perturbations,
converges exponentially with the system size to a quantized topological
invariant, even in the presence of interactions, and vanishes in the trivial
phase. We thus conjecture that it coincides with the edge-to-edge squashed
entanglement in the entire ground-state phase diagram of symmetry-protected
topological systems, and we provide some analytical evidence supporting the
claim. By comparing them with the entanglement negativity, we collect further
indications that the quantum conditional mutual information and the squashed
entanglement provide a very accurate characterization of nonlocal correlation
patterns in one-dimensional quantum matter.

##  I introduction

Identifying entanglement-based order parameters able to characterize a large
variety of quantum phases and at the same time to discriminate between
different forms of quantum orders has remained a major challenge in condensed
matter physics for the last two decades. The block von Neumann entanglement
entropy [1, 2, 3] and the block entanglement spectrum [4, 5, 6, 7] in simple
bipartite systems have become central tools for the characterization of
quantum collective behaviors, including topologically ordered phases [8, 9,
10, 11, 12, 13, 14, 15, 16, 17, 18]. Indeed, nontrivial topological order in
two-dimensional systems has been identified by means of the sub-leading term
to the block von Neumann entanglement entropy, the so called topological
entanglement entropy (TEE) [19, 13, 20].

When considering either one-dimensional or higher-dimensional systems with
bulk-edge correspondence, both the block von Neumann entanglement entropy and
the block entanglement spectrum fail to discriminate between topologically
ordered and standard Ginzburg-Landau ordered phases, associated with
spontaneous symmetry breaking and nonvanishing order parameter. In fact, these
measures based on simple bipartitions of a system into two parts (blocks)
cannot account for the different physical properties of the bulk and of the
edges between topologically trivial and topologically nontrivial phases of
quantum matter [6]. Moreover, for all open quantum systems in or out of
equilibrium in any dimension, block entropies and entanglement spectra
necessarily include contributions from both classical and quantum fluctuations
and thus cease to be valid and meaningful measures of nonlocal quantum
correlations.

Following some preliminary efforts to address the problem [21, 22, 23],
recently two well-defined multipartion-based measures of bipartite
entanglement and bipartite correlations, respectively the squashed
entanglement (SE) and the quantum conditional mutual information (QCMI), have
been introduced as possible efficient tools in the study of one-dimensional
quantum matter, given their ability to detect the fine-grain structures of
quantum systems and thus, in particular, to discriminate between the different
bulk and edge contributions to the long-distance correlation patterns in
topologically trivial and topologically nontrivial quantum many-body systems
[24]. These two quantities are intimately related, as the SE is defined as the
infimum of the QCMI over all possible state extensions. Therefore, the QCMI is
always an upper bound to the SE.

Specifically, for many-body systems with open boundary conditions we
introduced the edge-to-edge QCMI, thus being an upper bound to the SE between
the system edges, and we found that it defines the natural quantized, non-
local order parameter for Kitaev topological superconductors in one spatial
dimension and in quasi one-dimensional geometries [24]. For such systems, the
QCMI exhibits the correct scaling at the quantum phase transition, is stable
in the presence of interactions and robust against the effects of disorder and
local perturbations. We introduced two distinct multipartition-based forms of
the QCMI: the tripartition-based edge-edge QCMI
$I_{(3)}(A\\!\\!:\\!\\!B\,|\,C)$ corresponding to a edge $A$ \- entire bulk
$C$ \- edge $B$ tripartition, which leads to a phase diagram equivalent to
that of the corresponding spin chain obtained via a Jordan-Wigner
transformation, and the quadripartition-based edge-edge QCMI
$I_{(4)}(A\\!\\!:\\!\\!B\,|\,C_{1})$, corresponding to a edge $A$ \- bipartite
bulk $C=C_{1}C_{2}$ \- edge $B$ quadripartition and a partially traced-out
bulk (e.g. by tracing out $C_{2}$), which discriminates between symmetric
topological regimes and ordered phases with spontaneously broken symmetries
[25, 26, 27].

Motivated by these results, in the present work we provide an in-depth study
of the edge-to-edge QCMI and SE and show that they characterize topological
quantum phase transitions in one-dimensional systems. Specifically, we
investigate and compare the two paradigmatic models describing one-dimensional
topological insulators and superconductors, respectively the Su-Schrieffer-
Heeger (SSH) insulating chain [28] and the Kitaev superconducting wire [29].
For these systems we show that the edge-to-edge bipartite QCMI identifies the
correct nonlocal order parameter that singles out the phase transitions,
characterizes the topologically ordered phases of topological insulators and
superconductors, and discriminates between them.

The edge-to-edge QCMI is robust under variations of the sample conditions due
to disorder or local perturbations, and scales exponentially with the size of
the system, converging to a quantized, topologically invariant value even in
the presence of interactions. Crucially, the QCMI is sensitive to the
different nature of the edge modes. Indeed, it takes different quantized
values, respectively to Bell-state entanglement and half Bell-state
entanglement, depending on the statistics of the topological edge modes,
respectively Dirac fermions in the SSH chain and Majorana fermions (”half-
Dirac fermions”) in the Kitaev wire.

Having found that the edge-to-edge QCMI in low-dimensional topological systems
exhibits the same behaviour expected for the genuine SE between the system
edges, we conjecture that the two quantities do actually coincide, and for
small-size systems we provide further supporting analytical evidence to this
statement. In general, while the computational effort in the evaluation of the
SE is strongly dependent on the system size and generically a $NP$-hard
problem, evaluating the QCMI requires only a limited amount of computational
resources. This feature of the QCMI between the edges of a topological system
is due to the fact that in the latter the bulk does not contribute to the
quantum correlations between the edges. Therefore the QCMI is insensitive to
different partitions of the bulk and to the system size as soon as the latter
exceeds a (small) critical threshold value.

We also compare the edge-to-edge QCMI with the entanglement negativity, a
measure of bipartite entanglement that is widely used in the study of quantum
statistical mechanics and quantum matter because of its simplicity and
computability [30]. We identify some possible failures of the latter and
provide further evidence that multipartition-based measures such as the edge-
to-edge SE and QCMI are indeed the natural framework for the characterization
of one-dimensional quantum matter. Finally, we discuss the perspectives for
the experimental accessibility of SE/QCMI as well as their generalization to
many-body systems in higher spatial dimensions.

