# Topological squashed entanglement: nonlocal order parameter for one-
dimensional topological superconductors

Alfonso Maiellaro  Dipartimento di Ingegneria Industriale, Università di
Salerno, Via Giovanni Paolo II, 132, I-84084 Fisciano (SA), Italy  Antonio
Marino  Dipartimento di Ingegneria Industriale, Università di Salerno, Via
Giovanni Paolo II, 132, I-84084 Fisciano (SA), Italy  Fabrizio Illuminati
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

(July 4, 2022)

###### Abstract

Identifying entanglement-based order parameters characterizing topological
systems, in particular topological superconductors and topological insulators,
has remained a major challenge for the physics of quantum matter in the last
two decades. Here we show that the end-to-end, long-distance, bipartite
squashed entanglement between the edges of a many-body system, defined in
terms of the edge-to-edge quantum conditional mutual information, is the
natural nonlocal order parameter for topological superconductors in one
dimension as well as in quasi one-dimensional geometries. For the Kitaev chain
in the entire topological phase, the edge squashed entanglement is quantized
to $\log(2)/2$, half the maximal Bell-state entanglement, and vanishes in the
trivial phase. Such topological squashed entanglement exhibits the correct
scaling at the quantum phase transition, is stable in the presence of
interactions, and is robust against disorder and local perturbations. Edge
quantum conditional mutual information and edge squashed entanglement defined
with respect to different multipartitions discriminate topological
superconductors from symmetry breaking magnets, as shown by comparing the
fermionic Kitaev chain and the spin-1/2 Ising model in transverse field. For
systems featuring multiple topological phases with different numbers of edge
modes, like the quasi 1D Kitaev ladder, topological squashed entanglement
counts the number of Majorana excitations and distinguishes the different
topological phases of the system. In fact, we show that the edge quantum
conditional mutual information and the edge squashed entanglement remain valid
detectors of topological superconductivity even for systems, like the Kitaev
tie with long-range hopping, featuring geometrical frustration and a
suppressed bulk-edge correspondence.

##  I Introduction

Condensed matter physics is witnessing, among others, two groundbreaking and
concurring developments, respectively the application of concepts and methods
of quantum information science and the investigation of topological phases of
quantum matter.

Via the identification of entanglement boundary laws and their violations,
bipartite block entanglement (as measured by the reduced von Neumann entropy
in a bipartite system) has become a central tool for the characterization and
the diagnostics of large classes of phenomena in quantum many-body physics [1,
2, 3, 4, 5].

At the same time, by featuring perfectly conducting edge modes, patterns of
long-distance entanglement and robust ground-state degeneracy without symmetry
breaking, three traits that generalize the concept of ordered phase and phase
transition beyond the Ginzburg-Landau paradigm, topological states of matter
have attracted increasing attention both for their fundamental interest and
their potentiality for applications [1, 2, 3, 6, 7, 8, 9].

In particular, topological superconductors hosting edge Majorana zero energy
modes (MZEMs) [10] have been proposed as the working principle of various
disruptive quantum technologies, including fault-tolerant topological quantum
computation [11, 12, 13].

Topological order in two-dimensional systems is identified and detected by the
sub-leading contribution to the bipartite bulk-boundary von Neumann
entanglement entropy, the so-called topological entanglement entropy (TEE)
[14, 15, 16]. This is a great success of entanglement theory applied to the
investigation of quantum matter.

Yet, the approach based on block entropies and simple bipartitions of a system
into two connected parts (also often named ”halves” or ”blocks”) is limited
and cannot be applied to various important instances, thus showing the need
for more general and advanced methods of entanglement theory. Indeed, in one
dimension at zero temperature, no measure of bipartite entanglement based on
simple bipartitions, including the entanglement spectrum, can discriminate
topologically ordered phases from those corresponding to symmetry-breaking
order [17].

Moreover, irrespective of the spatial dimension, TEE cannot be generalized to
finite-temperature and nonequilibrium processes since the von Neumann entropy,
when defined on mixed states, includes contributions both from quantum and
classical correlations, thus ceasing to be a genuine, bona fide measure of
entanglement and nonclassicality.

A further limitation of the approach based on the elementary bipartition of
the system in two connected parts (or halves, or blocks) is that it is not
suitable for the characterization and quantification of the nonlocal
correlations that are in place between arbitrary subsystems, either connected
or disconnected; in particular, block entanglement entropy provides no
information on the physics of edge correlations and cannot characterize and
quantify the long-distance entanglement between edge modes. Therefore, going
beyond the approach based on simple bipartitions by introducing bipartite
entanglement measures defined on multipartitions would allow to identify
richer conceptual structures and investigate broader classes of complex
physical phenomena.

Thus motivated, in the present work we discuss a multipartition-based measure
of bipartite entanglement, the squashed entanglement (SE), previously
introduced in the general context of quantum information theory, and we apply
it to the study of one-dimensional and quasi one-dimensional topological
superconductors with model Hamiltonians supporting Majorana zero energy modes
(MZEMs) at the system edges. We show that the long-distance squashed
entanglement SE between the edges, that we name topological squashed
entanglement (TSE), is the nonvanishing, _nonlocal_ order parameter that 1)
characterizes unambiguously the topologically ordered phases of topological
superconductors, 2) discriminates topological order from Ginzburg-Landau order
associated to spontaneous symmetry breaking and nonvanishing local order
parameters, and 3) distinguishes between different types of topologically
ordered phases. In order to avoid confusion with the standard nomenclature
about (local) order parameters, from now onward we will denote by topological
order parameter or order parameter for topological superconductivity a
nonlocal quantity that fulfils all the above criteria, being also very robust
under even strong variations of the sample conditions (disorder or any local
perturbation) and scaling exponentially with the system size to an asymptotic
topological invariant. The latter is a numerical constant measurable with
arbitrary precision and defined in terms of bulk Hamiltonians. The nonlocal
and quantized nature of such a quantity is the main difference with order
parameters classifying other phases of matter and standard Ginzburg-Landau
order associated to spontaneous symmetry breaking.

As we will show, one can define two basic forms of upper bounds on bipartite
SE, respectively in terms of the tripartitions and quadripartitions that
correspond to two different forms of the bipartite quantum conditional mutual
information (QCMI). This general property of SE immediately allows to
introduce two forms of the corresponding TSE order parameter, the first one
based on edge-bulk-edge tripartitions, the second one based on edge-bulk-bulk-
edge quadripartitions. We will then show how the latter identifies
unambiguously the topological regime in the Kitaev fermionic chain,
discriminates it from standard Ginzburg-Landau order in the spin-$1/2$ Ising
chain, is stable in the presence of interactions, and is robust against the
effects of disorder. Further, we will discuss how the TSE discriminates
between different topological phases in fermionic systems defined on quasi
one-dimensional geometries of higher complexity such as two–leg Kitaev
ladders. Finally, we will consider models with long–range hopping and
consequently suppressed bulk–boundary correspondence, such as the Kitaev tie,
and we will show that even in this case the TSE is a good identifier of
topological features.

In the present work we focus on the thorough investigation of edge QCMI and
edge SE to three paradigmatic instances of $1D$ and quasi-$1D$ topological
systems. In the conclusions, concerning future perspectives, we will discuss
how to generalize the present framework to the study of the edge entanglement
structure of many-body systems in higher dimensions.

The paper is organized as follows. In Section II we introduce the QCMI and the
SE, review their main properties, and introduce the two fundamental forms of
upper bounds on the SE based on the QCMI. In Section III we review the basic
models of topological superconductivity in different lattice geometries and
define the different forms of the TSE order parameter corresponding to
tripartitions and quadripartitions. In Section IV we discuss in detail the TSE
of the $1D$ Kitaev model for topological superconductivity, compare it to the
end-to-end SE in the Ising chain, and discuss the effects of interactions and
disorder. In Section V we study the TSE of the two–leg Kitaev ladder and show
how it identifies and distinguishes between the different topological phases
of the model, while in Section VI we carry out the same investigation for the
case of the Kitaev tie with long–range hopping and suppressed bulk–boundary
correspondence. Finally, in Section VII we review our results and consider
possible generalizations and further applications of SE in the study of
quantum matter.

##  II Squashed entanglement

###  II.1 Definition and fundamental properties

We are looking for a measure of bipartite entanglement that has the following
properties:

  1. 1.

It is defined in any spatial dimension, at any temperature, and on all quantum
states (pure or mixed).

  2. 2.

It is bipartite but generically defined in terms of non-overlapping,
distinguishable multipartitions, so that it can be defined on pairs of any two
subsystems, either connected or disconnected, and must reduce to the bipartite
von Neumann entanglement entropy on pure states of simple bipartitions.

  3. 3.

It is a true, bona fide measure of bipartite entanglement, i.e. a convex
entanglement monotone that in addition is also asymptotically continuous,
monogamous, and additive on tensor products [18, 19].

In fact, such a measure exists and is the so-called squashed entanglement (SE)
[20, 21]. Given a quantum state $\rho_{AB}$ of a bipartite quantum system
$AB$, the SE $E_{sq}(\rho_{AB})$ between subsystems $A$ and $B$ in state
$\rho_{AB}$ is defined as:

$E_{sq}(\rho_{AB})=\inf_{\rho_{ABC}}\bigg{\\{}I(A:B|C)\bigg{\\}}\,,$ (1)

where the infimum is taken over all the quantum-state extensions of arbitrary
size $\rho_{ABC}$ such that $\rho_{AB}=Tr_{C}(\rho_{ABC})$, and $I(A:B|C)$ is
the quantum conditional mutual information (QCMI) between subsystems $A$ and
$B$ conditioned by extension $C$:

$I(A:B|C)=\frac{1}{2}\big{[}S(\rho_{AC})+S(\rho_{BC})-S(\rho_{C})-S(\rho_{ABC})\big{]},$
(2)

where $\rho_{AC}=Tr_{B}(\rho_{ABC})$, $\rho_{BC}=Tr_{A}(\rho_{ABC})$,
$\rho_{C}=Tr_{AB}(\rho_{ABC})$, and $S(\rho)$ denotes the von Neumann entropy
of a given quantum state $\rho$ [19]. Notice that here and in the following,
for a matter of convenience, we adopt a definition of the QCMI that differs
slightly, by a prefactor $1/2$, from the original one [22, 23]; consequently,
the same factor is conveniently absorbed in our definition of the SE.

SE owes its name by the construction Eqs. 1–2 that ”squashes” out the
classical correlations to leave only the quantum contributions to the mutual
information between parties $A$ and $B$ conditioned by party $C$ (that in
informatic terms can be seen as the ”conditioning environment”). SE is a lower
bound on the entanglement of formation and an upper bound on the distillable
secret key or distillable entanglement of a quantum state or a quantum channel
[21, 24], and reduces to the von Neumann entanglement entropy on pure states
of a bipartite system $AB$. Indeed, if $\rho_{AB}$ is pure, then
$\rho_{ABC}=\rho_{AB}\otimes\rho_{C}$, and thus
$E_{sq}(\rho_{AB})=[S(\rho_{A})+S(\rho_{B})]/2=S(\rho_{A})=S(\rho_{B})$.

SE is unique in that it is the only known entanglement quantifier that enjoys
all the desirable axiomatic properties required for a bona fide measure of
entanglement [19].

