# Random Matrix Theory Approach to Quantum Fisher Information in Quantum Many-
Body Systems

Venelin P. Pavlov  Center for Quantum Technologies, Department of Physics, St.
Kliment Ohridski University of Sofia, James Bourchier 5 blvd, 1164 Sofia,
Bulgaria  Yoana R. Chorbadzhiyska  Center for Quantum Technologies, Department
of Physics, St. Kliment Ohridski University of Sofia, James Bourchier 5 blvd,
1164 Sofia, Bulgaria  Charlie Nation  Department of Physics and Astronomy,
University College London, London WC1E 6BT, United Kingdom  Diego Porras
Institute of Fundamental Physics IFF-CSIC, Calle Serrano 113b, 28006 Madrid,
Spain  Peter A. Ivanov  Center for Quantum Technologies, Department of
Physics, St. Kliment Ohridski University of Sofia, James Bourchier 5 blvd,
1164 Sofia, Bulgaria

###### Abstract

We theoretically investigate parameter quantum estimation in quantum chaotic
systems. Our analysis is based on an effective description of non-integrable
quantum systems in terms of a random matrix Hamiltonian. Based on this
approach we derive an analytical expression for the time evolution of the
quantum Fisher information. We test our random matrix theory prediction with
the exact diagonalization of a non-integrable spin system, focusing on the
estimation of a local magnetic field by measurements of the many-body state.
Our numerical calculations agree with the effective random matrix theory
approach and show that the information on the local Hamiltonian parameter is
distributed throughout the quantum system during the quantum thermalization
process. Our analysis shows a first stage in which the initial information
spread is quadratic in time which quickly passes into linear increase with
slope determine by the decay rate of the measured spin observable. When the
information is fully spread among all degrees of freedom a second quadratic
time scale determines the long time behaviour of the quantum Fisher
information.

_Introduction.-_ Initially excited quantum systems typically equilibrate to
states exhibiting thermal properties, a process known as quantum
thermalization Polkovnikov2011 ; Popescu2006 ; Rigol2008 ; Eisert2014 ;
Alessio2016 ; Nandkishore2015 . At the core of our current understanding of
this intriguing phenomenon is the Eigenstate Thermalization Hypothesis (ETH),
which assumes that the many-body eigenstates of non-integrable Hamiltonians
yield the same expectation values of local observables as those calculated
with a microcanonical ensemble Deutsch1991 ; Deutsch2018 ; Srednicki1994 ;
Srednicki1996 . The ETH can be formally expressed as a conjecture on the
properties of matrix elements of local observables, which in turn can be
derived from an effective description of non-integrable systems in terms of
random matrix theory (RMT). The validity of the ETH has been confirmed for a
broad range of many-body systems by means of exact diagonalizations Rigol2009
; Rigol2012 ; Swan2019 ; Jansen2019 ; Kim2014 ; Kirkova2023 . Furthermore,
experimental quantum optical systems have allowed for investigation of quantum
thermalization and the emergence of statistical physics in isolated quantum
systems. Examples include recent experiments with ultracold atoms Gring2012 ;
Kaufman2016 , trapped ions Smith2016 ; Clos2016 ; Kranzl2023 , and
superconducting qubits Neill2016 .

An important fundamental issue in quantum many-body theory is how information
on local properties can be retrieved or estimated from observing the quantum
system’s dynamics. This problem is closely related to fundamental research on
the connection between quantum chaos and scrambling of quantum information
Swingle2018 ; Rampp2023  and also to applications like quantum metrology
Qian2010 ; Song2012 ; Fiderer2018 . For example, the exponential sensitivity
to small perturbation in imperfect time-reversal quantum dynamics is a widely
studied signal for irreversibility Gorin2006 ; Schmitt2019 ; Kirkorova2022 .
Here we address this question by investigating the dynamics of the quantum
Fisher information (QFI) in quantum ergodic systems. The QFI is a quantity of
central importance in quantum metrology. It quantifies the sensitivity of a
given input state to a unitary transformation and provides the fundamental
bound of the parameter estimation Paris2009 ; Pezze ; Liu2020 ;
Giovannetti2011 . The QFI also provides a sufficient condition to recognize
entanglement in multiparticle state Pezze2009 ; Toth2012 ; Brenes2020 ;
Pavlov2023 .

