# Quadrature protection of squeezed states in a one dimensional photonic
topological insulator

J. Medina Dueñas<EMAIL_ADDRESS>G. O’Ryan Pérez  Departamento de
Física, Facultad de Ciencias Físicas y Matemáticas, Universidad de Chile,
Santiago, Chile  Carla Hermann-Avigliano  Departamento de Física, Facultad de
Ciencias Físicas y Matemáticas, Universidad de Chile, Santiago, Chile  ANID -
Millenium Science Iniciative Program - Millenium Institute for Research in
Optics  L. E. F. Foa Torres  Departamento de Física, Facultad de Ciencias
Físicas y Matemáticas, Universidad de Chile, Santiago, Chile

###### Abstract

What is the role of topology in the propagation of quantum light in photonic
lattices? We address this question by studying the propagation of squeezed
states in a topological one-dimensional waveguide array, benchmarking our
results with those for a topologically trivial localized state, and studying
their robustness against disorder. Specifically, we study photon statistics,
one-mode and two-mode squeezing, and entanglement generation when the
localized state is excited with squeezed light. These quantum properties
inherit the shape of the localized state but, more interestingly, and unlike
in the topologically trivial case, we find that propagation of squeezed light
in a topologically protected state robustly preserves the phase of the
squeezed quadrature as the system evolves. We show how this latter topological
advantage can be harnessed for quantum information protocols.

During the last decades, a series of discoveries illuminated a way where
topological arguments allowed to explain or envision the behavior of electrons
in materials [1, 2, 3, 4, 5]. Originally discovered in solid state physics,
these properties are actually fundamental to wave phenomena within periodic
media, allowing for the extension of topological physics to photonics [6, 7,
8], mechanical systems [9], and circuits [10], among others [11, 12, 13];
where the exploitation of topologically protected boundary states stands as a
key aim across these diverse playgrounds. Topological phases have been found
in all kinds of crystal and photonic lattices: in one [14, 15], two [16, 17],
and three [18, 19, 20, 21] dimensions; exhibiting topological phases protected
by preserved [4, 17] or broken [3, 22, 23, 24, 25] time-reversal invariance,
or by crystal symmetries [26, 19]; among many others [27, 28, 29, 30, 31].

Not only serving as a convenient experimental playground, topological
photonics may bring new phenomena unique to quantum states of light. Topology
has been found to provide robustness in lasers [33, 32] and amplifiers [34,
35], while coupling between quantum states of light and topological degrees of
freedom allows for topological protection of photon statistics and quantum
correlation [36]. We focus on the latter topic. Research in this topic is only
recent, nevertheless, the behavior of single- and bi-photon states has been
studied in one- and two-dimensional lattices [37, 38, 39, 40, 41, 42, 43, 44,
45]. Promising experimental results show quantum interference of topological
states [40], and topological protection of quantum properties, such as
spectral [41] and spatial [42] correlation and entanglement [43, 44, 45].
Quantum topological photonics is thus positioned as a potential solution for
decoherence-free transport of quantum information even at room temperature
[36].

Research in this area has been focused on Fock-like states, while the
interplay between topology and squeezed light [46], which serves as a
fundamental building block for continuous variable quantum information [47,
48, 49], remains less explored. Photon correlations in squeezed light generate
lower than vacuum noise levels in one quadrature of the field at the expense
of an increment in the conjugate quadrature. Hence, squeezed light has been
central to important breakthroughs in metrology [50, 51]. Furthermore, these
correlations may present themselves between distinct modes of the field,
giving rise to entangled multi-mode squeezed states. In particular, two-mode
squeezed light allows for deterministic generation of entanglement [49, 52],
and in the high squeezing limit correspond to optic EPR states [47], acquiring
a central role in quantum information protocols, such as quantum teleportation
[53, 54] and quantum computation [55, 56]. However, the correlated nature of
these states makes them sensitive to losses and fabrication imperfections in
optic systems [57, 58]. It is therefore of high interest to elucidate the
effect of a lattices topology on squeezed light, in the search for topological
protection of squeezing.

Here we present a thorough report on the propagation of squeezed light in a
topological photonic lattice, and the effects of topology on quantum features
of light111After submission of this work, a related study was reported [59]..
We study the behavior of photon statistics, squeezing and entanglement in a
one-dimensional Su-Schrieffer-Heeger (SSH) [60, 61, 15] waveguide array when
exciting its topologically protected edge state with quadrature squeezed
light, employing analytic and numerical techniques. Our focus is to determine
the role that the lattices topology plays in this phenomena, establishing an
interplay between lattice symmetries and quantum correlations. To this goal we
benchmark our results with those of an impurity induced, topologically trivial
localized state, and study their response to disorder. A first conclusion that
we extract is that quantum properties inherit the localization of the edge
state and therefore follow its fate when disorder is introduced. But since
mere localization may be generated and preserved under topologically trivial
circumstances, one may wonder if any other more remarkable consequence of
topology or topological advantage exists. Interestingly, we find that the
phase of the squeezed quadrature is preserved in the topological state,
providing an advantage for propagation of squeezed light in photonic lattices
that may not be replicated by other topologically trivial localized states.

The article is organized as follows. In section 1 we present the SSH system
and briefly discuss its topological properties. We then study the system when
the edge state is excited with single-mode squeezed light under two different
scenarios. In section 2.1, we apply the squeezing operator on the localized
eigenmode, studying the distribution of squeezing across the lattice; while in
section 2.2, pursuing a more feasible method of squeezing the edge state, we
inject single mode squeezed light to the edge waveguide and study its
propagation. During section 3 we turn to the behavior of two-mode squeezing in
the topological lattice. We start by addressing the relation of two-mode
squeezing and entanglement (section 3.1), and later study the propagation of
two-mode squeezing along the lattices (section 3.2). Finally, in section 4 we
present an application of our results in a quantum teleportation protocol,
where the entanglement resource is topologically protected, which serves as a
proof of concept of the relevance of topological protection of quantum
information upon practical implementations. The article ends with final
remarks and perspectives on section 5.

