# Moving the epidemic tipping point through topologically targeted social
distancing

Sara Ansari  FutureLab on Game Theory and Networks of Interacting Agents,
Complexity Science Department, Potsdam Institute for Climate Impact Research,
Member of the Leibniz Association, PO Box 601203, 14412 Potsdam, Germany
Mehrnaz Anvari  Potsdam Institute for Climate Impact Research, Potsdam 14473,
Germany  Oskar Pfeffer  Potsdam Institute for Climate Impact Research, Potsdam
14473, Germany  Institute of Theoretical Physics, Technische Universität
Berlin, Hardenbergstr. 36, D-10623 Berlin,Germany  Nora Molkenthin  Potsdam
Institute for Climate Impact Research, Potsdam 14473, Germany  Frank Hellmann
Potsdam Institute for Climate Impact Research, Potsdam 14473, Germany  Jobst
Heitzig  FutureLab on Game Theory and Networks of Interacting Agents,
Complexity Science Department, Potsdam Institute for Climate Impact Research,
Member of the Leibniz Association, PO Box 601203, 14412 Potsdam, Germany
Jürgen Kurths  Potsdam Institute for Climate Impact Research, Potsdam 14473,
Germany  Institute of Physics, Humboldt University, Berlin 12489, German

###### Abstract

The epidemic threshold of a social system is the ratio of infection and
recovery rate above which a disease spreading in it becomes an epidemic. In
the absence of pharmaceutical interventions (i.e. vaccines), the only way to
control a given disease is to move this threshold by non-pharmaceutical
interventions like social distancing, past the epidemic threshold
corresponding to the disease, thereby tipping the system from epidemic into a
non-epidemic regime. Modeling the disease as a spreading process on a social
graph, social distancing can be modeled by removing some of the graphs links.
It has been conjectured that the largest eigenvalue of the adjacency matrix of
the resulting graph corresponds to the systems epidemic threshold. Here we use
a Markov chain Monte Carlo (MCMC) method to study those link removals that do
well at reducing the largest eigenvalue of the adjacency matrix. The MCMC
method generates samples from the relative canonical network ensemble with a
defined expectation value of $\lambda_{max}$. We call this the “well-
controlling network ensemble” (WCNE) and compare its structure to randomly
thinned networks with the same link density. We observe that networks in the
WCNE tend to be more homogeneous in the degree distribution and use this
insight to define two ad-hoc removal strategies, which also substantially
reduce the largest eigenvalue. A targeted removal of 80% of links can be as
effective as a random removal of 90%, leaving individuals with twice as many
contacts. Finally, by simulating epidemic spreading via either an SIS or an
SIR model on network ensembles created with different link removal strategies
(random, WCNE, or degree-homogenizing), we show that tipping from an epidemic
to a non-epidemic state happens at a larger critical ratio between infection
rate and recovery rate for WCNE and degree-homogenized networks than for those
obtained by random removals.

## Introduction

In the absence of pharmaceutical interventions, the prevention of infection
through a reduction of social contacts presents an effective method to slow or
even halt the spread of an epidemic greenstone2020does ;
maharaj2012controlling . However, limiting social contacts comes at a
significant psychological venkatesh2020social  and economical
koren2020business ; farboodi2020internal  cost. To reduce those socio-economic
adverse effects, it is therefore desirable to use targeted social distancing
measures, in order to remove fewer links for achieving the same effect. If the
infection patterns for a particular disease are known, for example, measures
can be taken to remove the most common routes of infection glass2006targeted ;
shim2013optimal ; enns2015link . Similarly, methods have been introduced for
targeted link removal if the infection status of individual nodes is known
nandi2016methods ; enns2015link ; matamalas2018effective  as well as systems
where the entire network structure is known in detail matamalas2018effective ;
yu2018identifying . This would be the case for example in more aggregated
settings, like the transport network, which has been shown to directly affect
disease spreading brockmann2013hidden . Moreover, in ghosh2020optimal  the
authors have introduced a test-kit based control strategy to avoid a strict
lock down or social distancing and, consequently, maintain economic stability
by partial opening of business centers.

Often, however, diseases without pharmaceutical interventions are new and we
lack detailed knowledge about their infection patterns, and in the absence of
functioning contact tracing many infections may go unnoticed, especially if
the symptoms are mild or unspecific. Therefore social distancing as a
behavioral change is the primary intervention for disease prevention
fong2020nonpharmaceutical .

In the extreme case of reducing the number of contacts in the social graph to
zero, by necessity the disease dies out and is fully controlled. If there are
no links left in the network, the disease can not spread. Intuitively we
expect that the more contacts are removed, the better a disease can be
controlled and prevented from becoming an epidemic. This can be made precise
given a model of the disease spread. For an approximation of SIS dynamics with
recovery rate $\delta$ and infectiousness $\beta$ it was shown in
Chakrabarti2008  that the epidemic threshold, $\tau$, is determined by the
inverse of the largest eigenvalue of the adjacency matrix of the underlying
contact network, i.e., $\tau=\lambda_{max}^{-1}$. If $\tau>\beta/\delta$, the
disease quickly dies out and remains confined to a small section of the
network, if $\tau<\beta/\delta$, however, the disease spreads as an epidemic
over large parts of the network and becomes endemic. At
$\tau^{c}=(\lambda^{c}_{max})^{-1}=\beta/\delta$ the system tips from epidemic
to non-epidemic regime. Decreasing $\lambda_{max}$ thus represents a
topological way of controlling an epidemic that is independent of the unknown
or unchangeable disease parameters $\beta$ and $\delta$.

By removing links from a network, one can generally reduce the value of
$\lambda_{max}$. However, the size of this reduction depends on the specific
links removed (see the example network in Chakrabarti2008 ). Here, we
introduce a way to remove links based on $\lambda_{max}$ in order to sample
from ensembles with a given disease-controlling property using Markov chain
Monte Carlo (MCMC) methods iba2014multicanonical . We thereby study what
characterizes contact reductions that perform well at controlling the disease
spread. We will see that well-controlling networks on average have
significantly lower $\lambda_{max}$ even at relatively low $\nu$. The largest
eigenvalue is a lower bound for SIS goltsev2012localization ;
pastor2015epidemic , and often, but not always indicative of the SIR
newman2002spread ; karrer2014percolation . Thus it is a natural candidate for
building well-controlling networks. In the last section we will see in
simulations that the resulting networks and methods do indeed work for SIR and
SIS models. We find that well-controlling networks have a more strongly peaked
degree distribution, as well as more homogeneously sized connected components
once the network is no longer connected. We therefore suggest and test two
heuristics for achieving a similar effect by homogenizing the node degrees in
the network through link removal and find them to lead to a similarly strong
control, suggesting that the strongly peaked degree distribution characterizes
well-controlling networks. Finally, by running both SIS and SIR models on
networks obtained from different removal strategies, we show that the tipping
point in the critical ratio of $\beta/\delta$ occurs for larger ratios in the
WCNE and degree-homogenized networks than in generic removals.

The results are robust across all initial network ensembles tested here,
namely Barabasi–Albert (BA) networks barabasi1999emergence , random geometric
(RG) graphs dall2002random  and a real-world network of friendships between
high-school students moody2001peer .