The paper is organized as follows. In Sec. II we introduce the QCMI, the SE
and the two fundamental forms of upper bounds of SE based on the QCMI, $I_{3}$
and $I_{4}$. We also review the main properties of the one-dimensional SSH and
Kitaev models. In Sec. III we study the main features of the QCMI in the two
systems, including the effects of interactions, and compare the behavior of
the QCMI with that of the entanglement negativity. In Sec. IV we investigate
and discuss the conjectured equivalence between the QCMI and the SE. In Sec. V
we discuss our results, consider possible experimental protocols to measure
the QCMI and the SE, and discuss generalizations and further applications in
the study of two-dimensional quantum matter. Technical details and
mathematical methods used troughout the paper are reported in the Appendices.
Finite-size scaling at phase transition boundaries and robustness against
disorder of the QCMI in the topological phases are reported respectively in
Appendices A and B, while in Appendix C we analyze the Jordan-Wigner
transformation applied to the two systems and discuss the phase diagrams of
the interacting SSH model.

##  II Theory

Consider a quantum system $G$, two arbitrary subsystems $A$ and $B$, and a
reminder $C$, such that $ABC$ defines a tripartition of $G$. Consider next a
partition of the reminder: $C=C_{1}C_{2}$, so that now $ABC_{1}C_{2}$ defines
a quadripartition of $G$. One can then introduce two inequivalent measures
$I_{(3)}=I(A\\!\\!:\\!\\!B\,|\,C)$ and $I_{(4)}=I(A\\!\\!:\\!\\!B\,|\,C_{1})$
of the QCMI between subsystems $A$ and $B$, respectively conditioned to $C$
and to $C_{1}$ once $C_{2}$ has been traced out [24]:

$I_{(3)}=\frac{1}{2}\biggl{[}S(\rho_{AC})+S(\rho_{BC})-S(\rho_{ABC})-S(\rho_{C})\biggr{]}\,,$
(1)

and

$I_{(4)}=\frac{1}{2}\biggl{[}S(\rho_{AC_{1}})+S(\rho_{BC_{1}})-S(\rho_{ABC_{1}})-S(\rho_{C_{1}})\biggr{]}\,,$
(2)

where $S(\rho)$ is the von Neumann entropy of the quantum state $\rho$. The
expressions $I_{(3)}$ and $I_{(4)}$ are the only two inequivalent QCMIs
corresponding to the same reduced state $\rho_{AB}$ obtained from the global
state $\rho_{G}$ [24]. On the other hand, the SE ($E_{sq}(\rho_{XY})$) [31,
32] of a bipartite system $X\cup Y$ is defined in terms of the QCMI as:

$\displaystyle
E_{sq}(\rho_{XY})=\inf_{\rho_{XYZ}}\bigg{\\{}I(X:Y|Z)\bigg{\\}},$ (3)

where $\inf_{\rho_{XYZ}}\bigg{\\{}I(X:Y|Z)\bigg{\\}}$ denotes the infimum of
$I(X:Y|Z)$ over all quantum states $\rho_{XYZ}$ of all possible extensions
$X\cup Y\cup Z$, with fixed subsystems $X$ and $Y$. Given a state $\rho_{G}$
of the entire system $G$, then $\rho_{AC}=Tr_{B}(\rho_{G})$,
$\rho_{BC}=Tr_{A}(\rho_{G})$, $\rho_{C}=Tr_{AB}(\rho_{G})$,
$\rho_{AC_{1}}=Tr_{BC_{2}}(\rho_{G})$, $\rho_{BC_{1}}=Tr_{AC_{2}}(\rho_{G})$,
$\rho_{ABC_{1}}=Tr_{C_{2}}(\rho_{G})$ and
$\rho_{C_{1}}=Tr_{ABC_{2}}(\rho_{G})$ are the reduced states of the subsystems
involved. SE is the ”perfect” measure of entanglement on all quantum states as
it is the only entanglement monotone that is also convex, additive,
asymptotically continuous, faithful, monogamous, and reduces to the von
Neumann entanglement entropy on pure states, thus satisfying all the required
properties for a _bona fide_ entanglement measure [33, 34, 24].

In complete generality, it trivially holds that $I_{(3)}\geq
E_{sq}(\rho_{AB})$ and $I_{(4)}\geq E_{sq}(\rho_{AB})$. On the other hand, if
$A$ and $B$ are the edges of a one-dimensional many-body system with open
boundary conditions, we will find that a further chain inequality appears to
hold: $I_{(3)}\geq I_{(4)}\geq E_{sq}(\rho_{AB})$. Actually, as discussed in
Sec. IV below, one can show numerically that equality holds at least for
symmetry-protected topological systems of limited size:
$I_{(3)}=I_{(4)}=E_{sq}(\rho_{AB})$. In Sec. IV we also propose further
arguments supporting the conjecture that the equality
$I_{(3)}=I_{(4)}=E_{sq}(\rho_{AB})$ between the edge-to-edge QCMIs and the
edge-to-edge SE holds for all one-dimensional symmetry-protected topological
systems of arbitrary size. We will thus discuss the topological properties of
the systems under investigation by considering $I_{(3)}$ and $I_{(4)}$
whenever they coincide and whenever they are expected to be equal to the exact
SE $E_{sq}$.

Figure 1: Panels (a) and (b): system multipartitions associated to the
definition of the corresponding bipartite, edge-to-edge SE and QCMIs. Panel
(a): tripartition of a quantum chain into two edges $A$ and $B$ and a bulk
$C$. Panel (b): quadripartition of the same chain, with a bipartite bulk
$C=C_{1}C_{2}$ and part $C_{2}$ of the bulk traced out. Panel (c): standard
bipartition of the same chain in two blocks (halves) for the study of the
block entanglement negativity. Panels (d) and (e): sketch of the tight-binding
models of the SSH and of the Kitaev chains. In panel (d) $a$ and $b$ are the
two fermionic on-site species in the unit cell of the SSH model Hamiltonian.

In Fig. 1, (a) and (b), we provide a sketch of the above multipartitions for a
quantum system on a one-dimensional lattice. Subsystems $A$ and $B$ identify
the system edges; $C$ identifies the full bulk. When the bulk is bipartite,
i.e. $C=C_{1}C_{2}$, we denote by $C_{1}$ the part of the bulk that remains
after part $C_{2}$ is traced out. Without loss of generality one can set the
size of $C_{1}$ as $L_{C_{1}}=1$ and identify it with the site adjacent to
edge $A$. We also fix the length of the edge partitions to $\lfloor
L/3\rfloor$, which is, for the system sizes considered, of the order of the
decay length of the edge modes in the bulk.