Firstly, SE is a full entanglement monotone, i.e. non increasing under local
operations and classical communication (LOCC), and convex [21]:
$E_{sq}(\lambda\rho+(1-\lambda)\sigma)\leq\lambda
E_{sq}(\rho)+(1-\lambda)E_{sq}(\sigma)$, with $\lambda\in[0,1]$.

Moreover, SE enjoys important additional properties that promote it to a full
entanglement measure. To begin with, SE is additive on tensor products [21]:
$E_{sq}(\rho\otimes\sigma)=E_{sq}(\rho)+E_{sq}(\sigma)$. Additivity is a very
important requirement: the total entanglement of a global state that is the
tensor product of states of independent (uncorrelated) systems must be the sum
of each individual entanglement. This property must hold for entanglement and
entropy just like any other extensive physical quantity. Entanglement
monotones that fail to be additive, like the entanglement negativity, can be
at best only approximate quantifiers of quantum entanglement.

SE is also continuous [25]: if two sequences of states $\rho_{m}$ and
$\sigma_{m}$ converge in trace norm:
$\lim_{m\rightarrow\infty}\|\rho_{m}-\sigma_{m}\|_{1}=0$, then
$\lim_{m\rightarrow\infty}E_{sq}(\rho_{m})-E_{sq}(\sigma_{m})=0$. Continuity
is also a very important requirement: two states that are infinitely close in
Hilbert space according to a contractive norm must differ infinitesimally in
their physical properties, including entanglement. Many entanglement
quantifiers that are widely popular and used because of their straightforward
computability, like the entanglement negativity, do not enjoy continuity.

Furthermore, SE is faithful, that is, $E_{sq}(\rho_{AB})\geq 0$, and
$E_{sq}(\rho_{AB})=0$ if and only if $\rho_{AB}$ is separable [26]. Again,
this is a very important property because it guarantees that a vanishing SE
implies with certainty that a quantum state is separable. Unfaithful
entanglement quantifiers can be zero on entangled states, the paramount
example being, once more, the entanglement negativity.

Finally, SE satisfies entanglement monogamy [27]; given three parties $A$,
$B$, and $C$, the following monogamy inequality holds [28]:
$E_{sq}(\rho_{ABC})\geq E_{sq}(\rho_{AB})+E_{sq}(\rho_{AC})$, that is,
bipartite SE cannot be freely shared among multiple parties; in particular, if
$A$ is maximally entangled with $B$, then it cannot be entangled with $C$.
This property assures that one can introduce generalizations of SE in order to
measure multipartite entanglement beyond the bipartite one. Moreover, the fact
that SE is monogamous has important consequences on the structure of the
bipartite SE between bulk and edge modes that we will study later on in this
work (We note in passing that the above monogamy inequality can be extended
straightforwardly to an arbitrary number $N$ of parties [29]).

SE is an entanglement measure with very important operational meaning. In
quantum communication theory it is defined in terms of the communication cost
in the distribution of quantum states among multiple parties [30] and it is a
tight upper bound on the length of a secret key shared by two parties holding
many copies of a quantum state [31, 24]. Moreover, it allows for multipartite
extensions with operational meaning [32]. Finally, SE plays a fundamental role
in channel theory, as the SE of a quantum channel is the optimal upper bound
on the quantum communication capacity of any channel assisted by unlimited
classical communication [33]. In particular, SE provides the tightest known
bound for this type of capacity (also called two-way assisted quantum
capacity) for the case of the amplitude damping channel [34].

Because of all the above properties, SE has been dubbed the ”perfect” measure
of entanglement [19], the only serious drawback concerning its computability.
In fact, although computing SE has been shown to be an NP-complete problem
[35], nevertheless it has been calculated analytically for some nontrivial
classes of states [36, 37]. Moreover, and perhaps more importantly, SE enjoys
a set of very useful lower bounds in terms of the reduced von Neumann
entropies, the relative entropy of entanglement and the relative 2-Rényi
entropy [38, 31, 39, 40]. These lower bounds can be combined with the natural
upper bounds in terms of the tripartite and quadripartite quantum conditional
mutual information (QCMI) that we will introduce in the following, in order to
provide a tight quantitative characterization of SE.

###  II.2 From bipartitions and two blocks to multipartitions and any pair of
subsystems

Given a physical system $G$ in a pure state $\rho_{G}=|G\rangle\langle G|$,
for instance a ground state in condensed matter physics and quantum
statistical mechanics or a vacuum state in quantum field theory, one typically
focuses on simple bipartitions $G=AB$ of system $G$ in two connected
subsystems (or blocks, or parts, or halves) $A$ and $B$, and considers then
the pure–state bipartite entanglement between the two subsystems as quantified
by the so-called von Neumann block entanglement entropy
$S(\rho_{A})=S(\rho_{B})$ of the reduced local states
$\rho_{A}=Tr_{B}\rho_{G}$ or, equivalently, $\rho_{B}=Tr_{A}\rho_{G}$.

Squashed entanglement greatly extends this picture to include all possible
different forms of bipartite entanglement, including the case of mixed states
and disconnected subsystems. To set the stage, consider a quantum system $G$
in an arbitrary state, pure or mixed, $\rho_{G}$. Next, consider partitioning
the global system $G$ in any two subsystems $A$ and $B$ plus a reminder $C$:
$G=ABC$. In turn, when suitable, one may consider further partitioning the
remainder $C$ as well, as we will see in the following. When $C$ is the empty
set and $\rho_{G}$ is a pure state, one recovers the standard two–block
bipartition and SE reduces to the von Neumann block entanglement entropy.

We now wish to determine the SE $E_{sq}(\rho_{AB})$ existing between any pair
of subsystems $A$ and $B$, possibly disconnected, in the reduced state
$\rho_{AB}=Tr_{C}\rho_{G}$. The first important observation here in order is
that there are always two, and only two, equivalent ways to obtain the same
reduced state $\rho_{AB}$ from the global state $\rho_{G}$ that however give
rise to two different (non equivalent) expressions for the QCMI $I$ of Eq. 2
prior to the extremization procedure in Eq. 1 that defines the (unique) SE
$E_{sq}(\rho_{AB})$. These two different expressions of the QCMI define two
fundamental upper bounds on the true SE that may in principle be different.
The two expressions obviously give rise to the same unique SE once
extremization is performed in Eq. 1.

Indeed, besides the immediate _tripartite_ form $\rho_{G}=\rho_{ABC}$ such
that $\rho_{AB}=Tr_{C}(\rho_{ABC})$, one can consider a further bipartite
splitting of the reminder: $C=C_{1}C_{2}$ and obtain the corresponding
_quadripartite_ form $\rho_{G}=\rho_{ABC}=\rho_{ABC_{1}C_{2}}$ such that
$\rho_{AB}=Tr_{C_{1}}(\rho_{ABC_{1}})$ (or, equivalently,
$\rho_{AB}=Tr_{C_{2}}(\rho_{ABC_{2}})$), where in turn
$\rho_{ABC_{1}}=Tr_{C_{2}}(\rho_{ABC_{1}C_{2}})$ (or, equivalently,
$\rho_{ABC_{2}}=Tr_{C_{1}}(\rho_{ABC_{1}C_{2}})$). The two different
procedures to derive the reduced state $\rho_{AB}$ give rise in Eq. 2 to two
different QCMIs $I_{(3)}(A:B|C)$ and $I_{(4)}(A:B|C_{1})$ (or, equivalently,
$I_{(4)}(A:B|C_{2})$) defined with respect to the tripartition $ABC$ and to
the quadripartition $ABC_{1}C_{2}$, respectively. In explicit form, the two
expressions read:

$\displaystyle
I_{(3)}=\frac{1}{2}\biggl{[}S(\rho_{AC})+S(\rho_{BC})-S(\rho_{ABC})-S(\rho_{C})\biggr{]}\,,$
(3) $\displaystyle
I_{(4)}=\frac{1}{2}\biggl{[}S(\rho_{AC_{1}})+S(\rho_{BC_{1}})-S(\rho_{ABC_{1}})-S(\rho_{C_{1}})\biggr{]},$
(4)

where $S(\rho_{ABC})=0$ if $\rho_{ABC}=\rho_{G}$ is pure.

The two inequivalent QCMIs $I_{(3)}$ and $I_{(4)}$ define two different upper
bounds (corresponding, respectively, to a tripartite and a quadripartite
multipartition of the system) to the same bipartite SE $E_{sq}(\rho_{AB})$
between subsystems $A$ and $B$ in the reduced state $\rho_{AB}$. It is
immediate to verify that any further splitting of the reminder $C$ in
multipartitions of higher order $C=C_{1}C_{2}C_{3}\dots C_{n}$ is redundant in
the sense that it defines a set of $n$ QCMIs of the form
$I_{(4)}(A:B|C_{i})\,,(i=1,...,n)$. Taking into account the mathematical
properties of the von Neumann entropies, including subadditivity and the
triangle inequality, one has the chain of inequalities

$I_{(3)}(A:B|C)\geq I_{(4)}(A:B|C_{1})\geq E_{sq}(\rho_{AB})\,.$ (5)

The two fundamental constructions, respectively via the tripartition $ABC$ and
the quadripartition $ABC_{1}C_{2}$, are summarized in panels (a) and (b) in
Fig. 1.

The exact $A$–$B$ bipartite SE in state $\rho_{AB}$ is obtained by computing
the infimum of the two expressions Eqs. 3–4 over all extensions of unbounded
dimension $\rho_{ABC}$ and $\rho_{ABC_{1}}$, respectively. This is in general
an exceedingly hard task; on the other hand, the true SE is readily obtained
whenever there are lower bounds available that coincide with some upper
bounds, for instance like the ones provided by Eqs. 3–4.

Summarizing what has been discussed so far, we have shown how by resorting to
the QCMI and the SE one moves from the elementary paradigm of bipartite
pure–state block entanglement entropy defined over minimal, irreducible
connected bipartitions of the global system to a general framework of
bipartite entanglement between any two subsystems, either connected or
disconnected, defined over arbitrary multipartitions of the global system.
Moreover, we have introduced two classes of upper bounds to the exact
bipartite SE that are defined, respectively, over tripartitions and
quadripartitions of the global system.

The transition from bipartitions to multipartitions and from block
entanglement to SE between generic subsystems is represented pictorially in
panels (c) and (d) of Fig. 1, where we illustrate schematically how such
transition allows in principle to discriminate systems with broken symmetries
and short-ranged entanglement from systems with a different type of global
order and long-distance entanglement between disconnected subsystems and thus
also between boundaries (edges) in the case of systems with open boundary
conditions (open chains, open ladders, etc). In the next sections we
investigate in detail some significant consequences of this paradigm shift in
the study of topological quantum matter.

Figure 1: Panels (a) and (b): Construction pattern of the two upper bounds,
the QCMIs $I_{(3)}(A:B|C)$ and $I_{(4)}(A:B|C_{1})$, on the true SE between
two arbitrary subsystems $A$ and $B$ of a given quantum system partitioned,
respectively as $ABC$ and as $ABC_{1}C_{2}$. In the given example the two
subsystems $A$ and $B$ are disconnected, but the construction holds in general
for any pair of non-overlapping subsystems, either connected or disconnected.
Panel (c): A system in a globally pure state is partitioned in two connected
subsystems (blocks); after tracing over the degrees of freedom of one of the
blocks, the von Neumann entropy of the reduced state defines the block
entanglement entropy quantifying the bipartite entanglement across the blocks
separation boundary. Panel (d): From block entanglement to SE between
subsystems. We illustrate the case of the long-distance SE and QCMIs
$I_{(3)}(A:B|C)$ and $I_{(4)}(A:B|C_{1})$ between two specific disconnected
subsystems, the system edges $A$ and $B$ separated by the bulk $C=C_{1}C_{2}$.
This is the case of interest, e.g., in the study of topological
superconductivity in quantum matter. The double–headed arrows represent
pictorially the spatial ranges of the different quantum correlation patterns.