## Basic notions and Model description

We start from an initial graph ensemble representing the social network
relevant for the spreading. The BA- and RG- network ensembles were chosen for
their depiction of different types and applications of networks, on which
epidemic spreading processes can happen. BA-Networks were introduced as a
simplified model for online social networks, which are relevant for the
spreading opinions, ideas or computer viruses. RG-networks on the other hand
capture the spatially embedded structure of most in-person social networks
most relevant for the spread of diseases.

We consider a generic reduction in which a certain percentage of links are
removed at random. Given a social network $N^{soc}$ with $E$ edges this
defines an ensemble of reduced networks $q^{\rho}(N^{soc})$ that are
equidistributed on the subgraphs of $N^{soc}$ with $(1-\rho)E$ edges, where
$0<\rho<1$ is the contact reduction rate. We then consider the canonical
network ensemble pfeffer2020RCNE  relative to this ensemble of generic
reductions that achieves a certain expectation value of the largest Eigenvalue
$\lambda_{max}$. That is, the ensemble of networks that is least
distinguishable from random contact removals at a given expectation value of
$\lambda_{max}$. The canonical network ensemble is given by the probabilities

$\displaystyle p(N)=\frac{1}{Z}e^{-\nu\lambda_{max}(N)}q^{\rho}(N).$ (1)

Here $Z$ is a normalization factor, $\nu$ is the inverse genericity as defined
in pfeffer2020RCNE , which interpolates between the generic ensemble (at
$\nu=0$) and one peaked at the best controlling networks at
$\nu\rightarrow\infty$ (optimized). We call the family of ensembles with
reasonably high values of $\nu$ the well-controlling network ensembles (WCNE)
and samples drawn from such an ensemble well-controlling networks (WCN). As in
pfeffer2020RCNE  we obtain well-controlling networks by employing an MCMC
Metropolis–Hastings method Metropolis1953EquationOS ; Hastings1970MonteCS .

In both synthetic cases, the initial networks $N^{soc}$ have 100 nodes and an
average degree of $\langle k\rangle\approx 18$, amounting to around 900 links.
The precise number of links for the random geometric networks fluctuates
slightly around that value due to the network construction algorithm used.

Figure 1: Example networks from network ensembles after removing $80\%$ of
contacts from an initial BA and RG graph with $N=100$ and $E\approx 900$. a)
and c) show the generic contact reductions at $\nu=0$ or, in the other word,
they show generic networks with random removed links. b) and d) present WCNE
at $\nu=1000$. In all figures nodes with the same color and size have the same
degree. To compare WCNE networks, i.e. b) and d), with the corresponding
generic networks with random removed links in a) and c), it is visually clear
that WCNE networks have more homogeneous degree distribution as well as more
homogeneous component sizes.

Starting from a reduced network drawn randomly from $q^{\rho}(N^{soc})$ at a
given contact reduction rate $\rho$, we keep the number of links constant
through all changes of the network. We then vary the set of removed links
using MCMC. The probability of increasing/decreasing $\lambda_{max}$ in this
step depends on the selected value of $\nu$.

Fig. 1 shows example networks from a generic removal and WCNE ensembles of BA
and RG starting ensembles with a removal rate of $\rho=0.8$. At first glance,
we see that the WCNE examples have fewer disconnected nodes and a generally
more homogeneous distribution of degrees and component sizes. In the
following, we describe the applied MCMC approach and its energy function in
detail.

The state of the system at a time step $t$ is given by the network structure
and thus the adjacency matrix $A_{t}$. The energy function for the process is
given by the largest eigenvalue of the adjacency matrix $A_{t}$, describing
the remaining network (i.e., the one obtained after removing the current
candidate set of edges),

$\varepsilon(A)=\lambda_{max}(A).$ (2)

As mentioned in the introduction, it was shown in Chakrabarti2008  that the
epidemic threshold $\tau$ is proportional to the inverse of $\lambda_{max}$
for an approximation of the SIS model. In the same paper it was also
conjectured that this proportionality also holds in the SIR model. Thus,
following that conjecture, we use $\lambda_{max}$ as our energy function.

At each Monte Carlo step $t\to t+1$, a proposal $A^{\prime}$ is generated to
swap one edge between the remaining set of edges in the network and the set of
currently removed edges, thus keeping the number of removed edges constant.
The proposal is then accepted with a probability

$\displaystyle P_{A\rightarrow
A^{\prime}}=\min\left(1,e^{\nu(\varepsilon(A)-\varepsilon(A^{\prime}))}\right),$
(3)

sampling from the canonical network ensemble at inverse genericity $\nu$.
Thus, a small $\nu$ results in almost every proposal being accepted and many
random changes being made. The final ensemble is then not different from the
one obtained by randomly removing the edges. In the other extreme of very
large $\nu$, almost only those proposals are accepted that lower the energy,
resulting in an ensemble with a lower average $\lambda_{max}$.

Figure 2: MCMC substantially decreases $\lambda_{max}$ at higher inverse
genericity $\nu$. (a) and (c) and show respectively the decrease of
$\lambda_{max}$ with MCMC for different initial networks for a range of values
of the genericity $\nu$. In all networks the removal rate is $\rho=0.4$. For
high genericity, such as $\nu\rightarrow 0$, $\lambda_{max}$ fluctuates just
around its initial value after link removal (yellow line), while it reaches a
low steady state for the smaller genericity (black line). (b) and (d) show the
average of $\lambda_{max}$ for a range of removal fractions
$\rho=\\{0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9\\}$ and genericities. As it is clear
in these figures, in the larger $\nu$, the curve is shifted more downwards,
indicating a significant increase in the epidemic
threshold$\lambda_{max}^{-1}$.

This procedure is repeated until a steady state in $\lambda_{max}$ is reached,
fluctuating around $\lambda_{max}(t\rightarrow\infty)$. We found this to be
the case after 11000 steps. It should be noted that for both the BA and RG
initial network ensembles we repeat the explained strategy for $n=100$
different network configurations and, then, average over all final values of
the largest eigenvalue to obtain
$\langle\lambda_{max}\rangle=\frac{1}{n}\sum_{i=1}^{n}\prescript{}{i}{\lambda_{max}(\text{final})}$.

## results

Examples for the evolution of $\lambda_{max}$ are shown in Figs. 2 a) and c)
for a range of $\nu$ and $\rho=0.4$.

While the ensemble after random link removal ($\nu=0$) has largest eigenvalues
fluctuating around $\lambda_{max}\approx 13.5$ for both BA and RG networks at
$\rho=0.4$, for the least general ensemble with $\nu=1000$ this goes down to
$\lambda_{max}\approx 11$ in case of BA networks and $\lambda_{max}\approx
10.5$ in case of RG networks. Values of $\nu>1000$ were not considered here
for several reasons. Firstly they require much longer (or more) runs to ensure
a proper mixing, also the gains between $\nu=500$ and $\nu=1000$ were already
minimal. Secondly our practical interest was in sampling the WCNE, rather than
finding the optimal link removal, because in practical scenarios, the
underlying network is neither known, nor always perfectly controllable.