##  1 The SSH lattice

The SSH system consists of a one dimensional dimerized chain of identical
modes, with a uniform onsite term $\beta$ across the entire lattice and
alternating hopping amplitudes $u$ and $v$, described by the hamiltonian

$\begin{split}\mathcal{H}_{\text{top}}=&\sum_{n}\beta
a_{2n}^{\dagger}a_{2n}+\beta a_{2n+1}^{\dagger}a_{2n+1}+\\\
&\left(ua_{2n}^{\dagger}a_{2n+1}+va_{2n+1}^{\dagger}a_{2n+2}+\text{h.c.}\right)\text{
,}\end{split}$ (1)

where $a_{n}$ is the annihilation operator at mode $n$. The system is formed
by two sublattices, even and odd modes, with a unit cell containing sites
$a_{2n}$ and $a_{2n+1}$. For a waveguide array, the operators $a_{n}$ are
bosonic as each mode represents a waveguide, and $\beta$ corresponds to the
propagation constant of the bare waveguides. We take $u$ and $v$ of the same
sign. This choice is physically motivated as the hoppings are typically
determined by the overlap of the evanescent tails of the modes located at each
waveguide [62]. The system presents a bulk bandgap of $|u-v|$, however,
exponentially localized states with a propagation constant equal to that of
the bare waveguides and perfect sublattice polarization appear at the edge of
a finite lattice terminated in a weak coupling. These states are topologically
protected by chiral symmetry [63], which is broken by onsite terms differing
on each sublattice or couplings between sites of the same sublattice for
example. Therefore, introducing disorder to the nearest neighbor hoppings
preserves the propagation constant of the edge state, and its sublattice
polarization. On the other hand, for symmetry breaking onsite disorder, the
topological phase of the lattice is destroyed and these properties fluctuate.
We note as well that perfect sublattice polarization can only arise for a
semi-infinite lattice as in the finite case there is always a small coupling
between states at opposite edges.

For a semi-infinite SSH lattice as depicted in Fig. 1-(a), with $|u|<|v|$, the
topological edge state has annihilation operator

$A_{\text{top}}=\sqrt{1-\alpha^{2}}\sum_{n\geq 0}(-\alpha)^{n}a_{2n}\text{ ,}$
(2)

where $\alpha=u/v$, and the absence of odd modes in $A_{\text{top}}$ reflects
sublattice polarization. The amplitude of the topological state at each
waveguide is shown in Fig. 1-(b).

Figure 1: Topological and impurity systems (without disorder) considered
through our work. The latter is used to benchmark our results for the
topological system. (a) Scheme of the lattices. Sites are represented by grey
circles and hoppings by the connecting lines. Unit cell is highlighted in
purple. (b) Localized eigenstates probability distribution hosted by each
lattice. The states show the same overall exponential decay when adjusting for
the different lattice constants, and differ by the sublattice polarization of
the topological state.

To benchmark our results we need a reference system. Since we are interested
in isolating the effect of topology, a natural candidate is a system hosting a
topologically trivial edge state. This motivates us to also consider a non-
dimerized one-dimensional chain of identical modes, with an onsite impurity
$\epsilon_{0}$, which induces an exponentially localized mode with no
topological protection. The hamiltonian of the semi-infinte topologically
trivial lattice, with the impurity located at its edge, is

$\begin{split}\mathcal{H}_{\text{impurity}}=\,&\epsilon_{0}a_{0}^{\dagger}a_{0}\,+\\\
&\sum_{n\geq 0}\beta
a_{n}^{\dagger}a_{n}+\left(wa_{n}^{\dagger}a_{n+1}+\text{h.c.}\right)\text{
,}\end{split}$ (3)

with $w$ the hopping amplitude between neighboring waveguides. The localized
state of this system is given by its annihilation operator

$\begin{split}A_{\text{impurity}}&=\sqrt{1-(w/\epsilon_{0})^{2}}\sum_{n\geq
0}(w/\epsilon_{0})^{n}a_{n}\\\ &=\sqrt{1-\alpha^{2}}\sum_{n\geq
0}(-\alpha)^{n}a_{n}\text{ ,}\end{split}$ (4)

where we have chosen $\epsilon_{0}=-w/\alpha$ so both localized states present
the same spatial distribution, with expception of sublattice polarization. We
also set $w=v\alpha/(1-\alpha)$ so both states present the same spectral
separation to the bulk bands. The semi-infinite topologically trivial lattice
is depicted in Fig. 1-(a), and the amplitude of its edge state in each
waveguide is shown in Fig. 1-(b). In contrast to the topological edge state,
the impurity induced one does not present any symmetry protected properties,
so its propagation constant and spatial distribution will vary upon any type
of disorder present in the lattice.

##  2 Single-mode squeezing in a topological state

We now study the behavior of single-mode squeezed light coupled to the
topological state under two perspectives: In section 2.1 we consider the state
of the system $|\psi\rangle$ as single-mode squeezed vacuum of the eigenmode,
that is $|\psi\rangle=S_{A_{top}}(\xi)|0\rangle$, with $S_{a}(\xi)$ the
single-mode squeezing operator of mode $a$, given by

$S_{a}(\xi)=\exp\left[\frac{1}{2}\left(\xi^{*}a^{2}-\xi(a^{\dagger})^{2}\right)\right]\text{
,}$ (5)

with $\xi$ the squeezing parameter. Experimentally it would be very
challenging to squeeze a collective mode of the lattice; nevertheless, this
scenario provides insight on how quantum properties present themselves when
exciting solely the topological state. In section 2.2 we turn to a more
feasible scenario, where we excite only the edge waveguide with single-mode
squeezed light, and study its evolution.