In this work we study the time-evolution of the QFI of non-integrable systems
by a RMT approach. We model the Hamiltonian of the ergodic system as the sum
of two contributions: a free non-interacting diagonal part and an interaction
term modelled by a Gaussian orthogonal random matrix. This approach is valid
as long as the coupling between a subsystem and the rest of a closed non-
integrable system can be modelled as random matrix. Such an approach was
originally proposed by J. Deutsch as a toy model to describe the emergence of
quantum thermalization in isolated quantum systems Deutsch1991 . Recently, it
was shown that this approach can be extended to predict the off-diagonal
elements of observables, recovering the ETH Nation2018 . The description on
RMT relies on strong assumptions that are basically equivalent to the ETH
itself, however it allows to make scaling predictions that can be tested in
experiments or exact numerical diagonalizations.

We test the results predicted from RMT by using an exact diagonalization of a
non-integrable spin chain. The model consists of a system Hamiltonian
describing one or a few non-interacting spins coupled with large spin system
which plays the role of a finite quantum many-body bath. We find three time
scales, which obey QFI. We show that in the beginning of the time evolution
the QFI increases _quadratically_. In this short time period the information
of the parameter is not locally lost and the behaviour of the QFI resembles
the standard quantum limit. After this time period the QFI quickly passes into
a _linear_ increase with slope defined by the width of the random wave
functions. Essentially, this width is the decay rate of the observable to the
microcanonical average. In this second stage the information of the parameter
propagates along the entire system. Remarkably, a second _quadratic_ time
scale appears that determines the long-time behavior of QFI. It occurs when
the information is spread among all quantum states involved in the evolution.
We show that in this third stage the QFI is inversely proportional to the
effective dimension of the system, a measure which quantifies the ergodicity
of the system. The transition between the linear to quadratic time regimes
occur at Heisenberg time determined by the density of states of the system.

For $N_{S}$ non-interacting spins one can expect that the QFI scales as $\sim
N_{S}$ which gives the standard shot-noise limit. Crucially, for a few spins
coupled to the quantum many-body heat bath, the system-bath interaction gives
rise to a spin-spin correlation within the small subsystem. Hence, we show the
quantum correlation may increase the QFI in a sense that it can exceeded the
QFI corresponding to the standard quantum limit (SQL) without any initial
entangled state preparation.

_Quantum parameter estimation.-_ We consider a quantum system described by a
non-integrable Hamiltonian, $\hat{H}(\lambda)=\hat{H}_{0}+\hat{H}_{I}$,
consisting of non-interacting Hamiltonian $\hat{H}_{0}=\hat{H}_{\rm
S}+\hat{H}_{\rm B}$ with $\hat{H}_{\rm S}$ and $\hat{H}_{\rm B}$ being the
Hamiltonians for the subsystem and the many-body environment and respectively
interaction part $\hat{H}_{I}$ describing the system-bath interaction. The
eigenvectors and eigenenergies of the total Hamiltonian are
$|\psi_{\mu}\rangle$ and $E_{\mu}$, such that
$\hat{H}|\psi_{\mu}\rangle=E_{\mu}|\psi_{\mu}\rangle$. We also define non-
interacting energy eigenbasis,
$\hat{H}_{0}|\varphi_{\alpha}\rangle=E_{\alpha}|\varphi_{\alpha}\rangle$. The
system is initially prepared in an out-of-equilibrium state
$|\Psi_{0}\rangle=\sum_{\mu}a_{\mu}|\psi_{\mu}\rangle$ with
$a_{\mu}=\langle\psi_{\mu}|\Psi_{0}\rangle$ which evolves under the action of
the unitary propagator,
$|\psi(\lambda)\rangle=e^{-i\hat{H}(\lambda)t}|\Psi_{0}\rangle$.

Figure 1: Estimates for the microcanonical average for varying values of $g$
and $N$. Analytic results (4) are compared with the predictions given by the
diagonal ensemble (numerical results). Average over $10$ realizations of the
random Hamiltonian is taken. The initial state is
$|\Psi_{0}\rangle=|\varphi_{\alpha}\rangle$ with $\alpha=N/2$.