Figure 3: Network measures in the ensembles. For both initial network
ensembles (BA and RG), the ensembles after link removal are analyzed. (a) and
(b) show the degree distributions after removal of 80% of links.(d) and (e)
show the average number of components after the removal of $\rho$ links. (g)
and (h) demonstrate the size of the giant component after removal of $90\%$
links.

In Fig. 2 b) and d) we see that the improvement from MCMC persists in both
network ensembles and over all removal fractions. We see that the impact of
removing 90% of links randomly can alternatively be achieved by removing only
80% in a targeted manner, leaving each individual with twice as many contacts
on average.

Figure 4: The degree-based strategies reduce $\lambda_{max}$ almost as well as
MCMC. The reduction is greater for smaller removal ratios but persists even if
90 % of edges are removed.

### Network analysis

We now analyze the resulting ensembles and thus compare several network
measures across random and WCNE removals in Fig. 3. We find several indicators
pointing to a homogenization of the networks through the reduction of
$\lambda_{max}$.

Across all initial network ensembles we observe a more strongly peaked degree
distribution for WCNE (blue diamonds), compared to the generic ensemble (red
circles), as shown in Fig. 3 a) and b).

Furthermore the number of connected components for the WCNE remains at 1 until
a removal ratio of $\rho=0.8$ compared to $\rho=0.6$ for the most generic
removals in BA networks (see Figs. 3(c)–(d)), blue lines. This indicates
homogeneity again, as no single nodes are split off from the giant component.
Investigating further the regime of large removal rates by looking at the size
of the giant component at $\rho=0.9$, we find in Figs. 3(e)–(f) that the
largest component is smaller in the WCNE ensemble than it is in the generic
one. This means that it is not single nodes or small sub-graphs that are
disconnected but the network splits into several networks of similar size, in
line with our ad-hoc observation from Fig. 1.

While the MCMC procedure is effective at increasing the epidemic threshold of
networks compared to similarly dense networks with randomly removed edges, it
is often not practically feasible to use. The procedure requires knowledge of
the original social network, which is not typically accessible when suggesting
social distancing measures. Moreover, there is not one fixed real network that
can be optimized, but rather an ever-changing, unknown interaction structure.
We need general characteristics of well-performing removal sets, that can be
applied to a time-dependent, unknown network. By understanding general
characteristics of removals that reduce the spreading we can formulate
policies for targeted reductions. Our above observations suggest that we need
to homogenize the degree distribution.

We therefore now introduce two ad-hoc methods for link removal, with this aim,
one operating with only local and one with global information.

i) The first one we call the _degree product method_. Starting from the
original network, we iteratively remove one of the links with the highest
degree product at random, which we define as the product of the degrees of the
two end nodes, until the fraction $\rho$ of removed links is reached. An
illustrative example is shown in Fig. 5 (top).

Figure 5: Top: Successively removing the link with the highest degree product
results in networks with a very narrow degree distribution. Bottom: Go through
nodes in order of increasing node number and remove any links exceeding the
cap $k_{max}=3$.

ii) The second one we call the _degree cap_. In order of ascending node
number, each node with degree $>k_{max}$ has all but $k_{max}$ of its links
removed at random. This is a random order in case of RG networks. In the BA
ensemble, low node numbers tend to be correlated with high degrees, thus high-
degree nodes are reduced first. However this discrepancy results in similar
effects on the network structure as shown in Fig. 3. An illustrative example
of the algorithm is shown in Fig. 5 (bottom).

Figure 6: Decreased $\lambda_{max}$ shifts the epidemic threshold to more
infectious values of the disease spreading models’ parameters in both SIR (a)
and SIS (b) simulations. This shift persists across removal rates $\rho$ and
tends to be larger for larger $\rho$. It also persists for the degree-
homogenizing methods. (c) shows the smallest value of $\beta/\delta$ for each
removal rate $\rho$, for which the cumulative number of infected individuals
exceeds 1% using the SIS model. It clearly indicates the shift and verifies
that it exists for both ad-hoc strategies as well as for the WCNE.

We have included networks generated with both degree-homogenizing methods in
the analyses of Fig. 3 and show their reduction of the largest eigenvalues in
Fig. 4. Subfigure c) also shows a similar level of reduction effect for a real
world high school contact network. Both methods result in networks with a more
peaked degree distribution than for random removals. The degree-product method
(black stars) reaches the most peaked one and correlates very well with the
network ensemble resulting from MCMC simulations in most network measures. The
degree cap (yellow crosses) results in a less pronounced peak. Both methods
mimic the delay of the onset of network fracturing, again with the degree
product method having a more pronounced effect. Interestingly for the size of
the giant component shown in Fig. 3(e)–(f), the results differ for the
different initial ensembles. While the degree product method consistently
results in smaller giant components at a similar range of $\lambda_{max}$ than
the WCNE, the degree cap method results in component sizes comparable to the
generic ensemble for BA and RG networks. In the High-school example it
produces the same component size as the degree product method, which is lower
than even those of WCNE.

### Simulation and real world networks

To validate our removal strategies’ disease-slowing property, we run both SIS
and SIR simulations on a real social network of high-school students. As a
measure for the disease becoming endemic, we compute the cumulative number of
infected nodes. This is plotted versus different disease parameter ratios
$\beta/\delta$ in Figs. 6(a) and (b) for SIR and SIS, respectively. The size
of the network is $N=1062$ and the figure averages over 640 realizations of
SIR and SIS simulations with 10 nodes initially infected at random. They were
simulated for 1000 steps for SIS and until equilibrium was reached for SIR. We
consistently find that tipping into the epidemic state occurs at lower values
of $\beta/\delta$ in the generic networks than in the WCNE across all removal
ratios for both SIR and SIS. However, the figure also indicates that there may
be a kind of trade-off here. While the tipping point is shifted, the
transition also becomes steeper, leading to larger numbers of total infected
people at very large $\beta/\delta$.

Finally, Fig. 6c) shows the smallest value of $\beta/\delta$ for each removal
rate $\rho$, for which the cumulative number of infected individuals exceeds
$1\%$ using the SIS model. It clearly indicates the shift and verifies that it
exists for both ad-hoc strategies as well as for the WCNE, supporting our
conjecture that the sharper peak in the degree distribution may be the cause
of the reduction in spreading.

## Discussion and Conclusion

We have studied the effects of targeted link removal on the epidemic threshold
in a network by comparing random removal, removal based on the largest
eigenvalue as an energy function and two types of degree-homogenizing removal.
We have found that at sufficiently low genericities (i.e., high $\nu$), all
three types of targeted removal have a significantly higher epidemic threshold
(i.e., lower $\lambda_{max}$) than the random removal. This also results in a
shift of the tipping point to the epidemic state in SIS and SIR simulations,
such that more infectious diseases can be controlled with fewer link removals
necessary.