In both cases, we study the distribution of one- and two-mode squeezing across
the lattice, and their response to disorder, comparing with the topologically
trivial state. We calculate one-mode squeezing in $dB$ at waveguide $n$
following

$S_{n}=\min_{\phi}\left[10\log_{10}\left(\frac{\langle(\Delta
X_{n}(\phi))^{2}\rangle}{\langle(\Delta
X)^{2}\rangle_{\text{vacuum}}}\right)\right]\text{ ,}$ (6)

where $\langle(\Delta X)^{2}\rangle_{\text{vacuum}}=1/4$ is the variance of
the quadratures for a vacuum state, and $\langle\left(\Delta
X_{n}(\phi)\right)^{2}\rangle=\langle X_{n}(\phi)^{2}\rangle-\langle
X_{n}(\phi)\rangle^{2}$ is the variance of the field in the $\phi$ rotated
quadrature at mode $n$, defined by

$X_{n}(\phi)=\frac{1}{2}\left[e^{-i\phi}a_{n}+e^{i\phi}a_{n}^{\dagger}\right]\text{
.}$ (7)

Non-zero squeezing is obtained when the field presents fluctuations lower than
those of vacuum on any quadrature.

Two-mode squeezing is observed in the superposition of quadratures of both
modes. We thus define the two-mode $\phi$ rotated quadrature between
waveguides $n$ and $m$ [65, 64]

$X_{n,m}(\phi)=\frac{1}{\sqrt{2}}\left[X_{n}(\phi)+X_{m}(\phi)\right]\text{
,}$ (8)

for which two-mode squeezing is calculated in an analogous form to Eq. (6).

When introducing disorder, we construct a lattice with hopping (onsite)
disorder $d$ by sampling each hopping (onsite) term from a uniform
distribution of width $d$, centered at the pristine value of the parameter.
The results presented for disordered systems correspond to the average over 50
mean values, each of them obtained after averaging the results for six
different (a sextet) random realizations of the disordered systems.

###  2.1 Squeezing the eigenstate

We consider semi-infinite lattices, and express their edge states as a linear
combination of the operators at each waveguide, $A=\sum c_{n}a_{n}$. For the
pristine systems the coefficients $c_{n}$ are given by Eq. (2) or (4)
depending on the situation, while for disordered systems they are obtained
numerically. For the state $|\psi\rangle=S_{A}(\xi)|0\rangle$ we find the
following expressions for one- and two-mode squeezing:

$\displaystyle S_{n}$
$\displaystyle=10\log_{10}\left(1-2|c_{n}|^{2}e^{-|\xi|}\sinh|\xi|\right)\text{
,}$ (9) $\displaystyle S_{n,m}$
$\displaystyle=10\log_{10}\left(1-|c_{n}+c_{m}|^{2}e^{-|\xi|}\sinh|\xi|\right)\text{
.}$ (10)

Figure 2: Distribution of squeezing when squeezing the localized eigenstates,
for $\alpha=0.3$ and $\xi=0.9$. Solid lines with markers show the statistical
average value over disordered systems, and dashed lines represent the
confidence interval associated with the standard deviation. Top (bottom) row
shows one-mode (two-mode) squeezing. The waveguides at which the observations
are made are shown in the legend, with T (I) indicating topological (impurity)
state. Zero squeezing at $S_{1}$(T) reflects sublattice polarization of the
topological state, which is broken when introducing onsite disorder as shown
in the top right panel. The bottom panels show that two-mode squeezing is
sensitive to the phase relation between the modes, obtaining higher squeezing
between waveguides 0 and 4 (2) for the topological (impurity) state.

Our results in Eq. (9) and (10), shown in Fig. 2 as disorder is introduced,
indicate that squeezing decays when distancing from the edge, following the
exponential decay of the eigenstates. For the pristine lattices, because the
topological state at even waveguides shows the same spatial distribution than
the impurity induced state, the measured quantities are identical in both
systems, but the values obtained at site $2n$ of the topological lattice are
obtained at site $n$ of the impurity one. Furthermore, Eq. (10) reveals that
two mode squeezing is sensitive to the phase relation between the individual
modes, generating higher squeezing between in-phase modes, being $S_{0,4}$
($S_{0,2}$) the highest, and $S_{0,2}$ ($S_{0,1}$) the lowest for the
topological (impurity induced) state, as shown in Fig. 2. For the topological
lattice, the absence of one-mode squeezing at odd waveguides reflects
sublattice polarization of the state, which is broken when introducing onsite
disorder, as occurs in Fig. 2, top right panel. We observe as well that for
high disorder values the states partially delocalize, which is reflected in a
decreasing magnitude of one-mode squeezing, while they also show larger
statistical fluctuations, represented by wider confidence intervals. Until
now, the propagation constant does not seem to take part in the behavior of
the systems. In order to observe its effect, we must study the propagation of
the states.

###  2.2 Propagation of single-mode squeezing

Since squeezing a collective mode of the system might be experimentally very
challenging, we now consider a more feasible state of the field, generated by
exciting only the edge waveguide with single mode squeezed vacuum, that is
$|\psi(z=0)\rangle=S_{a_{0}}(\xi)|0\rangle$, and study its evolution along the
propagation axis $z$ (see appendix A for numerical methods). We work in a
rotated frame, where the maximally squeezed quadrature of the edge waveguide
of the pristine lattices corresponds to $X(\phi=0)$ along the entire
propagation. We report mean photon number at each quadrature, squeezing at the
orthogonal quadratures $X_{1}=X(0)$ and $X_{2}=X(\pi/2)$, and maximal
squeezing at any rotated quadrature. To maintain this scenario as simple as
possible, we consider finite lattices of 15 waveguides each.

The propagation of the excitation across the pristine lattices is shown in
Fig. 3, top row, depicting the mean photon number in each waveguide as both
systems evolve. Part of the excitation remains permanently localized near the
edge due to coupling with the localized states, generating a distribution of
squeezing qualitatively equal to the one presented in section 2.1: Photon
number and one-mode squeezing decay while distancing from the edge, respecting
sublattice polarization in the topological lattice (see also Fig. 7 in
appendix B). Maximum two-mode squeezing is obtained between waveguides $0$ and
$4$ ($2$) in the topological (impurity) state (see Fig. 8 in appendix B).
These results show that when injecting one-mode squeezing to the edge
waveguide, not only does the excitation remain localized, but the quantum
properties of the field inherit the shape of the edge state. This is
particularly clear when analyzing two-mode squeezing, which proves to be
sensitive to the phase relation between the individual modes. We also observe
that part of the excitation does not couple to the edge state, but transports
across the lattice, which is explained by the non-perfect overlap between the
edge state and the zero-th waveguide.