The classical Fisher information (CFI) $F_{\rm
C}(\lambda)=\sum_{n}p(n|\lambda)^{-1}(\partial_{\lambda}p(n|\lambda))^{2}$
quantifies the amount of information on the parameter $\lambda$ which can be
derived by performing discrete measurements with probability distribution
$p(n|\lambda)$. The optimal strategy to measure the value of $\lambda$ is
however associated with a privileged observable that maximizes the CFI. The
CFI is upper bounded $F_{C}(\lambda)\leq F_{Q}(\lambda)$, where
$F_{Q}(\lambda)$ is the QFI. The ultimate achievable precision of the
parameter estimation is quantified via the quantum Cramér-Rao bound
$\delta\lambda^{2}\geq 1/(MF_{Q}(\lambda))$ where $M$ is the repetition
number. The QFI $F_{Q}(\lambda)$ is a measure of distinguishability of the
quantum states $|\psi(\lambda)\rangle$ and $|\psi(\lambda+d\lambda)\rangle$
with respect to the infinitesimal small variation of the parameter of interest
$\lambda$. For pure state, the QFI reads, Paris2009

$F_{Q}(\lambda)=4(\langle\partial_{\lambda}\psi(\lambda)|\partial_{\lambda}\psi(\lambda)\rangle-|\langle\psi(\lambda)|\partial_{\lambda}\psi(\lambda)\rangle|^{2}).$
(1)

The QFI can be expressed also by using the so-called symmetric logarithmic
derivative (SLD) operator $\hat{L}_{\lambda}$ through the relation
$F_{Q}(\lambda)=\langle\psi(\lambda)|\hat{L}_{\lambda}^{2}|\psi(\lambda)\rangle$.
For pure state the SLD operator can be written as
$\hat{L}_{\lambda}=2(|\partial_{\lambda}\psi(\lambda)\rangle\langle\psi(\lambda)|+|\psi(\lambda)\rangle\langle\partial_{\lambda}\psi(\lambda)|)$
and the optimal measurements that saturate the Cramér-Rao bound are projective
measurements formed by the eigenvectors of $\hat{L}_{\lambda}$.

Figure 2: Quantum Fisher information as a function of time for various $g$ and
$N$. We compare the results for the QFI derived from the random matrix
Hamiltonian using Eq. (1) and the analytical result Eq. (3). The random
Hamiltonian is averaged over $10$ realizations. The initial state is
$|\Psi_{0}\rangle=|\varphi_{\alpha}\rangle$ with $\alpha=N/2$ and
$(\hat{H}^{\prime}_{0})_{\alpha\beta}=\alpha\delta_{\alpha\beta}$.

Hereafter we assume that the dependence on $\lambda$ comes only from the non-
interacting Hamiltonian $\hat{H}_{0}(\lambda)$. This example corresponds to
cases where $\lambda$ represents spin frequency or external magnetic field
strength. Furthermore, we may express the QFI (1) in the basis of
$|\psi_{\mu}\rangle$ eigenvectors. We have sup

$\displaystyle F_{Q}(\lambda)$ $\displaystyle=$ $\displaystyle
4t^{2}\\{\sum_{\mu\nu\rho}a^{*}_{\mu}a_{\nu}(\partial_{\lambda}\hat{H}_{0})_{\mu\rho}(\partial_{\lambda}\hat{H}_{0})_{\rho\nu}e^{i\theta_{\mu\nu}t}{\rm
sinc}(\theta_{\mu\rho}t)$ (2) $\displaystyle\times{\rm
sinc}(\theta_{\rho\nu}t)$
$\displaystyle-|\sum_{\mu\nu}a^{*}_{\mu}a_{\nu}e^{i\theta_{\mu\nu}t}(\partial_{\lambda}\hat{H}_{0})_{\mu\nu}{\rm
sinc}(\theta_{\mu\nu}t)|^{2}\\},$

where
$(\partial_{\lambda}\hat{H}_{0})_{\mu\nu}=\langle\psi_{\mu}|\partial_{\lambda}\hat{H}_{0}|\psi_{\nu}\rangle$
are the matrix elements in the many-body interacting basis,
$\theta_{\mu\nu}=(E_{\mu}-E_{\nu})/2$, and ${\rm sinc}(x)=\sin(x)/x$. The
expression (2) is convenient for our further consideration because we can
apply a RMT approach to evaluate the respective matrix elements.

_Random Matrix Approach.-_ Our analysis of the QFI is based on the random
matrix model in which the non-interacting Hamiltonian $\hat{H}_{0}$ is
modelled by diagonal matrix of size $N$, with $\omega$ being the constant
spacing between the energy levels. The interaction term $\hat{H}_{I}$ is
modelled by a random matrix, $(\hat{H}_{I})_{\alpha\beta}=h_{\alpha\beta}$,
where $h_{\alpha\beta}$ are independent random numbers selected from the
Gaussian orthogonal ensemble with probability distribution $P(h)\propto
e^{-\frac{N}{4g^{2}}{\rm Tr}h^{2}}$, giving average $\langle
h_{\alpha\beta}\rangle=0$, and variance $\langle
h^{2}_{\alpha\beta}\rangle=g^{2}(1+\delta_{\alpha\beta})/N$ where $g$ is the
coupling strength Deutsch1991 ; Nation2018 .