In the present work we study and compare the SSH insulator and the Kitaev
superconductor according to the multipartitions shown in Fig. 1 using the
edge-to-edge QCMIs $I_{(3)}$ and $I_{(4)}$ to characterize the phase diagram
and the topologically ordered phases. Moreover, we will consider a further
estimator of bipartite entanglement, the so-called entanglement negativity
($\mathcal{N}$) [30], that essentially quantifies the Peres criterion for
separability [35]. The negativity is overwhelmingly used in quantum
information as well as in the study of quantum matter [36, 37] for its
simplicity and computability, notwithstanding that it fails to satisfy some of
the most important properties required for a valid entanglement measure. Given
a bipartition $\cal{L}\cal{R}$ into two adjacent blocks (halves) $\cal{L}$ and
$\cal{R}$, as shown in Fig. 1 (c), the negativity $\mathcal{N}$ is defined as
the trace norm of the partial transpose $\rho^{T_{\cal{L}}}$ of the bipartite
state $\rho(\cal{L}\cal{R})$ with respect to one of the two blocks:

$\mathcal{N}=\frac{||\rho^{T_{\cal{L}}}||_{1}-1}{2}=\sum_{i}|\lambda_{i}|\,,$
(4)

where $||\rho^{T_{\cal{L}}}||_{1}$ is the trace norm of $\rho^{T_{\cal{L}}}$
and the $\lambda_{i}$s are the negative eigenvalues of $\rho^{T_{\cal{L}}}$.
Next, we recall the SSH and Kitaev model Hamiltonians:

$\displaystyle
H_{SSH}=w\sum_{j=1}^{L}c^{\dagger}_{a,j}c_{b,j}+v\sum_{j=1}^{L-1}c^{\dagger}_{a,j+1}c_{b,j}+h.c.\,,$
(5) $\displaystyle\begin{split}H_{K}=&\sum_{j=1}^{L-1}\biggl{(}\Delta
c^{\dagger}_{j+1}c^{\dagger}_{j}-tc^{\dagger}_{j}c_{j+1}+h.c.\biggr{)}+\\\
&-\sum_{j=1}^{L}\mu\biggl{(}c^{\dagger}_{j}c_{j}-\frac{1}{2}\biggr{)}\,.\end{split}$
(6)

The SSH chain, Eq. 5, describes spinless fermions with staggered hopping
amplitudes $w$ and $v$. The two fermionic species $a$ and $b$ define the two
different degrees of freedom per unit cell. The Kiatev chain, Eq. 6, describes
spinless fermions with a $p$-wave superconducting pairing potential $\Delta$,
an hopping strength $t$ and an on-site chemical potential $\mu$.

These two models embed all the key properties of topological insulators and
superconductors: an insulating bulk with boundary conduction, a protecting
symmetry (chiral symmetry for the SSH model; particle-hole symmetry for the
Kitaev model) and a bulk-edge correspondence. The topologically ordered phase
occurs, respectively, when $w<v$ and when $\mu<2t$, $\Delta\neq 0$. The
crucial discriminant between the two models is that they belong to two
distinct classes of the ten-fold classification [38], featuring topological
edge modes of different physical nature: fermionic for the SSH insulator and
Majorana for the Kitaev superconductor. When interactions are included, the
additional interaction terms read:

$\displaystyle
H_{I,SSH}=\sum_{j=1}^{L}\left[U_{1}n_{a,j}n_{b,j}+U_{2}n_{b,j}n_{a,j+1}\right]\,,$
(7) $\displaystyle H_{I,K}=\sum_{j=1}^{L}U(2n_{j}-1)(2n_{j+1}-1)\,.$ (8)

where $n_{j}=c^{\dagger}_{j}c_{j}$. Unless otherwise stated, from now on we
set $U_{1}=U_{2}=U$.

##  III Results

In Fig. 2 (a) and (b), we report the phase diagrams of the two models
determined by the ratio between the edge-to-edge QCMI $I_{(4)}$ and the
reference Bell-state entanglement $E_{BS}=\ln 2$. The sizes of the two chains
are identified by the number of unit cells $L_{cell}$ for the SSH insulator
and by the number of fermionic sites $L$ for the Kitaev superconductor.
Throughout the entire topological phase $I_{(4)}/E_{BS}=1$ for the SSH chain
and $I_{(4)}/E_{BS}=1/2$ for the Kitaev chain, reflecting the different
statistics of the topological modes in the two models [39, 40, 41].

Figure 2: Phase diagrams as determined by the QCMI to Bell-state entanglement
ratio $I_{(4)}/E_{BS}$. Panel (a): SSH chain of size $L_{cell}=50$ unit cells.
Panel (b): Kitaev chain of size $L=60$ sites at $t=1$. Panels (c) and (d):
scaling in the topological and in the trivial regimes, with $t=\Delta=1$ in
panel (d). Insets: log-scale plots showing for both models the exponential
convergence to the quantized ground-state value of the edge-to-edge SE
$E_{sq}$.

The conjectured coincidence of the SE with the QCMI implies that
$E_{sq}=E_{BS}$ for the SSH insulator, identifying the presence of two
fermionic edge modes, while for the Kitaev superconductor $E_{sq}=E_{BS}/2$,
detecting the presence of two half-fermion Majorana edge modes. Finite-size
effects are clearly visible near the phase boundaries $|w|=|v|$ and
$|\mu|=2t$.

From Fig. 2 we see that in the topological phase the QCMI $I_{(4)}$ scales
exponentially to the quantized values $E_{BS}$ and $E_{BS}/2$, while it
remains pinned to zero in the trivial phase; since the SE is positive semi-
definite and bounded from above by the QCMI, it follows that throughout the
entire trivial phase it is certainly $I_{(4)}=E_{sq}=0$. On the other hand, at
the points of exact ground-state topological degeneracy, respectively $w=0$,
$v\neq 0$ and $|\mu|=0$, $t=\Delta$, both the topological fermionic modes and
the Majoranas decouple from the bulk and nucleate at the edge of the chains.
Correspondingly, the exact quantized value of the QCMI $I_{(4)}$ becomes
independent of the chain size.