When introducing disorder we do not observe a significant change in the mean
photon number of the localized portion of the excitation, which is coupled to
the edge state, for neither type of disorder. On the other hand, as shown in
Fig. 3, transport of the decoupled excitation is degraded due to Anderson
localization of the extended states. In the topological lattice, this causes a
persistent non-zero photon number at odd waveguides, even for symmetry
preserving hopping disorder.

Figure 3: Photon number $\langle N\rangle$ for $\alpha=0.3$ and $\xi=0.9$, in
units relative to the initial single-mode squeezed state. Top, middle and
bottom row correspond to pristine, hopping disordered ($d=0.6v$) and onsite
disordered ($d=0.6v$) lattices respectively, while left and right columns
correspond to the topological and impurity systems respectively. Part of the
excitation couples to the edge state and remains localized, reflected by a
persistent high photon number in the zero-th waveguide. The other part of the
excitation transports across the pristine lattices towards the opposite
border. Disorder induced Anderson localization obstructs transport across the
bulk of the lattices. Figure 4: Single-mode squeezing at waveguide zero, for
$\alpha=0.3$ and $\xi=0.9$. (a) Solid blue (red) curves with circular (square)
marker show squeezing at the maximally squeezed quadrature for the topological
(impurity) state. Dashed (dash-dotted) curves show squeezing at quadrature
$X_{1}$ ($X_{2}$), following the same color assignment as maximal squeezing.
The maximally squeezed quadrature of the pristine systems is always $X_{1}$,
as the dashed curves overlap with the solid ones. The maximally squeezed
quadrature of the topological state remains $X_{1}$ as long as chiral symmetry
is preserved (hopping disorder), but rotates when it is broken (onsite
disorder). The maximally squeezed quadrature of the impurity state rotates
upon any form of disorder. Fluctuations in the rotation angle of the squeezed
quadrature results in no average squeezing at $X_{1}$ and $X_{2}$. Panels (b)
to (g) show the Wigner function of the system at the zero-th waveguide (See
movie 1 for the evolution of the average Wigner functions). (b) Input single-
mode squeezed state, squeezed in $X_{1}$. (c) One realization of the hopping
disordered topological system at $z=10v^{-1}$, which remains squeezed in
$X_{1}$. (d) Average Wigner function for the hopping disordered topological
state at $z=10v^{-1}$, showing non-zero squeezing as all realizations remain
squeezed in $X_{1}$ due to topological protection. (e) and (f) Two
realizations of the hopping disordered impurity system at $z=10v^{-1}$, which
are squeezed at different rotated quadratures. (g) Average Wigner function for
the hopping disordered impurity state at $z=10v^{-1}$, showing zero average
squeezing as the squeezed quadrature differs between different random
realizations.

The most remarkable consequences of topological protection, or lack of it, are
observed when analyzing one-mode squeezing at the edge waveguide, as shown in
Fig. 4-(a). We observe that maximal squeezing in the pristine lattices, which
is always obtained in quadrature $X_{1}$, has an approximately equal value in
the topological and impurity state. The role of topology in the propagation of
squeezed light is revealed when introducing hopping disorder. Maximal
squeezing appears to be insensitive to disorder, however, a drastically
different behavior is observed for squeezing at quadratures $X_{1}$ and
$X_{2}$. Because hopping disorder preserves chiral symmetry, the propagation
constant of the topological state remains equal to that of the bare
waveguides, therefore its maximally squeezed quadrature continues to be
$X_{1}$ along the entire propagation (Fig. 4-(b)–(d)). In contrast, random
disorder generates fluctuations in the propagation constant of the impurity
state, rotating the maximally squeezed quadrature. Even though the behavior of
maximal squeezing remains similar to that of the pristine lattice, the
rotation angle of the squeezed quadrature differs between different random
realizations, thus when averaging them at a fixed quadrature, coherence of the
quantum state is lost, and no squeezing is obtained (Fig. 4-(e)–(g)).

This discussion is consistent with the results for the onsite disordered
lattices. Just as occurred for hopping disorder, the behavior of maximal
squeezing is similar to the pristine case. Nevertheless, introduction of
onsite disorder breaks chiral symmetry in the SSH lattice, therefore its edge
state looses topological protection, and its propagation constant ceases to be
topologically locked. The fluctuations in the propagation constant between
different random realizations causes different rotation angles of their
maximally squeezed quadratures, obtaining no squeezing when averaging over
them at a fixed quadrature. These results show that when propagating squeezed
light through a localized state in a photonic lattice, topological protection
of the state allows for phase coherence in the squeezed quadrature.

##  3 Two-mode squeezing of topological states

Having elucidated the role of topology in the propagation of single-mode
squeezed light, we now study two-mode squeezing coupled to respective
localized states. In contrast to single-mode squeezing, now the main quantum
correlations occur between different modes, giving rise to entanglement. As
seen in the previous section, the main consequence of the lattices topology is
protection of the squeezed quadrature. We therefore begin this section by
discussing the effects of rotating the squeezed quadrature on the generation
of two-mode squeezing and entanglement for a two-mode squeezed vacuum state
(section 3.1). We then study the propagation of two-mode squeezing across a
topological lattice, when the edge waveguides hosting the localized states are
initially squeezed (section 3.2).

###  3.1 Two-mode squeezing and entanglement

In this section we introduce some standard definitions. Although this can be
found in Refs. [47, 53, 66], we briefly introduce them here to make the
discussion self-contained and prepare the reader for the interpretation of our
results showed in section 3.2.

The two-mode squeezed vacuum state between modes $a$ and $b$ of the field is
defined by $|\xi_{a,b}\rangle=S_{a,b}(\xi)|0\rangle$, with $S_{a,b}$ the two-
mode squeezing operator

$S_{a,b}(\xi)=\exp\left[\xi^{*}ab-\xi a^{\dagger}b^{\dagger}\right]\text{ .}$
(11)

For such a state no one-mode squeezing is measured in the individual modes, as
$\langle(\Delta X_{\mu}(\phi))^{2}\rangle\geq\langle(\Delta
X)^{2}\rangle_{\text{vacuum}}$, with $\mu=a,b$. On the other hand, the
variance of the two-mode quadrature is $\langle(\Delta
X_{a,b}(\phi))^{2}\rangle=\left[\cosh 2|\xi|-\sinh
2|\xi|\cos(\theta-2\phi)\right]/4$, with $\xi=|\xi|e^{i\theta}$, thus maximal
squeezing is measured at $\phi=\theta/2$.