We expand the eigenstates of $\hat{H}$ in the non-interacting basis,
$|\psi_{\mu}\rangle=\sum_{\alpha}c_{\mu}(\alpha)|\varphi_{\alpha}\rangle$,
where $c_{\mu}(\alpha)$ are random variables whose statistical properties
depend on the properties of the random matrix $\hat{H}_{I}$. The probability
distribution of eigenstates is then described with Lorentzian,
$\langle|c_{\mu}(\alpha)|^{2}\rangle_{V}=\Lambda(\mu,\alpha)=(\omega\Gamma/\pi)((E_{\mu}-E_{\alpha})^{2}+\Gamma^{2})^{-1}$
Deutsch1991 ; Nation2018 , where $\Gamma=\pi g^{2}/N\omega$ is the width of
the wave function distribution and $\langle\ldots\rangle_{V}$ denotes an
average over realizations of the random matrix $(\hat{H}_{I})_{\alpha\beta}$.
The Lorentzian function is normalized such that
$\sum_{\mu}\Lambda(\mu,\alpha)=\sum_{\alpha}\Lambda(\mu,\alpha)=1$.
Furthermore, we assume a self-averaging condition where sum over random wave
functions are replace with their ensemble average
$\sum_{\alpha\ldots\beta}c_{\mu}(\alpha)\ldots
c_{\nu}(\beta)=\sum_{\alpha\ldots\beta}\langle c_{\mu}(\alpha)\ldots
c_{\nu}(\beta)\rangle_{V}$, see Supplemental Material (SM) sup .

The self-averaging condition is essential for the evaluation of the QFI (2),
and is shown to hold in the description of observables in Ref. Nation2019_1 ;
Dabelow2020 . Indeed, we can evaluate the sum of the matrix elements in (2) in
terms of averages of products of random-wave functions $c_{\mu}(\alpha)$. The
treatment of $c_{\mu}(\alpha)$ as an independent random Gaussian variables,
however, is not sufficient to determine the value of the off-diagonal matrix
elements of an observable which are consistent with ETH Nation2018 ;
Nation2019 ; Nation2019_1 ; Nation2020 . Because of that a non-Gaussian
corrections should be included which arise as a result of the orthonormality
condition. Based on a self-averaging condition the QFI (2) is given by (see SM
for more details sup )

$\displaystyle F_{Q}(\lambda)\approx
4t^{2}\left\\{\frac{\omega}{\pi\Gamma}(\hat{H}^{\prime 2}_{0})_{\rm
mc}+\frac{(\Delta\hat{H}_{0}^{\prime 2})_{\rm mc}}{2(\Gamma
t)^{2}}(e^{-2\Gamma t}-1+2\Gamma t)\right\\}.$ (3)

This is the main result of our work. Here
$\partial_{\lambda}\hat{H}_{0}=\hat{H}^{\prime}_{0}$ and
$(\hat{H}^{\prime}_{0})_{\rm mc}$ is the microcanonical average of an
observable $\hat{H}^{\prime}_{0}$ and $(\Delta\hat{H}_{0}^{\prime 2})_{\rm
mc}=(\hat{H}_{0}^{\prime 2})_{\rm mc}-(\hat{H}^{\prime}_{0})^{2}_{\rm mc}$ is
the microcanonical average of the variance of $\hat{H}^{\prime}_{0}$. The
result (3) is based on the following assumptions: (i) We assume _sparsity_ of
$\hat{H}^{\prime}_{0}$ which implies that its matrix elements in the non-
interacting basis is represented by a diagonal matrix or at least, by a matrix
with only a few non-diagonal elements. (ii) We define _smoothness_ of the
matrix elements of an observable in the following way:

$[(\hat{H}_{0}^{\prime})_{\alpha\alpha}]_{\mu}=\sum_{\alpha}\Lambda(\mu,\alpha)(\hat{H}^{\prime}_{0})_{\alpha\alpha},$
(4)

which represents the microcanonical average of the matrix elements
$(\hat{H}^{\prime}_{0})_{\alpha\alpha}$ around the energy $E_{\mu}$, namely
$(\hat{H}^{\prime}_{0})_{\rm
mc}=[(\hat{H}_{0}^{\prime})_{\alpha\alpha}]_{\mu}$. Such an average is well
defined as long as $\omega/\Gamma\ll 1$ which ensures that a large number of
matrix elements are averaged in
$[(\hat{H}_{0}^{\prime})_{\alpha\alpha}]_{\mu}$. Conditions for the validity
of these assumptions have been discussed in detail in Ref. Nation_thesis .