##  III Squashed entanglement and topological superconductors

###  III.1 The quest for order parameters

The archetypal model of topological $p$-wave superconductivity supporting edge
Majorana zero energy modes (MZEMs) is the Kitaev Hamiltonian of spinless
fermions on a 1D lattice [41]. Proposals for realistic implementations of the
Kitaev chain consider heterostructures made of semiconducting nanowires
coupled to $s$-wave superconducting substrates [42, 43, 44, 45].

Experimental evidence of MZEMs localized at the system edges has been obtained
in the study of the tunnel conductance of an InAs nanowire proximized by an
$s$-wave superconductor [46] and in the scanning tunneling microscopy of iron-
atom chains deposited on lead substrates [47].

The one-dimensional Kitaev model can be generalized to geometries of
significantly increasing complexity, e.g. via Kitaev ladders and ties, to
describe coupled superconducting nanowires with a phase diagram hosting a rich
variety of different topological phases [48, 49, 50, 51, 52, 53].

In view of its conceptual significance and potential realizability in
realistic systems of condensed matter physics, the Kitaev model has become a
central paradigm in the study of topologically ordered phases of matter
hosting MZEMs and, more generically, edge modes and edge states.

The problem of identifying unambiguous signatures of topologically ordered
phases, such as topological invariants and/or nonlocal order parameters, has
turned out to be a highly nontrivial task addressed in a number of different
approaches. One can look to momentum-space properties [54], when, after
imposing periodic boundary conditions, the translational invariance is
restored. This meaningful relation between the edges and the bulk of a system
is also known as bulk-edge correspondence [54] and leads to the definitions of
geometric indices $Q$ which signal the presence and the number of topological
zero-energy states of matter.

Unfortunately, for disordered systems or, in general, when translational
invariance cannot be restored, or also in the presence of interactions, the
topological invariants $Q$ are rarely readily accessible and hence useful both
analytically and experimentally [55, 56, 57, 58, 59].

In a more fundamental approach one tries then to identify topological
invariants based on the patterns of nonlocal quantum correlations and the
entanglement properties of the system, such as the bipartite entanglement
spectrum [17, 60, 61].

Indeed, the entanglement spectrum becomes two-fold degenerate when a quantum
system undergoes a topological phase transition [62, 63, 64]; however, this is
not a discriminating signature of topological order: exactly the same
degeneracy is featured by any system with the same Hamiltonian symmetries that
can support symmetry-breaking order [17, 65]. As a consequence, block
entanglement based on simple bipartitions of the system into two blocks does
not provide the information necessary to identify and discriminate
unambiguously topological order associated to edge modes and edge states.

Suitable _ad hoc_ combinations of Rényi entropies of disconnected and
partially overlapping multipartitions actually allow to define quantized
topological invariants able to detect topologically ordered phases in one-
dimensional topological superconductors [66]. Unfortunately, these invariants
are not entanglement monotones between distinguishable subsystems, let alone
entanglement measures, and thus are devoid of any clear meaning in terms of
nonlocal quantum correlations. Because of their _ad hoc_ nature and lack of
conceptual significance and physical motivation, one has no clue on how to
introduce such invariants systematically for the characterization of different
types of topologically ordered phases or to extend them to general situations
such as finite temperature and/or complex geometries and higher dimensions.

In the present work, we discuss SE in the context of topological quantum
matter featuring edge modes and edge states. We show that the edge-to-edge,
long-distance bipartite SE between the (disconnected) edges of one-dimensional
topological systems is a (nonlocal) order parameter, properly quantized,
featuring the correct scaling behavior at the critical point, characterizing
the ordered phases of topological superconductors in various one-dimensional
and quasi one-dimensional geometries, and discriminating them unambiguously
from other classes of ordered phases of matter.

Shifting focus from block entanglement entropy to SE between disconnected
subsystems separated by arbitrary distances finds a basic physical motivation
in the observation that systems with topological order display bulk band gaps
like those of ordinary insulators and conducting surface states that are
topologically protected by some symmetries. This naturally prompts to look for
nonlocal correlations between subsystems (e.g. the system edges) rather than
the block entanglement between two halves of the total system. In turn, this
implies replacing bipartitions with multipartitions, and resorting to SE as
the prospective quantity able to:

  * •

Quantify the prima facie bipartite, long-distance edge-edge entanglement in
the presence of edge modes.

  * •

Discriminate unambiguously topological order from symmetry-breaking order and
distinguish between different classes of topologically ordered phases.

In the following we will consider one-dimensional systems hosting MZEMs and
for such systems we will investigate the two bipartite edge-to-edge QCMIs,
$I_{(3)}(A:B|C)$ and $I_{(4)}(A:B|C_{1})$ that realize two upper bounds on the
true SE $E_{sq}(\rho_{AB})$ between edge $A$ and edge $B$. In particular, we
will show that the QCMI $I_{(4)}$ discriminates the one-dimensional Kitaev
model featuring edge modes and long-distance entanglement between the edges
from the Ising chain, its symmetry-breaking counterpart featuring no edge
modes and a short-distance entanglement structure.

For the Kitaev chain we will show that $I_{(3)}$ and $I_{(4)}$ coincide and
satisfy all the requirements for a genuine topological order parameter,
including: quantization as topological invariant at the exact topological
ground-state degeneracy point and throughout the topologically ordered phase;
scaling with the system size at the phase transition point; stability with
respect to interactions; and robustness against localized disorder and
imperfections.

Moreover, for the Kitaev chain the topological invariant in fact coincides
with a lower bound to the SE, and therefore the QCMIs $I_{(3)}$ and $I_{(4)}$
indeed coincide with the long-distance, bipartite SE $E_{sq}$ between the
chain edges at the exact topological degeneracy point and throughout the
entire topologically ordered phase. The transition from simple bipartitions
and block entanglement entropy to multipartitions and edge to edge long-
distance entanglement is illustrated pictorially in panels (c) and (d) of Fig.
1.

We will then generalize the method to consider models of topological
superconductors defined on geometries of higher complexity, the quasi one-
dimensional Kitaev ladder [52] and the Kitaev tie [53]. The Kitaev ladder
model is obtained by coupling two Kitaev chains by means of transverse hopping
and pairing terms. In fact, the Kitaev chain and the Kitaev ladder belong to
the two different topological classes $D$ and $BDI$ which are characterized by
two different topological bulk invariants, respectively the Pfaffian invariant
[67] and the winding number [52]. In the case of the Kitaev tie, the long-
range hopping term added to the Hamiltonian of the Kitaev chain define a
knotted–ring geometry which can be rearranged in the form of a tie. The model
is then the simplest realization of geometric frustration with no associated
bulk [68].

For all three models SE identifies and characterizes the different topological
behaviors of the systems, well characterized by the band topology for the
Kitaev chain and the Kitaev ladder and by the Majorana polarization and
topological transfer matrix for the Kitaev tie [68]. We will show that the
QCMI $I_{(4)}$ identifies the topological phase transitions in the Kitaev
ladder and distinguishes between the different topologically ordered phases of
the model corresponding to different numbers of Majorana excitations.

Remarkably, in the case of the Kitaev tie, a system which lacks a clearly
identifiable bulk, the SE is still very efficient in characterizing the
topological phases of the model even if, as expected, perfect quantization is
partially blurred due to the absence of a clear physical bulk-boundary
separation.

The theoretical framework based on multipartitions, bipartite QCMIs and
bipartite SE between generic subsystems (either connected or disconnected) can
be generalized to finite temperature and non-equilibrium, to multipartite
entanglement, and to many-body systems defined in higher-dimensional
geometries and with different classes of boundary conditions. In the
conclusions we will discuss some basic aspects of these future generalizations
and how, for each case study, they will depend crucially on the localization
and connection properties of edge modes and edge states.

###  III.2 Topological squashed entanglement

For the model systems defined in the following we consider tripartitions and
quadripartitions as sketched in Fig. 1 and we proceed to compute the QCMI
upper bounds $I_{(3)}$ and $I_{(4)}$ on the SE $E_{sq}(\rho_{AB})$ between the
edges $A$ and $B$ given a bulk $C=C_{1}C_{2}$.

In a condensed matter setting $I_{(3)}$ and $I_{(4)}$ are defined in terms of
the many body ground state, i.e. they are the QCMIs of the edges $A$ and $B$
conditioned, respectively, on the existence of the total bulk $C$ and the
partial bulk $C_{1}$. When going to sufficiently large system sizes such that
the edges are sufficiently distant from each other and fully decoupled from
the bulk, a nonvanishing $E_{sq}(\rho_{AB})$ establishes the existence of a
topological squashed entanglement (TSE), i.e. a nontrivial long-distance
quantum correlation between the edges.

Although the physical mechanisms are rather different, the edge-edge fermionic
TSE is reminiscent of other forms of long-distance entanglement (LDE) that are
established by entanglement monogamy between the end points of dimerized and
quasi-dimerized spin-$1/2$ chains with diverse patterns of nearest-neighbor
couplings or patterns of competing finite-range interactions [69, 70, 71].
Specifically, in these previous works we have shown that systems defined on
$1D$ chains with open boundary conditions will exhibit a nonvanishing end-to-
end, long-distance entanglement (LDE) whenever: $I)$ there is a weak coupling
between the edge regions and the bulk, or $II)$ there is a pattern of
alternating weak and strong nearest-neighbor couplings that leads to an
effective dimerization of the system. In turn, these two instances can be seen
as particular cases of more general patterns of modular entanglement [72] and
surface entanglement on networks [73].

Instance $I)$ yields an LDE between the chain ends that tends to decay very
slowly with the distance, while instance $II)$ realizes a perfect LDE between
the chain ends that does not decay with and is independent of the size of the
chain. The further form of LDE that we have now discovered is $III)$ the end-
to-end TSE in the topological phase of one-dimensional and quasi one-
dimensional fermionic systems. It is unclear at the moment whether forms $I)$
and $II)$ of LDE and form $III)$ are related and share some common
feature/origin. We plan to investigate the possible relations, also in
connection with the intriguing possibility that complex patterns of
interaction strengths might induce a kind of topological order also in some
classes of spin-$1/2$ systems.

In panels (a)–(c) of Fig. 2 we provide a sketch of the three model geometries
considered (Kitaev chain, Kitaev ladder, Kitaev tie), while in panels (d) and
(e) we draw a synthetic scheme of how the multipartitions reported in Fig. 1
are applied to these three explicit cases.

As detailed in the next sections, both the Kitaev ladder and the Kitaev tie
can be obtained, respectively, from two coupled Kitaev chains and by adding to
the Hamiltonian of a single chain a symmetric long–range hopping that couples
a single lattice site at position $d$ with its symmetric counterpart at
position $L-d+1$. In panels (d) and (e) of Fig. 2 we provide a schematics of
the two basic multipartitions for the model systems considered.