We quantify entanglement employing the Peres-Horodecki criterion for
continuous variables [66, 47], stating that the degree of non-separability is
measured by the negativity of the partially transposed density matrix.
Implementation of this criterion is as follows: We begin defining the $4\times
4$ correlation matrix $V$ by
$V_{i,j}=\langle\hat{\chi}_{i}\hat{\chi}_{j}+\hat{\chi}_{j}\hat{\chi}_{i}\rangle/2$,
with $\hat{\chi}=(X_{a}(0),X_{a}(\pi/2),X_{b}(0),X_{b}(\pi/2))$ the basis of
the phase-space representation of the system. The partially transposed
correlation matrix is obtained by $\tilde{V}=\Gamma V\Gamma$, with
$\Gamma=\text{diag}(1,1,1,-1)$, which acts by flipping the sign of
$X_{b}(\pi/2)$. Finally, negativity of the partially transposed density matrix
is determined by $\tilde{V}-\Lambda\ngeq 0$, with
$\Lambda_{i,j}=\langle\hat{\chi}_{i}\hat{\chi}_{j}-\hat{\chi}_{j}\hat{\chi}_{i}\rangle/2$.
Therefore, the minimum eigenvalue of $\tilde{V}-\Lambda$ serves as an
entanglement measure, revealing non-separability when it takes negative
values. For two-mode squeezed vacuum, the correlation matrix takes the form
$V=\mathcal{R}(\theta)V_{0}\mathcal{R}^{\dagger}(\theta)$, with

$V_{0}=\frac{1}{4}\begin{pmatrix}\cosh 2|\xi|&0&-\sinh 2|\xi|&0\\\ 0&\cosh
2|\xi|&0&\sinh 2|\xi|\\\ -\sinh 2|\xi|&0&\cosh 2|\xi|&0\\\ 0&\sinh
2|\xi|&0&\cosh 2|\xi|\end{pmatrix}\text{ ,}$ (12)

and $\mathcal{R}(\theta)=\text{diag}(R(\theta/2),R(\theta/2))$, with $R$ the
$2\times 2$ rotation matrix. We observe that, in agreement with the squeezing
measurement, any two-mode squeezed vacuum state can be represented as a
$\theta=0$ state upon a rotation of the phase space basis in $\theta/2$.
Logically, the degree of entanglement between the modes does not depend on the
basis choice of the system, but, as we will see next, the entangled variables
do depend on this.

The Wigner function of the state is a multivariate gaussian distribution

$W_{a,b}(\chi)=\frac{|V|^{-1/2}}{(2\pi)^{2}}\exp\left[-\frac{1}{2}\chi^{T}V^{-1}\chi\right]\text{
,}$ (13)

with $\chi=(q_{a},p_{a},q_{b},p_{b})$ the corresponding eigenvalues of
$\hat{\chi}$. For $\theta=0$

$\begin{split}W_{a,b}(\chi)&=\frac{4}{\pi^{2}}\exp\left\\{-e^{2|\xi|}\left[(q_{a}+q_{b})^{2}+(p_{a}-p_{b})^{2}\right]\right.\\\
&\left.-e^{-2|\xi|}\left[(q_{a}-q_{b})^{2}+(p_{a}+p_{b})^{2}\right]\right\\}\text{
.}\end{split}$ (14)

Taking the infinite squeezing limit $|\xi|\rightarrow\infty$, which is
unphysical as such a state would require infinite energy, but represents a
perfectly correlated maximally entangled state, the Wigner function approaches
$W\propto\delta(q_{a}+q_{b})\delta(p_{a}-p_{b})$. From this we conclude that
the entangled variables of the system are $q_{a}$ and $q_{b}$; and $p_{a}$ and
$p_{b}$.

If $\theta\neq 0$, the expression for the Wigner function of the state would
be the same as in Eq. (14), but for a rotated phase space basis, that is
$q_{\mu}\rightarrow q_{\mu}\cos(\theta/2)+p_{\mu}\sin(\theta/2)$, and
$p_{\mu}\rightarrow p_{\mu}\cos(\theta/2)-q_{\mu}\sin(\theta/2)$. Thus, the
entangled variables correspond to a rotation of the basis,
$\mathcal{R}(\theta)\chi$. From this analysis we conclude that, for a two-mode
squeezed vacuum state, rotation of the squeezed quadrature does not affect the
magnitude of entanglement in the system, but does change the entangled
variables. With this in mind, we now study the propagation of two-mode
squeezed light in the topological SSH lattice.

###  3.2 Propagation of two-mode squeezing

We consider two independent SSH lattices of 15 waveguides each, which we label
as lattices A and B, each one hosting a topologically protected edge state,
and study the propagation of two-mode squeezing along the system. For
comparison, we also consider topologically trivial lattices hosting impurity
induced edge states. The initial state of the field is given by the two-mode
squeezed vacuum state between the edge waveguide of both lattices, that is
$|\psi(z=0)\rangle=S_{a_{0},b_{0}}(\xi)|0\rangle$. We explore the response of
the systems to hopping and onsite disorder, while reporting two-mode squeezing
and entanglement between both edge waveguides.

Fig. 5, left column, shows the evolution of two-mode squeezing along the
lattices, whose results agree with those of section 2: Maximal squeezing
measured in both systems has an approximately equal value along the
propagation direction, which is neither sensitive to topology nor disorder.
For the topological system, the maximally squeezed quadrature is $X_{1}$ as
long as chiral symmetry is preserved, because the propagation constant of the
edge state is topologically locked. In contrast, if the symmetry is broken by
onsite disorder, squeezing measured at quadratures $X_{1}$ and $X_{2}$
averages to zero over the random realizations of the system due to
fluctuations in the rotation angle of the maximally squeezed quadrature. On
the other hand, as the impurity induced edge state does not present any
preserved properties, the squeezing values at fixed quadratures are highly
sensitive to both types of disorder.