It is important to observe that the same results are obtained resorting to the
QCMI $I_{(3)}$. Indeed, due to the exponential decoupling of the edges from
the insulating bulk, the definition of the edge-to-edge quantum conditional
mutual information in a topological system is insensitive to the different
partitions of the bulk, and therefore $I_{(3)}=I_{(4)}$ throughout the entire
phase diagram. In fact, at the exact points of topological degeneracy, the two
equal upper bounds $I_{(3)}=I_{(4)}$ on the edge-to-edge SE coincide with the
quantized values $E_{BS}$ and $E_{BS}/2$. Away from the points of exact
ground-state degeneracy but still inside the topologically ordered phases,
numerical analysis confirms that, for small-sized systems, $I_{(3)}$ and
$I_{(4)}$ still coincide and remain constant and equal, respectively, to the
Bell-state entanglement $E_{BS}$ for the SSH insulator and to half the Bell-
state entanglement $E_{BS}/2$ for the Kitaev superconductor. These findings,
together with the property of asymptotic continuity enjoyed by SE strongly
suggest that $I_{(3)}$ and $I_{(4)}$ coincide with the true SE
$E_{sq}(\rho_{AB})$ between the edges not only in the trivial phase but also
throughout the entire topological phase. Therefore, the edge-to-edge SE and,
equivalently, the two edge-to-edge QCMIs $I_{(3)}$ and $I_{(4)}$ all appear to
identify the same correct non-local order parameter for one-dimensional
topological insulators and superconductors.

In conclusion, all of the above leads us to conjecture that the edge-to-edge
SE coincides exactly with the edge-to-edge QCMIs throughout the entire ground-
state quantum phase diagrams of topological insulators and superconductors. We
discuss in more detail the conjecture in Sec. IV below, where we provide
analytical and numerical arguments that: i) the QMCIs coincide with the true
edge-to-edge SE in small systems; ii) the QMCIs scale exponentially to the SE
shared by the non-trivial edge modes in the topologically ordered phases; iii)
such SE between the system edges depends on the statistics of the edge modes;
iv) the QCMIs are robust against disorder and local perturbations. Details on
finite-size scaling at the phase transition boundaries are provided in
Appendix A, while point iv) is discussed at length in Appendix B.

Figure 3: Phase diagrams as determined by the entanglement negativity to Bell-
state entanglement ratio $\mathcal{N}/E_{BS}$. Panel (a): SSH chain of size
$L_{cell}=50$ unit cells. Panel (b): Kitaev chain of size $L=60$ sites at
$t=1$. Panels (c) and (d): scaling in the topological and in the trivial
regimes, with $t=1$ and $\Delta=1.8$ in panel (d).

As already mentioned, the entanglement negativity has become an increasingly
popular tool in the investigation of topological quantum matter [30, 36, 37,
42, 43]. Here we show that in fact at least some forms of $\mathcal{N}$ do not
provide the correct characterization of topological superconductors. In Fig. 3
we report the phase diagrams of the SSH and Kitaev chains as determined by the
ratio $\mathcal{N}/E_{BS}$. For the SSH insulator we have that
$\mathcal{N}=E_{BS}/\log 4$ in the nontrivial phase and $\mathcal{N}=0$
otherwise. On the other hand, $\mathcal{N}$ fails to reproduce the correct
phase diagram of the Kitaev superconductor: from Fig. 3 we see that
$\mathcal{N}$ vanishes asymptotically with increasing size of the system. This
feature is a manifestation of the possible unfaithfulness of the negativity.
On the contrary, the SE $E_{sq}$ is faithful, and thus $E_{sq}=0$ is a
necessary and sufficient condition for separability. The above analysis
confirms that, at variance with bipartition-based estimators of bipartite
entanglement, the paradigm of multipartition-based bipartite SE appears to
represent the correct framework for the investigation of one-dimensional
quantum matter. It remains to be seen whether a more careful and sophisticated
approach to the definition of the entanglement negativity in fermionic
systems, such as that pioneered by Shinsei Ryu and coworkers [44, 45], can
lead to a characterization of topological systems that not only detects the
transition, but also features the correct invariant topological quantization
in the ordered phase, discriminates among different topological systems and
edge modes, and discriminates between symmetric topological order and ordered
phases associated to spontaneous symmetry breaking.

When interactions are included, it is convenient to map fermionic systems into
interacting spin chains via the Jordan-Wigner mapping [25] (see Appendix C for
details). Although this procedure extends the Hilbert space dimension from
$2L$ to $2^{L}$, resorting to the QCMI $I_{(3)}$ allows to investigate the
effects of interactions without numerical approximations. Therefore, for small
system size, one can perform a direct comparison between well-known results
coming from exact diagonalization and from numerical techniques [46, 47]. In
Fig. 4 we report the contour plots of the QCMI to Bell-state entanglement
ratio $I_{(3)}/E_{BS}$ for the SSH and Kitaev model Hamiltonians with the
addition of the interaction terms $H_{I,SSH}$ and $H_{I,KC}$. From Fig. 4 (a)
we see that in the presence of repulsive interactions the mean field
topologically ordered phase $|w|\leq|v|$ is progressively reduced, until a
trivial phase is reached independently of $w$ for $U>1$. On the other hand,
when we consider attractive interactions, a new region with $I_{(3)}/E_{BS}=1$
progressively reopens, being independent of $w$ for strong enough attraction
($U<-1$). The detailed analysis is illustrated in Appendix C.

Figure 4: Phase diagrams of the interacting SSH and Kitaev chains determined
by the QCMI to Bell-state entanglement ratio $I_{(3)}/E_{BS}$ as a function of
the interaction strength $U$. In panel (a) the SSH chain size is $L_{cell}=14$
and $v=1$. In panel (b) the length of the Kitaev chain is $L=18$, while
$t=\Delta=1$. The drawn-in phase diagram of the interacting Kitaev chain is
taken from Ref. [48].

In Fig. 4 (b) we compare the phase diagram of the interacting Kitaev chain
determined by the ratio $I_{(3)}/E_{BS}$ with the standard reference one
obtained by a variety of analytic and numerical methods in the recent
literature [48]. We find an excellent qualitative and quantitative agreement
between both diagrams when the system behaves as a band insulator and a
topological superconductor. Indeed, both diagrams match the same band
insulator-topological superconductor phase boundary. $I_{(3)}$ also signals
the presence of an incommensurate charge density wave phase (ICDW), in fair
agreement with the most recent DRMG results [49, 50].