In Fig. 1 we show the microcanonical average of observable
$\hat{H}^{\prime}_{0}=\alpha\delta_{\alpha\beta}$ according to (4) compared
with the diagonal average $\langle\bar{H}^{\prime}_{0}\rangle={\rm
Tr}(\hat{H}^{\prime}_{0}\hat{\rho}_{\rm DE})$, where $\hat{\rho}_{\rm
DE}=\sum_{\mu}|a_{\mu}|^{2}|\psi_{\mu}\rangle\langle\psi_{\mu}|$ is the
density matrix of the diagonal ensemble. In Fig. 2 we show the time evolution
of the QFI where we set $\lambda=\omega$. We compare the exact result based on
Eq. (1) using the random matrix model and analytical expression (3). The
numerical and analytical results are averaged over $10$ realizations of the
random Hamiltonian. We set the initial state to be an eigenstate of the non-
interacting Hamiltonian, $|\Psi_{0}\rangle=|\varphi_{\alpha}\rangle$ with
$\alpha$ selected at the middle of the energy spectrum. We compare the results
for various $g$ and $N$. We see a good agreement between both results except
for the initial time evolution. We observe however that the relative error
between both results decreases as we increase the matrix size $N$ sup . We
also test the result for other initial states $|\varphi_{\alpha}\rangle$. We
find that the relative error convergences faster for initial states at the
lower half of the non-interacting energy spectrum and respectively slower at
the upper half.

_Exact Diagonalization.-_ We now turn to the comparison of our main result (3)
with exact diagonalization of non-integrable spin chain. We consider 1D spin
system with a Hamiltonian of the form

$\hat{H}=\hat{H}_{\rm S}+\hat{H}_{\rm B}+\hat{H}_{\rm SB}.$ (5)

The system Hamiltonian describes a single spin in a presence of a $B$-field

$\hat{H}_{\rm S}=B\sigma^{z}_{1},$ (6)

where $\sigma^{q}_{j}$ ($q=x,y,z$) are the Pauli matrices acting on $j$-th
site and $B$ is the parameter we wish to estimate, namely $\lambda=B$. The
bath Hamiltonian describes a spin chain with Ising interaction

$\hat{H}_{\rm B}=\sum_{k>1}^{N}B_{x}^{(\rm
B)}\sigma_{k}^{x}+\sum_{k>1}^{N-1}J_{x}(\sigma^{+}_{k}\sigma^{-}_{k+1}+\sigma^{-}_{k}\sigma^{+}_{k+1}),$
(7)

where $B^{(B)}_{x}$ is the magnetic field along the $x$-axis and $J_{x}>0$ is
the spin-spin coupling. The interaction Hamiltonian describes a coupling
between the system spin and a single bath spin of index $r$

$\hat{H}_{\rm SB}=J_{z}^{(\rm SB)}\sigma^{z}_{1}\sigma^{z}_{r}+J_{x}^{(\rm
SB)}(\sigma^{+}_{1}\sigma^{-}_{r}+\sigma^{-}_{1}\sigma^{+}_{r}),$ (8)

with coupling strengths $J^{(\rm SB)}_{z}$ and $J^{(\rm SB)}_{x}$.

As long as the non-integrable system is well described by RMT we expect that
expression Eq. (3) holds with the modification $\omega\rightarrow 1/D(E_{0})$
where $D(E_{0})$ is the density of states at the initial energy $E_{0}$
Nation2018 . This assumption holds when the energy scale over which the
density of states changes is large with respect to $\Gamma$, the energy width
of the random wave functions. This limits the above approach to a weak
coupling regime.