We have found this to coincide with a more sharply peaked degree distribution
as well as fewer and more homogeneously sized connected components, which we
have interpreted as an overall homogenization. Consequently, we have proposed
the two degree-homogenizing methods, which are also effective at decreasing
$\lambda_{max}$, as well as shift the onset of epidemic spreading. It is worth
mentioning that we have also tested a removal strategy based on edge-
betweenness. However, it was found to be less effective at increasing the
epidemic threshold than random removal and thus omitted in the results.

While running a full MCMC may be infeasible in practice, where the social
network is fluctuating and unknown, the two ad-hoc methods provide a simple
topological way of targeting links. This shows that a cap in the number of
permitted contacts per person, as already practised in many countries as part
of social distancing measures against Covid-19, is actually close to optimal
in terms of topological link targeting in static networks, at least in rather
simple model simulations such as ours.

Realistically however, measures may also include non-binary changes (such as
wearing masks, shortening contacts or keeping a distance) as well as temporal
changes in infectiousness and fluctuating interaction patterns. Although we
have chosen the epidemic threshold in this work as our measure to reduce
outbreak sizes, it would be a very interesting follow-up of this work to
compare our results to ensembles resulting from other network measures, such
as giant connected component, commonly associated with reduced outbreak sizes.
While we have here exemplified the method using simple static SIR and SIS
models, a more complex energy function would in principle make such an
analysis possible also in the case of inhomogeneous (weighted) or temporally
fluctuating transmission probabilities $\beta$, as well as node, rather than
edge removal (vaccinations). Changes of connection strengths or edge weights
could be used instead of complete removals of edges as a model of transmission
reductions through measures, such as mask wearing, meeting outdoors, improved
ventilation and keeping a distance.

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# Metrology and multipartite entanglement in measurement-induced phase
transition

Giovanni Di Fresco<EMAIL_ADDRESS>Dipartimento di Fisica e
Chimica “Emilio Segrè”, Group of Interdisciplinary Theoretical Physics,
Università degli studi di Palermo, Viale delle Scienze, Ed. 18, I-90128
Palermo, Italy  Bernardo Spagnolo<EMAIL_ADDRESS>Dipartimento di
Fisica e Chimica “Emilio Segrè”, Group of Interdisciplinary Theoretical
Physics, Università degli studi di Palermo, Viale delle Scienze, Ed. 18,
I-90128 Palermo, Italy  Davide Valenti<EMAIL_ADDRESS>Dipartimento
di Fisica e Chimica “Emilio Segrè”, Group of Interdisciplinary Theoretical
Physics, Università degli studi di Palermo, Viale delle Scienze, Ed. 18,
I-90128 Palermo, Italy  Angelo Carollo<EMAIL_ADDRESS>Dipartimento
di Fisica e Chimica “Emilio Segrè”, Group of Interdisciplinary Theoretical
Physics, Università degli studi di Palermo, Viale delle Scienze, Ed. 18,
I-90128 Palermo, Italy

###### Abstract

Measurement-induced phase transition arises from the competition between a
deterministic quantum evolution and a repeated measurement process. We explore
the measurement-induced phase transition through the quantum Fisher
information in two different metrological scenarios. We demonstrate through
the scaling behavior of the quantum Fisher information the transition of the
multi-partite entanglement across the phases. In analogy with standard quantum
phase transition, we reveal signature of a measurement-induced phase
transition in the non-analytic behaviour of the quantum Fisher information as
the measurement strength approaches the critical value. Our results offer
novel insights into the features of a quantum systems undergoing measurement-
induced phase transition and indicate potential avenues for further
exploration in the field of quantum physics.

A many-body quantum system subject to a measurement process may undergo abrupt
changes, similar to quantum phase transitions (QPTs), as a function of the
monitoring rate [1, 2, 3, 4, 5, 5, 6, 7]. Unlike conventional QPT that are
driven by external factors, these criticalities, called measurement-induced
phase transition (MIPT), arise from the interplay between the deterministic
unitary evolution and the measurement process. This phenomenon has been found
to have important implications for understanding the properties of many-body
quantum systems, such as the onset of quantum entanglement [8, 9, 10, 11, 12,
13, 14] and the emergence of topological order [15].

In this letter we propose to explore the concept of MIPT through the use of
QFI. The reason to characterise this phenomenon through QFI is two-fold. On
the one hand, the QFI is a measure of the sensitivity of a quantum system to
small changes in a parameter. As such, the QFI has been widely exploited to
signal and characterise the critical behaviour of equilibrium [16, 17, 18, 19,
20, 21, 22, 23, 24, 25, 26, 27, 28, 29] as well as non-equilibrium QPTs [30].
It is thus natural to ask whether and how the QFI may respond to critical
transitions when the latter are driven by measuring processes. On the other
hand, one of the characterising features of MIPT is an observable qualitative
change of the system’s entanglement properties, which has been observed
through the scaling behavior of the entanglement entropy [1, 2, 3, 4, 5, 6].
The QFI is a witness of multi-partite entanglement [31, 32, 33, 34], as well
known from quantum metrology [35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46,
47, 48, 49]. It thus provides a complementary characterisation of quantum
correlation accross MIPT, beyond entanglement entropy: on the one hand QFI
reveals the multi-partite nature of entanglement, on the other it
distinguishes metrologically useful quantum correlation from other forms of
entanglements. Moreover, QFI provides a means to characterize entanglement of
systems which are not necessarily pure states, a technical limitation which
affects other measures, such as the entanglement entropy.

Motivated by the above arguments, we examine the QFI in two complemetary
metrological scenarios. In both of them we consider as a prototypical example
the MIPT of a one-dimensional Ising chain in a transverse field and subject to
a continuous observation. In the first scenario, we investigate the QFI
associate to the system’s response to a single-parameter unitary
transformation. This metrological scheme, is the standard scenario in which
the scaling of QFI can be used as a witness of multipartite entanglement. In
the second scenario, we study the time-dependent behavior of the QFI after the
system undergoes a quench in the monitoring rate. Our analysis uncovers an
exponential scaling of the QFI, where we identify a non-analyticity of the QFI
when the measurement strength approaches the critical value. Our findings
provide new insights into the behavior of quantum systems subjected to MIPT,
and the role of the QFI in characterizing the associated quantum criticality.

Model— We consider a one-dimensional Ising chain with a transverse magnetic
field [50, 51, 52]

$H=-\sum_{i}\left[\sigma_{i}^{x}\sigma_{i+1}^{x}+h\sigma_{i}^{z}\right]$ (1)

which is coupled to an apparatus that measures $n=\frac{1+\sigma_{z}}{2}$,
with $\gamma$ as a coupling, such that the evolution of the system is given by
the stochastic quantum jump differential equation [5]

$\begin{split}&d\left|\psi\right\rangle=-iHdt\left|\psi\right\rangle-\frac{\gamma}{2}dt\sum_{i}\left(n_{i}-\langle
n_{i}\rangle_{t}\right)\left|\psi_{t}\right\rangle+\\\
&\sum_{i}dN_{t}^{i}\left(\frac{n_{i}}{\sqrt{\langle
n_{i}\rangle_{t}}}-1\right)\left|\psi\right\rangle.\end{split}$ (2)

The variable $dN_{t}^{i}$ is a stochastic Poisson variable ($dN_{t}^{i}=0,1$),
with each noise source being delta-correlated and ensemble average
$\overline{dN_{t}^{i}}=\gamma\langle n_{i}\rangle_{t}dt$. In this paper,
however, we are interested in trajectories in which no jump occurs (no-click
limit), which corresponds to a deterministic post-selected dynamics fixed by
the non-Hermitian Hamiltonian

$H_{\mathrm{eff}}=H-\frac{i\gamma}{2}\sum_{i}n_{i}.$ (3)

The system has a critical point at $\gamma_{c}=4\sqrt{1-h^{2}}$ [53, 4, 6],
that marks two regions in which the entanglement goes from a logarithmic
scaling to a constant scaling in the system size[5, 6].