The phase diagram of the interacting Kitaev chain is currently understood by
means of combination of several numerical techniques, like DMRG and
bosonization [47, 46, 51, 52, 53], that involve various adjustments on the
ground state parity, oscillatory behaviors of the ground-state wave functions
and interpolation schemes on few numerical points [54, 49, 50], so that
discrepancies and uncertainties emerge, even concerning a possible additional
phase with odd ground-state parity between the ICDW and the Mott insulator
phases [50], so that a definite phase boundary for the ICDW phase has yet to
be identified.

In the $I_{(3)}$-based phase diagram, the QCMI $I_{(3)}=0$, and therefore
$E_{sq}=0$ both in the trivial insulator and in the ICDW phases, signaling the
absence of long-distance entanglement between the edges. On the other hand,
$I_{(3)}/E_{BS}\sim 1/2$ in the Mott insulator phase, implying a nontrivial
long-distance edge-to-edge squashed entanglement $E_{sq}$, provided the
conjectured coincidence between QCMI and SE holds. This finding yields strong
support to recent studies suggesting the presence of topological order in the
Mott regime of one-band fermionic systems [55]. In this picture, the metal-
insulator transition is equivalent to a topological transition via a mid-gap
pole in the self-energy that matches the spectral pole of the localized
surface state in a topological insulator. The QCMI and SE paradigm
encapsulates such property of the Mott insulating phase by yielding a
nontrivial quantized value of the QCMI to Bell-state entanglement ratio
$I_{(3)}/E_{BS}$. In this respect, the difference between the numerical
evaluation and the expected value $1/2$ for the ratio $I_{3}/E_{BS}$ in the
Mott phase could originate from residual finite-size effects, obeying a slower
finite-size scaling compared to the other phases.

##  IV Equivalence between QCMI and SE: some approximate analytical results

The SE ($E_{sq}(\rho_{XY})$) introduced in Eq. 3 measures the bipartite
entanglement between subsystems $X$ and $Y$ in the joint quantum state
$\rho_{XY}$. It is defined in terms of nonoverlapping distinguishable
multipartitions, for any spatial dimension, at any temperature, and on all
quantum states (pure or mixed). The SE is defined as the infimum of the QCMI,
taken over all the quantum-state extensions of arbitrary size $\rho_{XYZ}$
such that $\rho_{XY}=Tr_{Z}(\rho_{XYZ})$.

Figure 5: (a)-(b): The edge-to-edge QCMI evaluated as a function of the number
$\\#$ of possible bulk quantum-state extensions $C$ inserted between edges $A$
and $B$ ($1\leq\\#\leq N$). We have considered an SSH chain of $L_{cell}=15$
unit cells, panel (a), and a Kitaev chain of $L=15$ sites, panel (b). For the
two systems, we fix the edge lengths respectively at $L_{A}=L_{B}\sim
L_{cell}/3$ and at $L_{A}=L_{B}\sim L/3$. For the Kitaev chain we also set
$t=\Delta$. The finite-size scalings of the approximate $E_{sq}$ and the exact
$I_{(4)}$ are plotted in panels (c) and (d), respectively for the case of the
SSH and of the Kitaev chain. In all cases, we find $E_{sq}=I_{(4)}$.

Limited to a one-dimensional lattice of finite size $L$, the number $N$ of all
the possible quantum-state extensions $Z$ is expressed by
$N=\sum_{k=1}^{L_{Z}}\binom{L_{Z}}{k}$, $L_{Z}$ being the maximum size of $Z$
allowed, once the edge partitions have been fixed ($L_{X}$, $L_{Y}$),
$L_{Z}=L-L_{X}-L_{Y}$. Since $N$ increases exponentially with $L_{Z}$
($N=2^{L_{Z}}$), the computational resources required for the minimization of
the QCMI explode exponentially as well, and the problem is indeed $NP$-hard.
On the other hand, considering lattices of moderate size, one can determine an
analytical approximant to the true SE restricted to one-dimensional quantum-
state extensions.

In Fig. 5(a)-(b) we report the QCMI as a function of every one-dimensional
quantum-state extension for four selected points of the phase diagram of the
SSH chain, panel (a), and of the Kitaev chain, panel (b). One can see from
Figs. 5(a) and 5(b) that for topological systems the actual value of the QCMI
does not depend on the choice of the bulk partition. Indeed, topological
states of matter represent special cases where the bulk is uncorrelated from
the edges and the full, end-to-end QCMI is carried entirely by the edges
alone. In this case, tracing out a connected or disconnected part of the bulk
has no effects on the edge-to-edge correlations and all the possible different
forms of the QCMI are expected to coincide. This is actually what occurs both
in topological insulators and in topological superconductors.

In Fig. 5(c)-(d) we compare the finite-size scaling of the approximate SE
$E_{sq}$ obtained via a succession of finite one-dimensional state extensions
and the QCMI $I_{(4)}$ on a set of selected points of the topological phase
diagram, respectively for the SSH model, see Fig. 5(c), and for the Kitaev
chain, see Fig. 5(d). We observe that the QCMI $I_{(4)}$ coincides exactly
with the approximate analytical expression obtained for the SE $E_{sq}$, but
for a negligible numerical discrepancy originated by finite-size effects. For
a topological insulator of moderate system sizes the approximate analytical
expression of the SE $E_{sq}$ between the system edges converges to the Bell
state entanglement: $E_{sq}(\rho_{AB})=\ln 2$, while for a topological
superconductor it converges to half the Bell state entanglement:
$E_{sq}(\rho_{AB})=\ln 2/2$.

Hereafter, we provide an approximate analytical argument that $E_{BS}/2$
represents the bulk limit of the edge-to-edge SE in the Kitaev superconducting
chain. First of all, the constant quantized value $E_{BS}/2$ taken by the
QCMIs can be analytically determined at the points of exact ground-state
topological degeneracy, i.e. at $\mu=0$, $t=\Delta$. In this limit, the
topological edge modes decouple from the bulk and all QCMIs become independent
of the chain size. Due to the above properties, $I_{(3)}$ and $I_{(4)}$ are
expected to coincide with the true SE without size constraints, see Fig. 5.
For this reason, without loss of generality, we consider the collection of the
many-body ground-state density matrices $\rho_{ABC}$ of the tripartite system
$ABC$ as the set of possible state-extensions for the approximate computation
of the edge-to-edge SE.