Then using Eq. (3) with $\hat{H}_{0}^{\prime}=\sigma^{z}_{1}$ and thereby
$(\hat{H}^{\prime 2}_{0})_{\rm mc}=1$ and $(\Delta\hat{H}_{0}^{\prime 2})_{\rm
mc}=1-(\sigma^{z}_{1})^{2}_{\rm mc}$, the QFI becomes

$F_{Q}(B)\approx 4t^{2}\left\\{\frac{1}{\pi
D(E_{0})\Gamma}+\frac{1-(\sigma_{1}^{z})^{2}_{\rm mc}}{2(\Gamma
t)^{2}}(e^{-2\Gamma t}-1+2\Gamma t)\right\\}.$ (9)

Note that in applying the QFI as above we are describing a local observable of
the spin system in terms of RMT. In the SM sup  we show that such local
observables are indeed well described in terms of RMT as long as certain
energy scales of the system are sufficiently separated. Notably, the sparsity
condition above follows trivially for a local observable Nation_thesis .

In order to find the value of $\Gamma$ we use that the time dependence of an
observable $\hat{O}$ obeys Nation2019 ; Dabelow2020

$\langle\hat{O}(t)\rangle=\langle\hat{O}(t)\rangle_{0}e^{-2\Gamma
t}+\langle\bar{O}\rangle(1-e^{-2\Gamma t}),$ (10)

where $\langle\hat{O}(t)\rangle_{0}$ is the evolution of the observable
according the non-interaction Hamiltonian $\hat{H}_{0}$ and
$\langle\bar{O}\rangle$ is the time-average value defined by
$\langle\bar{O}\rangle={\rm Tr}(\hat{O}\hat{\rho}_{\rm DE})$. Thermalization
of a closed system implies the equality
$\langle\bar{O}\rangle\approx(\hat{O})_{\rm mc}$.

Figure 3: (a) Density of states, as a function of energy for $N=13$ and
initial state $|\Psi_{0}\rangle=|\varphi_{\alpha}\rangle$ with $\alpha=5500$.
The dashed lines show the point $D(E_{0})$, where
$E_{0}=\langle\Psi_{0}|\hat{H}|\Psi_{0}\rangle$. (b) Time evolution of the
spin system observable $\sigma^{z}_{1}$ both numerically and analytically
using Eq. (10). (c) Quantum Fisher information as a function of time for
initial state $|\Psi_{0}\rangle=|\varphi_{\alpha}\rangle$ with $\alpha=5500$
for $N=13$ and for antiferromagnetic initial state
$|\Psi_{0}\rangle=|\uparrow\rangle_{S}|\downarrow\uparrow\downarrow...\rangle_{B}$
for $N=15$. We compare the exact result for the QFI (1) with Hamiltonian (5)
and the analytical expression (9). The parameters are set to $B=0.01$,
$B_{x}^{(\rm B)}=0.3$, $J_{z}^{(\rm SB)}=0.2$, $J_{x}^{(\rm SB)}=0.4$,
$J_{x}=1$, and $r=5$. The decay rate is $\Gamma=0.15$ for both $N=13$ and
$N=15$ is calculated by using Eq (10) to fit the exact time evolution of the
$\sigma^{z}_{1}$ operator for the system spin. The density of states
$D(E_{0})$ is evaluated by interpolation of $D(E)$.

In Fig. 3(a) we plot the density of states as a function of the energy. From
here we can extract the value of $D(E_{0})$ at the initial energy $E_{0}$. We
also fit the time evolution of observable $\sigma^{z}_{1}$ to obtain the value
of $\Gamma$, see inset of Fig. 3(b). In Fig. 3(c) we show the comparison
between the exact result for the QFI using (1) and the analytical expression
(9) for various initial states. Since the small spin sub-system thermalizes
the information for the parameter $B$ is locally lost. What we see, however,
is that because of the spin-spin interaction the information has been not
lost, but spread among the other degrees of freedom. At the beginning of the
time evolution for $t\lesssim(2\Gamma)^{-1}$ the information flow is
quadratic, where $F_{Q}(B)\approx 4t^{2}(\Delta\hat{H}_{0}^{\prime 2})_{\rm
mc}$. In this first stage the information for the parameter is still not
locally lost and it can be determined with uncertainty bounded by the SQL.
After this time period the growth of QFI becomes linear in time with slope
determined by the decay rate $\Gamma$, namely
$F_{Q}(B)\approx(4t/\Gamma)(\Delta\hat{H}_{0}^{\prime 2})_{\rm mc}$. In this
second stage the information flow propagates along the entire system.
Remarkably, a second quadratic time scale defines the long time behaviour of
the QFI, that occurs when the information has fully spread between all degrees
of freedom. In this third case we have $F_{Q}(B)\approx(4t^{2}/\pi
D(E_{0})\Gamma)(\hat{H}^{\prime 2}_{0})_{\rm mc}$. In fact, we may connect the
third time scaling of QFI with the effective dimension of the system. The
effective dimension is defined by $d_{\rm
eff}=(\sum_{\mu}|\langle\Psi(0)|\psi_{\mu}\rangle|^{4})^{-1}$ and provides an
estimation for the ergodicity of a system. Also the mean amplitude of time
fluctuations of an observable are bounded by $d_{\rm eff}^{-1/2}$ Linden2009 .
The condition $d_{\rm eff}\gg 1$ implies that the initial state is composed of
a large number of energy eigenstates which leads to suppression of temporal
fluctuations of an observable and equilibration of the system. Using RMT
approach it can be shown that $d_{\rm eff}=(2\pi/3)D(E_{0})\Gamma$ Nation2018
. Hence the long-time behaviour of the QFI becomes
$F_{Q}(B)\approx(8t^{2}/3d_{\rm eff})(\hat{H}^{\prime 2}_{0})_{\rm mc}$. This
relation indicates that the final quadratic behaviour of the QFI occurs when
the information of the parameter has been distributed over all quantum states
involved in the evolution of the quantum system. The crossover between the
linear to quadratic time regimes occurs at the Heisenberg time $\tau\approx\pi
D(E_{0})((\Delta\hat{H}_{0}^{\prime 2})_{\rm mc}/(\hat{H}^{\prime 2}_{0})_{\rm
mc})$, which is longest time scale for the system Schiulaz2019 . We point out
that the density of states is related with the microcanonical entropy $S$ via
the relation $1/D(E_{0})=e^{-S}$. Since the entropy is extensive quantity, the
transition time $\tau$ increases with the number of spins.