The model can be diagonalized in the fermionic formalism as [53, 6]

$H=\sum_{k}\left(\begin{matrix}c^{\dagger}_{k}&c_{-k}\end{matrix}\right)M_{k}\left(\begin{matrix}c_{k}\\\
c_{-k}^{\dagger}\end{matrix}\right)$ (4)

where

$M_{k}=\left(\begin{matrix}\alpha_{k}&\beta_{k}\\\
\beta_{k}&-\alpha_{k}\end{matrix}\right)$ (5)

and $\alpha_{k}=-2\cos{k}-2h-\frac{i\gamma}{2}$, $\beta_{k}=2\sin{k}$. The
eigenvalues of Eq. (5), $\epsilon_{k}=\pm\sqrt{\alpha_{k}^{2}+\beta_{k}^{2}}$,
have both real and imaginary part, which we denote as
$E_{k}=\Re{(\epsilon_{k})}$ and $\Gamma_{k}=\Im{(\epsilon_{k})}$. We choose as
our ground state for each $k$ the value with negative imaginary part [53]. The
spectrum has a critical mode for $k=\arccos\left(-h\right)$ whose eigenvalue
is real for $\gamma<\gamma_{c}$ and imaginary for $\gamma>\gamma_{c}$.

Entanglement properties— This section aims to show that the scaling properties
of QFI can also be used to detect the entanglement properties of MIPT. To
achieve this goal, we follow the procedure outlined in Ref. [5] to compute the
time evolution of our system. It is worth noting that due to the system’s
quadratic nature at all times, the ground state can be expressed as

$\left|\psi\right\rangle\left(t\right)=\frac{1}{\sqrt{\det\left(U\right)}}\exp\left(\frac{1}{2}\sum_{j_{1}j_{2}}Z_{j_{1}j_{2}}c_{j_{1}}^{\dagger}c_{j_{2}}^{\dagger}\right)\left|0\right\rangle,$
(6)

where $Z=-\left(U^{\dagger}\right)^{-1}V^{\dagger}$. $U$ and $V$ are the
elements of the $2\times 2$ matrix

$\mathcal{U}_{t}=\left(\begin{matrix}U_{t}&V_{t}^{\dagger}\\\
V_{t}&U_{t}^{\dagger}\end{matrix}\right).$ (7)

This matrix is the rotation matrix that transforms our original fermionic
operators $\left(c_{i},c_{j}^{\dagger}\right)$ to a new set
$\left(\chi_{i},\chi_{j}^{\dagger}\right)$

$\left(\begin{matrix}\boldsymbol{\chi}\\\
\boldsymbol{\chi}^{\dagger}\end{matrix}\right)=\mathcal{U}_{t}\left(\begin{matrix}\boldsymbol{c}\\\
\boldsymbol{c}^{\dagger}\end{matrix}\right)$ (8)

such that $\chi_{i}\left|0\right\rangle=0$ $\forall i$ . The dynamics of the
system is all encoded in the operator $\mathcal{U}$ and its Heisenberg
equation of motion is [5]

$d\mathcal{U}=\left(-iH_{\mathrm{eff}}dt\right)\mathcal{U}.$ (9)

To utilize QFI as an entanglement witness, meticulous preparation of the
metrology scheme is needed [34]. Specifically, we opt for open boundary
conditions, and set $h=0$ in Eq. (1). This choice is made because the presence
of a transverse magnetic field does not affect deeply the aspects of our
interest. Moreover, for the simpler quantum Ising chain, selecting the
appropriate generator of the unitary transformation is more straightforward.
After initiating the state evolution in accordance with Eq.(9), we introduce
the estimation parameter via unitary evolution at time $t$

$\rho_{\varphi}=e^{-i\varphi S_{x}}\rho\left(t\right)e^{i\varphi S_{x}},$ (10)

where $S_{x}$ is the spin operator111We have also explored different
directions, but our results indicate that the x-direction is the most
compelling.. The QFI associated to Eq. (10) is [55]

$F=4\Delta^{2}\left(S_{x}\right).$ (11)

where $\Delta^{2}S_{x}$ is the variance of the spin operator (we follow the
results of Ref. [51] to evaluate this quantity). Fig. 1 shows the scaling of
the QFI in the two different phases of the system. Specifically, panel (b) of
the figure depicts the scaling of the QFI for $\gamma<\gamma_{c}$, which
exhibits a super extensive dependence on the system size $N$ ($F_{Q}\propto
N^{\eta}$) with $\eta>1$ (e.g., $\eta=1.5$ in the illustrated case). In
contrast, panel (a) shows that for $\gamma>\gamma_{c}$, the QFI follows a
normal scaling behavior with $\eta=1$. These distinct scaling behaviors not
only enable QFI to serve as a valuable metric for characterizing MIPT, but
also offer a means to identify the presence of multipartite entanglement in
the state, which is beyond the scope of the entanglement entropy. Indeed, it
is well known that if

$\frac{F}{N}\leq m,$ (12)

in the system there is at least a $\left(m+1\right)$-partite entanglement [31,
32, 33, 34].

Figure 1: The scaling behavior of the QFI is presented in these log-log plots,
with $x=1$ and $h=0$ in both panels: (a) for $\gamma=4.5$ the system is in a
phase where the entanglement remains constant with respect to the system size,
resulting in a linear scaling of the QFI; (b) for $\gamma=0.75$ the system is
in a phase where the entanglement varies with the system size, leading to an
enhanced scaling of the QFI.

