# Growing Schrödinger’s cat states by local unitary time evolution of product
states

Saverio Bocini  Université Paris-Saclay, CNRS, LPTMS, 91405, Orsay, France
Maurizio Fagotti<EMAIL_ADDRESS>Université
Paris-Saclay, CNRS, LPTMS, 91405, Orsay, France

###### Abstract

We envisage many-body systems that can be described by quantum spin-chain
Hamiltonians with a trivial separable eigenstate. For generic Hamiltonians,
such a state represents a quantum scar. We show that, typically, a
macroscopically-entangled state naturally grows after a single projective
measurement of just one spin in the quantum scar. Moreover, we identify a
condition under which what is growing is a “Schrödinger’s cat state”. Our
analysis does not reveal any particular requirement for the entangled state to
develop, provided that the quantum scar does not minimise/maximise a local
conservation law. We study two examples explicitly: systems described by
generic Hamiltonians and a model that exhibits a $U(1)$ hidden symmetry. The
latter can be reinterpreted as a 2-leg ladder in which the interactions along
the legs are controlled by the local state on the other leg through
transistor-like building blocks.

Quantum superpositions of two macroscopically distinct states, known as
“Schrödinger’s cat states”, from the famous Gedankenexperiment proposed by
Schrödinger in 1935 [1, 2], or also as “(generalized) GHZ states”, from the
observation by Greenberger, Horne, and Zeilingerin [3] on quantum spin models
escaping the original Bell’s inequalities [4], are precious for their broad
applications, e.g., in quantum metrology and quantum computation, but rare and
generally short-lived for their fragility under real experimental conditions
(which include decoherence, noise, particle loss, etc.). Several protocols
have been proposed to contain the latter problems [5, 6, 7, 8, 9, 10, 11, 12,
13] and some of them have been experimentally realised [14, 15, 16, 17]. While
such an instability undermines applications, it is, in fact, the defining
feature of cat states, or, more generally, of states with extensive
multipartite entanglement [18]. These are somehow unnatural states of matter
in which fundamental physical properties such as cluster decomposition are
lost. Engineering a cat state requires therefore a lot of control of the
system, which, in turn, results in fine-tuned protocols designed with the
clear goal of generating such exceptional states.

We propose a theoretical protocol that stands out for its naturalness, in the
sense that the growth of macroscopic entanglement is a manifestation of an
intriguing physical phenomenon rather than of clever manipulations of a
system. The most general system that we consider is described by a quantum
spin-$\frac{1}{2}$ chain Hamiltonian $\boldsymbol{\mathbf{H}}$ with local
interactions and a trivial separable eigenstate. Without loss of generality,
the latter can be set equal to

$\ket{\Psi(0)}=\ket{\Uparrow}\equiv\ket{\uparrow\cdots\uparrow}\,,$ (1)

where $\ket{\uparrow}$ denotes the eigenvector of the Pauli matrix
$\sigma^{z}$ with eigenvalue $1$ and $\Uparrow$ will represent in this paper a
generic number of adjacent $\uparrow$. Often the presence of such a trivial
eigenstate is a consequence of a $U(1)$ symmetry, e.g., the conservation of
$\boldsymbol{\mathbf{S}}^{z}=\frac{1}{2}\sum_{\ell}\boldsymbol{\mathbf{\sigma}}_{\ell}^{z}$,
where $\boldsymbol{\mathbf{\sigma}}_{\ell}^{\alpha}$ are local operators
acting like the Pauli matrices $\sigma^{\alpha}$ on site $\ell$ and like the
identity elsewhere. As we will see before long, the most interesting cases are
however those in which $\boldsymbol{\mathbf{S}}^{z}$ is not conserved. If
there are no additional relevant conserved operators, either $\ket{\Uparrow}$
is the ground state or it is an exact _quantum scar_ [19, 20, 21]: its
properties contrast with those of the eigenstates with similar energy. Quantum
scars have recently attracted a lot of attention [22, 23, 24], also in
connection with their unusual entanglement properties, displaying low
bipartite entanglement, still potentially featuring extensive multipartite
entanglement [25]. In our case the quantum scar is exceptional, as it is fully
separable.

The basic idea is that the quantum scar is essentially metastable and could be
transmuted into a state with extensive multipartite entanglement just by a
local perturbation. Specifically, we consider the effect of a quantum
measurement of a spin in a tilted direction with respect to $z$, so that the
state is projected into

$\ket{\Psi_{\theta}(0^{+})}=e^{i\theta\boldsymbol{\mathbf{\sigma}}_{0}^{y}}\ket{\Uparrow}=\cos\theta\ket{\Uparrow}+\sin\theta\ket{\Uparrow\downarrow\Uparrow}$
(2)

with probability $\cos^{2}\theta$. Time evolution affects only the second term
on the right hand side of Eq. [2], hence

$\ket{\Psi_{\theta}(t)}=\cos\theta\ket{\Uparrow}+\sin\theta
e^{-i\boldsymbol{\mathbf{H}}t}\ket{\Uparrow\downarrow\Uparrow},\quad t>0\,.$
(3)

The Lieb-Robinson bounds [26] ensure that the perturbation is irrelevant
outside a light cone emerging from the space-time point of the measurement.
Since the state before the measurement is an eigenstate, the measurement
triggers a nontrivial time evolution only within a region whose length grows
linearly with the time. Hence, neglecting the part of the system outside that
region does not significantly affect the purity of the state. We call
$\Omega_{\epsilon}(t)$ the smallest subsystem that contains the spin that is
measured and that, as clarified later, can be considered pure at time $t$ with
accuracy $1-\epsilon$. In the following, most of our considerations, including
the use of terms such as “macroscopic” and “extensive”, will be referred to
subsystem $\Omega_{\epsilon}(t)$. This allows us to treat on the same footing
infinite systems, which are arguably more interesting from a theoretical point
of view, and finite ones (provided that the actual system’s size is larger
than $\Omega_{\epsilon}(t)$), which are instead more relevant for the
experiments. The situation is pictorially described in Fig. 1: after the spin
flip at the origin, the component $\ket{\Uparrow\downarrow\Uparrow}$ of the
state evolves in time, while the component $\ket{\Uparrow}$ stays constant;
the effective size of the system is given by the spins that are inside the
light cone originating from the local perturbation, while the rest of the
spins are neglected.

First we show that, quite generally, in the settings considered the
perturbation has everlasting effects on the expectation values of local
observables arbitrarily far from the position of the measurement. That is to
say,
$e^{-i\boldsymbol{\mathbf{H}}t}\ket{\Uparrow\downarrow\Uparrow}\equiv\ket{\Psi_{\pi/2}(t)}$
is macroscopically different from $\ket{\Uparrow}$, hence there exists an
observable
$\boldsymbol{\mathbf{A}}=\sum_{\ell}\boldsymbol{\mathbf{a}}_{\ell}$, where
$\boldsymbol{\mathbf{a}}_{\ell}$ are finite-support observables, such that
$\braket{\Uparrow}{\boldsymbol{\mathbf{A}}}{\Uparrow}-\braket{\Psi_{\pi/2}(t)}{\boldsymbol{\mathbf{A}}}{\Psi_{\pi/2}(t)}$
is extensive, i.e., it scales linearly with the (effective) system size
$|\Omega_{\epsilon}(t)|$ [27]. Such an effect is already remarkable, as,
generally, local perturbations in quantum many-body systems either fade away
or remain localised. We take it as the starting point of our investigation
into the multipartite entanglement properties of $\ket{\Psi_{\theta}(t)}$. We
remind the reader that a state is said to have extensive multipartite
entanglement if and only if the variance scales as the square of the system’s
length for at least one extensive operator [28, 29, 30]. Being
$\ket{\Psi_{\pi/2}(t)}$ macroscopically different from $\ket{\Uparrow}$, the
reader is already in the position to infer that the projective measurement on
the quantum scar $\ket{\Uparrow}$ will generate a state with extensive
multipartite entanglement. But the situation is even more interesting. We
provide evidence and then conjecture that, if the model has a hidden $U(1)$
symmetry (see below for the definition), $\ket{\Psi_{\pi/2}(t)}$ is not
macroscopically entangled; under those conditions, the projective measurement
generates a genuine Schrödinger cat’s state consisting of an (ideally)
arbitrarily large number of sites.

Figure 1: Cartoon of the protocol. The state with a single spin down time
evolves into a state that is macroscopically different from the pre-
measurement state. If its fluctuations remain ordinary, the resulting
superposition is a Schrödinger’s cat state. The scissors represent the fact
that we ignore the region outside $\Omega_{\epsilon}(t)$.

##  1 The system

We investigate a variety of Hamiltonians with local densities. Our requirement
of $\ket{\Uparrow}$ being eigenstate of the Hamiltonian allows for any kind of
interactions that commute with the total spin in the $z$ direction,
$\boldsymbol{\mathbf{S}}^{z}=\frac{1}{2}\sum_{\ell}\boldsymbol{\mathbf{\sigma}}^{z}_{\ell}$,
(such as the hopping term
$\boldsymbol{\mathbf{\sigma}}_{\ell}^{x}\boldsymbol{\mathbf{\sigma}}_{\ell+n}^{x}+\boldsymbol{\mathbf{\sigma}}_{\ell}^{y}\boldsymbol{\mathbf{\sigma}}_{\ell+n}^{y}$
or the longitudinal Dzyaloshinskii–Moriya interaction
$\boldsymbol{\mathbf{\sigma}}_{\ell}^{x}\boldsymbol{\mathbf{\sigma}}_{\ell+n}^{y}-\boldsymbol{\mathbf{\sigma}}_{\ell}^{y}\boldsymbol{\mathbf{\sigma}}_{\ell+n}^{x}$)
but not only that. The constraint is much weaker than a $U(1)$ symmetry and
the Hamiltonian density can feature any term of the form
$\frac{\boldsymbol{\mathbf{1}}-\boldsymbol{\mathbf{\sigma}}_{\ell}^{z}}{2}\boldsymbol{\mathbf{O}}_{\ell}\frac{\boldsymbol{\mathbf{1}}-\boldsymbol{\mathbf{\sigma}}_{\ell}^{z}}{2}$,
where $\boldsymbol{\mathbf{O}}_{\ell}$ is a local operator with support in a
region around $\ell$. Among the allowed operators, those with a hidden $U(1)$
symmetry (such as
$\boldsymbol{\mathbf{\sigma}}_{\ell-1}^{x}(\boldsymbol{\mathbf{1}}-\boldsymbol{\mathbf{\sigma}}_{\ell}^{z})\boldsymbol{\mathbf{\sigma}}_{\ell+1}^{x}$
or
$\boldsymbol{\mathbf{\sigma}}_{\ell-1}^{x}(\boldsymbol{\mathbf{\sigma}}_{\ell}^{x}\boldsymbol{\mathbf{\sigma}}_{\ell+1}^{x}+\boldsymbol{\mathbf{\sigma}}_{\ell}^{y}\boldsymbol{\mathbf{\sigma}}_{\ell+1}^{y})\boldsymbol{\mathbf{\sigma}}_{\ell+2}^{x}$)
will play a special role.

###  1.1 Hidden $U(1)$ symmetry

The first class of Hamiltonians we consider has recently sparked some
attention because it describes systems that are macroscopically sensitive to
local perturbations [31, 32]. Such a sensitivity is triggered by so-called
“semilocal conservation laws”, whose densities act as local observables only
in a restricted space of operators with a particular symmetry. This unusual
property could seem, at first glance, innocuous, but it allows the state to
retain memory of so-called “string order” [33, 34, 35], making in turn
symmetry-protected topological order survive the limit of infinite time [32].
An example of operator with a semilocal density in a system that is symmetric
under the spin flip
$\boldsymbol{\mathbf{\sigma}}_{\ell}^{x,y}\rightarrow-\boldsymbol{\mathbf{\sigma}}_{\ell}^{x,y}$
is

$\tilde{\boldsymbol{\mathbf{S}}}^{z}=\frac{1}{2}\sum_{\ell}\boldsymbol{\mathbf{\Pi}}^{z}(\ell)\,,$
(4)

where $\boldsymbol{\mathbf{\Pi}}^{z}(\ell)$ can be thought of as a semi-
infinite string of $\boldsymbol{\mathbf{\sigma}}^{z}$ and is such that
$[\boldsymbol{\mathbf{\Pi}}^{z}(\ell)]^{2}=\boldsymbol{\mathbf{I}}$,
$[\boldsymbol{\mathbf{\Pi}}^{z}(\ell),\boldsymbol{\mathbf{\sigma}}_{j}^{z}]=0$
and
$\boldsymbol{\mathbf{\Pi}}^{z}(\ell)\boldsymbol{\mathbf{\sigma}}_{j}^{x,y}=\mathrm{sgn}(\ell-j-\frac{1}{2})\boldsymbol{\mathbf{\sigma}}_{j}^{x,y}\boldsymbol{\mathbf{\Pi}}^{z}(\ell)$.
In spin-flip invariant settings, the density
$\boldsymbol{\mathbf{\Pi}}^{z}(\ell)$ of $\tilde{\boldsymbol{\mathbf{S}}}^{z}$
is a local observable and, if $\tilde{\boldsymbol{\mathbf{S}}}^{z}$ commutes
with the Hamiltonian, it has important consequences on the evolution of the
system, similarly to any other local conservation law — we refer the reader to
Ref. [32] for additional details.