In Fig. 4 we plot the short and long time behaviour of the QFI. We observe a
good agreement between the exact and the analytical results. As we see
increasing the time the QFI makes a transition to quadratic time regime.
Increasing the spin-bath coupling $J^{(\rm SB)}_{x}$ leads to higher decay
rate $\Gamma$ which lower the QFI according to Eq (9). We note that for larger
spin-bath couplings, we expect the same general phenomena, however with
differing functional forms of the random wave function distribution. For
example, at intermediate couplings it has been observed that the random wave
function takes a Gaussian form Atlas2017 , and for strong couplings where a
full random matrix Hamiltonian is valid, the density of states dominates the
energy dependence and hence leads to decay in the form of a Bessel function
Herrera2014 ; Herrera2018 . In each case a RMT approach holds, however the
assumption here of Lorentzian wave functions is strictly valid for weak
couplings.

An important issue is whether we can recover the behaviour of the QFI by
measuring a suitable observable. An optimal measurement that provides equality
between CFI and QFI is given by the eigenvectors of the SLD operator
$\hat{L}_{B}$. We numerically diagonalize $\hat{L}_{B}$ and respectively
calculate the CFI as is shown in Fig. 3. Such a basis, however, is composed by
an entangled states and is not suitable for measurement. A more convenient
approach is to detect the spin populations $p_{s_{1},\ldots,s_{N}}={\rm
Tr}(\hat{\rho}(t)\hat{\Pi}_{s_{1},\dots,s_{N}})$, where
$\hat{\rho}(t)=|\psi(t)\rangle\langle\psi(t)|$ is the density operator and
$\hat{\Pi}_{s_{1},\dots,s_{N}}$ is the projection operator with
$s_{l}=\uparrow_{l},\downarrow_{l}$. In Fig. (3) we plot the CFI. We see that
although the CFI is lower than QFI the main properties of the QFI are captured
by detection the spin populations.

Figure 4: a) Short time evolution of QFI for chain with $N=13$ spins. b) The
QFI for various $J^{\rm SB}_{x}$ for $t=10^{4}$. c) Long time evolution of the
QFI. The initial state is $|\Psi_{0}\rangle=|\varphi_{\alpha}\rangle$ for
$\alpha=5500$. We compare the numerical results (solid lines) with the
analytical expression (9).