A Kitaev chain Hamiltonian with $\mu=0$ and $t=\Delta$ can be expressed via
the Jordan-Wigner mapping as
$H_{spin}^{KC}=-t\sum_{j=1}^{L-1}\sigma^{x}_{j}\sigma^{x}_{j+1}$. The density
matrix of the parity-preserving many-body ground state reads:

$\displaystyle\begin{split}\rho_{ABC}=&\frac{1}{2}\
\biggl{[}\ket{\uparrow\dots\uparrow}\\!\\!\bra{\uparrow\dots\uparrow}_{L}+\ket{\uparrow\dots\uparrow}\\!\\!\bra{\downarrow\dots\downarrow}_{L}+\\\
&\ket{\downarrow\dots\downarrow}\\!\\!\bra{\uparrow\dots\uparrow}_{L}+\ket{\downarrow\dots\downarrow}\\!\\!\bra{\downarrow\dots\downarrow}_{L}\biggr{]}\,,\end{split}$
(9)

where $L$ is the length of the chain and the notation
$\ket{\alpha\dots\alpha}\\!\\!\bra{\beta\dots\beta}_{L}$ has been introduced
to denote a matrix with $L$ spins where
$\ket{\alpha}/\ket{\beta}=\ket{\uparrow},\ket{\downarrow}$ and with
$\ket{\uparrow}=1/\sqrt{2}\ (1,1)^{T}$ and $\ket{\downarrow}=1/\sqrt{2}\
(1,-1)^{T}$. It is straightforward to show that, independently on the
partition lengths, tracing out a part of the system leads to the following
reduced density matrices:

$\displaystyle\rho_{AC}\\!=\\!\frac{1}{2}\biggl{[}\ket{\uparrow\dots\uparrow}\\!\\!\bra{\uparrow\dots\uparrow}_{L_{A}+L_{C}}\\!\\!+\\!\ket{\downarrow\dots\downarrow}\\!\\!\bra{\downarrow\dots\downarrow}_{L_{A}+L_{C}}\\!\\!\biggr{]},$
$\displaystyle\rho_{BC}\\!=\\!\frac{1}{2}\biggl{[}\ket{\uparrow\dots\uparrow}\\!\\!\bra{\uparrow\dots\uparrow}_{L_{B}+L_{C}}\\!\\!+\\!\ket{\downarrow\dots\downarrow}\\!\\!\bra{\downarrow\dots\downarrow}_{L_{B}+L_{C}}\\!\\!\biggr{]},$
$\displaystyle\rho_{C}\\!=\\!\frac{1}{2}\biggl{[}\ket{\uparrow\dots\uparrow}\\!\\!\bra{\uparrow\dots\uparrow}_{L_{C}}\\!\\!+\\!\ket{\downarrow\dots\downarrow}\\!\\!\bra{\downarrow\dots\downarrow}_{L_{C}}\biggr{]}\,,$
(10)

where $L_{A}$, $L_{B}$ and $L_{C}$ refer respectively to the lengths of the
edges $A$, $B$ and of the bulk $C$. Via the expressions in Eq. 10, the reduced
von Neumann entropies are easily computed. It turns out that
$S_{AC}=S_{BC}=S_{C}=\log 2$, while of course for the pure ground state
projector $S_{ABC}=0$. Therefore, the end-to-end (edge-edge) SE of the reduced
two-edge state $\rho_{AB}$ reads $E_{sq}(\rho_{AB})=\log 2/2$. A similar
derivation holds for the SSH chain with $w=0$, $v\neq 0$, leading to
$E_{sq}(\rho_{AB})=\log 2$.

In conclusion, we have shown how to compute an approximate form of the SE
between the edges restricted only to one-dimensional state extensions and for
systems of finite size, and we have found that it coincides with the two
distinct forms $I_{(3)}$ and $I_{(4)}$ of the QCMI between the edges. It turns
out that he approximate SE saturates to the bulk value expected for the true
edge-to-edge SE already for systems of limited size. We have shown that the
aforementioned bulk value can be explicitly computed according to the
analytical proof provided above. In view of these observations, we conclude
that the conjecture $E_{sq}=I_{(3)}=I_{(4)}$ is strongly corroborated and
should hold without restrictions on the system size because the bulk part of
the system does not contribute to the quantum correlations between the edges
within topologically ordered phases.

##  V Discussion

In the present work we have shown that the squashed entanglement between the
edges of a quantum many-body system, together with its natural upper bounds
defined by two inequivalent forms of the edge-to-edge quantum mutual
information conditioned by the system bulk, realize the natural framework of
non-local order parameters characterizing topological quantum matter in one
dimension. Such quantities discriminate between topological insulators and
superconductors by identifying the different physical nature and statistics of
the edge modes in the two cases. Use of the edge-to-edge quantum conditional
mutual information yields the exact quantum phase diagram for the non-
interacting models; in the presence of interactions, it reproduces both
qualitatively and quantitatively the consensus results obtained so far by
various analytical and numerical approaches. The latter highly nontrivial
result makes the quantum conditional mutual information between the system
edges the natural benchmark to test the accuracy of numerical approximations.

The fundamental property of the quantum conditional mutual information and
squashed entanglement that singles them out when compared to standard
quantifiers of pure-state bipartite entanglement such as the block
entanglement entropy and the entanglement spectrum is that the former,
although being bipartite measures, are defined, at variance with the latter,
in terms of multipartitions. This property in turn warrants that quantum
conditional mutual information and squashed entanglement are sensitive to and
can detect the different physical nature of the bulks (e.g. conducting or
insulating) and of the edges for different systems and can thus distinguish
and discriminate topologically ordered phases from ordinary Ginzburg-Landau
symmetry-breaking orders as well as identify and discern different
topologically ordered regimes. In essence, this fine-grain ability keeps track
of the different parts of a system, either edge-bulk-edge or edge-partial
bulk-partial bulk-edge, and it is responsible for the tremendous effectiveness
of edge-state quantum conditional mutual information and edge-state squashed
entanglement in detecting, identifying and discriminating different types of
quantum phase transitions and collective quantum orders.