The (two) symmetry sectors of the spin-flip symmetry are connected by any odd
local operator, e.g., $\boldsymbol{\mathbf{\sigma}}_{j}^{x}$. Acting with it
on the state splits the latter in two parts where the expectation value of
semilocal operators takes opposite signs. This is at the origin of the
macroscopic effects of local perturbations pointed out in Ref. [31]. The
behaviour of fluctuations in these systems has not yet been investigated but
the sensitivity of local observables to a single spin flip is a strong
indication that the quantum measurement of a local observable such as
$\boldsymbol{\mathbf{\sigma}}_{j}^{x}$ could result in macroscopically
entangled states. As a specific example, we consider the following Hamiltonian

$\boldsymbol{\mathbf{H}}_{1}=\sum_{\ell}\frac{\boldsymbol{\mathbf{1}}-\boldsymbol{\mathbf{\sigma}}_{\ell}^{z}}{8}[J\vec{\boldsymbol{\mathbf{\sigma}}}_{\ell-1}\overset{S}{\cdot}\vec{\boldsymbol{\mathbf{\sigma}}}_{\ell+1}+\\\
+\vec{D}\cdot(\vec{\boldsymbol{\mathbf{\sigma}}}_{\ell-1}\times\vec{\boldsymbol{\mathbf{\sigma}}}_{\ell+1})]-\frac{\vec{h}}{2}\cdot\vec{\boldsymbol{\mathbf{\sigma}}}_{\ell}$
(5)

where $\vec{a}\overset{S}{\cdot}\vec{b}=\vec{a}\cdot(S\vec{b})$. This
Hamiltonian commutes with the semilocal charge
$\tilde{\boldsymbol{\mathbf{S}}}^{z}$ — Eq. [4] — provided that

$S=\begin{bmatrix}\frac{1+\gamma}{2}&w&0\\\ w&\frac{1-\gamma}{2}&0\\\
0&0&\Delta\end{bmatrix}\quad\vec{D}=\begin{bmatrix}0\\\ 0\\\
D^{z}\end{bmatrix}\quad\vec{h}=\begin{bmatrix}0\\\ 0\\\ h^{z}\end{bmatrix}\,.$
(6)

It describes a system that can be interpreted as a quantum transistor chain,
in which the building blocks
$\boldsymbol{\mathbf{T}}^{\alpha\beta}_{\ell}=\boldsymbol{\mathbf{\sigma}}_{\ell-1}^{\alpha}\frac{\boldsymbol{\mathbf{I}}-\boldsymbol{\mathbf{\sigma}}_{\ell}^{z}}{2}\boldsymbol{\mathbf{\sigma}}_{\ell+1}^{\beta}$,
with $(\alpha,\beta)\in\\{(x,x),(x,y),(y,x),(y,y),(z,z),(0,0)\\}$
($\boldsymbol{\mathbf{\sigma}}^{0}_{\ell}\equiv\boldsymbol{\mathbf{1}}$),
commute with $\tilde{\boldsymbol{\mathbf{S}}}^{z}$ — see Fig. 2.

Figure 2: Pictorial representation of the quantum transistor chain,
constructed as a sequence of building blocks
$\boldsymbol{\mathbf{T}}^{\alpha\beta}_{\ell}=\boldsymbol{\mathbf{\sigma}}_{\ell-1}^{\alpha}\frac{\boldsymbol{\mathbf{I}}-\boldsymbol{\mathbf{\sigma}}_{\ell}^{z}}{2}\boldsymbol{\mathbf{\sigma}}_{\ell+1}^{\beta}$.

It is known that $\boldsymbol{\mathbf{H}}_{1}$ has infinitely many conserved
operators with quasilocal [36] or semilocal [32] densities in the following
regions of the parameter space:

  1. 1.

If the only nonzero coupling constants are $J$, $\gamma$, and $h^{z}$, and
$\gamma=\pm 1$, the model is dual to the (integrable) Heisenberg XXZ one [37,
31];

  2. 2.

If the only nonzero coupling constants are $J$, $\Delta$, and $h^{z}$, the
model belongs to the family of hard-rod deformations of XXZ studied in Ref.
[38]. In particular, for $\Delta=0$ it is a special case of the Bariev model
[39] and was recently dubbed “dual folded XXZ” [37]. These are special
integrable models exhibiting Hilbert-space fragmentation and quantum jamming
[37, 40, 41, 42].

For the first family of integrable models, Ref. [31] has already shown that a
subsystem of $\ket{\Psi_{\pi/2}(t)}$, linearly growing in time, is
macroscopically different from the corresponding one in $\ket{\Uparrow}$. An
analogous conclusion was drawn in Refs [41, 42] for the second family of
integrable models, though starting from slightly different product states (in
those systems $\ket{\Uparrow\downarrow\Uparrow}$ is an excited eigenstate as
well). Ref. [41] also observed that a similar behaviour could be seen even in
generic systems with a jamming sector. We announce that a part of the
phenomenology common to all these settings is much more general than the
specific models studied so far.

###  1.2 Generic model

We establish contact with the systems currently studied in quantum simulators,
for example with trapped ions [43], by considering the following Hamiltonian

$\boldsymbol{\mathbf{H}}_{2}=\frac{1}{4}\sum_{\ell}\Bigl{[}\sum_{r=1}J_{r}\vec{\boldsymbol{\mathbf{\sigma}}}_{\ell}\overset{S_{r}}{\cdot}\vec{\boldsymbol{\mathbf{\sigma}}}_{\ell+r}+\vec{D}_{r}\cdot(\vec{\boldsymbol{\mathbf{\sigma}}}_{\ell}\times\vec{\boldsymbol{\mathbf{\sigma}}}_{\ell+r})\Bigr{]}-\frac{\vec{h}}{2}\cdot\vec{\boldsymbol{\mathbf{\sigma}}}_{\ell}.$
(7)

This is the most general Hamiltonian where spins interact in pairs: there’s a
Heisenberg exchange term, a Dzyaloshinskii-Moriya interaction, and a coupling
with an external field. The trivial quantum scar $\ket{\Uparrow}$ appears when

$S_{r}=\begin{bmatrix}1&0&\frac{1}{2}\gamma_{r}^{x}\\\
0&1&\frac{1}{2}\gamma_{r}^{y}\\\
\frac{1}{2}\gamma_{r}^{x}&\frac{1}{2}\gamma_{r}^{y}&1+\gamma_{r}^{z}\end{bmatrix}\quad\vec{h}=\begin{bmatrix}\frac{1}{2}\sum_{r=1}J_{r}\gamma^{x}_{r}\\\
\frac{1}{2}\sum_{r=1}J_{r}\gamma^{y}_{r}\\\ h^{z}\end{bmatrix}.$ (8)

The parameter $\vec{\gamma}$ incorporates both an anisotropy in the Heisenberg
interaction and a rotation of the axes (which is relevant because we have
fixed the orientation of the spins of the separable eigenstate). As will be
clarified before long, the effect we exploit to generate macroscopically
entangled states requires a $U(1)$-breaking interaction, therefore we will
only consider systems in which some coupling constants among $D_{r}^{x}$,
$D_{r}^{y}$, $\gamma_{r}^{x}$, $\gamma_{r}^{y}$ are nonzero.

In such a generic setting there is no a-priori reason to expect the localised
perturbation to produce macroscopic effects; for example, the system does not
exhibit semilocal charges or special constraining interactions. Is the
presence of a simple quantum scar sufficient to trigger the phenomenon?

Before reporting the results of our investigation, we remind the reader of
some special regions of the parameter space:

  * •

If the only nonzero coupling constants are $J_{1}$, $\gamma_{1}^{z}$, and
$h^{z}$, the system is integrable and known as XXZ model, which is arguably
the most important paradigm of quantum magnetism in 1D [44]111Sometimes,
especially in experimental works, the name “XXZ model” is referred also to the
higher range model with nonzero $J_{r}$ and $\gamma_{r}^{z}=\gamma_{1}^{z}$..

  * •

More generally, if $\vec{D}_{r}=0$ and $\vec{\gamma}_{r}=\vec{\gamma}_{1}$,
the system describes an XYZ Heisenberg model with a tilted orientation, the
integrable case corresponding to $\vec{h}=0$ and $J_{r}=\delta_{r1}J_{1}$. See
also Ref. [45] for a special region of the parameter space in which
$\ket{\Uparrow}$ becomes the symmetry-breaking ground state.

  * •

If the only nonzero coupling constants are $J_{1}$, $\gamma_{1}^{x}$,
$D_{1}^{y}$, and $h^{z}$, in the limit $J_{1}\rightarrow 0$ at fixed
$\gamma_{1}^{x}J_{1}=-2D_{1}^{y}$ the system approaches the so-called quantum
East model [46], which has recently attracted a lot of attention for its
unusual properties [47].

##  2 Results

###  2.1 The effective system $\Omega(t)$

We have introduced $\Omega_{\epsilon}(t)$ as the smallest subsystem,
containing the measured spin, that can be considered pure at time $t$ with
accuracy $1-\epsilon$. We clarify here this point. First of all, it can be
readily proven that the initial perturbation is irrelevant outside a light
cone emerging from the space-time point of the measurement using a corollary
of the Lieb-Robinson bounds derived in Ref. [48]. Indeed, the Heisenberg
representation of a local operator is exponentially close to an operator with
support in a finite region including the support of the operator, the argument
of the exponential being proportional to $d-v_{LR}|t|$, where $d$ is the
smallest distance between the observable and the boundary of the region and
$v_{LR}$ is the Lieb-Robinson velocity. In our specific case, this means that
the following decomposition holds:
$\exp[i\theta\boldsymbol{\mathbf{\sigma}}_{0}^{y}(-t)]=\boldsymbol{\mathbf{U}}_{\Omega}+\boldsymbol{\mathbf{\Delta}}_{\Omega}$,
where $\boldsymbol{\mathbf{U}}_{\Omega}$ is a unitary operator with support in
a region $\Omega$ centered at the measurement position and
$\parallel\boldsymbol{\mathbf{\Delta}}_{\Omega}\parallel\lesssim\exp[-(|\Omega|/2-vt)/\xi]$,
with $\xi$ a non-universal constant. This gives

$\parallel\ket{\Psi(t)}\bra{\Psi(t)}-\boldsymbol{\mathbf{U}}_{\Omega}\ket{\Uparrow}\bra{\Uparrow}\boldsymbol{\mathbf{U}}_{\Omega}^{\dagger}\parallel=\\\
\parallel\boldsymbol{\mathbf{U}}_{\Omega}\ket{\Uparrow}\bra{\Uparrow}\boldsymbol{\mathbf{\Delta}}_{\Omega}^{\dagger}+\boldsymbol{\mathbf{\Delta}}_{\Omega}\ket{\Uparrow}\bra{\Uparrow}\boldsymbol{\mathbf{U}}_{\Omega}+\boldsymbol{\mathbf{\Delta}}_{\Omega}\ket{\Uparrow}\bra{\Uparrow}\boldsymbol{\mathbf{\Delta}}_{\Omega}^{\dagger}\parallel\\\
\leq
2\parallel\boldsymbol{\mathbf{\Delta}}_{\Omega}\parallel+\parallel\boldsymbol{\mathbf{\Delta}}_{\Omega}\parallel^{2}\lesssim
e^{-(|\Omega|/2-vt)}\,;$ (9)

that is to say, up to exponentially small corrections the reduced density
matrix of $\Omega$ is pure.

We call $\Omega_{\epsilon}(t)$ the smallest spin block for which
$\parallel\mathrm{tr}_{\Omega_{\epsilon}(t)}[e^{-i\boldsymbol{\mathbf{H}}t}\ket{\Uparrow\downarrow\Uparrow}\bra{\Uparrow\downarrow\Uparrow}e^{-i\boldsymbol{\mathbf{H}}t}]-(\ket{\Uparrow}\bra{\Uparrow})_{\overline{\Omega_{\epsilon}(t)}}\parallel$
is smaller than $\epsilon$, where
$(\ket{\Uparrow}\bra{\Uparrow})_{\overline{\Omega_{\epsilon}(t)}}$ is the
state $\ket{\Uparrow}$ restricted to the complement of $\Omega_{\epsilon}(t)$.
Note that $\Omega_{\epsilon}(0)$ contains only the spin at the origin for any
choice of $\epsilon$.