Figure 5: Two-mode squeezing and entanglement between both edge waveguides,
for $\alpha=0.3$ and $\xi=0.9$. Left column: solid blue (red) curves with
circular (square) marker show squeezing at the maximally squeezed quadrature
for the topological (impurity) state. Dashed (dash-dotted) curves show
squeezing at quadrature $X_{1}$ ($X_{2}$), following the same color assignment
as maximal squeezing. Right column: solid blue (dashed red) curves show
entanglement for the topological (impurity) state, in units relative to the
initial two-mode squeezed vacuum state. Similar to maximal squeezing, the
magnitude of entanglement is not sensitive to the topology of the edge state
nor the type of disorder, but squeezing at quadratures $X_{1}$ and $X_{2}$
reveals topological protection of the maximally entangled variables when
chiral symmetry is preserved.

The behavior of entanglement in the systems, portrayed in Fig. 5, right
column, is qualitatively similar to that of maximal two-mode squeezing: it
takes an approximately constant value along the propagation axis, and does not
appear to be sensitive to disorder. From this perspective, the results
indicate that the total amount of quantum correlations present in the system
are independent of the topology of the involved edge states. However, they
must be interpreted alongside the information provided by squeezing at the
non-rotated quadratures, and the discussion of section 3.1. Despite the fact
that the amount of entanglement is independent of the topology of the edge
state, its topological protection allows to control the maximally entangled
variables, protecting them from any symmetry preserving disorder.

##  4 Quantum teleportation employing a topological lattice

In this section we demonstrate the previously highlighted consequences of the
topological protection of squeezing. For this we implement quantum
teleportation of a single-photon, where the two-mode squeezed states studied
in section 3.2 serve as the shared entanglement resource. As we will show
later in this section, carrying out the teleportation protocol in a
topological lattice turns out to be advantageous compared with a topologically
trivial lattice bearing a localized state. This topological advantage is
quantified by a robust fidelity in the presence of disorder. In the following
we describe the teleportation protocol and expose our main results.

The teleportation protocol has been developed in Ref. [53]. To make the
discussion self-contained, in this paragraph we outline the main steps. We
wish to to transfer an arbitrary quantum state $|\psi_{\text{in}}\rangle$ from
Alice’s location to Bob’s, for which we rely on a shared entanglement resource
between both parties, given by the state $|\psi_{a,b}\rangle$. We must
consider three modes of the field: $a$ and $b$ corresponding to the entangled
modes sent to Alice and Bob respectively, and mode $c$ containing the input
state. Defining $\chi_{\mu}=(q_{\mu},p_{\mu})$ (with $\mu=a,b,c$) the phase
space basis of the respective modes, the Wigner function of the system in this
initial setup is

$W_{0}\left(\chi_{a},\chi_{b},\chi_{c}\right)=W_{a,b}\left(\chi_{a},\chi_{b}\right)W_{\text{in}}\left(\chi_{c}\right)\text{
,}$ (15)

where $W_{a,b}$ and $W_{\text{in}}$ represent the Wigner function of
$|\psi_{a,b}\rangle$ and $|\psi_{\text{in}}\rangle$ respectively. Alice
couples her entangled mode with the input state at a 50:50 beam splitter,
obtaining the Wigner function

$W_{1}\left(\chi_{a},\chi_{b},\chi_{c}\right)=W_{0}\left(\frac{\chi_{a}-\chi_{c}}{\sqrt{2}},\chi_{b},\frac{\chi_{c}+\chi_{a}}{\sqrt{2}}\right)\text{
.}$ (16)

Alice then measures the quadratures corresponding to $q_{a}$ and $p_{c}$,
which become classically determined random variables, following the
probability distribution

$\mathcal{P}(q_{a},p_{c})=\int
dp_{a}\,dq_{b}\,dp_{b}\,dq_{c}\,W_{1}\left(\chi_{a},\chi_{b},\chi_{c}\right)\text{
.}$ (17)

Given a pair of measured values $q_{a}$ and $p_{c}$, the state of the system
at Bob’s mode collapses to

$\begin{split}W_{2}(\chi_{b}&|q_{a},p_{c})=\mathcal{N}\int
dp_{a}\,dq_{c}\,W_{1}(\chi_{a},\chi_{b},\chi_{c})\\\ &=2\mathcal{N}\int
dx\,dy\,W_{\text{in}}\left(x,y\right)\\\
&W_{a,b}\left(\sqrt{2}q_{a}-x,y-\sqrt{2}p_{c},q_{b},p_{b}\right)\text{
,}\end{split}$ (18)

where normalization must be added explicitly as a result of the non-unitary
measurement operation, and we defined $\sqrt{2}x=q_{a}+q_{c}$ and
$\sqrt{2}y=p_{a}+p_{c}$ for the last equality. If the entanglement resource is
non-rotated two-mode squeezed vacuum, then Bob’s output state approaches the
exact input state in the infinite squeezing limit, aside from a phase space
displacement, that is $W_{2}(\chi_{b}|q_{a},p_{c})\rightarrow
W_{\text{in}}(q_{b}+\sqrt{2}q_{a},p_{b}+\sqrt{2}p_{c})$. To correct this, Bob
displaces his mode, obtaining the final output state, given by the Wigner
function

$W_{\text{out}}(q,p|q_{a},p_{c})=W_{2}(q-\sqrt{2}q_{a},p-\sqrt{2}p_{c}|q_{a},p_{c})\text{
.}$ (19)

A scheme of the teleportation circuit can be seen in Fig. 1 of Ref. [53].

For the implementation of this protocol, we consider that the entangled two-
mode squeezed state is sent to each party through a topological or trivial
photonic lattice, as presented in section 3.2. Thus, at the output of lattice
A, Alice extracts her share of the entangled state from the edge waveguide of
the lattice, tracing out all other modes; and so does Bob with lattice B.
Finally, we take the input state as a single-photon,
$|\psi_{\text{in}}\rangle=c^{\dagger}|0\rangle$. Note however, that the
teleportation protocol is valid for any input, which might even be unknown to
our protagonists.