Figure 2: Schematic of a Kitaev chain (a), a two–leg Kitaev ladder (b) and a
Kitaev tie (c). The Kitaev ladder is obtained coupling two Kitaev chains by
superconducting and hopping terms. For a Kitaev chain of length $L$, the
Kitaev tie is obtained by adding a long–range hopping term coupling a site at
position $d$ with the symmetric one at the position $L-d+1$. The basic
tripartition and quadripartition are reported, respectively, in panels (d) and
(e). In both panels the edges are denoted by $A$ and $B$ (red color); in panel
(d) the total bulk $C=C_{1}C_{2}$ is reported in blue; in panel (e) the bulk
portions $C_{1}$ and $C_{2}$ are reported in blue and in green, respectively.
The diamonds are generic and can correspond to a single fermionic site when
referred to a chain or a tie and to two fermionic sites when referred to a
ladder. In the quadripartition, when not otherwise specified, the length
$L_{C_{1}}$ of the bulk portion $C_{1}$ is fixed at $L_{C_{1}}=1$.

In the following, the lengths of the various subsystems will be denoted by
$L_{\alpha}$, with $\alpha=A$, $B$, $C$, $C_{1}$, $C_{2}$. Throughout, the
lengths of the two edges are assumed to coincide, $L_{A}=L_{B}$. Denoting by
$L$ the chain length, when not otherwise specified we will set
$L_{A}=L_{B}=L_{e}$, with $L_{e}=L/3$. The latter choice allows to take into
account the exponential bulk–edge decoupling behavior as a function of the
system size that is typical of topological modes.

Moreover, as explained later on (see Fig. 5, panel (a)), our results will be
independent of the relative size of the two sub-bulks, so that without loss of
generality we will take $L_{C_{1}}=1$ throughout, as reported in panel (e) of
Fig. 2.

A crucial step in order to compute the TSE consists in diagonalizing the
reduced density matrices of the various subsystems, once the many body ground
state density matrix $\rho=|\Psi\rangle_{G}{}_{G}\langle\Psi|$ is assigned.

When interactions are neglected, the Kitaev-type Hamiltonians are quadratic in
the fermionic degrees of freedoms and can be diagonalized by a Bogoliubov
transformation. In this case, an appropriate approach to compute the von
Neumann entropies of the subsystems has been introduced by Peschel [74].

The Bogoliubov transformation ensures a direct access to fermionic
correlations on the many body ground state $|\Psi\rangle_{G}$ and the reduced
density matrix of a given subsystem $\alpha$ can be recast in the form of a
thermal density matrix, $\rho_{\alpha}=(1/Z)e^{-\mathcal{H}_{\alpha}}$, where
$\mathcal{H}_{\alpha}=\sum_{l=1}^{L_{\alpha}}\epsilon_{l}f^{\dagger}_{l}f_{l}$
is the effective entanglement Hamiltonian of the reduced system [75, 76, 77,
78], and the constant factor $Z$ (the ”partition function”) ensures the
correct normalization $Tr(\rho_{\alpha})=1$.

Given the lattice fermionic creation and annihilation operators
$\\{c_{i},c^{\dagger}_{i}\\}$, the spectrum of the entanglement Hamiltonian
can be evaluated numerically by solving the eigenvalue problem:

$\big{(}2T-1-2D\big{)}\big{(}2T-1+2D\big{)}\phi_{l}=\tanh^{2}\bigg{(}\frac{\epsilon_{l}}{2}\bigg{)}\phi_{l}\,,$
(6)

where $T_{ij}=\langle\Psi|c^{\dagger}_{i}c_{j}|\Psi\rangle$ and
$D_{ij}=\langle\Psi|c^{\dagger}_{i}c^{\dagger}_{j}|\Psi\rangle$ are the matrix
elements of the fermionic two-point correlations in a quantum state
$|\Psi\rangle$ (for instance, the ground state $|\Psi\rangle_{G}$), and $1$ is
the identity matrix. The index $l$ runs over all the lattice sites belonging
to subsystem $\alpha$.

Due to the form of $\rho_{\alpha}$, the von Neumann entropy of the reduced
state $S(\rho_{\alpha})$ can be obtained in terms of the eigenvalues of
entanglement Hamiltonian:

$S(\rho_{\alpha})=\sum_{l}\ln(1+e^{-\epsilon_{l}})+\sum_{l}\frac{\epsilon_{l}}{e^{\epsilon_{l}}+1}.$
(7)

The Peschel algorithm for free fermionic systems reduces the computational
efforts and allows to access the entanglement properties for systems of very
large sizes.

When interactions are included in the models, the mapping to the effective
entanglement Hamiltonian and the Gaussian thermal structure of the reduced
density matrix are no longer applicable and one must resort to numerical
techniques, e.g. the density matrix renormalization group (DMRG) method,
particularly well suited to treat one-dimensional systems [79, 80, 81].

Complementary to numerical methods, one can resort to direct numerical matrix
diagonalization in the parameter region that allows for exact analytic
expressions of the state vectors, such as along the exact factorization lines
of interacting spin models and Kitaev–type fermionic systems [82, 83, 84, 85],
as we will discuss in the following.

##  IV Topological squashed entanglement of the Kitaev chain

###  IV.1 The nonlocal order parameter

Here and in the following we investigate the paradigmatic case of the
fermionic Kitaev chain [41], considering first the non-interacting model:

$H_{K}=\sum_{j=1}^{L}-\mu c^{\dagger}_{j}c_{j}+\sum_{j=1}^{L-1}(\Delta
c^{\dagger}_{j+1}c^{\dagger}_{j}-tc^{\dagger}_{j}c_{j+1}+h.c.).$ (8)

The Hamiltonian $H_{K}$ in Eq. 8 describes a system of spinless fermions
confined to a one dimensional lattice of length $L$ with on site creation and
annihilation operators $c_{j}^{\dagger}$, $c_{j}$ ($j=1,\dots,L$), subject to
a $p$-wave superconducting coupling. The coefficients $\Delta$, $t$ and $\mu$
are respectively the strength of the superconducting pairing, the nearest-
neighbor hopping amplitude and the on-site energy offset.

This model features a topological phase for $\mu<2t$ and $\Delta\neq 0$ with
two robust Majorana zero-energy modes localized at the two edges. The energy
$E_{M}$ of such MZEMs scales exponentially to zero with the system’s size and
is expected to vanish asymptotically in the thermodynamic limit.

Figure 3: Panel (a): Phase diagram of the non-interacting Kitaev chain as
measured by the two-dimensional contour plot of the ratio between the QCMI
$I_{(4)}$ and the quantized TSE $E^{0}_{sq}=\log(2)/2$ on the $\Delta$–$\mu$
plane with a grid of $25\times 25$ points, reference hopping amplitude $t=1$,
chain length $L=60$, and edge length $L_{e}=L/3$. Panel (b): Scaling-size
effects on $I_{(4)}/E^{0}_{sq}$ as a function of $L$ for different values of
the chemical potential $\mu$. Panel (c): Scaling of $I_{(4)}/E^{0}_{sq}$ as a
function of $\mu$ for different values of $L$.

In the limit of small on site energy offset (chemical potential), precisely at
the analytically solvable point $\mu=0$ and $t=\Delta$, the model features
exact two-fold topological ground-state degeneracy and two MZEMs with exactly
$E_{M}=0$, independently of the chain length. The remaining non-topological
modes of the spectrum are gapped and form a band. In the opposite regime of
large on site energy offset $\mu>2t$ the Kitaev wire is a trivial band
insulator.

By construction the Hamiltonian satisfies the particle-hole symmetry and
belongs to the class $\mathcal{D}$ of the ten-fold classification [86], with
topological invariant given by the Pfaffian invariant [67].

For the non-interacting Kitaev chain, we have computed the two QCMIs $I_{(3)}$
and $I_{(4)}$ and verified that $I_{(3)}=I_{(4)}$ throughout the entire phase
diagram, so that in the reminder of the present subsection we will focus only
on $I_{(4)}$. The physical origin of the equality between the two QCMIs in the
topological case will become clear in the following.

In panel (a) of Fig. 3 we report the phase diagram obtained by means of the
ratio between the QCMI $I_{(4)}$ and the constant quantity
$E^{0}_{sq}=\log(2)/2$. One finds that throughout the entire topological phase
the ratio $I_{(4)}/E^{0}_{sq}=1$ so that $I_{(4)}$ is quantized exactly at the
value $I_{(4)}=E^{0}_{sq}=\log(2)/2$. Moreover, one has that $I_{(4)}=0$
throughout the entire trivial phase.

Finite-size effects are clearly visible near the phase transition line at
$\mu=2t$, as also shown in panel (b) of Fig. 3, where the ratio
$I_{(4)}/E^{0}_{sq}$ is plotted as a function of the length $L$ of the chain
for $t=\Delta=1$ and three different values of the chemical potential $\mu$,
respectively close to the exact topological point, in the topological phase
close to the critical point, and well into the trivial phase.

In particular, $I_{(4)}$ scales exponentially to the quantized value
$E^{0}_{sq}$ inside the topological phase (red curve) and it coincides with
$E^{0}_{sq}$ independently of the size of the chain for values of $\mu$
sufficiently close to the analytic point (blue curve). The QCMIs vanish
identically at all values of $\mu$ in the trivial phase (green curve).

Finally, in panel (c) of Fig. 3 we report the behavior of the ratio
$I_{(4)}/E^{0}_{sq}$ as a function of $\mu$ for different values of the length
of the chain $L$, showing the correct approach to quantization inside the
topological phase and the correct scaling behavior approaching the critical
point. These results show that the QCMIs $I_{(3)}$ and $I_{(4)}$ provide a
completely equivalent detection and characterization of the topologically non-
trivial regime of a superconducting wire.

In fact, by applying the lower-bound inequalities holding on SE [38, 31, 39,
40], it turns out that the constant value $E^{0}_{sq}=\log(2)/2$ is a lower
bound to the true SE $E_{sq}(\rho_{AB})$ between the edges at the exact
topological point $\mu=0$. By continuity, the bound extends to neighboring
values of $\mu$. Since the upper bound $I_{(4)}$ and the quantized lower bound
$E^{0}_{sq}$ coincide in this region, both coincide with the true long-
distance SE $E_{sq}(\rho_{AB})$ between the two disconnected edges $A$ and
$B$.

In conclusion, collecting all the above results, the quantized TSE

$E_{sq}(\rho_{AB})=E^{0}_{sq}=\log(\sqrt{2})=\frac{\log(2)}{2}$ (9)

is the nonlocal order parameter for a Kitaev superconducting wire.

The physical origin of a quantized non-vanishing long-distance entanglement
between the system edges in the topological phase arises from the interplay
between the exponential bulk-edge separation due to the topological nature of
the system and the fundamental monogamy property of nonlocal quantum
correlations [28].

As the two edges progressively decouple from the bulk their mutual information
and correlation are enhanced by the monogamy constraint on the shared
information and on the amount of shareable entanglement among the different
subsystems. When the system moves away from the topological phase,
correlations become once again short-ranged and the long-distance entanglement
between the two extremes of the chain collapses.

The quantized value of the edge–edge TSE is $\log(2)/2$ rather than the
maximal Bell–pair entanglement $\log(2)$. This fact shows that the TSE is
fully sensitive to and takes correctly into account the nature of the
topological modes, distinguishing between the entanglement of half-fermions,
like MZMEs, from the maximal Bell–pair entanglement.