###  2.2 $U(1)$ symmetry

We point out here that, if the total magnetisation
$\boldsymbol{\mathbf{S}}^{z}$ is conserved, the states $\ket{\Uparrow}$ and
$\ket{\Psi_{\pi/2}(t)}$ can not become macroscopically different and their
linear combination is not macroscopically entangled. Specifically, Appendix C
provides a proof that a state $\ket{\Psi}$ obtained by time evolution under a
$U(1)$-symmetric Hamiltonian after a local perturbation to $\ket{\Uparrow}$ is
macroscopically equivalent to $\ket{\Uparrow}$; that is to say, the quantity
$\braket{\Psi}{\boldsymbol{\mathbf{O}}}{\Psi}-\braket{\Uparrow}{\boldsymbol{\mathbf{O}}}{\Uparrow}$
is subextensive for any translational-invariant operator
$\boldsymbol{\mathbf{O}}=\sum_{\ell}\boldsymbol{\mathbf{O}}_{A_{\ell}}$, with
$\boldsymbol{\mathbf{O}}_{A_{\ell}}$ local operators. We also show that the
variance of any of such operators $\boldsymbol{\mathbf{O}}$ with respect to
the state $\ket{\Psi}$ grows at most linearly with system size
$|\Omega_{\epsilon}(t)|$, ruling out the possibility to obtain a
macroscopically entangled state from time evolution of a locally-perturbed
$\ket{\Uparrow}$. And this holds true for any local perturbation. Physically,
we can understand this result from the fact that, first, a local operator can
only flip a finite number of spins, and, second, the initial state,
$\ket{\Uparrow}$, has maximal $\boldsymbol{\mathbf{S}}^{z}$. As a result, the
Hilbert space accessible to time evolution is too small for macroscopic
entanglement to develop. This is the first clue supporting the expectation
that, in order to build up macroscopic entanglement, $\ket{\Uparrow}$ can not
be the ground state of a local conservation law. In generic systems, we can
read this as a statement that $\ket{\Uparrow}$ should be a quantum scar.

###  2.3 Hidden $U(1)$ symmetry

We start with $\boldsymbol{\mathbf{H}}_{1}$ — Eq. [5] — in the integrability
region in which, as discussed before, we can take for granted that
$\ket{\Psi_{\pi/2}(t)}$ becomes macroscopically different from
$\ket{\Uparrow}$:

$|\braket{\Uparrow\downarrow\Uparrow}{e^{i\boldsymbol{\mathbf{H}}_{1}t}\boldsymbol{\mathbf{O}}e^{-i\boldsymbol{\mathbf{H}}_{1}t}}{\Uparrow\downarrow\Uparrow}-\braket{\Uparrow}{\boldsymbol{\mathbf{O}}}{\Uparrow}|\sim
t\,,$ (10)

with $\boldsymbol{\mathbf{O}}$ a generic (spin-flip invariant) extensive
operator. This is expected as long as the initial state $\ket{\Uparrow}$ is in
the middle of the spectrum of $\boldsymbol{\mathbf{H}}_{1}$. Incidentally, we
can understand that this latter condition is necessary by considering the dual
XXZ model — Eq. [1] — in which, for $|h^{z}|$ large enough, $\ket{\Uparrow}$
becomes the ground state (or the maximum energy state). By applying the
Kramers-Wannier duality mapping proposed in Ref. [31] the system is mapped
into the time evolution of a domain wall in the XXZ model, where the magnetic
field $h^{z}$ plays the role of the anisotropy. As proven in Ref. [49], the
domain wall does not spread when the anisotropy is larger than a critical
value, which is equivalent to say that the perturbation remains localised when
$|h^{z}|$ is large enough to move $\ket{\Uparrow}$ at the boundaries of the
energy spectrum.

In the integrability region of $\boldsymbol{\mathbf{H}}_{1}$ there is just one
piece of information that we cannot retrieve from the scientific literature:
the behaviour of fluctuations. This is important to us because the Lieb-
Robinson bound guarantees $|\Omega_{\epsilon}(t)|=\mathcal{O}(t)$, and hence,
if the variance of extensive observables in $\ket{\Psi_{\pi/2}(t)}$ were
$\mathcal{O}(t)$, we could readily conclude that the projective measurement
generates a genuine cat state.

We measure macroscopic entanglement with the so called quantumness, which
characterises the asymptotic behaviour of the maximum quantum Fisher
information among all extensive observables (see Materials and Methods). In
fact, we only compute a lower bound of it, $\mathcal{N}_{\mathrm{eff}}^{(1)}$,
resulting from reducing the space of observables. An example is reported in
Fig. 3, where we plot $\mathcal{N}_{\mathrm{eff}}^{(1)}$ as a function of the
effective system size $|\Omega_{\epsilon}(t)|$. To study the asymptotic
behavior of $\mathcal{N}_{\mathrm{eff}}^{(1)}$ we fit the data with a curve
parametrised as
$\beta_{0}+\beta_{1}|\Omega_{\epsilon}(t)|+\beta_{2}\frac{1}{|\Omega_{\epsilon}(t)|}$,
where the term $\frac{1}{|\Omega_{\epsilon}(t)|}$ stands for a potential sub-
leading term. The estimated leading order is consistent with a constant
($\beta_{1}\approx 0$) in the case $\theta=\pi/2$ and with a linear growth
($\beta_{1}\neq 0$) in the generic case. The figure also shows how the
difference of magnetization between flipping or not the spin is proportional
to $|\Omega_{\epsilon}(t)|$, meaning that the two states $\ket{\Uparrow}$ and
$\ket{\Psi_{\pi/2}(t)}$ are macroscopically different and therefore the state
$\ket{\Psi_{\theta}(t)}$ is macroscopically entangled for generic $\theta$.
Fig. 3 also reports the probability density $P_{\theta}(m)$ to get $m$ from a
measurement of $\boldsymbol{\mathbf{S}}^{z}$ given that the system is in the
state $\ket{\Psi_{\theta}(t)}$; the case $\theta=\frac{\pi}{2}$ shows standard
fluctuations, while the more general case $\theta=\frac{\pi}{4}$ shows the
bimodal distribution that is characteristic of a cat state. We have shifted
$m$ by the expectation value of $\boldsymbol{\mathbf{S}}^{z}(t)$ so as to
exhibit plots that are independent of the system’s size (incidentally, this
also makes it equivalent to consider either the full chain or
$\Omega_{\epsilon}(t)$). The probability distribution in the GHZ state
$(\ket{\Uparrow}+\ket{\Downarrow})/\sqrt{2}$ would exhibit two Kronecker
deltas at the maximum and minimum magnetisations; here one delta is replaced
by a Gaussian centred at a different magnetization, but it is still well
separated from the other peak.

Figure 3: Time evolution of $\ket{\Psi_{\theta}(t)}$ under Hamiltonian
$\boldsymbol{\mathbf{H}}_{1}$ with $J=2.8$, $\gamma=1$,
$w=\Delta=D^{z}=h^{z}=0$. From the top we have: 1) Difference between the
total magnetization in $\ket{\Uparrow}$ and $\ket{\Psi_{\pi/2}(t)}$, i.e.
$\Delta\boldsymbol{\mathbf{S}}^{z}(\theta)=\braket{\Uparrow}{\boldsymbol{\mathbf{S}}^{z}}{\Uparrow}-\braket{\Psi_{\theta}(t)}{\boldsymbol{\mathbf{S}}^{z}}{\Psi_{\theta}(t)}$;
inset: local magnetization profile for the last four times of the main plot.
2) Macroscopic quantumness $N_{\mathrm{eff}}^{(1)}$ as a function of the
effective system size $|\Omega_{\epsilon}(t)|$, with $\epsilon=0.001$. 3) and
4) Probability density $P_{\theta}(m)$ for $\theta=\tfrac{\pi}{2}$ and
$\theta=\tfrac{\pi}{4}$ respectively.

We have checked that, as long as the initial state is in the bulk of the
spectrum, the qualitative behaviour remains the same even moving away from
integrability. An example is reported in Fig. 4. Remarkably, even in this non-
integrable case the profiles of local magnetization (the insets in the plots
of the total magnetization) computed at different times collapse to the same
curve in the ray coordinate $\ell/t$, which manifests the ballistic change in
the total magnetization. Note also that from the profile of the local
magnetization we get a visual definition of $\Omega_{\epsilon}(t)$: it
essentially coincides with the region with
$\braket{\sigma^{z}_{\ell}(t)}\not\approx 1$.

Figure 4: The same as in the top plots of Fig. 3 for Hamiltonian
$\boldsymbol{\mathbf{H}}_{1}$ with
$J=1,\gamma=0.5,w=0.7,\Delta=0,D^{z}=0.6,h^{z}=0$.

We mention that the tidy structure of the transistor Hamiltonian
$\boldsymbol{\mathbf{H}}_{1}$ allows for an alternative interpretation of the
protocol. The state before the measurement has all transistor’s switches open,
and time evolution is blocked. The measurement has the effect of turning a
switch in a quantum superposition of open and closed (the switch is exactly
closed only for $\theta=\frac{\pi}{2}$). Closing a switch enables Heisenberg
and Dzyaloshinskii-Moriya interactions between the neighboring spins of the
opposite leg of the chain, which, in turn, partially close other switches, and
so on and so forth, leading to a complete reconfiguration of the state that
affects a region whose size grows linearly in time.

###  2.4 Generic model

We have considered several models without a hidden $U(1)$ symmetry in which
$\ket{\Uparrow}$ is still a quantum scar. We see quite generally that, also in
those cases, the spin flip has everlasting effects arbitrarily far from the
origin. In Fig. 5 we report an example using the Hamiltonian
$\boldsymbol{\mathbf{H}}_{2}$. The plot of the magnetization shows that the
states $\ket{\Uparrow}$ and $\ket{\Psi_{\pi/2}(t)}$ are macroscopically
different, indeed the difference of the magnetization in the two states grows
linearly with the effective system size. Again, we conclude that our protocol
leads to the formation of a macroscopically entangled state. Concerning the
quantumness, we have fitted the data for the lower bound
$\mathcal{N}_{\mathrm{eff}}^{(1)}$ with the same Ansatz as before
$\beta_{0}+\beta_{1}|\Omega_{\epsilon}(t)|+\beta_{2}\frac{1}{|\Omega_{\epsilon}(t)|}$;
the analysis points to an asymptotic linear growth for any $\theta\neq 0$,
including this time also $\theta=\frac{\pi}{2}$. Therefore,
$\ket{\Psi_{\theta}(t)}$ still exhibits macroscopic entanglement for generic
$\theta$, but, in contrast to the quantum transistor chain, also
$\ket{\Psi_{\pi/2}(t)}$ is macroscopically entangled, undermining in turn the
formation of a cat state.

Figure 5: The same as in the top plots of Fig. 3 for Hamiltonian
$\boldsymbol{\mathbf{H}}_{2}$ with $J_{1}=J_{2}=1$, $J_{r>2}=0$,
$\gamma^{z}_{1}=\gamma^{z}_{2}=-0.6$, $D_{r>1}^{y}=h^{z}=\gamma^{z}_{r>2}=0$,
$D_{1}^{y}=-0.9$, $\gamma^{x}_{r}=\gamma^{y}_{r}=D^{x}_{r}=D^{z}_{r}=0,\forall
r$. The widest profile in the inset corresponds to the largest time.

###  2.5 Imperfections in the preparation of the state

We investigate the effect of perturbing the state by time evolution under a
homogeneous Hamiltonian $\boldsymbol{\mathbf{H}}_{0}$ for a fixed time
$t_{0}$, i.e.,
$\ket{\Psi_{0}(0^{-})}=e^{-i\boldsymbol{\mathbf{H}}_{0}t_{0}}\ket{\Uparrow}$.
We then study the time evolution of
$\ket{\Psi_{\theta}(0^{+})}=e^{i\theta\boldsymbol{\mathbf{\sigma}}_{0}^{y}}\ket{\Psi_{0}(0^{-})}$,
which is the analog of Eq. [2]. We warn the reader of a complication: the
initial evolution affects the whole chain, so, strictly speaking, there is no
approximately pure subsystem. If we insist on considering the subsystem
associated with the spreading of the local perturbation (corresponding to the
sharp change of behaviour in the profile of the magnetization), we should
analyse the quantum Fisher information of a mixed state, which is a harder
quantity to compute. In this preliminary study of the stability of our
protocol, we limit ourselves to check the qualitative behaviour of the
probability distribution of the magnetization.

We distinguish two classes of perturbations: in the first class, the
perturbation is spin-flip symmetric (thus the hidden $U(1)$ symmetry is
preserved), e.g., $\boldsymbol{\mathbf{H}}_{0}$ could be the integrable Ising
Hamiltonian

$\boldsymbol{\mathbf{H}}_{0,1}=-\frac{1}{4}\sum_{\ell}\left(\boldsymbol{\mathbf{\sigma}}_{\ell}^{x}\boldsymbol{\mathbf{\sigma}}_{\ell+1}^{x}+h_{0}^{z}\boldsymbol{\mathbf{\sigma}}_{\ell}^{z}\right)\,;$
(11)

in the second class, the pre-measuerement Hamiltonian breaks the spin-flip
invariance behind the hidden $U(1)$ symmetry, e.g.