Figure 6: Panel (a) shows input Wigner function to be teleported,
$W_{\text{in}}$. Panels (b) and (d) ((c) and (e)) show the averaged output
Wigner function $\mathcal{W}_{\text{out}}$ for the pristine and hopping
disordered $d=0.3v$ topological (topologically trivial) lattices. Wigner
functions are expressed in units relative to the peak value of
$W_{\text{in}}$. The fidelity of the output states is indicated on each panel,
and the colorbar on the right applies for panels (b)–(e). Teleportation in the
topological system proves to be robust against hopping disorder, preserving
the fidelity of the output state, and preserving the quantum nature of the
single-photon input state represented by negative values in the Wigner
function.

We perform the teleportation for pristine and hopping disordered lattices
($d=0.3v$). The fidelity of a single teleportation event is calculated by
$F=\pi\int dq\,dp\,W_{\text{in}}(q,p)W_{\text{out}}(q,p|q_{a},p_{c})$. Since
the output state depends on the measured values of $q_{a}$ and $p_{c}$, and on
the random disorder present in the lattice, we report the average fidelity
over all measurements, and all random realizations, that is

$\mathcal{F}=\frac{1}{N}\sum_{n=1}^{N}\int dq_{a}\,dp_{c}\mathcal{P}F\text{
,}$ (20)

where $N=300$ is the total number of random realizations. We may construct an
averaged output Wigner function $\mathcal{W}_{\text{out}}$ defined by:

$\begin{split}\mathcal{W}_{\text{out}}(q,p)=\frac{1}{N}\sum_{n=1}^{N}&\int
dq_{a}\,dp_{c}\,\mathcal{P}(q_{a},p_{c})\\\
&W_{\text{out}}(q,p|q_{a},p_{c})\text{ ,}\end{split}$ (21)

so the average fidelity is $\mathcal{F}=\pi\int
dq\,dp\,W_{\text{in}}\mathcal{W}_{\text{out}}$. Even though the output state
never takes the exact form of $\mathcal{W}_{\text{out}}$, it allows to
visualize the proximity of the output state to the input one in the general
case.

When teleporting the single photon state employing the pristine lattices, we
find practically identical results between the topological and trivial
lattices, as shown in Fig. 6-(b) and (d), characterized by an average fidelity
of $0.483$ and $0.486$ respectively. Comparing with the input Wigner function
(Fig. 6-(a)), we observe that the fidelity losses are mainly explained by a
reduction of the peak values, and a consequent widening of the function.
Negativity of the output function serves as a landmark of the preservation of
the input state’s quantum nature, which can only be achieved by means of
quantum teleportation schemes. Quantitatively, the reduction of the negative
peak at the origin with respect to the input state is of $13.9\%$ and $14.3\%$
in the topological and trivial systems respectively.

The averaged output state of the topological system remains practically
unchanged when introducing hopping disorder, as shown Fig. 6-(c), maintaining
the fidelity of the pristine lattice. The output Wigner function continues to
take negative values, with a peak reduction of $14.1\%$, proving that
teleportation in the topological system robustly preserves the quantum
coherence of the input state. In contrast, teleportation in the topologically
trivial system rapidly deteriorates as disorder is added. Even for low levels
of disorder, the average fidelity falls to 0.242, roughly half of that of the
the pristine system, and even though the output function presents a local
minimum at the origin, it does not take negative values.

The drastic difference in the teleportation results between the topological
and trivial systems is a consequence of topological protection of the phase of
the squeezed quadrature. Transmission of the entangled two-mode state to the
parties through a topological state ensures that Alice measures the maximally
entangled variables, regardless of the magnitude of hopping disorder present
in the lattice, resulting in robust preservation of the quantum coherence of
the input state.

##  5 Final remarks

We have studied the propagation of squeezing and entanglement along a
topological SSH waveguide array, focusing on the effects of the lattices
topology in this phenomena. We found that topological ordering of the lattice
robustly preserves the phase of the squeezed quadrature when propagating
squeezed light through a topologically protected localized state, for both
single- and two-mode squeezed light. We also showed that the distribution of
squeezing across the lattice inherits the spatial distribution of the
localized state to which it is coupled, inheriting its topological protection
as well. We discussed the interplay between entanglement, squeezing, and the
phase of the squeezing; concluding that the topological phase of the lattice
fixes the entangled variables. We finally provided a practical implementation
of this system in a quantum teleportation protocol, where transmission of the
entanglement resource to the parties through a topological lattice grants
robustness and preservation of the quantum coherence of the teleported state.

We expect our findings on topological protection of squeezing to be of
interest to any development in continuous variable quantum information, mainly
as an alternative to protecting quantum coherence in photonic platforms.
However, exploitation of topological phenomena in quantum optic systems, in
particular continuous variable ones, is only recent, and its full
possibilities are yet to be discovered. Our research is based on the SSH
lattice, whose topological phase is broken by onsite disorder, which is always
present to some extent in realistic devices; and being a one-dimensional
lattice, this system is particularly sensitive to Anderson localization. Thus,
a natural interest arises in studying the behavior of quantum light in two-
dimensional lattices, with propagating edge states which enjoy better
topological protection.

##  6 Acknowledgments

We thank P. Solano for useful comments and suggestions. J.M.D. is supported by
CONICYT Grant CONICYT-PFCHA/MagísterNacional/2019-22200526. C.H-A.
acknowledges support by CONICYT-PAI grant N°77180003, FONDECYT grant
N°11190078, and ANID - Millenium Science Inititive Program - ICN17_012.
L.E.F.F.T. acknowledges the support of FONDECYT grant N° 1211038, the Abdus
Salam International Centre for Theoretical Physics and the Simons Foundation.