The TSE can be compared with the so-called entanglement topological invariant
(ETI), a nonlocal order parameter for one-dimensional topological
superconductors introduced in Ref. [66]. This quantity is defined in terms of
a combination of reduced von Neumann entropies associated to a pseudo-
quadripartition obtained by considering two disconnected and partially
overlapping sets, $A_{1}$ and $A_{2}$, their union $A_{1}\cup A_{2}$, and
their intersection $A_{1}\cap A_{2}$. As such, the ETI has no sensible
relation to a real physical multipartition of the system and the corresponding
nonlocal quantum correlations between separated and non-overlapping
subsystems. In particular, it is not an entanglement monotone, let alone a
genuine entanglement measure between any two separated and distinguishable
subsystems like, e.g., the system edges.

One particularly limiting tenet of the ETI as introduced in Ref. [66] is that
a key ingredient for its derivation is the existence of disconnected
partitions of the system. In fact, we have just shown and explained in detail
why a fully connected edge-bulk-edge system tripartition and the associated
QCMI $I_{(3)}$ yield exactly the same characterization of the edge modes and
the topologically ordered phases as the one provided by the quadripartition
edge-bulk-bulk-edge and the associated QCMI $I_{(4)}$.

Also, in Ref. [66] it is found that the ETI takes the value $\log(2)$, i.e.
twice the value of the TSE. As discussed above, the value $\log(2)/2$ assumed
by the TSE is the one that takes correctly into account the statistics of the
Majorana excitations versus that of full Dirac fermions; moreover, the factor
$1/2$ provides the correct reduction on pure states to the unique entanglement
measure, i.e. the von Neumann entanglement entropy. Such correct
identifications are a direct consequence of the precise physical
identification of the system multipartition and general definition and
properties of the edge-edge QCMI and TSE. No such comprehensive framework
exists for the ETI. In conclusion, the scheme based on the edge-edge bipartite
QCMI and TSE defined on multipartitions provides a well-defined, general
physical framework to determine the correct nonlocal order parameter for
topologically ordered phases in one dimension. Moreover, this framework can be
extended in principle to the study of topological systems in any dimension and
at any temperature, either at equilibrium or in non-equilibrium.

Concerning the experimental accessibility of topological squashed
entanglement, the problem boils down to that of measuring quantum entropies of
a set of reduced states in quantum many–body systems. A recent proposal relies
on the thermodynamic study of the entanglement Hamiltonian for the direct
experimental probing of von Neumann entropies via quantum quenches [87].
Another possibility, specific for systems featuring topological order,
consists in identifying minimum entropy states and then experimentally
simulating the behavior of the associated von Neumann entropies via the
classical microwave analogs of such states. In this way it is in principle
possible to simulate various non-trivial instances of reduced entropies and
topological order, as shown explicitly for the transition from a trivial phase
to a $\mathcal{Z}_{2}$–symmetric topological phase [88].

Another 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 [39, 40]. A possible strategy is then to adapt to fermionic systems
[89] the schemes previously proposed for the experimental measurement of Rényi
entropies in bosonic and spin systems [90, 91, 92] and the corresponding
experimental techniques that led to the first experimental measurement of the
2-Rényi entropy in a many-body system [93]. This possibility has been already
suggested in Ref. [66] for the experimental measurement of the ETI.

###  IV.2 SE and TSE: discriminating between symmetry breaking magnets and
topological superconductors

A core question concerns the ability of a proposed nonlocal order parameter to
discriminate topological superconductors from systems featuring local order
parameters, spontaneous symmetry breaking and Ginzburg-Landau type of ordered
phases, addressing their different entanglement properties and patterns.

We focus our attention on the comparison between the spin-$1/2$ Ising chain
and the fermionic Kitaev chain. Setting $t=\Delta=1$ and applying the global
Jordan-Wigner transformations [65], the Hamiltonian in Eq. (8) can be mapped
into that of the $1D$ Ising model in transverse field:

$H_{I}=J\sum_{j=1}^{L-1}\sigma_{j}^{x}\sigma_{j+1}^{x}+h\sum_{j=1}^{L}\sigma_{j}^{z},$
(10)

with $J=-t$ and $h=\mu/2$. Both models share a two-fold degenerate ground
state and the parity symmetry $\mathcal{Z}_{2}$, but the physics of the order
displayed is radically different. While in the Ising model the
$\mathcal{Z}_{2}$ spin reflection symmetry is spontaneously broken with the
appearance of a nonvanishing local order parameter, the topological degeneracy
in the Kitaev chain is realized with perfect conservation of the Hamiltonian
symmetry and no local order parameters.

Despite the different type of order that they support, the two models share
the same bipartite entanglement spectrum and bipartite von Neumann block
entanglement entropy. This is not surprising, since the nonlocal Jordan-Wigner
transformation (a particular case of the general Klein transformation in
quantum field theory) does not affect the short-distance entanglement
structure across the separation between the system’s halves for Hamiltonians
sums of local (nearest-neighbor or short-ranged) interaction terms.

The landscape is quite different when dealing with the structure of long-
distance entanglement (LDE) between the end boundaries of a many-body system:
clearly, we expect LDE to be significantly affected by global transformations
acting on the entire system.

Figure 4: Ratios of the end-to-end QCMIs $I_{(3)}$ and $I_{(4)}$ to the
quantized TSE unit $E^{0}_{SQ}=\log(2)/2$ for the Ising chain (panels (a) and
(c)) as functions of the transverse field $h$, and for the Kitaev chain
(panels (b) and (d)) as functions of the chemical potential $\mu$. While for
the Kitaev chain the equality $I_{(3)}=I_{(4)}$ always holds, for the Ising
chain $I_{(3)}\neq I_{(4)}$ and $I_{(4)}$ tends to vanish as the system size
increases. The QCMI $I_{(4)}$ thus discriminates between topological and
symmetry-breaking systems. Throughout, the Hamiltonian parameters are set at
$t=\Delta=1$ for the Kitaev chain and $J=-1$ for the Ising chain.

In panels (a) and (b) of Fig. 4 we report the behavior of the end-to-end QCMI
$I_{(3)}$ for the Ising and the Kitaev chains for two lengths $L=6$ and $L=9$.
The two curves intersect at the critical points $h=1$ and $\mu=2$. In the
thermodynamic limit, the following behaviors are expected:
$I_{(3),Ising}/E^{0}_{sq,Ising}=1$ ($I_{(3),Kitaev}/E^{0}_{sq,Kitaev}=1$) for
$0<h<1$ ($0<\mu<2$) and $I_{(3),Ising}/E^{0}_{sq,Ising}=0$
($I_{(3),Kitaev}/E^{0}_{sq,Kitaev}=0$) for $h>1$ ($\mu>2$). This analysis
shows that the QCMI $I_{(3)}$ detects and characterizes both topological and
symmetry breaking orders but is not capable of distinguishing between them.

In panels (c) and (d) of Fig. 4 we report the behavior of the end-to-end QCMI
$I_{(4)}$ for the Ising and Kitaev chains; we see that $I_{(4)}$ distinguishes
topological superconducting phases from ordered phases with spontaneously
broken symmetries.

For the Ising chain, we see that removing any finite part $C_{2}$ of the bulk
$C=C_{1}C_{2}$ before performing the partial trace with respect to the
reminder $C_{1}$ returns a strongly suppressed end-to-end QCMI $I_{(4)}$ that
is expected to vanish in the thermodynamic limit.

At variance with the symmetry-breaking case, the topology of the Kitaev chain
is essentially concentrated on the boundaries, so that in the computation of
the end-to-end QCMIs performing the partial trace with respect to the entire
bulk $C$ is entirely equivalent to removing a part of the bulk ($C_{2}$) and
then performing the partial trace with respect to the remaining part of the
bulk ($C_{1}$). As a consequence, $I_{(3)}=I_{(4)}$ for a topological system,
while in general $I_{(3)}\neq I_{(4)}$ for a system with symmetry-breaking
order.

Ultimately, these results depend on and highlight the different global
properties of the two systems. Indeed, the bulk of the Kitaev chain is
insulating and exponentially decoupled from the boundaries, forcing (by
monogamy) the onset of the long-distance correlation between the ends of the
chain. Vice versa, the bulk of the Ising chain is conductive, so that cutting
one part of it before performing the partial trace on the reminder abruptly
interrupts the information flow and destroys the correlation structure between
the two ends of the chain. These physical differences determine the different
structure of the end-to-end QCMIs and of the end-to-end SE in the two models.

Concerning quadripartitions, an interesting, although not surprising, feature
is the independence of the results on the relative sizes of the two parts of
the bulk $C_{2}$ (the cut) and $C_{1}$ (the reminder), so that one can always
adopt the most convenient choice $C_{1}=1$ in the actual calculations.

In Fig. 5 panel (a) we report the behavior of the difference
$I_{(4)}-E^{0}_{sq}$ as a function of $L_{C_{1}}$ with $\mu=0.5$, $t=1$ and
$L=60$, and for $\Delta=t$ (green curve) and $\Delta=0.1t$ (red curve). We see
that $I_{(4)}-E^{0}_{sq}$ is constant in both cases, regardless of the length
$L_{C_{1}}$.

Figure 5: Panel (a): The difference $I_{(4)}-E^{0}_{sq}$ as a function of the
size $L_{C_{1}}$ of the reminder of the bulk in a Kitaev chain of $L=60$
sites. We have considered both the balanced case $t=\Delta=1$ (green curve)
and the unbalanced case $t=1$, $\Delta=0.1$ (red curve). Different choices of
$L_{C_{1}}$ have no effect on the evaluation of $I_{(4)}$. The inset shows the
scaling of $I_{(4)}$ towards the quantized value $E^{0}_{SQ}=\log(2)/2$ of the
TSE in the balanced regime as the size $L$ of the chain is increased. Panel
(b): Schematic representation of the chain in the Majorana basis $\\{a_{j}$,
$b_{j}\\}$) for $t\neq\Delta$. Panel (c): Same, with $t=\Delta$.

In the balanced hopping-pairing case $t=\Delta$, the SE is fully quantized at
the fundamental unit $E^{0}_{sq}=\log(2)/2$ already for an intermediate chain
length of $L=60$ (green curve), while in the strongly unbalanced case
$\Delta=0.1t$ one can observe (red curve) a finite-size scaling on
$I_{(4)}-E^{0}_{sq}$.

These behaviors show that the balanced case $t=\Delta$ is way more insensitive
to finite size effects than the unbalanced case $t\neq\Delta$. On the other
hand, as the inset in panel (a) of Fig. 5 shows, for the unbalanced case
$t\neq\Delta$ the difference $I_{(4)}-E^{0}_{sq}$ decreases with increasing
sizes of the system, tending to vanish, as expected, in the thermodynamic
limit.

The different robustness to finite size effects between the balanced and
unbalanced configurations is illustrated in panels (b) and (c) of Fig. 5,
where we draw sketches of the Kitaev chain in the basis of Majorana fermions
$a_{j}=c_{j}+c^{\dagger}_{j}$ and $b_{j}=-i(c_{j}-c^{\dagger}_{j})$ for the
two aforementioned cases. The balanced setting $t=\Delta$ illustrated in panel
(c) corresponds to turning off the coupling $(t-\Delta)$ $a_{j}b_{j+1}$
between third-neighboring Majorana fermions. In this configuration the
Majorana fermions are thus more weakly coupled than in the unbalanced case
$t\neq\Delta$ illustrated in panel (b). On the other hand, the setting
$t=\Delta$ and $\mu=0$ corresponds to the exact analytic, topological
degeneracy point of the model, where the edge modes $a_{1}$ and $b_{L}$ at the
two ends of the chain decouple from the lattice and become exact zero energy
modes.