A crucial step in order to extend our results to include all topological
quantum matter is to generalize the concept of edge-state quantum conditional
mutual information and edge-state squashed entanglement to generic many-body
systems in dimension $D\geq 2$. For such higher-dimensional cases, the main
challenge lies in the correct identification of the appropriate bulk and edge
parts for a given multipartition. Among two-dimensional systems, second order
topological materials ($HOTM_{2}$) [56, 57] play a fundamental role, both as
novel platforms for topological insulators or to realize topological
superconducting braiding dynamics [58]. $HOTM_{2}$ are systems with gapped
one-dimensional boundaries and zero-dimensional localized modes (corner
states). For such systems, the identifications of edges $A$, $B$ and bulks $C$
$C_{1}$, $C_{2}$ is clear and easily connected to that of the one-dimensional
case. This class of systems is thus the first prominent candidate in order to
implement and test the scheme of topological squashed entanglement to many-
body systems in $D=2$.

Concerning the experimental accessibility of topological squashed
entanglement, the problem boils down to that of measuring quantum entropies of
a set of reduced states. A recent proposal relies on the thermodynamic study
of the entanglement Hamiltonian for the direct experimental probing of von
Neumann entropies via quantum quenches [59]. A possibility specifically
taylored for systems featuring topological order consists in identifying
minimum entropy states and then experimentally simulating the behaviour of the
associated von Neumann entropies via the classical microwave analogues of such
states [60]. A further intriguing possibility arises from the observation that
highly informative bounds on von Neumann entropies, quantum conditional mutual
information, and squashed entanglement can be constructed in terms of Rényi
entropies [61, 62]. The strategy is then to adapt to fermionic systems [63]
the schemes previously proposed for the experimental probe of Rényi entropies
in bosonic and spin systems [64, 65, 66] and the corresponding techniques that
led to the first measurement of the quadratic Rényi entropy in a many-body
system [67].

##  Appendix A Finite-size scaling

As already showed SE scales exponentially to the entanglement shared by the
non-trivial edge modes in the topological ordered phases, while it is pinned
to zero for points out of the such regime. At phase boundaries the edge modes
are less localized at the ends of the chain and possible nontrivial effects on
SE should appear more transparent.

Figure 6: Finite-size scaling at phase transition boundaries of the QCMI to
Bell-state entanglement ratio $I_{(4)}/E_{BS}$ for the SSH chain (a) and the
KC chain (b).

In panels (a) and (b) of Fig. 6 we show that, for fixed size, the QCMI to
Bell-state entanglement ratio $I_{(4)}/E_{BS}$ is lowered when topological
phase transition points are approached: $w=v$, $\mu=2t$. However an increasing
scaling with size towards $E_{BS}$ and $E_{BS}/2$ respectively for the SSH
chain, panels (a), and the Kitaev model, panel (b), is observed, signaling the
robustness of the entanglement non-local order parameter even close to the
phase transition. Significant deviations from the quantized entanglement
values are only obtained when $w$ and $t$ differ form aforementioned phase
transition points by the order of $10^{-2}$.

##  Appendix B Effects of disorder

Robustness against disorder is a specific property of topological materials
[68, 69, 70, 71]. In fact, disorder may even induce localization effects as in
the case of Anderson insulators [72], thus favouring topological phases of
matter, whereas in other systems such as, e.g., semiconductor-based Majorana
nanowires and topological insulator nanoribbons, it can yield detrimental
effects [69, 73]. The topologically ordered phases of the SSH and Kitaev
chains are robust to the effects of disorder and local perturbations [68, 74,
75, 71, 23].

Figure 7: The QCMI to Bell-state entanglement ratio $I_{(4)}/E_{BS}$ in the
presence of disordered hopping integrals in the SSH and Kitaev chains. Panel
(a): behaviour of $I_{(4)}/E_{BS}$ for the SSH chain with random hopping
amplitudes $w_{i}=w+W_{1}\delta_{i}$ and $v_{i}=v+W_{2}\delta_{i}$, with
$\delta_{i}$ randomly generated in the interval $(-0.5,0.5)$ and with
$2W_{1}=W_{2}=W$, $v=1$, $L_{cell}=50$. Panel (b): behaviour of
$I_{(4)}/E_{BS}$ for The Kitaev chain with random hopping amplitudes $t_{i}$.
The random realizations are generated by a uniform probability distribution
defined in the interval $t_{i}\in(t-\tau,t+\tau)$. The reference values of the
parameters have been fixed as $t=1$, $\Delta=0.1$, $L=50$.

For both models, we study the response of the QCMI to the disorder induced on
the hopping integrals. This choice provides a model of the effective mass
gradient and random doping along the system that originate from the growth
process of the one-dimensional nanowires. Specifically, we consider uniform,
chirality-preserving disorder on the hoppings of the SSH chain:

$\displaystyle w_{i}=w+W_{1}\delta_{i}\,,$ (11) $\displaystyle
v_{i}=v+W_{2}\delta_{i}\,,$ (12)

with $2W_{1}=W_{2}=W$, and $\delta_{i}$ randomly generated in the interval
$(-0.5,0.5)$. For the Kitaev chain we consider the following random hopping
integrals:

$\displaystyle t_{i}=t+\tau_{i}^{dis}\,,$ (13)

with $\tau_{i}^{dis}$ uniformly distributed in the interval $(-\tau,\tau)$.

In Fig. 7, we let $W$ and $\tau$ range from perturbative values $W=v/10000$
and $\tau=\Delta/4$ up to a regime for which the strength of the disorder
becomes comparable with the band gaps of the two models, respectively $W=v/2$
and $\tau=5\Delta$.

For the SSH chain, we see that the QCMI to Bell-state entanglement ratio
$I_{(4)}/E_{BS}$ remains resilient over all the examined regimes, only being
affected by some fluctuations in the regime of very high disorder, especially
near the phase boundary. A similar phenomenology holds for the superconducting
chain from $\tau=\Delta/4$ up to $\tau=\Delta$. For higher values of $\tau$,
e.g. when $\tau=5\Delta$ the phase boundary defining the topological
transition tends to be suppressed. This effect suggests that random hopping is
more effective in perturbing the topologically ordered phase in the Kitaev
chain then in the SSH chain. At any rate, the above analysis yields that
realistic values of the disorder strength do not cause any significant
disruption of the topologically ordered phases as measured by the SE.

##  Appendix C Interacting systems

Figure 8: Phase diagrams of the interacting SSH chain in the $w-v$ plane at
different fixed values of the interaction strength $U$. The light blue lines
correspond to the phase boundaries of the non-interacting case ($w=v$). The
contour plots reproduce the behavior of the function
$Sign[(I_{(3)}-I_{(3)}^{w=v})/E_{BS}$], where $I_{(3)}^{w=v}/E_{BS}=0.15$ is
the value of $I_{(3)}$ when $U=0$ and $w=v$. The length of the chain is fixed
at $L_{cell}=14$.