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##  Appendix A Numerical simulation of propagation of squeezed states

In this section we detail how the propagation of squeezed states is realized
by numerical simulation. Our method relies on the fact that squeezed vacuum
states are Gaussian with zero mean value of the quadratures, and are therefore
fully described by their correlation matrix, which can be obtained from
matrices $N$ and $M$ defined by their matrix elements $N_{n,m}=\langle
a_{n}^{\dagger}a_{m}\rangle$ and $M_{n,m}=\langle a_{n}a_{m}\rangle$. Thus the
problem reduces to simulating the evolution of both matrices.

Let $\hat{H}$ be the hamiltonian operator of the system, so the annihilation
operators $a_{n}$ satisfy the Heisenberg equation

$-i\frac{da_{n}}{dz}=\left[\hat{H},a_{n}\right]$ (22)

The solution to this equation is
$a_{n}(z)=\hat{U}^{\dagger}(z)a_{n}(0)\hat{U}(z)$, with
$\hat{U}(z)=\exp(-i\hat{H}z)$ the $z$ evolution operator. We represent the
hamiltonian operator in matrix form, under the basis of the annihilation
operators of the waveguides $a_{n}$, as

$\hat{H}=\sum_{n,m}a_{n}^{\dagger}H_{n,m}a_{m}\text{ .}$ (23)

Let $Q$ be the matrix containing the eigenvectors of $H$ in its columns, so
$Q^{\dagger}HQ=\tilde{H}$, with $\tilde{H}$ the diagonal form of $H$,
containing the propagation constant of each eigenmode. The $z$ evolution
operator is represented in matrix form by $U=Q\exp(-i\tilde{H}z)Q^{\dagger}$,
and the $z$ evolution of the waveguide operators is

$a_{n}(z)=\sum_{m}U_{n,m}(z)a_{m}(0)$ (24)

We finally obtain the evolution of the $N$ and $M$ matrices, described by

$\displaystyle N_{n,m}(z)=\sum_{k,l}U_{n,k}^{*}(z)U_{m,l}(z)N_{k,l}(0)\text{
,}$ (25) $\displaystyle
M_{n,m}(z)=\sum_{k,l}U_{n,k}(z)U_{m,l}(z)M_{k,l}(0)\text{ .}$ (26)

##  Appendix B Extended results of section 2.2

In this section we present a more complete report of the results discussed in
section 2.2 of the main text, that could help the reader follow the
corresponding discussions. For all figures we have used $\alpha=0.3$, and
$\xi=0.9$.

In Fig. 7 we show one-mode squeezing along the propagation axis at waveguides
$1$, $2$ and $4$ of the topological lattice, and waveguides $1$ and $2$ of the
impurity one. These results allow to envision the spatial distribution of one-
mode squeezing across the lattices.

Figure 7: One-mode squeezing along the propagation axis, for $\alpha=0.3$ and
$\xi=0.9$. Top, middle and bottom row shows one-mode squeezing for pristine,
hopping disordered ($d=0.6v$), and onsite disordered ($d=0.6v$) lattices
respectively, while left and right columns refer to the topological and
impurity states respectively. The site at which the measurement is taken is
indicated in the legend, which is valid for the entire column.

In Fig. 8 we show two-mode squeezing along the propagation axis, measured
between the edge waveguide and the first two waveguides that take part in the
localized eigenmode, namely, waveguides 2 and 4 of the topological lattice,
and waveguides 1 and 2 of the impurity one. This shows the effects of the
phase relation between the modes on two-mode squeezing.

Figure 8: Two-mode squeezing along the propagation axis, for $\alpha=0.3$ and
$\xi=0.9$. Top, middle and bottom row shows two-mode squeezing for pristine,
hopping disorder ($d=0.6v$), and onsite disordered ($d=0.6v$) lattices
respectively, while left and right columns refer to the topological and
impurity states respectively. The site at which the measurement is taken is
indicated in the legend, which is valid for the entire column.

##  Appendix C On the statistical fluctuations for Fig. 4 and 5

In this section we refer to the statistical fluctuations for the data
presented in Fig. 4 and 5 of the main text. In Fig. 9 we show maximal one-mode
squeezing corresponding to the data in Fig. 4, and maximal two-mode squeezing
and entanglement corresponding to the data in Fig. 5, along with the
confidence intervals of the corresponding quantities. We observe that after
the transient response of the system when the initial state is injected at the
edge waveguide, the curves of the topological and impurity state take
approximately equal values, falling within the confidence interval of each
other. We also observe that hopping disorder induced fluctuations are
considerably larger than those generated by onsite disorder in both systems.
Furthermore, fluctuations in the hopping disordered topological state are
larger than those of the impurity one. However, the magnitude of the
fluctuations is consistently lower than the magnitude of the physical
quantities themselves in all cases. We have checked as well that increasing
the number of random realizations yields narrower confidence intervals, and
that higher disorder values yield larger confidence intervals, in tune with th
behavior observed in Fig. 2.

Figure 9: Maximal one-mode squeezing corresponding to the data in Fig. 4, and
maximal two-mode squeezing and entanglement corresponding to the data in Fig.
5. Statistical average values are represented by solid lines, while dashed
lines represent the confidence interval associated with the standard
deviation, with blue (red) lines referring to the topological (impurity
state).

##  Appendix D Movie 1: Caption

Evolution of the average Wigner function at the zero-th waveguide when the
lattices are initially excited with single-mode squeezed state, as studied in
section 2.2, for $\alpha=0.3$ and $\xi=0.9$. Top, middle and bottom row refer
to the pristine, hopping disordered ($d=0.6v$), and onsite disordered
($d=0.6v$) lattices respectively, while left and right columns refer to the
topological, and impurity induced (topologically trivial) states respectively.
The bar at the bottom indicates the distance along the propagation axis $z$.
When chiral symmetry is preserved (pristine and hoping disorder) the maximally
squeezed quadrature of the topological state is always $X_{1}$ as its
propagation constant is that of the bare waveguides, therefore the ellipse
remains oriented as the initial state. In contrast, when the symmetry is
broken (onsite disorder), the maximally squeezed quadrature differs between
different random realizations, thus no squeezing is obtained when averaging
over them. Because the impurity induced state does not present any symmetry
protected properties, upon any type of disorder the maximally squeezed
quadrature fluctuates between the different random realizations and no average
squeezing is measured in either case.