$\boldsymbol{\mathbf{H}}_{0,2}=-\frac{1}{4}\sum_{\ell}\left(\boldsymbol{\mathbf{\sigma}}_{\ell}^{x}\boldsymbol{\mathbf{\sigma}}_{\ell+1}^{x}+h_{0}^{z}\boldsymbol{\mathbf{\sigma}}_{\ell}^{z}+h_{0}^{x}\boldsymbol{\mathbf{\sigma}}_{\ell}^{x}\right)\,,$
(12)

which is a non-integrable version of the Ising model. Note that, in both
cases, the larger $h_{0}^{z}$, the smaller the expected effect, therefore
$h_{0}^{z}$ can be used as a control parameter.

The first class of perturbations is not expected to destabilize the growth of
a cat state: On the one hand, Ref. [31] showed that the perturbation has
everlasting macroscopic effects even after so-called “global quenches” from a
spin-flip invariant initial state. On the other hand, Ref. [50] indirectly
confirms clustering in $\ket{\Psi_{\pi/2}(t)}$ at long times with
noninteracting transistor Hamiltonians, indeed the mutual information
approaches zero at large distances. Our numerical analysis is consistent with
these expectations, indeed we still see the formation of a bimodal probability
distribution (with two well separated peaks).

On the contrary, for the second class of perturbations the probability density
does not present two well separated peaks, as shown e.g. in the bottom row of
Fig. 6. This rules out the formation of a cat state, but the basic features
pointing to macroscopic entanglement (such as the linear growth in time of the
difference of magnetization between flipping or not the spin) are still
present for times longer than our maximum simulation times.

Figure 6: Time evolution under $\boldsymbol{\mathbf{H}}_{1}$ after perturbing
the state $\ket{\Uparrow}$ with $\boldsymbol{\mathbf{H}}_{0,2}$ for a time
$t_{0}=0.4$. The parameters are set to $J=2.8$, $\gamma=1$,
$w=\Delta=D^{z}=h^{z}=0$, for $\boldsymbol{\mathbf{H}}_{1}$, and
$h^{z}_{0}=0.7$ and $h^{x}_{0}=0.5$ for $\boldsymbol{\mathbf{H}}_{0,2}$. Top:
the difference in magnetization with or without the spin flip grows linearly
in time, at least for the times reached by our simulation. Bottom: more than
two peaks appear and they are not well separated.

##  3 Discussion

###  3.1 Local perturbations with macroscopic effects

In ordinary 1D systems a spin flip does not change the macroscopic properties
of a state whatever long the time is. The naive expectation is that its effect
spreads out across the chain as $t^{\alpha}$ but fades away as $t^{-\alpha}$,
with $\alpha=1$ in integrable models and $\alpha=\tfrac{1}{2}$ in most of the
generic systems. Recently, three situations have been pointed out in which
this picture can break down:

  * •

ground states in symmetry-broken phases [51, 52].

  * •

quantum jammed states in systems with Hilbert space fragmentation [41].

  * •

systems with semilocal conservation laws [31].

To these three known cases, we have added models with a fully separable
quantum scar. This novel scenario has two advantages: first, a product state
can be easily prepared experimentally, and, second, its stationarity gives the
freedom to stop the (virtual) experiment at any time without compromising the
outcome.

Remarkably, the norm is that the perturbation generates macroscopic
entanglement, indeed, the possible but improbable operators breaking that rule
can be cured by the addition of a local operator that does not change the
excited state — e.g., $\epsilon\boldsymbol{\mathbf{\sigma}}_{\ell}^{z}$ for
our initial state $\ket{\Uparrow}$.

In the special case in which there is a hidden $U(1)$ symmetry our analysis of
the fluctuations reveals that the entangled part of the system is in a
Schrödinger’s cat state.

###  3.2 On the spreading

The existence of the semilocal charge
$\sum_{\ell}\boldsymbol{\mathbf{\Pi}}^{z}(\ell)$ ensures that the effect of
the spin flip does not fade away at long time [31, 32], but it is not
sufficient to exclude the effect to remain confined around the position of the
measurement. This is indeed what happens when $h$ is too large, in which case
the Hamiltonian exhibits localised excitations (see Appendix B). In the
generic case, however, the perturbation spreads out. Using the underlying
integrable structure, Ref. [31] showed that the spreading is ballistic for
$\gamma=1$, $w=\Delta=D^{z}=0$, and $|h^{z}|<\frac{1}{4}$. And for slightly
different initial product states, ballistic behaviour was observed [41, 42]
also for $\gamma=w=\Delta=D^{z}=h^{z}=0$, for which the model is integrable as
well. A priori one would not expect ballistic spreading in generic models, but
rather diffusive. In Ref. [41], in particular, a preliminary analysis of the
effect of some integrability-breaking perturbations pointed to a non-ballistic
spreading. In contrast, we find numerical evidence that ballistic behaviour
extends to a wide range of parameters, at least, within the time window
investigated. In order to rule integrability out, we have checked the
statistics of the energy levels, and, for generic parameters, our data are in
excellent agreement with a Wigner-Dyson distribution (see the Appendix A),
supporting, in turn, the non-integrability of the generic models investigated.

##  4 Materials and Methods

###  4.1 Simulations of the dynamics

Numerical simulations are performed with Julia ITensor library [53]. We use a
time-evolving block decimation (TEBD) algorithm and, in all data reported,
time evolution is discretized in time steps $\delta t=0.01$ with 2nd order
Trotter-Suzuki gates [54].

###  4.2 Full counting statistics

The full probability distribution of a given observable encodes the entire
information about its moments. Let $L$ be the chain’s length. We define
$P_{\theta}(m)$ as the probability to get $m$ from a measurement of
$\boldsymbol{\mathbf{S}}^{z}$ given that the system is in the state
$\ket{\Psi_{\theta}(t)}$. Note that
$m\in\\{-\frac{L}{2},-\frac{L}{2}+1,...,\frac{L}{2}\\}$. To set a convention,
we assume $L$ to be divisible by $4$, so that the maximum possible value $m$
of the magnetization is an integer even number. The generating function of the
moments of the probability distribution is defined as

$G_{\theta}(k)=\braket{\Psi_{\theta}(t)}{e^{i\frac{2\pi
k}{L+1}\boldsymbol{\mathbf{S}}^{z}}}{\Psi_{\theta}(t)}.$ (13)

The probability $P_{\theta}(m)$ is then the Fourier transform of the
generating function:

$P_{\theta}(m)=\frac{1}{L+1}\sum_{k=-L/2}^{L/2}e^{-i\frac{2\pi
k}{L+1}m}G_{\theta}(k).$ (14)

We compute the generating function numerically using Julia ITensor library
[53].

We point out that, if the model is invariant under spin flip
$\boldsymbol{\mathbf{\sigma}}^{x,y}\rightarrow-\boldsymbol{\mathbf{\sigma}}^{x,y}$,
the state $\ket{\Psi_{\pi/2}(t)}$ belongs to the sector in which the parity
operator
$\boldsymbol{\mathbf{\Pi}}^{z}=\prod_{\ell}\boldsymbol{\mathbf{\sigma}}^{z}_{\ell}$
has eigenvalue $-1$, which means $P_{\pi/2}(m)=0$ for any even $m$. This
implies that, in the generic case of $\ket{\Psi_{\theta}(t)}$ with
$\theta\neq\frac{\pi}{2}$, $P_{\theta}(m)=0$ for any even $m$ except for
$P_{\theta}(L/2)=\cos^{2}\theta$. This is why, in the plots of the probability
density for models with spin-flip invariance, we report only odd values of $m$
and $m=L/2$.

###  4.3 Quantumness and Quantum Fisher information

Ref. [55] proposed the quantum Fisher information as a measure of the
“macroscopicity” of quantum effects in lattice systems. Its computation is
however rather complicated when addressed in full generality, both
analytically and numerically. For the sake of simplicity, we follow the
approach of Refs [56, 28, 29] and restrict ourselves to examining a sufficient
condition. Specifically, we investigate the quantum Fisher information
$\mathcal{F}(\boldsymbol{\mathbf{O}})$ of extensive operators whose densities
have support on a single site, i.e.,
$\boldsymbol{\mathbf{O}}[\\{\vec{n}\\}]=\sum_{j\in\Omega_{\epsilon}(t)}\vec{n}_{j}\cdot\vec{\boldsymbol{\mathbf{\sigma}}}_{j}$,
where
$\vec{\boldsymbol{\mathbf{\sigma}}}_{j}\equiv\\{\boldsymbol{\mathbf{\sigma}}^{x}_{j},\boldsymbol{\mathbf{\sigma}}^{y}_{j},\boldsymbol{\mathbf{\sigma}}^{z}_{j}\\}$
and the coefficients are normalised as $|\vec{n}_{j}|^{2}=1$. In pure states,
which is the case we consider, $\mathcal{F}(\boldsymbol{\mathbf{O}})$ equals
four times the variance of $\boldsymbol{\mathbf{O}}$. Using the notation of
[30], we introduce the quantumness $N_{\mathrm{eff}}$ of a state as its
maximal quantum Fisher information with respect to all extensive observables:

$N_{\mathrm{eff}}=\frac{1}{4|\Omega_{\epsilon}(t)|}\max_{\boldsymbol{\mathbf{O}}}\mathcal{F}(\boldsymbol{\mathbf{O}}).$
(15)

We denote by $N_{\mathrm{eff}}^{(1)}$ the maximisation restricted to the
observables with single-site density as introduced above. $N_{\mathrm{eff}}$
was interpreted as an effective size of the macroscopic quantum state, thus
$N_{\mathrm{eff}}^{(1)}$ is a lower bound for the effective size. We introduce
the covariance matrix

$[\mathrm{K}(t)]_{n,\alpha;m,\beta}=\frac{1}{2}\braket{\Psi_{\pi/2}(t)}{\\{\boldsymbol{\mathbf{\sigma}}_{n}^{\alpha},\boldsymbol{\mathbf{\sigma}}_{m}^{\beta}\\}}{\Psi_{\pi/2}(t)}+\\\
-\braket{\Psi_{\pi/2}(t)}{\boldsymbol{\mathbf{\sigma}}_{n}^{\alpha}}{\Psi_{\pi/2}(t)}\braket{\Psi_{\pi/2}(t)}{\boldsymbol{\mathbf{\sigma}}_{m}^{\beta}}{\Psi_{\pi/2}(t)}\,,$
(16)

where
$\\{\boldsymbol{\mathbf{\sigma}}_{n}^{\alpha},\boldsymbol{\mathbf{\sigma}}_{m}^{\beta}\\}=\boldsymbol{\mathbf{\sigma}}_{n}^{\alpha}\boldsymbol{\mathbf{\sigma}}_{m}^{\beta}+\boldsymbol{\mathbf{\sigma}}_{m}^{\beta}\boldsymbol{\mathbf{\sigma}}_{n}^{\alpha}$.
In pure states, $\mathcal{N}_{\mathrm{eff}}^{(1)}$ satisfies

$\mathcal{N}_{\mathrm{eff}}^{(1)}=\frac{\mathrm{tr}[\mathrm{D}(t)]}{|\Omega_{\epsilon}(t)|}\,,\quad\text{with}\quad\mathrm{K}(t)\vec{v}(t)=[\mathrm{D}(t)\otimes\mathrm{I}_{3}]\vec{v}(t)\,,$
(17)

where $n,m\in\Omega_{\epsilon}(t)$ (the vector space is
$3|\Omega_{\epsilon}(t)|$-dimensional) and $\mathrm{D}(t)$ is required to be
diagonal ($[\vec{v}]_{n}$ are normalised). We solve Eq. [17] using a Lanczos
algorithm with three basic iterative steps:

  1. 1.

$\vec{w}^{(n)}=\mathrm{K}\vec{v}^{(n)}$

  2. 2.

$[\mathrm{D}^{(n)}]_{j}=\parallel\vec{w}_{j}^{(n)}\parallel^{-1}$

  3. 3.

$\vec{v}^{(n+1)}=(\mathrm{D}^{(n)}\otimes\mathrm{I}_{3})\vec{w}^{(n)}$

In most of the cases this procedure worked well without particular
stabilizers.

##  5 Conclusion

We have shown that, quite generally, macroscopic entanglement emerges from a
localised inhomogeneity in a state that would be otherwise stationary. This
can not happen in an ordinary excited state that maximises the entropy under
the constraints of the local conservation laws. In generic systems, indeed, we
observe the phenomenon in quantum scars with anomalously low bipartite
entanglement. We also argue that, in the specific case in which the excited
state maximises instead the entropy under the constraints of also semi-local
conservation laws, macroscopic entanglement takes the simple form of a cat
state, providing in turn a novel way to generate cat states in a class of
quantum many-body systems with local interactions.

This work leaves several open questions. First, a rigorous proof of the
generation of cat states is missing. Second, the observation of ballistic
behaviour in generic systems could be questioned as a finite-time effect, so
additional investigations are imperative. Third, we have only provided a
preliminary and incomplete check of the stability of our protocol, but this is
clearly an important issue and does require additional investigations.