Non-Hermitian quantum quench— This section aims to investigate the QFI of a
system initially prepared in the ground state of the unperturbed Ising
Hamiltonian and quenched at time $t$ by the application of the continuous
measurement process, leading to the deterministic non-Hermitian Hamiltonian
evolution $H_{\mathrm{eff}}$. Our goal is to demonstrate that the effect of
MIPT on the QFI is similar to that of a second order QPT (i.e., a divergence
at the critical point). To this end, we select the ground state of Eq. (1) as
initial state with periodic boundary conditions, which allows us to use the
Hamiltonian expression in $k-$ space. The final state can be expressed in a
general form

$\left|\psi_{t}\right\rangle=N\left(t\right)e^{-iH_{\mathrm{eff}}t}\left|\psi_{0}\right\rangle,$
(13)

where $N\left(t\right)$ is a normalization factor. To compute the QFI for the
state in Eq. (13), it is necessary to extend the results of Ref. [56] for a
non-Hermitian Hamiltonian. The QFI can be evaluated using the expression

$F=4\left(\left\langle\partial_{\lambda}\psi\right|\left.\partial_{\lambda}\psi\right\rangle-\left|\left\langle\partial_{\lambda}\psi\right|\left.\psi\right\rangle\right|^{2}\right).$
(14)

We found that in a non-Hermitian scenario the QFI will depend on the
covariance of the operator $O_{\lambda}$ (the detailed derivation is provided
in Appendix B)

$\begin{split}&\frac{F}{4}=\left\langle\psi\right|O_{\lambda}^{\dagger}O_{\lambda}\left|\psi\right\rangle-\left|\left\langle\psi\right|O_{\lambda}\left|\psi\right\rangle\right|^{2}\end{split},$
(15)

which is defined as

$O_{\lambda}=\int_{0}^{t}ds\exp{\left(-iH_{\mathrm{eff}}s\right)}\partial_{\lambda}\left(-iH_{\mathrm{eff}}\right)\exp{\left(iH_{\mathrm{eff}}s\right)}.$
(16)

If we want to estimate the parameter $\gamma$, this expression can be cast as
a quadratic function of the fermionic operator $c_{k}$ and $c_{k}^{\dagger}$
in the following closed form

$O_{\gamma}=-\sum_{k}\left(\begin{matrix}c^{\dagger}_{k}&c_{-k}\end{matrix}\right)R_{k}\left(\begin{matrix}c_{k}\\\
c^{\dagger}_{-k}\end{matrix}\right)-\frac{Nt}{4}.$ (17)

where $R_{k}$ is a $2\times 2$ time dependent matrix (see appendix B)

The state at time $t$ [6] can be written as

$\left|\psi\right\rangle=\prod_{k}\frac{u_{k}\left(t\right)+v_{k}\left(t\right)c^{\dagger}_{k}c^{\dagger}_{-k}}{\sqrt{|u_{k}|^{2}+|v_{k}|^{2}}}\left|0\right\rangle,$
(18)

where

$\left\\{\begin{split}&u_{k}\left(t\right)=u_{k}\left(0\right)\cos\left(\varepsilon
t\right)-i\left(\frac{\beta}{\epsilon}v_{k}\left(0\right)-\frac{\alpha}{\epsilon}u_{k}\left(0\right)\right)\sin\left(\varepsilon
t\right)\\\ &v_{k}\left(t\right)=v_{k}\left(0\right)\cos\left(\varepsilon
t\right)-i\left(\frac{\beta}{\epsilon}u_{k}\left(0\right)+\frac{\alpha}{\epsilon}v_{k}\left(0\right)\right)\sin\left(\varepsilon
t\right).\end{split}\right.$ (19)

If we now evaluate the expression of the QFI in Eq.(15), after some tedious
but straightforward algebra, we end up with Eq. (37) in Appendix B.

The QFI as a function of time is depicted in Fig. 2. It is observed that,
following an initial transient period whose duration is dependent on $\gamma$,
an exponential scaling regime emerges, with a scaling behavior approaching
$2\gamma$. The exponential divergence observed is related to the non-
Hermitianity of the model and it arises from the imaginary part of the
spectrum. It is however not directly related to the critical features of the
system.

Figure 2:  The figure presents the time evolution of the quantum Fisher
information (QFI) for various parameters of the Hamiltonian. In panels (a) and
(b) we fix $h=0.3$. We observe that an exponential scaling emerges, which
depends on $\gamma$ and occurs at a time that varies with the parameter
values. Specifically, in panel (a), with $\gamma=0.2$, the exponential scaling
occurs at a time that differs from panel (b), where $\gamma=2$.

The effects of MIPT on QFI, which can potentially manifest itself in a non-
analytic behaviour of QFI accross the MIPT, is partially concealed by this
exponential time dependence. In order to unveil the singular part of QFI one
can suitably factor out the exponential scaling. This procedure provides a
smooth mapping which leaves the analytic properties of the QFI unaltered.
Indeed, when examining the long-time behavior of the QFI, one can decompose
the QFI in terms of the eigenmodes $k$, each dominated by a specific time-
dependent exponential factor 222Strictly speaking, there is a minor misuse of
notation. In the case of $\gamma<\gamma_{c}$, the critical mode $k_{c}$ has a
real eigenvalue $\epsilon_{k_{c}}$, resulting in a time dependence which is
not exponential, but proportional to $t^{2}$. However, a careful analysis of
this case does not impact the bounds in Eq.(21). This behaviour is indeed
taken into account explicitly in our examination of the critical mode’s
scaling.

$F=\sum_{k}F_{k}e^{4\Gamma_{k}t}.$ (20)

where, we recall, $\Gamma_{k}$’s are the imaginary part of the eigenvalues
$\epsilon_{k}$ while $F_{k}$’s are time-independent coefficients, which are
expressed in detail in appendix B. The $\Gamma_{k}$ functions are continuous
with respect to $\gamma$, hence, any divergences of QFI due to the MIPT are to
be found in the $F_{k}$ terms. Moreover, one can explicitly relate any
divergent behaviour of the QFI to that of an auxiliary function
$\bar{F}=\sum_{k}F_{k}$ thanks to the following chain of inequalities

$\bar{F}<F<e^{\frac{\Gamma_{\rm{max}}}{4}t}\bar{F},$ (21)

where $\Gamma_{\rm{max}}$ is the maximum value of the imaginary part of the
spectrum, which is a continuous function of $\gamma$. This shows that
$\bar{F}$ retains the same divergent behaviour of $F$, but its study is
considerably more amenable to numerical analysis, due to the absence of the
exponential factors $e^{4\Gamma_{k}t}$. In Fig. 3 we plot the dependence of
$\bar{F}$ on $\gamma$, which clearly displays a peak at the critical value
$\gamma=\gamma_{c}$, with a distinctive asymmetric behaviour as the
criticality is approached from either above or below.

We analytically derive the singular part of the QFI (see Appendix B for
details), which scales as

$F\sim F_{k_{c}}\propto\left\\{\begin{split}&(\gamma-\gamma_{c})^{-3}\hskip
14.22636pt\gamma>\gamma_{c}\\\ &(\gamma_{c}-\gamma)^{-2}\hskip
14.22636pt\gamma<\gamma_{c},\end{split}\right.$ (22)

where $k_{c}=\arccos\left(-h\right)$ is a critical mode, i.e. the mode which
is gapless at the MIPT. These features are in accordance with the numerical
results shown in Fig. 3 and confirm them as a distinctive signature of the
MIPT in the QFI behavior. The non-analytical behaviour of QFI is strictly
related to the closing of the gap at criticality. In particular, the different
scaling laws as the critical point is approached from above or below
$\gamma_{c}$ can be pin-pointed to the different character of the spectrum at
the two sides of the criticality. The critical point is in fact an exceptional
point [53] of the model. This is associated to a gap of the model which is
real for $\gamma<\gamma_{c}$ and imaginary for $\gamma>\gamma_{c}$.
Heuristically, these qualitatively different behaviours of the spectrum
reflect the distinct characters of the two phases: (i) $\gamma<\gamma_{c}$ in
which, as in standard QPT, the dynamics is dominated by coherent phenomena,
with a real energy gap, and (ii) $\gamma>\gamma_{c}$ where the dynamics is
driven by measurement processes, entailing a dissipative, Zeno-like gap. In
both cases, the divergence of the QFI manages to capture the closing of the
gap, and discriminates the distinctive features of the two phases through the
asymmetry of scaling laws.