###  IV.3 TSE in the presence of interactions

The standard interacting Kitaev chain is obtained by adding a nearest-neighbor
density-density interaction term to the free Kitaev Hamiltonian:

$H_{IK}=H_{K}+U\sum_{j=1}^{L-1}n_{j}n_{j+1},$ (11)

where $U$ is the interaction coupling constant and
$n_{j}=c^{\dagger}_{j}c_{j}$ is the on site fermion number operator. The
intervals $U>0$ and $U<0$ correspond, respectively, to a repulsive and to an
attractive interaction. The phase diagram of the model has been investigated
in depth by a variety of numerical and approximate analytical methods [94, 95,
66, 85, 96] and is reported schematically in Fig. 6.

We see that, although very strong repulsive interactions eventually force the
system into a Mott insulating phase, a topological superconducting state is
established in between the trivial band insulator phase and the Mott
localization even for fairly strong repulsive interactions. The black dashed
line inside the topological phase is the factorization line of equation
$\mu=\mu^{*}$ described in the following.

Figure 6: Phase diagram of the interacting Kitaev chain on the $\mu$–$U$ plane
for $\Delta=t$. The dashed black curve is the factorization line
$\mu=\mu^{*}=4\sqrt{U^{2}+t\ U}$. It starts at the exact topological
degeneracy point $\mu=U=0$, develops entirely inside the topological phase,
and pinches asymptotically into the trivial phase for sufficiently large
values of $\mu$ and $U$. The acronym $ICDW$ stands for incommensurate charge
density wave.

The interacting Kitaev chain can be mapped exactly onto the interacting $XYZ$
spin-1/2 model in an external magnetic field

$\displaystyle H_{xyz}$ $\displaystyle=$
$\displaystyle\sum_{j=1}^{L-1}\left[J_{x}\sigma_{j}^{x}\sigma_{j+1}^{x}+J_{y}\sigma_{j}^{y}\sigma_{j+1}^{y}+J_{z}\sigma_{j}^{z}\sigma_{j+1}^{z}\right]$
(12) $\displaystyle+$ $\displaystyle h\sum_{j=1}^{L}\sigma_{j}^{z}\,,$

where $J_{x}=-(\Delta+t)/2$, $J_{y}=(\Delta-t)/2$, $J_{z}=-U$, and $h=\mu/2$.

The model admits exact ground state solutions factorized in the product of
single-spin (single-site) wave functions along the so-called factorization
line $h=h^{*}=z_{d}\sqrt{(J_{x}+J_{z})(J_{y}+J_{z})}$ in any dimension [82,
83, 97], with $z_{d}$ the coordination number (so that, e.g., $z_{d}=2$ in the
1D case).

For the Kitaev chain, this corresponds to the exact solution for the many-body
ground states along the factorization line [85]:

$\mu=\mu^{*}=4\sqrt{U^{2}+t\ U+(t^{2}-\Delta^{2})/4}\,,$ (13)

which reduces to $\mu^{*}=4\sqrt{U(U+t)}$ for $t=\Delta$.

In general, along any factorization line (provided it exists) an interacting
model Hamiltonian $H$ becomes frustration–free and can be written as the sum
of commuting local terms $H_{j}$: $H=\sum_{j}H_{j}$ with $[H_{i},H_{j}]=0$
[84].

The exact ground states of definite symmetry of the interacting Kitaev chain
for $\mu=\mu^{*}$ are the following entangled linear combinations [85]:

$\displaystyle|\Psi^{even}\rangle=\frac{1}{\sqrt{2}}\bigl{(}|\Psi^{+}\rangle+|\Psi^{-}\rangle\bigr{)}$
(14)
$\displaystyle|\Psi^{odd}\rangle=\frac{1}{\sqrt{2}}\bigl{(}|\Psi^{+}\rangle-|\Psi^{-}\rangle\bigr{)},$
(15)

where the non-orthogonal states $|\Psi^{+}\rangle$ and $|\Psi^{-}\rangle$ are
fully factorized in the product of single-site wave functions and read [85]:

$|\Psi^{\pm}\rangle=\frac{1}{(\alpha+1)^{L/2}}e^{\pm\alpha
c^{\dagger}_{1}}\dots\ e^{\pm\alpha c^{\dagger}_{L}}|vac\rangle,$ (16)

where

$\alpha=\sqrt{\cot(\theta^{*}/2)},\ \ \theta^{*}=\arctan(2\Delta/\mu^{*}),$
(17)

with $|vac\rangle$ denoting the lattice vacuum state.

The ground states $|\Psi^{even}\rangle$ and $|\Psi^{odd}\rangle$ are
orthogonal and are the only possible ground state of $H_{IK}$ for
$\mu=\mu^{*}$. Ground state factorization holds in the topological phase with
a repulsive interaction $U\geq 0$, and the even and odd ground states are
degenerate with energy $E_{0}=-(L-1)(U+t)$.

The behavior of the QCMI $I_{(4)}$ normalized to the quantized unit of TSE
$E^{0}_{sq}=log(2)/2$ along the Illuminati-Katsura factorization line
$\mu=\mu^{*}$ is reported in Fig. 7 panel (a) as a function of the chain
length $L$ for different values of the interaction strength, and in panel (b)
as a function of the interaction strength for different values of the chain
length $L$.

The QCMI $I_{(4)}$ exhibits the correct scaling, saturating to the quantized
value $log(2)/2$ of the SE for sufficiently large values of the chain size
$L$, and vanishing asymptotically for very large values of the interaction
strength $U$ as the factorization line pinches the boundary of the topological
phase and the system enters in the trivial phase.

In conclusion, we see that the quantized TSE $E^{0}_{sq}$ identifies the
correct nonlinear order parameter for the topological phase of the fermionic
Kitaev chain, either free or interacting.

Figure 7: The QCMI to TSE ratio $I_{(4)}/E^{0}_{sq}$ for the exact ground
state solutions of the interacting Kitaev chain on the Illuminati-Katsura
factorization line $\mu=\mu^{*}$ as a function of the chain length $L$ for
different values of the interaction strength $U$ (panel (a)), and as a
function of $U$ for different values of $L$ (panel (b)). In both cases the
scaling behavior of $I_{(4)}$ converges to the quantized value $log(2)/2$ of
the TSE for the non-interacting Kitaev chain. Eventually, for increasingly
large values of $U$, the factorization line pinches into the trivial phase and
the QCMI $I_{(4)}$ vanishes asymptotically. Throughout, $t=\Delta$ and
$L_{e}=L/3$.

###  IV.4 Robustness of TSE against disorder

Disorder plays a fundamental role in low-dimensional electronic systems [98,
99, 100, 55] and robustness against disorder is a defining property of
topological materials. In fact, disorder may even induce localization effects
as in the case of Anderson insulators [101], thus favoring topological phases
of matter, whereas in other systems such as, e.g., semiconductor Majorana
nanowires and topological insulator nanoribbons, it can yield detrimental
effects [99, 102]. It has been shown that topological phases of the Kitaev
chain are robust to the effects of disorder and local perturbations [98, 103,
104, 55]. Such immunity is a fundamental property that is expected to hold
whenever topological superconductors are involved. This observation suggests
that the edge-to-edge QCMIs and SE should also be as robust to disorder as
other topological indicators defined in terms of spectral or transport
properties [105, 106].

We study two distinct classes of configurations, each characterized by a
different source of disorder with a clear tracking of its physical origin.
Since one of the prominent schemes to realize a topological Kitaev chain is by
proximity effects between a semiconducting nanowire and a conventional
superconductor [42, 43], we consider two distinct sources of disorder: random
site-dependent hopping amplitudes $t_{i}$ and random site-dependent pairing
potentials $\Delta_{i}$. The former provides a model of the effective mass
gradient and random doping along the nanowire that originate from the growth
process of the nanowire itself; the latter emulates unwanted spatial
variations along the wire which can affect the nanowire-superconducting
coupling and therefore the induced superconducting gap.

Figure 8: The QCMI to TSE ratio $I_{(4)}/E^{0}_{sq}$ as a function of the
chemical potential $\mu$ for a chain of length $L=70$ and two different types
of disorder. Panel (a): behavior of $I_{(4)}/E^{0}_{sq}$ in the presence of
random hopping amplitudes $t_{i}$. Panel (b): behavior of $I_{(4)}/E^{0}_{sq}$
in the presence of random pairing potentials $\Delta_{i}$. The random
realizations are generated by a continuous probability distribution defined in
the interval $t_{i}\in(t-\tau$, $t+\tau)$, $\Delta_{i}\in(\Delta-\delta$,
$\Delta+\delta)$. The reference values of the parameters have been fixed as
$t=1$, $\Delta=0.1$. The different curves are parameterized by disorder of
increasing strength.

In Fig. 8 we report the behavior of the QCMI to TSE ratio $I_{(4)}/E^{0}_{sq}$
of the Kitaev chain Eq. 8 as a function of the chemical potential $\mu$ for
the aforementioned two different sources of disorder. In panels (a) and (b) we
consider respectively random hopping integrals $t_{i}=t+\tau^{dis}_{i}$ and
random pairing potentials $\Delta_{i}=\Delta+\delta_{i}^{dis}$ with the random
strengths of disorder $\tau^{dis}_{i}$ and $\delta_{i}^{dis}$ uniformly
distributed in the intervals $(-\tau$, $\tau)$ and $(-\delta$, $\delta)$.

In Fig. 8, for the sake of completeness disorder effects are investigated
spanning the spectrum of noise strengths from the perturbative regime up to
strongly disordered configurations. It is worth mentioning here that the
maximum disorder strength considered in panel (b) of Fig. 8 is comparable with
the mean value of the pairing potential $\Delta$. Such configuration
corresponds to an extremely disordered system, a situation that is not
expected in realistic experimental conditions. A similar phenomenology can be
observed by studying disorder effects induced by random values of the hopping
integral, see Fig. 8 panel (a), from $\tau=\Delta/4$ up to $\tau=\Delta$ while
for $\tau=5\Delta$ the phase boundary defining the topological transition is
reduced. This behaviour suggests that a random hopping is more effective in
perturbing topologically ordered phases than a random superconducting pairing.

The actual values of disorder experimentally achievable with currently
available technologies are in general significantly smaller than those
simulated in Fig. 8; the latter must thus be taken as a theoretical-numerical
test/prediction of the robustness of edge QCMI and edge TSE. In particular,
the value $\tau=5\Delta$, although not comparable with the maximally allowed
disorder on the energy gap, is of the order of half of the band width;
therefore, it must certainly be considered nonperturbative.

Indeed, Fig. 8 shows, as it should be expected, that within the typical
regimes of weak to moderate disorder the TSE nonlocal order parameter is
strongly resilient in the entire superconducting phase.

##  V Systems with multiple topological phases and edge modes: TSE of the
Kitaev ladder

In this section we generalize the previous investigations to consider systems
enjoying multiple topological phases and multiple MZEMs, i.e. more than one
Majorana zero mode per edge.

Various such generalizations of the $1D$ Kitaev model have been introduced
[107, 48, 49, 50, 51, 108]; the simplest instance can be realized by coupling
multiple Kitaev chains with transverse hopping and pairing terms to form quasi
one-dimensional $n$-leg Kitaev ladders [52, 109]. The case $n=2$ defines the
two-leg Kitaev ladder. This is the minimal model featuring multimode
topological phases, which usually arise in models of significant complexity
that involve higher-dimensional platforms [110, 111, 112, 113].