S.B. thanks Tommaso Roscilde and Augusto Smerzi for useful discussions. This
work was supported by the European Research Council under the Starting Grant
No. 805252 LoCoMacro.

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##  Appendix A Energy level statistics

As a test of integrability or otherwise, we investigate the energy level
statistics of the Hamiltonian of the spin-$\frac{1}{2}$ nearest-neighbor
transistor chain

$\boldsymbol{\mathbf{H}}=\sum_{\ell}\frac{\boldsymbol{\mathbf{1}}-\boldsymbol{\mathbf{\sigma}}_{\ell}^{z}}{8}[J\vec{\boldsymbol{\mathbf{\sigma}}}_{\ell-1}\overset{S}{\cdot}\vec{\boldsymbol{\mathbf{\sigma}}}_{\ell+1}+\vec{D}\cdot(\vec{\boldsymbol{\mathbf{\sigma}}}_{\ell-1}\times\vec{\boldsymbol{\mathbf{\sigma}}}_{\ell+1})]-\frac{\vec{h}}{2}\cdot\vec{\boldsymbol{\mathbf{\sigma}}}_{\ell},$
(18)

where $\vec{a}\overset{S}{\cdot}\vec{b}=\vec{a}\cdot(S\vec{b})$ and

$S=\begin{bmatrix}\frac{1+\gamma}{2}&w&0\\\ w&\frac{1-\gamma}{2}&0\\\
0&0&\Delta\end{bmatrix}\quad\vec{D}=\begin{bmatrix}0\\\ 0\\\
D^{z}\end{bmatrix}\quad\vec{h}=\begin{bmatrix}0\\\ 0\\\ h^{z}\end{bmatrix}\,.$
(19)

We enforce periodic boundary conditions and restrict ourselves to the sector
characterised by zero momentum, zero semilocal charge
$\tilde{\boldsymbol{\mathbf{S}}}^{z}$ (the largest eigenspace), and
$\prod_{j}\boldsymbol{\mathbf{\sigma}}_{j}^{z}=\prod_{j}\boldsymbol{\mathbf{\sigma}}_{2j}^{z}=1$.

Such a sector is equivalent to the sector with zero momentum, zero
magnetization, and
$\prod_{j}\boldsymbol{\mathbf{\tau}}_{j}^{x}=\prod_{j}\boldsymbol{\mathbf{\tau}}_{j}^{z}=1$,
of the Hamiltonian

$\boldsymbol{\mathbf{H}}_{\tau}=\sum_{\ell}J\frac{(1+\gamma)\boldsymbol{\mathbf{I}}+(1-\gamma)\boldsymbol{\mathbf{\tau}}_{\ell-1}^{z}\boldsymbol{\mathbf{\tau}}_{\ell+2}^{z}}{4}\frac{\boldsymbol{\mathbf{\tau}}_{\ell}^{x}\boldsymbol{\mathbf{\tau}}_{\ell+1}^{x}+\boldsymbol{\mathbf{\tau}}_{\ell}^{y}\boldsymbol{\mathbf{\tau}}_{\ell+1}^{y}}{4}+\\\
+\frac{(D^{z}+Jw)\boldsymbol{\mathbf{\tau}}_{\ell+2}^{z}+(D^{z}-Jw)\boldsymbol{\mathbf{\tau}}_{\ell-1}^{z}}{2}\frac{\boldsymbol{\mathbf{\tau}}_{\ell}^{x}\boldsymbol{\mathbf{\tau}}_{\ell+1}^{y}-\boldsymbol{\mathbf{\tau}}_{\ell}^{y}\boldsymbol{\mathbf{\tau}}_{\ell+1}^{x}}{4}-\frac{h^{z}}{2}\boldsymbol{\mathbf{\tau}}_{\ell}^{z}\boldsymbol{\mathbf{\tau}}_{\ell+1}^{z}$
(20)

with periodic boundary conditions. Indeed, $\boldsymbol{\mathbf{H}}_{\tau}$ is
related to $\boldsymbol{\mathbf{H}}$ by a Kramers-Wannier duality mapping
which transforms semilocal operators into odd local ones [31]

$\boldsymbol{\mathbf{\sigma}}_{j}^{x}=\boldsymbol{\mathbf{\Pi}}_{\tau,-}^{x}(j)\qquad\quad\boldsymbol{\mathbf{\sigma}}_{j}^{y}=\begin{cases}\boldsymbol{\mathbf{\tau}}_{1}^{x}\boldsymbol{\mathbf{\tau}}_{2}^{z}&j=1\\\
\boldsymbol{\mathbf{\Pi}}^{x}_{\tau,-}(j-1)\boldsymbol{\mathbf{\tau}}_{j}^{y}\boldsymbol{\mathbf{\tau}}_{j+1}^{z}&1<j<L\\\
-\boldsymbol{\mathbf{\tau}}_{L}^{z}&j=L\end{cases}\qquad\quad\boldsymbol{\mathbf{\sigma}}_{j}^{z}=\begin{cases}\boldsymbol{\mathbf{\tau}}_{j}^{z}\boldsymbol{\mathbf{\tau}}_{j+1}^{z}&1\leq
j<L\\\
\boldsymbol{\mathbf{\Pi}}^{x}_{\tau}\boldsymbol{\mathbf{\tau}}_{1}^{z}\boldsymbol{\mathbf{\tau}}_{L}^{z}&j=L\,,\end{cases}$
(21)

where

$\displaystyle\boldsymbol{\mathbf{\Pi}}^{x}_{\tau,-}(j)=-\boldsymbol{\mathbf{\tau}}_{1}^{y}\prod_{\ell=2}^{j}\boldsymbol{\mathbf{\tau}}_{\ell}^{x},\qquad\boldsymbol{\mathbf{\Pi}}^{x}_{\tau}=\prod_{\ell=1}^{L}\boldsymbol{\mathbf{\tau}}_{\ell}^{x}\,.$
(22)

Fig. 7 shows quite clearly that the Hamiltonian of Fig. 3 in the main text is
not integrable. Indeed, the cumulative distribution of the energy level
spacing is in excellent agreement with the Wigner-Dyson distribution.

Figure 7:  Cumulative distribution of the unfolded energy level spacing in a
sector of $\boldsymbol{\mathbf{H}}$ with
$J=1,\gamma=0.5,w=0.7,\Delta=0,D^{z}=0.6,h^{z}=0$ in a chain with $20$ spins.
The data are shown as a solid curve; the dotted and dashed curves are the
(normalised) Poisson and Wigner-Dyson predictions expected in integrable and
generic models, respectively.

##  Appendix B Localised excitations

The main exception to the development of macroscopic entanglement after a
local projective measurement is when the pre-measurement state is the ground
state in a disordered phase. We consider, for example, the dual XXZ
Hamiltonian
$H=\sum_{\ell}\boldsymbol{\mathbf{\sigma}}^{x}_{\ell-1}\frac{1-\boldsymbol{\mathbf{\sigma}}^{z}_{\ell}}{8}\boldsymbol{\mathbf{\sigma}}^{x}_{\ell+1}-\frac{h^{z}}{2}\boldsymbol{\mathbf{\sigma}}^{z}_{\ell}$.
If the intensity of the magnetic field $|h^{z}|$ is larger than $\frac{1}{2}$,
the pre-measurement state becomes the ground state, and a spin flip excites
only local modes that do not spread with time — see Fig. 8.

Figure 8: The magnetization does not change for any accessible time. The fact
that the magnetization is maximal except for a few sites around the origin
tells us that most of the system is in the $\ket{\Uparrow}$ state.

##  Appendix C Absence of macroscopic effects in models with a $U(1)$ symmetry

In this section we prove that a state $\ket{\Psi}$ obtained by time evolution
under a $U(1)$-symmetric Hamiltonian after a local perturbation to
$\ket{\Uparrow}$ is macroscopically equivalent to $\ket{\Uparrow}$; that is to
say, the quantity
$\braket{\Psi}{\boldsymbol{\mathbf{O}}}{\Psi}-\braket{\Uparrow}{\boldsymbol{\mathbf{O}}}{\Uparrow}$
is subextensive for any translational-invariant operator
$\boldsymbol{\mathbf{O}}=\sum_{\ell}\boldsymbol{\mathbf{O}}_{A_{\ell}}$, with
$\boldsymbol{\mathbf{O}}_{A_{\ell}}$ local operators. We also prove that the
variance of any of such operators $\boldsymbol{\mathbf{O}}$ with respect to
the state $\ket{\Psi}$ grows at most linearly with system
$|\Omega_{\epsilon}(t)|$, ruling out the possibility to obtain a
macroscopically entangled state from time-evolution of a locally-perturbed
$\ket{\Uparrow}$.

Let us denote by $\ket{\Psi_{s}}$ a state obtained by flipping $s$ spins in
$\ket{\Uparrow}$, or linear combination of them. We start by observing that,
if $\boldsymbol{\mathbf{S}}^{z}$ is conserved, the state time-evolves as

$\ket{\Psi_{s}(t)}=\sum_{n_{1}<n_{2}<...<n_{s}}w_{\\{n\\}}(t)\boldsymbol{\mathbf{\sigma}}_{n_{1}}^{-}\boldsymbol{\mathbf{\sigma}}_{n_{2}}^{-}...\boldsymbol{\mathbf{\sigma}}_{n_{s}}^{-}\ket{\Uparrow}\,,$

for some coefficients such that
$\sum_{n_{1}<n_{2}<...<n_{s}}|w_{\\{n\\}}|^{2}=1$. Let
$\boldsymbol{\mathbf{O}}_{A_{\ell}}$ be an operator with support in $A$, where
$A_{\ell}$ is a set of $|A|$ adjacent sites starting with the $\ell$-th spin:
$A=\\{\ell,...,\ell+|A|-1\\}$. We have

$\braket{\Psi_{s}(t)}{\boldsymbol{\mathbf{O}}_{A_{\ell}}}{\Psi_{s}(t)}-\braket{\Uparrow}{\boldsymbol{\mathbf{O}}_{A_{\ell}}}{\Uparrow}\\\
=\sum_{i=1}^{s}\sum_{\begin{subarray}{c}n_{1}<...<n_{s}\\\ n_{k}\in
A_{\ell}\Leftrightarrow
k=i\end{subarray}}\biggl{(}\sum_{\begin{subarray}{c}n_{i}^{\prime}\\\
n_{k}^{\prime}\in A_{\ell},\forall
k\in\\{i\\}\end{subarray}}w_{\\{n\\}}^{*}(t)w_{\\{n\\},n_{i}\rightarrow
n^{\prime}_{i}}(t)\braket{\Uparrow}{\boldsymbol{\mathbf{\sigma}}^{+}_{n^{\prime}_{i}}\boldsymbol{\mathbf{O}}_{A_{\ell}}\boldsymbol{\mathbf{\sigma}}^{-}_{n_{i}}}{\Uparrow}-|w_{\\{n\\}}(t)|^{2}\braket{\Uparrow}{\boldsymbol{\mathbf{O}}_{A_{\ell}}}{\Uparrow}\biggr{)}\\\
+\sum_{i=1}^{s-1}\sum_{\begin{subarray}{c}n_{1}<...<n_{s}\\\ n_{k}\in
A_{\ell}\Leftrightarrow
k\in\\{i,i+1\\}\end{subarray}}\biggl{(}\sum_{\begin{subarray}{c}n_{i}^{\prime}<n^{\prime}_{i+1}\\\
n^{\prime}_{k}\in A_{\ell},\forall
k\in\\{i,i+1\\}\end{subarray}}w_{\\{n\\}}^{*}(t)w_{\begin{subarray}{c}\\{n\\},n_{i}\rightarrow
n^{\prime}_{i}\\\ n_{i+1}\rightarrow
n^{\prime}_{i+1}\end{subarray}}(t)\braket{\Uparrow}{\boldsymbol{\mathbf{\sigma}}^{+}_{n^{\prime}_{i}}\boldsymbol{\mathbf{\sigma}}^{+}_{n^{\prime}_{i+1}}\boldsymbol{\mathbf{O}}_{A_{\ell}}\boldsymbol{\mathbf{\sigma}}^{-}_{n_{i}}\boldsymbol{\mathbf{\sigma}}^{-}_{n_{i+1}}}{\Uparrow}-|w_{\\{n\\}}(t)|^{2}\braket{\Uparrow}{\boldsymbol{\mathbf{O}}_{A_{\ell}}}{\Uparrow}\biggr{)}\\\
+...+\sum_{\begin{subarray}{c}n_{1}<...<n_{s}\\\ n_{k}\in A_{\ell},\forall
k\in\\{1,...,s\\}\end{subarray}}\biggl{(}\sum_{\begin{subarray}{c}n^{\prime}_{1}<...<n^{\prime}_{s}\\\
n^{\prime}_{k}\in A_{\ell},\forall
k\in\\{1,...,s\\}\end{subarray}}w_{\\{n\\}}^{*}(t)w_{\\{n^{\prime}\\}}(t)\braket{\Uparrow}{\boldsymbol{\mathbf{\sigma}}^{+}_{n_{1}^{\prime}}...\boldsymbol{\mathbf{\sigma}}^{+}_{n_{s}^{\prime}}\boldsymbol{\mathbf{O}}_{A_{\ell}}\boldsymbol{\mathbf{\sigma}}^{-}_{n_{1}}...\boldsymbol{\mathbf{\sigma}}^{-}_{n_{s}}}{\Uparrow}-|w_{\\{n\\}}(t)|^{2}\braket{\Uparrow}{\boldsymbol{\mathbf{O}}_{A_{\ell}}}{\Uparrow}\biggr{)}\,.$
(23)