Figure 3:  The main figure illustrates the variation of the $\bar{F}$ for
different values of $\gamma$, where $\gamma_{c}$ represents the critical value
of the measurement rate (h = 0.6). In the inset, presented on a log-log plot,
we depict the scaling behavior of the critical mode as reported in Eq. (22)
($F_{k_{c}}\propto 1/|\gamma-\gamma_{c}|^{\eta}$), in magenta the scaling for
$\gamma<\gamma_{c}$ and in red for $\gamma>\gamma_{c}$.

Conclusion— We have studied the impact of the MIPT on the QFI of a one-
dimensional Ising chain subjected to a transverse magnetic field. Our analysis
reveals a change in the scaling behaviour of the QFI accross the transition
point, with a super-extensive (sub-extensive) scaling law for values of the
measuring rate $\gamma$ below (above) its critical value
$\gamma_{c}=\sqrt{h-1}$. This is revealed by the presence of multi-partite
entanglement for $\gamma<\gamma_{c}$, not signalled for $\gamma>\gamma_{c}$.
This behaviour is consistent with the entanglement phase transition observed
through the entanglement entropy [4, 5, 6]. The QFI, however, provides a new
insight into the multipartite nature of the entaglement associated with the
two phases, demostrating its usefulness as a resource for quantum metrology.
Additionally, we have shown that the MIPT can be revealed through the
divergent behaviour of the QFI with respect to the measuring rate. This non-
trivial finding parallels the analogous non-analyticity of the QFI accross
standard QPTs, when the QFI is evaluated with respect to the parameter driving
the transition [22]. Our findings provide new insights into the behavior of
quantum systems subjected to MIPT and highlight the potential use of MIPT for
quantum technological purposes.
Future research should explore the impact of the MIPT on the QFI of other
types of quantum systems and investigate the role of measurement in other
physical quantities. These efforts will contribute to the development of
quantum computing and metrology, and provide a deeper understanding of the
behavior of quantum systems under measurement.
During the preparation of this manuscript the authors became aware of the
related work [58], which analyses multipartite entanglement across MIPT.

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##  Appendix A Appendix A

To ensure clarity and self-consistency, in this Appendix we provide a set of
information and details of the non-Hermitian Hamiltonian in Eq. (3). Firstly,
we employ the Jordan-Wigner transformation

$\left\\{\begin{split}&\sigma_{i}^{x}=\prod_{j<i}\left(2n_{i}-1\right)\left(c_{i}+c_{i}^{\dagger}\right)\\\
&\sigma_{i}^{z}=1-2c_{i}^{\dagger}c_{i}\end{split}\right.$ (23)

to rewrite our Hamiltonian in fermionic language. Subsequently, we move to the
momentum space through the Fourier transform
$c_{j}=\frac{e^{i\pi/4}}{\sqrt{N}}\sum_{k}e^{ikj}c_{k}$. This results in the
expression of Eq. (4)

$H=\sum_{k}\left(\begin{matrix}c^{\dagger}_{k}&c_{-k}\end{matrix}\right)M_{k}\left(\begin{matrix}c_{k}\\\
c_{-k}^{\dagger}\end{matrix}\right),$ (24)

where

$M_{k}=\left(\begin{matrix}\alpha_{k}&\beta_{k}\\\
\beta_{k}&-\alpha_{k}\end{matrix}\right).$ (25)

In this work, every time we considered the expression of the Hamiltonian in
the momentum space, we worked with periodic boundary conditions on the spin
chain and an even number of fermions. This corresponded to use anti-periodic
boundary conditions for the fermionic chain with

$k=\left\\{\frac{\left(2n-1\right)\pi}{N}\left|n=1,...,\frac{L}{2}\right.\right\\},$
(26)

where only the positive values were used due to the symmetries of the
Hamiltonian. We also note that in Eq. (4), a constant term was neglected as it
does not affect the system’s dynamics. The eigenvalues of Eq.(25), denoted as
$\epsilon_{k}=\pm\sqrt{\epsilon_{k}^{2}+\Delta_{k}^{2}}$, possess both real
and imaginary parts. The real and imaginary parts are denoted as
$E_{k}=\Re{(\epsilon_{k})}$ and $\Gamma_{k}=\Im{(\epsilon_{k})}$,
respectively. It is important to note that the convention on the sign can be
chosen independently for each $k$ in the system’s spectrum. In this study, we
have chosen the convention such that $\Gamma_{k}\leq 0$ always. Furthermore,
it is worth noting that there exists a critical value of $k$, given by
$k_{c}=\arccos\left(-h\right)$, where $\epsilon_{k}$ is real for
$\gamma\leq\gamma_{c}$ and imaginary for $\gamma>\gamma_{c}$. This peculiar
behavior of non-Hermitian systems is illustrated in Fig.4, where the behavior
of the spectrum is reported for various values of $\gamma$.

Figure 4:  The figure presents the spectrum of the Hamiltonian in Eq. (4) for
different values of the measurement rate $\gamma$. The upper panel displays
the real part of the spectrum, gapped at $\pm k_{c}$ for
$\gamma/\gamma_{c}<1$, while it becomes gapless for $\gamma/\gamma_{c}\geq 1$.
On the other hand, the lower panel shows the behavior of the imaginary part,
which attains its maximum value $\Gamma=0$ at $\pm k_{c}$ for
$\gamma\leq\gamma_{c}$, whereas it remains non-zero for $\gamma>\gamma_{c}$.

The time-dependent state can be computed easily due to the initial state’s
well-defined parity and translation invariance. As a result, the dynamics can
be decomposed into $N/2$ independent ones

$\left|\psi\right\rangle=\prod_{k}\left|\psi_{k}\right\rangle.$ (27)

If we denote with
$\left|\tilde{\psi}\right\rangle=\left(u_{k}+v_{k}c^{\dagger}_{k}c^{\dagger}_{-k}\right)\left|0\right\rangle$
the unnormalized $k$ state we can easily see that its dynamics is given by

$i\frac{d}{dt}\left(\begin{matrix}u_{k}\\\
v_{k}\end{matrix}\right)=M_{k}\left(\begin{matrix}u_{k}\\\
v_{k}\end{matrix}\right)$ (28)

whose solution is given in Eq. (19).