###  C.1 Mapping to interacting spin Hamiltonians

The Jordan-Wigner transformation [25] is an highly non-local mapping between
fermionic operators and spin $1/2$ operators. On each site, an empty state is
mapped into a spin up and an occupied one to a spin down. The non-local part
of this mapping is called the Jordan-Wigner string and fixes the
(anti)commutation relation between sites, by counting the parity of overturned
sites to the left of the spin on which it is applied.

This transformation explicitly breaks the translational invariance of the
model, by singling out a particular site as a starting point for the string.
Denoting by $c_{j,\alpha}$ and $c_{j,\alpha}^{\dagger}$ the generic
annihilation and creation fermionic operators, the Jordan-Wigner mapping is
defined by:

$\displaystyle c_{j,\alpha}$ $\displaystyle=$ $\displaystyle
e^{-i\pi\sum_{\alpha^{\prime}}\sum_{l=1}^{j-1}c^{\dagger}_{l,\alpha^{\prime}}c_{l,\alpha^{\prime}}}\sigma^{+}_{j,\alpha}\,,$
(14) $\displaystyle c^{\dagger}_{j,\alpha}$ $\displaystyle=$
$\displaystyle\sigma^{-}_{j,\alpha}e^{i\pi\sum_{\alpha^{\prime}}\sum_{l=1}^{j-1}c^{\dagger}_{l,\alpha^{\prime}}c_{l,\alpha^{\prime}}}\,,$
(15) $\displaystyle n_{j,\alpha}$ $\displaystyle=$
$\displaystyle\frac{1-\sigma^{z}_{j,\alpha}}{2}\,,$ (16)

where $j$ singles out the explicit lattice site while $\alpha$ denotes the
remaining degrees of freedom of the system. The aforementioned parity string
of the overturned sites is
$e^{-i\pi\sum_{\alpha^{\prime}}\sum_{l=1}^{j-1}c^{\dagger}_{l,\alpha^{\prime}}c_{l,\alpha^{\prime}}}$.
The operators $\sigma_{j,\alpha}^{(+,-)}=(\sigma_{j,\alpha}^{x}\pm
i\sigma_{j,\alpha}^{y})/2$ are the well-known combination of Pauli matrices
and the last relation in Equation (16) allows to express the parity operator
of the fermionic site $j$ with degrees of freedom $\alpha$ as $e^{-i\pi
c^{\dagger}_{j,\alpha}c_{j,\alpha}}=\sigma_{j,\alpha}^{z}$.

Using the algebra of spin $1/2$ operators and the constraint that on different
sites Pauli matrices commute, it is straightforward to derive the following
spin-$1/2$ representations of the interacting SSH insulator and Kitaev
superconductor:

$\displaystyle H_{spin}^{SSH}=$
$\displaystyle\frac{1}{2}\biggr{[}U\bigl{(}L-\frac{1}{2}\bigr{)}+w\sum_{j=1}^{L}\bigl{(}\sigma^{x}_{aj}\sigma^{x}_{bj}+\sigma^{y}_{aj}\sigma^{y}_{bj}\bigr{)}+$
(17)
$\displaystyle+v\sum_{j=1}^{L-1}\bigl{(}\sigma^{x}_{bj}\sigma^{x}_{aj+1}+\sigma^{y}_{bj}\sigma^{y}_{aj+1}\bigr{)}+\frac{U}{2}\bigl{(}\sum_{1}^{L}\sigma^{z}_{aj}\sigma^{z}_{bj}+$
$\displaystyle+\sum_{1}^{L-1}\sigma^{z}_{bj}\sigma^{z}_{aj+1}\bigr{)}-U\bigl{(}\sum_{j=2}^{L}\sigma^{z}_{aj}+\sum_{j=1}^{L-1}\sigma^{z}_{bj}\bigr{)}+$
$\displaystyle-\frac{U}{2}\bigl{(}\sigma^{z}_{a1}+\sigma^{z}_{bL}\bigr{)}\biggl{]}\,,$

$\displaystyle\begin{split}H_{spin}^{KC}\\!=&\\!\\!\sum_{j=1}^{L-1}\\!\biggl{(}-(t\\!+\\!\Delta)\sigma^{x}_{j}\sigma^{x}_{j+1}\\!+\\!(t\\!-\\!\Delta)\sigma^{y}_{j}\sigma^{y}_{j+1}+U\sigma_{j}^{z}\sigma_{j+1}^{z}\\!\biggr{)}\\\
&+\frac{\mu}{2}\sum_{j=1}^{L}\sigma^{z}_{j}\,.\end{split}$ (18)

When the SSH Hamiltonian in Eq. 17 is considered, $\alpha=a$, $b$ represent
the degree of freedom of the unit cell, while the Kitaev chain Hamiltonian in
Eq. 18 describes exactly one fermionic degree of freedom per lattice site. We
see that the interacting SSH model transforms into a staggered $XYZ$ chain
with external magnetic field along the $z$-direction, while the Kitaev chain
is described by a $XYZ$ chain with external magnetic field along the
$z$-direction. Moving to the spin representation, the Hilbert space dimension
grows exponentially from $2L$ to $2^{L}$.

###  C.2 Phase diagram of the interacting SSH chain

By means of the Jordan-Wigner mapping, the edge-to-edge QCMI can be computed
exactly for interacting systems of relatively modest size. In the main text we
have discussed the phase diagrams of the interacting SSH and Kitaev models by
means of the $I_{(3)}$ in the $U-w$ and $U-\mu$ plane of the phase space. Here
we discuss the phase diagram of the interacting SSH chain in the $w-v$ plane.

In Fig. 8 (a)-(d) we report four different phase contour plots, each
corresponding to a fixed value of the Coulomb interaction strength $U$. As the
Coulomb repulsion $U$ is increased, the topological phase boundary gets
progressively reduced in comparison to that of a non interacting case $w=v$
(blue light line). On the other hand, for increasing Coulomb attraction the
topological phase reopens, becoming the dominant contribution of the entire
phase diagram.

## Acknowledgements

F.I. and A.M. acknowledge support by MUR (Ministero dell’Università e della
Ricerca) via the project PRIN 2017 ”Taming complexity via QUantum Strategies:
a Hybrid Integrated Photonic approach” (QUSHIP) Id. 2017SRNBRK.

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