An upper bound for the absolute value of the left hand side of the equation is
obtained using
$\braket{v}{\boldsymbol{\mathbf{O}}}{v}\leq\parallel\boldsymbol{\mathbf{O}}\parallel\braket{v}{v}$,
where $\parallel\boldsymbol{\mathbf{O}}\parallel$ denotes the operator norm,
indeed we have

$|\braket{\Psi_{s}(t)}{\boldsymbol{\mathbf{O}}_{A_{\ell}}}{\Psi_{s}(t)}-\braket{\Uparrow}{\boldsymbol{\mathbf{O}}_{A_{\ell}}}{\Uparrow}|\leq\\\
\leq
2\parallel\boldsymbol{\mathbf{O}}_{A_{\ell}}\parallel\left(\sum_{i=1}^{s}\sum_{\begin{subarray}{c}n_{1}<...<n_{s}\\\
n_{k}\in A_{\ell}\Leftrightarrow
k=i\end{subarray}}|w_{\\{n\\}}(t)|^{2}+\sum_{i=1}^{s-1}\sum_{\begin{subarray}{c}n_{1}<...<n_{s}\\\
n_{k}\in A_{\ell}\Leftrightarrow
k\in\\{i,i+1\\}\end{subarray}}|w_{\\{n\\}}(t)|^{2}+...+\sum_{\begin{subarray}{c}n_{1}<...<n_{s}\\\
n_{k}\in A_{\ell},\forall
k\in\\{1,...,s\\}\end{subarray}}|w_{\\{n\\}}(t)|^{2}\right)\leq\\\ \leq
2\parallel\boldsymbol{\mathbf{O}}_{A_{\ell}}\parallel\sum_{i=1}^{s}\sum_{\begin{subarray}{c}n_{1}<...<n_{s}\\\
n_{i}\in A_{\ell}\end{subarray}}|w_{\\{n\\}}(t)|^{2}\,,$ (24)

where the last step is not an equality because there are some coefficients $w$
that are counted more than once in the last line (e.g., if we take $s=2$ and
$A_{\ell}=\\{\ell,\ell+1\\}$, there is originally just one term
$|w_{\ell,\ell+1}|^{2}$, coming from the sum
$\sum_{\begin{subarray}{c}n_{1}<...<n_{s}\\\ n_{k}\in A_{\ell},\forall
k\in\\{1,...,s\\}\end{subarray}}$, but the sum
$\sum_{i=1}^{s}\sum_{\begin{subarray}{c}n_{1}<...<n_{s}\\\ n_{i}\in
A_{\ell}\end{subarray}}$ has two of such terms, that we get for $i=\ell$ and
$i=\ell+1$). Let us now consider the corresponding translational-invariant
extensive operator
$\boldsymbol{\mathbf{O}}=\sum_{\ell\in\Omega_{\epsilon}(t)}\boldsymbol{\mathbf{O}}_{A_{\ell}}$,
where here $\Omega_{\epsilon}$ has a generalised definition with respect to
the main text, since we want it to include all the light cones steaming from
each spin flip. This gives

$|\braket{\Psi_{s}(t)}{\boldsymbol{\mathbf{O}}}{\Psi_{s}(t)}-\braket{\Uparrow}{\boldsymbol{\mathbf{O}}}{\Uparrow}|\leq\sum_{\ell\in\Omega_{\epsilon}(t)}|\braket{\Psi_{s}(t)}{\boldsymbol{\mathbf{O}}_{A_{\ell}}}{\Psi_{s}(t)}-\braket{\Uparrow}{\boldsymbol{\mathbf{O}}_{A_{\ell}}}{\Uparrow}|\leq\\\
\leq
2\parallel\boldsymbol{\mathbf{O}}_{A_{1}}\parallel\sum_{\ell\in\Omega_{\epsilon}(t)}\sum_{i=1}^{s}\sum_{\begin{subarray}{c}n_{1}<...<n_{s}\\\
n_{i}\in A_{\ell}\end{subarray}}|w_{\\{n\\}}(t)|^{2}\lesssim
2\parallel\boldsymbol{\mathbf{O}}_{A_{1}}\parallel|A|\sum_{i=1}^{s}\sum_{n_{1}<...<n_{s}}|w_{\\{n\\}}(t)|^{2}=2s\parallel\boldsymbol{\mathbf{O}}_{A_{1}}\parallel|A|\,,$
(25)

where we used translational invariance and $\lesssim$ rather than $\leq$
because $\Omega_{\epsilon}(t)$ is only approximately pure. This shows that
$\ket{\Psi_{s}(t)}$ and $\ket{\Uparrow}$ are macroscopically indistinguishable
for any time $t$.

We want now to extend the result to the square of the operator. We follow the
same steps:

$\braket{\Psi_{s}(t)}{\boldsymbol{\mathbf{O}}_{A_{\ell}}\boldsymbol{\mathbf{O}}_{A_{\ell^{\prime}}}}{\Psi_{s}(t)}-\braket{\Uparrow}{\boldsymbol{\mathbf{O}}_{A_{\ell}}\boldsymbol{\mathbf{O}}_{A_{\ell^{\prime}}}}{\Uparrow}\\\
=\sum_{i=1}^{s}\sum_{\begin{subarray}{c}n_{1}<...<n_{s}\\\ n_{k}\in
A_{\ell}\cup A_{\ell^{\prime}}\Leftrightarrow
k=i\end{subarray}}\biggl{(}\sum_{\begin{subarray}{c}n_{i}^{\prime}\\\
n_{k}^{\prime}\in A_{\ell}\cup A_{\ell^{\prime}},\forall
k\in\\{i\\}\end{subarray}}w_{\\{n\\}}^{*}(t)w_{\\{n\\},n_{i}\rightarrow
n^{\prime}_{i}}(t)\braket{\Uparrow}{\boldsymbol{\mathbf{\sigma}}^{+}_{n^{\prime}_{i}}\boldsymbol{\mathbf{O}}_{A_{\ell}}\boldsymbol{\mathbf{O}}_{A_{\ell^{\prime}}}\boldsymbol{\mathbf{\sigma}}^{-}_{n_{i}}}{\Uparrow}-|w_{\\{n\\}}(t)|^{2}\braket{\Uparrow}{\boldsymbol{\mathbf{O}}_{A_{\ell}}\boldsymbol{\mathbf{O}}_{A_{\ell^{\prime}}}}{\Uparrow}\biggr{)}\\\
+\sum_{i<j=1}^{s}\sum_{\begin{subarray}{c}n_{1}<...<n_{s}\\\ n_{k}\in
A_{\ell}\cup A_{\ell^{\prime}}\Leftrightarrow
k\in\\{i,j\\}\end{subarray}}\biggl{(}\sum_{\begin{subarray}{c}n_{i}^{\prime}<n^{\prime}_{j}\\\
n^{\prime}_{k}\in A_{\ell}\cup A_{\ell^{\prime}},\forall
k\in\\{i,j\\}\end{subarray}}w_{\\{n\\}}^{*}(t)w_{\begin{subarray}{c}\\{n\\},n_{i}\rightarrow
n^{\prime}_{i}\\\ n_{j}\rightarrow
n^{\prime}_{j}\end{subarray}}(t)\braket{\Uparrow}{\boldsymbol{\mathbf{\sigma}}^{+}_{n^{\prime}_{i}}\boldsymbol{\mathbf{\sigma}}^{+}_{n^{\prime}_{j}}\boldsymbol{\mathbf{O}}_{A_{\ell}}\boldsymbol{\mathbf{O}}_{A_{\ell^{\prime}}}\boldsymbol{\mathbf{\sigma}}^{-}_{n_{i}}\boldsymbol{\mathbf{\sigma}}^{-}_{n_{j}}}{\Uparrow}+\\\
-|w_{\\{n\\}}(t)|^{2}\braket{\Uparrow}{\boldsymbol{\mathbf{O}}_{A_{\ell}}\boldsymbol{\mathbf{O}}_{A_{\ell^{\prime}}}}{\Uparrow}\biggr{)}+\dots+\sum_{\begin{subarray}{c}n_{1}<...<n_{s}\\\
n_{k}\in A_{\ell}\cup A_{\ell^{\prime}},\forall
k\in\\{1,...,s\\}\end{subarray}}\biggl{(}\sum_{\begin{subarray}{c}n^{\prime}_{1}<...<n^{\prime}_{s}\\\
n^{\prime}_{k}\in A_{\ell}\cup A_{\ell^{\prime}},\forall
k\in\\{1,...,s\\}\end{subarray}}w_{\\{n\\}}^{*}(t)w_{\\{n^{\prime}\\}}(t)\times\\\
\times\braket{\Uparrow}{\boldsymbol{\mathbf{\sigma}}^{+}_{n_{1}^{\prime}}...\boldsymbol{\mathbf{\sigma}}^{+}_{n_{s}^{\prime}}\boldsymbol{\mathbf{O}}_{A_{\ell}}\boldsymbol{\mathbf{O}}_{A_{\ell^{\prime}}}\boldsymbol{\mathbf{\sigma}}^{-}_{n_{1}}...\boldsymbol{\mathbf{\sigma}}^{-}_{n_{s}}}{\Uparrow}-|w_{\\{n\\}}(t)|^{2}\braket{\Uparrow}{\boldsymbol{\mathbf{O}}_{A_{\ell}}\boldsymbol{\mathbf{O}}_{A_{\ell^{\prime}}}}{\Uparrow}\biggr{)}\,,$
(26)

from which

$|\braket{\Psi_{s}(t)}{\boldsymbol{\mathbf{O}}_{A_{\ell}}\boldsymbol{\mathbf{O}}_{A_{\ell^{\prime}}}}{\Psi_{s}(t)}-\braket{\Uparrow}{\boldsymbol{\mathbf{O}}_{A_{\ell}}\boldsymbol{\mathbf{O}}_{A_{\ell^{\prime}}}}{\Uparrow}|\leq
2\parallel\boldsymbol{\mathbf{O}}_{A_{\ell}}\parallel^{2}\sum_{i=1}^{s}\sum_{\begin{subarray}{c}n_{1}<...<n_{s}\\\
n_{k}\in A_{\ell}\cup A_{\ell^{\prime}}\Leftrightarrow
k=i\end{subarray}}|w_{\\{n\\}}(t)|^{2}+\\\
+2\parallel\boldsymbol{\mathbf{O}}_{A_{\ell}}\parallel^{2}\sum_{i<j=1}^{s}\sum_{\begin{subarray}{c}n_{1}<...<n_{s}\\\
n_{k}\in A_{\ell}\cup A_{\ell^{\prime}}\Leftrightarrow
k\in\\{i,j\\}\end{subarray}}|w_{\\{n\\}}(t)|^{2}+\dots+2\parallel\boldsymbol{\mathbf{O}}_{A_{\ell}}\parallel^{2}\sum_{\begin{subarray}{c}n_{1}<...<n_{s}\\\
n_{k}\in A_{\ell}\cup A_{\ell^{\prime}},\forall
k\in\\{1,...,s\\}\end{subarray}}|w_{\\{n\\}}(t)|^{2}\\\ \leq
2\parallel\boldsymbol{\mathbf{O}}_{A_{\ell}}\parallel^{2}\sum_{i=1}^{s}\sum_{\begin{subarray}{c}n_{1}<...<n_{s}\\\
n_{i}\in A_{\ell}\cup A_{\ell^{\prime}}\end{subarray}}|w_{\\{n\\}}(t)|^{2}\,,$
(27)

and finally

$|\braket{\Psi_{s}(t)}{\boldsymbol{\mathbf{O}}^{2}}{\Psi_{s}(t)}-\braket{\Uparrow}{\boldsymbol{\mathbf{O}}^{2}}{\Uparrow}|\leq\sum_{\ell,\ell^{\prime}\in\Omega_{\epsilon}(t)}|\braket{\Psi_{s}(t)}{\boldsymbol{\mathbf{O}}_{A_{\ell}}\boldsymbol{\mathbf{O}}_{A_{\ell^{\prime}}}}{\Psi_{s}(t)}-\braket{\Uparrow}{\boldsymbol{\mathbf{O}}_{A_{\ell}}\boldsymbol{\mathbf{O}}_{A_{\ell^{\prime}}}}{\Uparrow}|\\\
\leq
2\parallel\boldsymbol{\mathbf{O}}_{A_{1}}\parallel^{2}\sum_{\ell,\ell^{\prime}\in\Omega_{\epsilon}(t)}\sum_{i=1}^{s}\sum_{\begin{subarray}{c}n_{1}<...<n_{s}\\\
n_{i}\in A_{\ell}\cup A_{\ell^{\prime}}\end{subarray}}|w_{\\{n\\}}(t)|^{2}\leq
4\parallel\boldsymbol{\mathbf{O}}_{A_{1}}\parallel^{2}\sum_{\ell,\ell^{\prime}\in\Omega_{\epsilon}(t)}\sum_{i=1}^{s}\sum_{\begin{subarray}{c}n_{1}<...<n_{s}\\\
n_{i}\in A_{\ell}\end{subarray}}|w_{\\{n\\}}(t)|^{2}\\\ \lesssim
4\parallel\boldsymbol{\mathbf{O}}_{A_{1}}\parallel^{2}|A||\Omega_{\epsilon}(t)|\sum_{i=1}^{s}\sum_{n_{1}<...<n_{s}}|w_{\\{n\\}}(t)|^{2}=4s\parallel\boldsymbol{\mathbf{O}}_{A_{1}}\parallel^{2}|A||\Omega_{\epsilon}(t)|\,.$
(28)

This shows that the variance of the any operator $\boldsymbol{\mathbf{O}}$ on
the state $\ket{\Psi_{s}(t)}$ coincides with the one on the state
$\ket{\Uparrow}$ modulo corrections of order
$\mathcal{O}(|\Omega_{\epsilon}(t)|)$. Since $\ket{\Uparrow}$ is a state for
which clustering of correlations holds and his variance cannot be larger than
system’s size, we obtain that the state $\ket{\Psi_{s}(t)}$ has the same
properties, implying that it is not macroscopically entangled. Our proof can
be easily generalised to any power of the operator, but higher powers are not
needed to our purposes.