##  Appendix B  Appendix B

This Appendix presents the derivation of the QFI expression for the non-
Hermitian quantum quench. As mentioned in the main text, the state at time $t$
can be expressed as a function of the initial state
$\left|\psi_{0}\right\rangle$, using the general form

$\left|\psi\left(t\right)\right\rangle=N\left(t\right)e^{-iH_{\mathrm{eff}}t}\left|\psi_{0}\right\rangle.$
(29)

To obtain the QFI, it is necessary to compute the derivative of the state with
respect to a parameter of interest, which can be expressed as

$\left|\partial_{\lambda}\psi\right\rangle=\left(\partial_{\lambda}N\left(t\right)e^{-iH_{\mathrm{eff}}t}+N\left(t\right)\partial_{\lambda}e^{-iH_{\mathrm{eff}}t}\right)\left|\psi_{0}\right\rangle.$
(30)

It is important to note that the derivative of the exponential term in the
last term of Eq. (30) should be carefully considered, as $\partial_{\lambda}H$
and $H$ do not commute. By using Sneddon’s formula [59] allows us to express
Eq. (30) as

— ∂λψ⟩=(∂λN(t)N(t)+∫0tds exp(-iHeffs)∂λ(-iHeff)exp(iHeffs))— ψ⟩= =(∂λN(t)N(t)+
^Oλ)— ψ⟩,

(31)

and applying the general definition of the QFI, we can easily obtain

$\begin{split}&\frac{F}{4}=\left(\left\langle\partial_{\lambda}\psi\right|\left.\partial_{\lambda}\psi\right\rangle-\left|\left\langle\partial_{\lambda}\psi\right|\left.\psi\right\rangle\right|^{2}\right)=\\\
&=\left(\left\langle\psi\right|\left(\left(\frac{\partial_{\lambda}N}{N}\right)^{*}+O_{\lambda}^{\dagger}\right)\left(\frac{\partial_{\lambda}N}{N}+O_{\lambda}\right)\left|\psi\right\rangle\right)\\\
&-\left(\frac{\partial_{\lambda}N}{N}+\left\langle\psi\right|O_{\lambda}\left|\psi\right\rangle\right)\left(\left(\frac{\partial_{\lambda}N}{N}\right)^{*}+\left\langle\psi\right|O_{\lambda}^{\dagger}\left|\psi\right\rangle\right)=\\\
&=\left\langle\psi\right|O_{\lambda}^{\dagger}O_{\lambda}\left|\psi\right\rangle-\left\langle\psi\right|O_{\lambda}\left|\psi\right\rangle\left\langle\psi\right|O_{\lambda}^{\dagger}\left|\psi\right\rangle.\end{split}$
(32)

The closed expression for the operator $O_{\gamma}$ in Eq. (17) can be
obtained by explicitly computing the derivative of the Hamiltonian with
respect to $\gamma$, which yields the following result

$\displaystyle O_{\gamma}=-\int_{0}^{t}ds\hskip
2.84544pte^{-iH_{\mathrm{eff}}s}\sum_{k}\left(\begin{matrix}c^{\dagger}_{k}&c_{-k}\end{matrix}\right)\left(\begin{matrix}1&0\\\
0&-1\end{matrix}\right)\left(\begin{matrix}c_{k}\\\
c^{\dagger}_{-k}\end{matrix}\right)e^{iH_{\mathrm{eff}}s}$
$\displaystyle-\frac{Nt}{4}.$ (33)

After evaluating the equation of motion for the operator $c_{k}$ we can
rewrite Eq. (B) as

Oγ= ∑k(c†kc-k)[∫0tds e-iMks(1 0 0 -1 )ei Mks](ckc†-k)- Nt4.

(34)

At this point, following a series of tedious but straightforward algebraic
manipulations, we arrive at the conclusion that

$O_{\gamma}=-\sum_{k}\left(\begin{matrix}c^{\dagger}_{k}&c_{-k}\end{matrix}\right)R_{k}\left(\begin{matrix}c_{k}\\\
c^{\dagger}_{-k}\end{matrix}\right)-\frac{Nt}{4},$ (35)

where

Rk=(AkBkCk-Ak).==(tαk2ϵk2\+ sin(2ϵkt)2ϵk3βk2αkβktϵk2\+ 2βksin2(ϵkt)ϵki -
αkβksin(2ϵkt)2ϵk3αkβktϵk2\- 2βksin2(ϵkt)ϵki + αkβksin(2ϵkt)2ϵk3-tαk2ϵk2\-
sin(2ϵkt)2ϵk3βk2)

(36)

Using the expression of Eq.(18), we can evaluate the expectation value of
Eq.(32), resulting in the following expression

F4= ⟨ψ— Oλ†Oλ— ψ⟩- ⟨ψ— Oλ— ψ⟩⟨ψ— Oλ†— ψ⟩= =∑k[[—uk—2(—Ak—2\+ —Ck—2)-2ℜ(u*kvk(-
AkB*k\+ CkA*k))+ —vk—2(—Ak—2+—Bk—2)]—uk—2\+ —vk—2+-—(-—uk2—Ak\- Ckuk*vk\-
Bkukvk*\+ Ak—vk—2)—uk—2\+ —vk—2—2].

(37)

When examining the long-time behavior of the QFI, it becomes evident that we
can factorize the exponential scaling. This can be seen from

$\left\\{\begin{split}&A_{k}=\pm\frac{i\beta^{2}_{k}}{4\epsilon_{k}^{3}}e^{2i\tilde{\varepsilon_{k}}}=\tilde{A}_{k}e^{2i\tilde{\varepsilon_{k}}}\\\
&B_{k}=i\frac{-\beta_{k}\epsilon_{k}\mp\alpha_{k}\beta_{k}}{4\epsilon_{k}^{3}}e^{2i\tilde{\varepsilon_{k}}}=\tilde{B}_{k}e^{2i\tilde{\varepsilon_{k}}}\\\
&C_{k}=i\frac{\beta_{k}\epsilon_{k}\mp\alpha_{k}\beta_{k}}{4\epsilon_{k}^{3}}e^{2i\tilde{\varepsilon_{k}}}=\tilde{B}_{k}e^{2i\tilde{\varepsilon_{k}}}\end{split}\right.$
(38)

where $\tilde{\epsilon}_{k}$ is the eigenvalue with positive imaginary part
(these scaling laws are valid for all values of $k$ and $\gamma$, except for
$k_{c}$ when $\gamma<\gamma_{c}$.). To compute the long time behavior of the
QFI we can evaluate the expectation value using the ground state of the
effective Hamiltonian [53]. Then follows that the long time behavior of the
quantum Fisher information is

F = ∑k[(—~uk—2(—~Ak—2\+ —~Ck—2)- 2ℜ(~uk~vk*(~A*k~Ck\-
~Ak~B*k))+—~vk—2(—~Ak—2\+ —~Bk—2))e4Γkt(—~uk—2\+ —~vk—2)+- —(-~Ak—~uk—2\-
~Ck~uk*~vk-~Bk~uk~v*k+~Ak—~v—2)—2e4Γkt((—~uk—2\+ —~vk—2))2]== ∑kFke4Γkt.

(39)

The critical mode’s expression of the QFI can be determined analytically by
assessing Eq.(39) for $k_{c}=\arccos(-h)$. When $\gamma>\gamma_{c}$, the QFI
can be computed using the expressions provided in Eq.(38). In contrast, when
$\gamma<\gamma_{c}$, the linear term in $t$ becomes the leading term in the
long time approximation of Eq. (36). This phenomenon is responsible for the
difference in the scaling laws of Eq. (22).