The Hamiltonian $H_{KL}$ of the two-leg Kitaev ladder can be written as
follows:

$H_{KL}=H_{K_{1}}+H_{K_{2}}+H_{K_{12}}\,,$ (18)

where the two Kitaev chains $H_{K_{1}}$ and $H_{K_{2}}$ define the legs of the
ladder and

$H_{K_{12}}=\sum_{i=1}^{L}[-t_{1}c^{\dagger}_{i,1}c_{i,2}+\Delta_{1}c_{i,1}c_{i,2}+h.c.]$
(19)

describes the coupling between the two legs $H_{K_{1}}$ and $H_{K_{2}}$. The
coupling is realized by means of a transverse hopping with amplitude $t_{1}$
and a transverse pairing with amplitude $\Delta_{1}$; these two transverse
Hamiltonian terms provide the rungs of the ladder.

The model has been introduced and discussed at length in Ref. [52]. The Kitaev
ladder satisfies particle-hole, time reversal and chiral symmetry, belonging
to the $BDI$ class of the ten-fold classification.

Figure 9: Panels (a) and (c): phase diagram of the two-leg Kitaev ladder
determined by means of the winding number in $k$-space. Panel (a): phase
diagram in the $\mu$–$t_{1}$ plane, with the remaining Hamiltonian parameters
fixed at $t=1$, $\Delta=0.8$, and $\Delta_{1}=0.8$. Panel (c): phase diagram
in the $\mu$–$t_{1}$ plane, with the remaining Hamiltonian parameters fixed at
$t=1$, $\Delta=0.8$, and $\Delta_{1}=0.09$. Panel (b): phase diagram of the
two-leg Kitaev ladder determined by means of the QCMI to TSE ratio
$I_{(4)}/E^{0}_{sq}$. The phase diagram is in the same $\mu$–$t_{1}$ plane,
with the values of the remaining Hamiltonian parameters set as in panel (a),
and it is obtained on a grid of $25\times 25$ points with the order of the
interpolating polynomial equal to $1$. The dashed red lines in panel (c)
denote the straight transition lines (”cuts”) that cross the different phase
boundaries. The horizontal cut is at constant transverse hopping $t_{1}$; the
vertical cut is at constant chemical potential $\mu$. In all panels, the
length $L$ per ladder leg and the length $L_{e}$ per edge partition are fixed,
respectively, at $L=50$ and $L_{e}=12$.

The phase diagram of the model can be investigated by means of the winding
number:

$W=Tr\int^{\pi}_{-\pi}\frac{dk}{2\pi i}\ \ A_{k}^{-1}\partial_{k}A_{k}=$
$-\int^{\pi}_{-\pi}\frac{dk}{2\pi i}\ \ \partial_{k}\ln DetA_{k}$ (20)

where $A_{k}$ can be expressed in terms of the model parameters once we impose
periodic boundary conditions and the bulk-edge correspondence is invoked:

$\displaystyle
A_{k}=\begin{pmatrix}\epsilon_{k}-\Delta_{k}&-t_{1}-\Delta_{1}\\\
-t_{1}+\Delta_{1}&\epsilon_{k}-\Delta_{k}\\\ \end{pmatrix},$ (21)

where $\epsilon_{k}=2t\cos k+\mu$, $\Delta_{k}=2i\Delta\sin k$ and
$k\in[-\pi,\pi]$.

In Fig. 9 we report the phase diagrams of the two-leg Kitaev ladder obtained
respectively by means of the winding number, as shown in panels (a) and (c),
and by means of the QCMI $I_{(4)}$ normalized by the quantized edge-edge TSE
$E^{0}_{sq}$, as shown in panel (b) for the same choice of Hamiltonian
parameters as in panel (a). The diagram in panel (b) is obtained for a ladder
of $L=50$ sites per leg (chain), with the length of each edge partition fixed
at $L_{e}=12$, and via an interpolation on a grid of $25\times 25$ points.

The digits drawn on the phase diagrams in panels (a) and (c) count the number
of MZEMs per ladder edge in each phase. Indeed, the system possesses a richer
phase diagram compared to that of the single Kitaev chain. Denoting by $N_{m}$
the number of MZEMs per edge, the two-leg Kitaev ladder features two
alternating topological phases, respectively endowed with $N_{m}=1$ and
$N_{m}=2$ edge modes, and a trivial phase with no edge modes ($N_{m}=0$).

Despite the fact that the phase diagrams in panels (a) and (c) are defined in
terms of bulk properties, while the phase diagram in panel (b) is obtained in
terms of the topological boundary entanglement, they show an excellent
qualitative and quantitative agreement already at moderate system sizes.

In panel (c), we report the contour plot of the winding number for a different
set of values of the Hamiltonian parameters with respect to the ones of panels
(a) and (b). Indeed, with the choice of Hamiltonian parameters as in panel
(c), the system features the possibility of moving along straight transition
lines that cross all the three different phases $2$-$1$-$0$, once either
$t_{1}$ or $\mu$ is kept constant. The dashed red lines drawn in panel (c)
denote such horizontal (constant $t_{1}$) and vertical (constant $\mu$) cuts
across the different phase boundaries.

Figure 10: Panel (a): QCMI-TSE ratio $I_{(4)}/E^{0}_{sq}$ as a function of the
chemical potential $\mu$ along the horizontal cut at fixed transverse hopping
$t_{1}$ drawn in panel (c) of Fig. 9, for different lengths $L$ of the ladder.
Panel (b): $I_{(4)}/E^{0}_{sq}$ as a function of $t_{1}$ along the vertical
cut at constant $\mu$ drawn in panel (c) of Fig. 9, for different values of
the length $L$ of any of the two equal legs of the ladder. The plots at
different lengths of the ladder legs show the fast scaling behavior of the
QCMI-TSE ratio with the system size. Irrespective of the choice of the
transition line crossing the different phase boundaries, in each phase
$I_{(4)}/E^{0}_{sq}$ is quantized to an integer counting the number of edge
MZEMs featured by the phase, and jumps by one unit at each transition,
realizing a series of Hall-like plateaux.

In Fig. 10 we report the behavior of the QCMI-TSE ratio $I_{(4)}/E^{0}_{sq}$,
respectively along the horizontal and vertical cuts crossing the different
phases of the ladder, as drawn in panel (c) of Fig. 9. In both cases, we
observe the three plateaux $I_{(4)}/E^{0}_{sq}=2,1,0$ corresponding,
respectively, to the two topological phases and to the trivial phase.

Remarkably, we find that TSE not only discriminates topological phases from
trivial ones but also distinguishes different topological phases by counting
the corresponding number of Majorana edge modes $N_{m}$. Indeed, for the two-
leg Kitaev ladder we have:

$\frac{I_{(4)}}{E^{0}_{sq}}=N_{m}\,.$ (22)

If the above relation has general validity, then in principle QCMI and TSE
could apply to topological systems of arbitrary complexity featuring any
number of topological modes. Besides its conceptual importance, this property
bears a clear practical advantage, since while the definition of the TSE is
unique and system-independent, different invariants have to be defined each
time, following the ten-fold classification, depending on the symmetries
obeyed by each specific system under investigation.

Another paradigmatic model belonging to the same BDI topological class of the
two-leg Kitaev ladder is the Su-Schrieffer-Heeger (SSH) fermionic chain [114].
For such system the disconnected Rényi entropy has been shown to reproduce
correctly the phase diagram of the model [115].

Despite belonging to the same topological universality class and therefore
sharing the same topological band invariant (i.e., the winding number), the
two models feature different phase diagrams and, perhaps more importantly,
differ in the statistics of the edge modes. In particular, while the SSH model
exhibits a trivial phase and one topological phase with one fermionic mode per
edge, the two-leg Kitaev ladder features a trivial phase and two different
topological phases, respectively with one and two Majorana modes per edge.
Therefore, in one case (SSH) we have a simple phase diagram with one
topological phase and fermionic edge statistics, while in the other case
(ladder) we have a richer phase diagram with two different topological phases
and nonabelian anyon edge statistics.

What motivates the investigation of the relatively simple example of the two-
leg superconducting ladder is precisely that it allows to study the transition
between different multi-mode topological phases and to verify that our scheme
indeed discriminates between them by the occurrence of different quantized
plateaux of the TSE that count the integer number of edge modes present in
each phase. These results indicate that our scheme is in principle also
applicable to any multi-mode system with multiple topological phases. Thus, in
conclusion, edge-edge QCMI and TSE not only discriminate topological regimes
from ordered phases with broken symmetries and from trivial phases, but also
distinguish different types of topologically ordered phases.

##  VI Systems with suppressed bulk edge correspondence: TSE of the Kitaev tie

In this section we study by means of the edge-to-edge QCMI and the quantized
TSE the effects of the geometrical frustration induced by adding an hopping
term of arbitrary spatial range to a Kitaev chain. The model has been
originally introduced and investigated in Refs. [53, 68]. The tight-binding
Hamiltonian of the model can be written as

$H_{T}=H_{K}+H_{d}\,,$ (23)

where $H_{K}$ denotes the Kitaev chain Hamiltonian and

$H_{d}=-t_{d}(c^{\dagger}_{d}c_{L-d+1}+h.c.)$ (24)

is an extra long-range hopping term, connecting two symmetrical sites of the
chain $d$ and $L-d+1$ with an hopping amplitude $t_{d}$. The long-range
hopping, identified by the parameter $d$, can vary along the length of the
chain, playing the role of a movable knot which induces a geometric
frustration on the original chain Hamiltonian. This results in a legged-ring
system with no clearly identifiable bulk, and consequently a suppressed bulk-
edge correspondence, referred to as a Kitaev tie.

The phase diagram of the system in the $\mu$–$d$ plane shows an interstitial
character with topological phases nucleating inside trivial regions as the
knot position $d$ is varied while keeping $\mu<2t$. This phenomenon is
referred to as topological frustration.

For small values of $d$, the interstitial character of the non-trivial phases
is more evident, since the system resembles a ring with very short legs. On
the other hand, for large values of $d$ there is a significant growth of the
topological phase domains, since the ring is reduced and the system approaches
the limiting regime of a perturbed Kitaev chain.

The Kitaev tie is realizable in single-walled carbon nanotubes [53, 68]; these
are flexible ballistic conductors [116] where superconducting proximity effect
can be easily implemented [117, 118, 119].

Figure 11: Panel (a), red curve: behavior of the complement
$1-I_{(4)}/I_{max}$ of the ratio between the QCMI and its maximum value as a
function of the chemical potential $\mu$. Panel (a), blue curve: behavior of
the lowest energy eigenvalue $E$ as a function of $\mu$. Both curves are drawn
for a Kitaev tie of $L=121$ sites, edges of $L_{e}=40$ sites, and the long-
range hopping term positioned at $d=30$, i.e. within the system edges. Panel
(b), red curve: behavior of the complement of the QCMI-TSE ratio
$1-I_{(4)}/E^{0}_{sq}$ as a function of $\mu$. Panel (b), blue curve: behavior
of the lowest energy eigenvalue $E$ as functions of $\mu$. Both curves are
drawn for the same Kitaev tie as in panel (a), but this time with the long-
range hopping positioned at $d=50$, i.e. outside the system edges. The QCMI
$I_{(4)}$ is evaluated for an edge length $L_{e}=40$. In all plots the
Hamiltonian parameters are fixed at $t=t_{d}$, $\Delta=0.02$ and in the unit
of the hopping $t=1$.