We now extend the results above to more general states. We consider, first,
the quantity $\braket{\Psi_{s^{\prime}}}{\boldsymbol{\mathbf{O}}}{\Psi_{s}}$,
$s\neq s^{\prime}$, for two different states $\ket{\Psi_{s}}$ and
$\ket{\Psi_{s^{\prime}}}$. Without loss of generality, we can assume
$s>s^{\prime}$. We start from

$|\braket{\Psi_{s^{\prime}}}{\boldsymbol{\mathbf{O}}}{\Psi_{s}}|\leq\sum_{\ell\in\Omega_{\epsilon}(t)}\sum_{n^{\prime}_{1}<...<n^{\prime}_{s^{\prime}}}\sum_{n_{1}<...<n_{s}}|\braket{\Uparrow}{\boldsymbol{\mathbf{\sigma}}^{+}_{n^{\prime}_{1}}...\boldsymbol{\mathbf{\sigma}}^{+}_{n^{\prime}_{s^{\prime}}}\boldsymbol{\mathbf{O}}_{A_{\ell}}\boldsymbol{\mathbf{\sigma}}^{-}_{n_{1}}...\boldsymbol{\mathbf{\sigma}}^{-}_{n_{s}}}{\Uparrow}||w^{\prime}_{\\{n^{\prime}\\}}(t)||w_{\\{n\\}}(t)|,$
(29)

and rewrite the sums regrouping them, for fixed $\ell$, according to how many
of the primed indices $\\{n^{\prime}\\}$ are outside the set $A_{\ell}$. This
is a number $p$ that goes from 0 to $s^{\prime}$. In order to get a non-zero
contribution, the number of indices $\\{n\\}$ outside $A_{\ell}$ should also
be $p$. The number of indices $\\{n^{\prime}\\}\in A_{\ell}$ is then
$s^{\prime}-p\in\\{0,...,s^{\prime}\\}$, while the number of indices
$\\{n\\}\in A_{\ell}$ is $s-p\in\\{s-s^{\prime},...,s\\}$; note that there is
no contribution from $|A_{\ell}|<s-p$. For given $p$, there are $p+1$ distinct
subsets of consecutive $n$ that can be in $A_{\ell}$ (they are characterised
by the position of the first element of the subset); to keep track of this
fact, we introduce a sum over an index $j$, such that the index $n_{j}$ is the
smallest of the indices $\\{n\\}\in A_{\ell}$. Such a decomposition leads to

$|\braket{\Psi_{s^{\prime}}}{\boldsymbol{\mathbf{O}}}{\Psi_{s}}|\leq\sum_{\ell\in\Omega_{\epsilon}(t)}\sum_{p=0}^{s^{\prime}}\sum_{j=1}^{p+1}\sum_{\begin{subarray}{c}n_{1}<...<n_{s}\\\
n_{k}\in A_{\ell}\Leftrightarrow j\leq
k<j+s-p\end{subarray}}\sum_{\begin{subarray}{c}n^{\prime}_{1}<...<n^{\prime}_{s^{\prime}-p}\\\
n^{\prime}_{k}\in A_{\ell},\forall
k\in\\{1,...,s^{\prime}-p\\}\end{subarray}}\\\
|\braket{\Uparrow}{\boldsymbol{\mathbf{\sigma}}^{+}_{n^{\prime}_{1}}...\boldsymbol{\mathbf{\sigma}}^{+}_{n^{\prime}_{s^{\prime}-p}}\boldsymbol{\mathbf{O}}_{A_{\ell}}\boldsymbol{\mathbf{\sigma}}^{-}_{n_{j}}...\boldsymbol{\mathbf{\sigma}}^{-}_{n_{j+s-p-1}}}{\Uparrow}||w^{\prime}_{\\{n\\}\setminus\\{n_{j},...,n_{j+s-p-1}\\}\cup\\{n^{\prime}_{1},...,n^{\prime}_{s^{\prime}-p}\\}}(t)||w_{\\{n\\}}(t)|\leq\\\
\leq\parallel\boldsymbol{\mathbf{O}}_{A_{\ell}}\parallel\sum_{p=0}^{s^{\prime}}\sum_{j=1}^{p+1}\sum_{\ell\in\Omega_{\epsilon}(t)}\sum_{\begin{subarray}{c}n_{1}<...<n_{s}\\\
n_{k}\in A_{\ell}\Leftrightarrow j\leq
k<j+s-p\end{subarray}}\sum_{\begin{subarray}{c}n^{\prime}_{1}<...<n^{\prime}_{s^{\prime}-p}\\\
n^{\prime}_{k}\in A_{\ell},\forall
k\in\\{1,...,s^{\prime}-p\\}\end{subarray}}|w^{\prime}_{\\{n\\}\setminus\\{n_{j},...,n_{j+s-p-1}\\}\cup\\{n^{\prime}_{1},...,n^{\prime}_{s^{\prime}-p}\\}}(t)||w_{\\{n\\}}(t)|.$
(30)

It is now convenient to write the sum over the indices $n,n^{\prime}$ in
$A_{\ell}$ in another way. We introduce $P_{k}^{K}$ as the set of all the
$k$-tuples of increasing numbers in $\\{0,...,K-1\\}$. Note that $P_{k}^{K}$
has $\binom{K}{k}$ elements. We then have

$|\braket{\Psi_{s^{\prime}}}{\boldsymbol{\mathbf{O}}}{\Psi_{s}}|\leq\parallel\boldsymbol{\mathbf{O}}_{A_{\ell}}\parallel\sum_{p=0}^{s^{\prime}}\sum_{j=1}^{p+1}\sum_{\ell\in\Omega_{\epsilon}(t)}\sum_{(k_{1},...,k_{s-p})\in
P_{s-p}^{|A|}}\sum_{(k^{\prime}_{1},...,k^{\prime}_{s^{\prime}-p})\in
P_{s^{\prime}-p}^{|A|}}\sum_{n_{1}<...<n_{j-1}<A_{\ell}<n_{j+s-p}<...<n_{s}}\\\
|w^{\prime}_{\\{n_{1},...,n_{j-1},\ell+k^{\prime}_{1},...,\ell+k^{\prime}_{s^{\prime}-p},n_{j+s-p},...,n_{s}\\}}(t)||w_{\\{n_{1},...,n_{j-1},\ell+k_{1},...,\ell+k_{s-p},n_{j+s-p},...,n_{s}}(t)|,$
(31)

which we rewrite isolating the term with $p=s^{\prime}$ (the case in which all
the indices $\\{n^{\prime}\\}$ are outside $A_{\ell}$), which is special:

$|\braket{\Psi_{s^{\prime}}}{\boldsymbol{\mathbf{O}}}{\Psi_{s}}|\leq\parallel\boldsymbol{\mathbf{O}}_{A_{\ell}}\parallel\sum_{p=0}^{s^{\prime}-1}\sum_{j=1}^{p+1}\sum_{(k_{1},...,k_{s-p})\in
P_{s-p}^{|A|}}\sum_{(k^{\prime}_{1},...,k^{\prime}_{s^{\prime}-p})\in
P_{s^{\prime}-p}^{|A|}}\sum_{\ell\in\Omega_{\epsilon}(t)}\sum_{n_{1}<...<n_{j-1}<A_{\ell}<n_{j+s-p}<...<n_{s}}\\\
|w^{\prime}_{\\{n_{1},...,n_{j-1},\ell+k^{\prime}_{1},...,\ell+k^{\prime}_{s^{\prime}-p},n_{j+s-p},...,n_{s}\\}}(t)||w_{\\{n_{1},...,n_{j-1},\ell+k_{1},...,\ell+k_{s-p},n_{j+s-p},...,n_{s}}(t)|\\\
+\parallel\boldsymbol{\mathbf{O}}_{A_{\ell}}\parallel\sum_{j=1}^{s^{\prime}+1}\sum_{(k_{1},...,k_{s-s^{\prime}})\in
P_{s-s^{\prime}}^{|A|}}\sum_{\ell\in\Omega_{\epsilon}(t)}\sum_{n_{1}<...<n_{j-1}<A_{\ell}<n_{j+s-s^{\prime}}<...<n_{s}}\\\
|w^{\prime}_{\\{n_{1},...,n_{j-1},n_{j+s-s^{\prime}},...,n_{s}\\}}(t)||w_{\\{n_{1},...,n_{j-1},\ell+k_{1},...,\ell+k_{s-s^{\prime}},n_{j+s-s^{\prime}},...,n_{s}}(t)|.$
(32)

Now, for fixed $p,j,\\{k\\},\\{k^{\prime}\\}$, we want to maximize the sums
over $\ell,\\{n\\}$ independently, which gives an upper bound to the full
quantity. To that aim, we introduce the following compact notation: for the
terms with $p<s^{\prime}$

$\sum_{\ell\in\Omega_{\epsilon}(t)}\sum_{n_{1}<...<n_{j-1}<A_{\ell}<n_{j+s-p}<...<n_{s}}|w^{\prime}_{\\{n_{1},...,n_{j-1},\ell+k^{\prime}_{1},...,\ell+k^{\prime}_{s^{\prime}-p},n_{j+s-p},...,n_{s}\\}}(t)|\times\\\
\times|w_{\\{n_{1},...,n_{j-1},\ell+k_{1},...,\ell+k_{s-p},n_{j+s-p},...,n_{s}}(t)|\quad\equiv\sum_{b\in
B}W^{\prime}_{b}W_{b}\,,$ (33)

and for the term $p=s^{\prime}$

$\sum_{\ell\in\Omega_{\epsilon}(t)}\sum_{n_{1}<...<n_{j-1}<A_{\ell}<n_{j+s-s^{\prime}}<...<n_{s}}|w^{\prime}_{\\{n_{1},...,n_{j-1},n_{j+s-s^{\prime}},...,n_{s}\\}}(t)||w_{\\{n_{1},...,n_{j-1},\ell+k_{1},...,\ell+k_{s-s^{\prime}},n_{j+s-s^{\prime}},...,n_{s}}(t)|\\\
\equiv\sum_{\ell\in\Omega_{\epsilon}(t)}\sum_{c\in
C}W^{\prime}_{c}W_{c,\ell}\,,$ (34)

where $b$ and $c$ regroup all the indices that $w$ and $w^{\prime}$ have in
common and $W,W^{\prime}$ are a synthetic expression for the coefficients that
report only the indices we are interested in; note that in the first case
there are as many independent $w$ as $w^{\prime}$, while, for $p=s^{\prime}$,
$w^{\prime}$ does not depend on $\ell$ anymore. In order to maximize the
expressions above, we should take into accounts the constraints
$\sum_{n^{\prime}_{1}<...<n^{\prime}_{s^{\prime}}}|w^{\prime}_{\\{n^{\prime}\\}}|^{2}=1$
and $\sum_{n_{1}<...<n_{s}}|w_{\\{n\\}}|^{2}=1$. In our search for the
maximum, we can use the stronger conditions $\sum_{b\in B}W_{b}^{2}=1$ and
$\sum_{b\in B}(W^{\prime}_{b})^{2}=1$ in the first case and $\sum_{c\in
C}(W_{c}^{\prime})^{2}=1$ and $\sum_{c\in
C,\ell\in\Omega_{\epsilon}(t)}W_{c,\ell}^{2}=1$ in the second case, which
amount to require that the coefficients that are not involved in the sum are
zero (given the original constraint, this maximises the sum because all the
terms are positive). Using the method of Lagrange multipliers, one can show
that, under these constraints, $\sum_{b\in B}W^{\prime}_{b}W_{b}$ is maximised
by $W_{b}=W^{\prime}_{b}$ and equals 1, while
$\sum_{\ell\in\Omega_{\epsilon}(t)}\sum_{c\in C}W^{\prime}_{c}W_{c,\ell}$ has
degenerate points of maximum in
$W_{c,\ell}=|\Omega_{\epsilon}(t)|^{-1/2}W^{\prime}_{c}$ and gives
$|\Omega_{\epsilon}(t)|^{1/2}$. In the end

$|\braket{\Psi_{s^{\prime}}}{\boldsymbol{\mathbf{O}}}{\Psi_{s}}|\leq\parallel\boldsymbol{\mathbf{O}}_{A_{\ell}}\parallel\sum_{j=1}^{s^{\prime}+1}\sum_{(k_{1},...,k_{s-s^{\prime}})\in
P_{s-s^{\prime}}^{|A|}}|\Omega_{\epsilon}(t)|^{1/2}+\mathcal{O}(|\Omega_{\epsilon}(t)|^{0})=\\\
=\parallel\boldsymbol{\mathbf{O}}_{A_{\ell}}\parallel(s^{\prime}+1)\binom{|A|}{s-s^{\prime}}|\Omega_{\epsilon}(t)|^{1/2}+\mathcal{O}(|\Omega_{\epsilon}(t)|^{0})\,.$
(35)

This result will be used below.

We finally consider a finite sum of states obtained by flipping different
numbers of spins: $\ket{\Psi^{(S)}}=\sum_{s=1}^{S}a_{s}\ket{\Psi_{s}}$, where
$a_{j}\in\mathbb{R},\forall j$ (any potential phase can be absorbed in the
states), and $\sum_{s=1}^{S}a_{j}^{2}=1$. From Eq. [35] it readily follows

$|\braket{\Psi_{S+1}}{\boldsymbol{\mathbf{O}}}{\Psi^{(S)}}|\leq\sum_{s=1}^{S}|\braket{\Psi_{S+1}}{\boldsymbol{\mathbf{O}}}{\Psi_{s}}|\lesssim\mathcal{O}(|\Omega_{\epsilon}(t)|^{1/2})\,.$
(36)

Let us now consider the state
$\ket{\Psi^{(2)}}=a_{1}\ket{\Psi_{1}}+a_{2}\ket{\Psi_{2}}$, with
$a_{1},a_{2}\in\mathbb{R}$ and $a_{1}^{2}+a_{2}^{2}=1$. First of all, note
that such a state is macroscopically equivalent to $\ket{\Uparrow}$. Indeed,
using
$\braket{\Psi_{2}}{\boldsymbol{\mathbf{O}}}{\Psi_{2}}=\braket{\Psi_{1}}{\boldsymbol{\mathbf{O}}}{\Psi_{1}}+\mathcal{O}(|\Omega_{\epsilon}(t)|^{0})$
(a corollary of Eq. [25]), we have

$\braket{\Psi^{(2)}}{\boldsymbol{\mathbf{O}}}{\Psi^{(2)}}-\braket{\Uparrow}{\boldsymbol{\mathbf{O}}}{\Uparrow}=a_{1}^{2}\braket{\Psi_{1}}{\boldsymbol{\mathbf{O}}}{\Psi_{1}}+a_{2}^{2}\braket{\Psi_{2}}{\boldsymbol{\mathbf{O}}}{\Psi_{2}}+2a_{1}a_{2}Re(\braket{\Psi_{1}}{\boldsymbol{\mathbf{O}}}{\Psi_{2}})-\braket{\Uparrow}{\boldsymbol{\mathbf{O}}}{\Uparrow}\\\
=2a_{1}a_{2}Re(\braket{\Psi_{1}}{\boldsymbol{\mathbf{O}}}{\Psi_{2}})+\mathcal{O}(|\Omega_{\epsilon}(t)|^{0})\lesssim\mathcal{O}(|\Omega_{\epsilon}(t)|^{1/2})\,,$
(37)

which is sub-extensive. Similarly, we can conclude that
$\braket{\Psi^{(S)}}{\boldsymbol{\mathbf{O}}}{\Psi^{(S)}}-\braket{\Uparrow}{\boldsymbol{\mathbf{O}}}{\Uparrow}\lesssim\mathcal{O}(|\Omega_{\epsilon}(t)|^{1/2})$,
meaning that $\ket{\Psi^{(S)}}$ is macroscopically equivalent to
$\ket{\Uparrow}$ for any finite $S$. Let us now look at the variance of
$\boldsymbol{\mathbf{O}}$ in $\ket{\Psi^{(2)}}$:

$\braket{\Psi^{(2)}(t)}{\boldsymbol{\mathbf{O}}^{2}}{\Psi^{(2)}(t)}-\braket{\Psi^{(2)}(t)}{\boldsymbol{\mathbf{O}}}{\Psi^{(2)}(t)}^{2}=a_{1}^{2}\braket{\Psi_{1}}{\boldsymbol{\mathbf{O}}^{2}}{\Psi_{1}}+a_{2}^{2}\braket{\Psi_{2}}{\boldsymbol{\mathbf{O}}^{2}}{\Psi_{2}}+2a_{1}a_{2}Re\left(\braket{\Psi_{1}}{\boldsymbol{\mathbf{O}}^{2}}{\Psi_{2}}\right)+\\\
-\left(a_{1}^{2}\braket{\Psi_{1}}{\boldsymbol{\mathbf{O}}}{\Psi_{1}}+a_{2}^{2}\braket{\Psi_{2}(t)}{\boldsymbol{\mathbf{O}}}{\Psi_{2}(t)}+2a_{1}a_{2}Re\left(\braket{\Psi_{1}}{\boldsymbol{\mathbf{O}}}{\Psi_{2}(t)}\right)\right)^{2}=\\\
=\braket{\Psi_{1}}{\boldsymbol{\mathbf{O}}^{2}}{\Psi_{1}}+2a_{1}a_{2}Re\left(\braket{\Psi_{1}}{\boldsymbol{\mathbf{O}}^{2}}{\Psi_{2}(t)}\right)+\mathcal{O}(|\Omega_{\epsilon}(t)|)+\\\
-\left(\braket{\Psi_{1}}{\boldsymbol{\mathbf{O}}}{\Psi_{1}}+2a_{1}a_{2}Re\left(\braket{\Psi_{1}}{\boldsymbol{\mathbf{O}}}{\Psi_{2}(t)}\right)+\mathcal{O}(|\Omega_{\epsilon}(t)|^{0})\right)^{2}\,,$
(38)

where we used
$\braket{\Psi_{2}}{\boldsymbol{\mathbf{O}}}{\Psi_{2}}=\braket{\Psi_{1}}{\boldsymbol{\mathbf{O}}}{\Psi_{1}}+\mathcal{O}(|\Omega_{\epsilon}(t)|^{0})$
and
$\braket{\Psi_{2}}{\boldsymbol{\mathbf{O}}^{2}}{\Psi_{2}}=\braket{\Psi_{1}}{\boldsymbol{\mathbf{O}}^{2}}{\Psi_{1}}+\mathcal{O}(|\Omega_{\epsilon}(t)|)$
(corollaries of Eq. [25] and Eq. [28]). Using that the variance of any state
$\ket{\Psi_{s}}$ can not grow faster than $|\Omega_{\epsilon}(t)|$, we get

$\braket{\Psi^{(2)}(t)}{\boldsymbol{\mathbf{O}}^{2}}{\Psi^{(2)}(t)}-\braket{\Psi^{(2)}(t)}{\boldsymbol{\mathbf{O}}}{\Psi^{(2)}(t)}^{2}\\\
=2a_{1}a_{2}\left[Re\left(\braket{\Psi_{1}}{\boldsymbol{\mathbf{O}}^{2}}{\Psi_{2}(t)}\right)-\braket{\Psi_{1}}{\boldsymbol{\mathbf{O}}}{\Psi_{1}}Re\left(\braket{\Psi_{1}}{\boldsymbol{\mathbf{O}}}{\Psi_{2}(t)}\right)\right]+\mathcal{O}(|\Omega_{\epsilon}(t)|)\,.$
(39)

If the leading contribution to the variance were larger than
$\mathcal{O}(|\Omega_{\epsilon}(t)|)$, by changing the sign of $a_{1}$ we
could make the variance negative; since the variance is by definition
positive, by contradiction, we have that the leading order cannot be larger
than $\mathcal{O}(|\Omega_{\epsilon}(t)|)$. We can now assume that the
variance in the state $\ket{\Psi^{(S-1)}}$ is of order
$\mathcal{O}(|\Omega_{\epsilon}(t)|)$ and show that this implies that also the
variance in $\ket{\Psi^{(S)}}$ is. The proof is essentially the same as above
with just a small modification, namely,
$\braket{\Psi^{(S-1)}}{\boldsymbol{\mathbf{O}}}{\Psi^{(S-1)}}-\braket{\Uparrow}{\boldsymbol{\mathbf{O}}}{\Uparrow}$
is $\mathcal{O}(|\Omega_{\epsilon}(t)|^{1/2})$ rather than
$\mathcal{O}(|\Omega_{\epsilon}(t)|^{1/2})$. This gives an extra term in the
potential leading order, which however does not affect the final result:
calling $\ket{\Psi^{(S)}}=a_{1}\ket{{\Psi_{S}}}+a_{2}\ket{\Psi^{(S-1)}}$, we
have

$\braket{\Psi^{(S)}(t)}{\boldsymbol{\mathbf{O}}^{2}}{\Psi^{(S)}(t)}-\braket{\Psi^{(S)}(t)}{\boldsymbol{\mathbf{O}}}{\Psi^{(S)}(t)}^{2}\\\
=a_{1}^{2}\braket{\Psi_{S}}{\boldsymbol{\mathbf{O}}^{2}}{\Psi_{S}}+a_{2}^{2}\braket{\Psi^{(S-1)}}{\boldsymbol{\mathbf{O}}^{2}}{\Psi^{(S-1)}}+2a_{1}a_{2}Re\left(\braket{\Psi_{S}}{\boldsymbol{\mathbf{O}}^{2}}{\Psi^{(S-1)}}\right)\\\
-\left(a_{1}^{2}\braket{\Psi_{S}}{\boldsymbol{\mathbf{O}}}{\Psi_{S}}+a_{2}^{2}\braket{\Psi^{(S-1)}(t)}{\boldsymbol{\mathbf{O}}}{\Psi^{(S-1)}(t)}+2a_{1}a_{2}Re\left(\braket{\Psi_{S}}{\boldsymbol{\mathbf{O}}}{\Psi^{(S-1)}(t)}\right)\right)^{2}\\\
=\braket{\Psi_{S}}{\boldsymbol{\mathbf{O}}^{2}}{\Psi_{S}}+2a_{1}a_{2}Re\left(\braket{\Psi_{S}}{\boldsymbol{\mathbf{O}}^{2}}{\Psi^{(S-1)}(t)}\right)+\mathcal{O}(|\Omega_{\epsilon}(t)|)\\\
-\left(\braket{\Psi_{S}}{\boldsymbol{\mathbf{O}}}{\Psi_{S}}+2a_{1}a_{2}Re\left(\braket{\Psi_{S}}{\boldsymbol{\mathbf{O}}}{\Psi^{(S-1)}(t)}\right)+\mathcal{O}(a_{1}^{0}|\Omega_{\epsilon}(t)|^{1/2})\right)^{2}\\\
\lesssim
2a_{1}a_{2}\left[Re\left(\braket{\Psi_{S}}{\boldsymbol{\mathbf{O}}^{2}}{\Psi^{(S-1)}(t)}\right)-\braket{\Psi_{S}}{\boldsymbol{\mathbf{O}}}{\Psi_{S}}Re\left(\braket{\Psi_{S}}{\boldsymbol{\mathbf{O}}}{\Psi^{(S-1)}(t)}+\mathcal{O}(a_{1}^{0}|\Omega_{\epsilon}(t)|^{1/2}\right)\right]+\mathcal{O}(|\Omega_{\epsilon}(t)|)\,.$
(40)

Since $a_{1}$ appears linearly as in the case $S=2$, we can use the same trick
and we have by contradiction that the variance in $\ket{\Psi^{(S)}}$ cannot
grow faster than the system’s size.

In conclusion, we have proved that the state obtained by any local
perturbation of $\ket{\Uparrow}$ is macroscopically equivalent to
$\ket{\Uparrow}$ and the variance of any operator in this state does not grow
faster than system size, where the system size is defined by
$\Omega_{\epsilon}(t)$.