In Fig. 11, we report the behavior of different indicators based on the edge-
to-edge QCMI $I_{(4)}$ and the TSE $E^{0}_{sq}$ as functions of the chemical
potential $\mu$ for a Kitaev tie of $L=121$ sites (blue curves) and we compare
it with the behavior of the spectral energy function of the system (orange
curves). Specifically, denoting by $I_{max}$ the maximum value achieved by
$I_{(4)}$ and by $E$ the lowest energy eigenvalue, in panel (a) we compare $E$
and the complement $1-I_{(4)}/I_{max}$ of the QCMI normalized to its maximum
value as functions of $\mu$, while in panel (b) we compare $E$ and the
complement of the QCMI to TSE ratio $1-I_{(4)}/E^{0}_{sq}$. With respect to
the extension of the edges, $L_{e}=L/3=40$, we consider two different
positions of the long-range hopping. In panel (a) we fix $d=30$, i.e. with the
knot included within the edges $A$ and $B$. In panel (b) we fix $d=50$, i.e.
with the knot included in the traced out bulk $C_{2}$.

In the presence of topological frustration the occurrence of unpaired Majorana
modes can be related, respectively, to minima or vanishing values of the
energy eigenvalue [68]. Indeed, by means of Majorana polarization [120, 121]
it can be shown that energy minima are associated to localized Majorana modes,
while energy maxima correspond to hybridized modes [68].

All plots in Fig. 11 clearly show signatures of the topological frustration,
pinpointed by an oscillating behavior of the various quantities as functions
of the chemical potential. For systems with an associated bulk, the QCMI
$I_{(4)}$ converges to the quantized TSE $E^{0}_{sq}$ already for moderate
lattice sizes, as shown in the discussion of the Kitaev chain and of the
Kitaev ladder. For the Kitaev tie, the absence of an associated bulk is
signaled by the fact that the complement of the QCMI to TSE ratio does not
vanish, reaching instead nonvanishing minima, even for lattices of relatively
large sizes ($L=121$).

On the other hand, the perfect correspondence of the positions of both the
minima and the maxima of the different functions shows that the QCMI and the
TSE capture the main features of the topological frustration phenomenon just
as well as the energy. In particular, although the energy eigenvalue provides
a stronger marker of the presence of MZEMs, remarkably the QCMI to TSE ratio
$I_{(4)}/E^{0}_{sq}$ is a good quantifier of the emerging topological features
even for systems, like the Kitaev tie, that do not allow the existence of a
bulk.

Fig. 11 shows that the actual value of the absolute maximum $I_{max}$ reached
by the QCMI $I_{(4)}$ strongly depends on whether the scope $d$ of the long-
range hopping falls inside the edge extension, i.e.$d<L_{e}$, as in panel (a),
or outside the edge extension, i.e. $d>L_{e}$, as in panel (b). Remarkably,
$I_{(4)}$ exceeds $E^{0}_{sq}$ ($I_{max}\approx 0.62$) when $d<L_{e}$, while
when $d>L_{e}$, the QCMI converges to the TSE $E^{0}_{sq}$ as in the cases of
the Kitaev chains and ladders. In the case of the Kitaev tie the
inapplicability of the bulk-edge correspondence is responsible for the partial
loss of quantization of the QCMI that is recovered in the limit of very large
scopes $d$ of the long-range hopping.

From Fig. 11 we see that both indicators, the energy and the QCMI to TSE ratio
$I_{(4)}/E^{0}_{sq}$, are reliable indicators of the tie phenomenology, as
both detect and capture the essentials of the topological frustration. As both
indicators are sensitive to finite-size effects, the effect is stronger for
the QCMI to TSE ratio: while the energy approaches the zero value, the QCMI
remains within a smal but finite range from the TSE.

##  VII Discussion and outlook

In the present work we have discussed the application of squashed entanglement
to the study of one-dimensional topological systems. Squashed entanglement is
a measure of bipartite entanglement defined on multipartitions that
generalizes the von Neumann entanglement entropy; it allows to quantify the
bipartite entanglement between any two subsystems, connected or disconnected,
in any quantum state, either pure or mixed, and reduces to the von Neumann
entanglement entropy on pure states. Introducing generic tripartitions and
quadripartitions of a quantum system, we have identified two general classes
of upper bounds on the bipartite squashed entanglement in terms of the quantum
conditional mutual information between subsystems.

For models of topological superconductors that admit Majorana zero energy
modes at the system edges, we have shown that in the topological phase the
upper bound provided by the edge-edge quantum conditional mutual information
associated to the system quadripartition in fact coincides with the squashed
entanglement between the system edges in the ground state at the exact
topological point. The squashed entanglement in the topological phase is
quantized at $\log(2)/2$, half the value of the maximal Bell entanglement,
counting the Majorana splitting of the Dirac fermions.

The long-distance topological squashed entanglement between the edges is
constant throughout the topological phase, exhibits the correct scaling when
approaching the critical point, and vanishes identically in the trivial phase.
Moreover, it is stable in the presence of interactions, resilient to the
effects of disorder and local perturbations, and discriminates topological
superconductivity from orders associated to spontaneous symmetry breaking. It
thus realizes the desired, and long-sought for, entanglement-based order
parameter for symmetry-protected topological superconductivity.

For systems with geometries of higher complexity with respect to the $1D$
Kitaev chain, featuring multiple topological phases and more than one Majorana
zero energy mode per edge, like e.g. the quasi one-dimensional Kitaev ladder,
topological squashed entanglement distinguishes between the multiple phases by
plateaux that count the number of Majorana edge modes. For systems with
suppressed bulk-edge correspondence, like e.g. the Kitaev tie, topological
squashed entanglement, though ceasing to be perfectly quantized, is anyway
able to identify the interstitial topological phases and discriminate them
from the trivial ones.

The long-distance TSE between the edges of a fermionic chain is reminiscent of
other forms of long-distance entanglement (LDE) that are established by
entanglement monogamy between the end points of one-dimensional spin-$1/2$
chains with complex interaction patterns [69, 70, 71, 72, 73]. It is unclear
at the moment whether TSE and LDE in modulated spin are related and share some
common feature/origin. We plan to investigate the problem, also in connection
with the intriguing possibility that complex patterns of interaction strengths
might induce a kind of topological order also in some classes of spin-$1/2$
systems. Finally, we would like to remark that end-to-end LDE and Wen’s long-
range entanglement patterns in the bulk are in principle quite distinct
concepts; as such, making use of both of them and their interplay might lead
to a deeper understanding and finer classification of topological phases of
matter in higher dimensions.

The generality of the concept of squashed entanglement between arbitrary
subsystems allows for several possible future research directions. Leaving
aside for the moment being the tantalizing possibility of monogamy-induced
multipartite extensions, here we wish to discuss the application of bipartite
SE to the study of topological systems in higher dimensions as well as at
finite temperature and out of equilibrium.

A crucial challenge in order to generalize the framework introduced in the
present work to systems in $D\geq 2$ spatial dimensions concerns the correct
identification of the different bulk and edge parts in the multipartitions.
Here we wish to mention that among topological two-dimensional systems, second
order topological superconductors ($HOTSC_{2}$) [112, 113, 122, 123] are of
particular relevance both for their fundamental properties and for the
possibility that they offer to implement braiding dynamics in a rather
straightforward way [124]. Second-order topological superconductors are two-
dimensional systems with gapped one-dimensional boundaries and zero-
dimensional localized modes (corner modes). For these 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
represents provides a suitable arena to generalize TSE, albeit numerically, in
$D=2$ dimensions. We plan to report on these and related problems in upcoming
future work.

In order to extend even further the concept of edge QCMI and edge SE to the
full generality of arbitrary systems in any spatial dimension, the central
issue concerns the correct identification of the system edges, their
localization properties, and their degree of connectivity or lack thereof.
Clearly, this type of analysis will be heavily model-dependent. For instance,
regarding the role of open boundary conditions in higher dimensions,
identification of the bulk and of the edges will depend on the form of the
boundaries, for instance on the number and form of the corners and vertices
for different lattice geometries.

When considering periodic or partially periodic boundary conditions, including
e.g. ring, cylinder, and torus geometries, the straightforward choices appear
to be tripartitions and quadripartitions of the system in adjacent arches in
order to realize and isolate effectively disconnected edges. To set the stage,
consider for instance a ring geometry fixed by imposing periodic boundary
conditions (PBCs). In such a geometrical configuration, the two edge Majorana
Zero Energy Modes (MZEMs) of the original open chain recombine with each other
and annihilate in a full fermion. Despite the fact that MZEMs are no longer
present, the topological invariants continue to yield signatures of the
topological nature of the system, provided one suitably modifies the choice of
partitions. Considering an elongated ring (think for instance of an
autodrome), in order to realize a tripartition suitable for long-distance
entanglement (LDE) of the topological squashed entanglement (TSE) type, we
need to cut the two antipodal curves that will then play the role of the long-
distance subsystems $A$, $B$, and consider as the bulk $C$ the two remaining
disconnected segments connecting the two arches $A$, $B$. By taking each of
the two segments separately, i.e. as parts $C_{1}$, $C_{2}$ of the full bulk
$C$, we can then realize the desired quadripartition as well.

Finally, in a very trivial sense, given that on pure quantum states, e.g.
ground states of many–body systems, the squashed entanglement reduces exactly
to the von Neumann block entanglement entropy when the system is partitioned
into just two simple blocks, the subleading contribution to the von Neumann
block entanglement entropy, the block topological squashed entanglement,
trivially coincides with the topological entanglement entropy in the case of
true topological order in two-dimensional systems. Restricted to this obvious
meaning, squashed entanglement already provides a framework unifying different
classes of topological order in higher dimensions.

Another relevant line of research concerns the generalization of TSE to open
topological systems, i.e. topological systems coupled to an environment. In
this case a first possible step forward can be based on the self-energy
formalism, by reducing the original model to an effective system structurally
equivalent to the original but where the edges and the bulk are renormalized
by the effects of the environment. Indeed, in such case it is possible to
recover the full partition structure of a closed system. We plan to apply this
method, via the corresponding non-Hermitian Hamiltonians, to the investigation
of edge QCMI and edge SE for open topological systems in one dimension.

Extending further the above line of research leads to the problem of applying
QCMI and SE to the study in full generality of topological quantum matter at
finite temperature and out of equilibrium [125]. In fact, even for non
topological matter it would be interesting to study finite temperature quantum
criticality resorting to squashed entanglement and compare it systematically
with other measures of nonclassicality such as, e.g., the entropic discord
[126] or the Bell nonlocality [127]. In this perspective, hierarchies of
quantum complexity according to different layers of quantumness, ranging from
discord and coherence to entanglement and nonlocality, could realize the
scaffold of a unified research framework applicable to a wide spectrum of
problems, ranging from quantum information and quantum matter to elementary
particle physics and quantum gravity.

Acknowledgments – We 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. We
thank an anonymous referee for several enlightening comments that helped to
improve on the original manuscript.

Author contributions – F.I. conceived the fundamental idea, organized the
project and supervised the work. F. I. and A. Marino performed part of the
calculations, while A. Maiellaro performed the full calculations, produced the
numerical plots and realized the figures. F. I. and A. Maiellaro drafted the
manuscript, discussed the results and edited the paper.

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