We proceed with an application of our result (9) to a spin system Hamiltonian
consisting of two spins, $\hat{H}_{\rm S}=B(\sigma^{z}_{1}+\sigma^{z}_{2})$.
In that case we have $(\hat{H}^{\prime 2}_{0})_{\rm
mc}=(\Delta\hat{H}_{0}^{\prime 2})_{\rm
mc}=2+2(\sigma^{z}_{1}\sigma^{z}_{2})_{\rm mc}$. We see that as long as there
is no correlation between the system spins,
$(\sigma^{z}_{1}\sigma^{z}_{2})_{\rm mc}=(\sigma^{z}_{1})_{\rm
mc}(\sigma^{z}_{2})_{\rm mc}\approx 0$, the QFI is twice the QFI for a single
system spin, $F^{(\rm SQL)}_{Q}(B)=2F_{Q}(B)$, which corresponds to the SQL.
Remarkably, we find a special system-bath interaction which leads to
correlation between the two spins in a sense that
$(\sigma^{z}_{1}\sigma^{z}_{2})_{\rm mc}>0$. Therefore, the corresponding QFI
becomes $F^{(1)}_{Q}(B)>F^{(\rm SQL)}_{Q}(B)$ and thus one can overcome the
SQL without initial entangled state preparation. We give additional details on
this application in the SM sup .

_Conclusion.-_ We derive a random matrix approach to the time evolution of the
QFI in a quantum ergodic system. The QFI is a measure for distinguishability
of two quantum states with respect to infinitesimal variation of some
parameter. Our analysis is based on a random Hamiltonian that yields an
approximate description of a quantum non-integrable system. We show that the
initial time growth of the QFI is quadratic, which quickly passes into a
linear with slope defined by the width of the random wave function.
Furthermore, we have found a second quadratic time scale which determines the
long time behaviour of the QFI. This timescale is shown to correspond to the
Heisenberg time, after which information of the local observable has spread
throughout all accessible degrees of freedom of the system.

We compared our RMT result with exact diagonalization of non-integrable spin
chain, confirming the RMT prediction of three separate timescales. We have
shown that the information for a parameter describing single spin system is
locally lost but propagates among the other degrees of freedom of the spin
system. The transition time between the linear and quadratic time regimes
depends on the density of states, and increases with the number of spins.

_Acknowledgments.-_ V. P. P. and P. A. I. acknowledge the Bulgarian national
plan for recovery and resilience, contract BG-RRP-2.004-0008-C01 (SUMMIT:
Sofia University Marking Momentum for Innovation and Technological Transfer),
project number 3.1.4. C. N. acknowledges support from the Engineering and
Physical Sciences Research Council (EPSRC UK) and the Gordon and Betty Moore
Foundation grant GBMF8820. D. P. acknowledges support from the Spanish project
PID2021-127968NB-I00 funded by MCIN/AEI/10.13039/501100011033/FEDER, UE.

## References

  * (1) A. Polkovnikov, K. Sengupta, A. Silva, and M. Vengalattore, Rev. Mod. Phys. 83, 863 (2011).
  * (2) S. Popescu, A. J. Short, and A. Winter, Nat. Phys. 2, 754 (2006).
  * (3) M. Rigol, V. Dunjko, and M. Olshanii, Nature (London) 452, 854 (2008).
  * (4) J. Eisert, M. Friesdorf, and C. Gogolin, Nat. Phys. 11, 124 (2014).
  * (5) L. D’Alessio, Y. Kafri, A. Polkovnikov, and M. Rigol, Adv. Phys, 65, 239 (2016).
  * (6) R. Nandkishore and D. A. Huse, Annu. Rev. Condens. Matter Phys. 6, 15 (2015).
  * (7) J. M. Deutsch, Phys. Rev. A 43, 2046 (1991).
  * (8) J. M. Deutsch, Rep. Prog. Phys. 81, 082001 (2018).
  * (9) M. Srednicki, Phys. Rev. E 50, 888 (1994).
  * (10) M. Srednicki, J. Phys. A: Mathematical and General 29, L75 (1996).
  * (11) M. Rigol, Phys. Rev. Lett. 103, 100403 (2009).
  * (12) M. Rigol and M. Srednicki, Phys. Rev. Lett. 108, 110601 (2012).
  * (13) R. J. Lewis-Swan, A. Safavi-Naini, J. J. Bollinger, and A. M. Rey, Nat. Commun. 10, 1581 (2019).
  * (14) D. Jansen, J. Stolpp, L. Vidmar, and F. Heidrich-Meisner, Phys. Rev. B 99, 155130 (2019).
  * (15) H. Kim, T. N. Ikeda, and D. A. Huse, Phys. Rev. E 90, 052105 (2014).
  * (16) A. V. Kirkova and P. A. Ivanov, Phys. Scr. 98, 045105 (2023).
  * (17) M. Gring, M. Kuhnert, T. Langen, T. Kitagawa, B. Rauer, M. Schreitl, I. Mazets, D. A. Smith, E. Demler, and J. Schmiedmayer , Science 337, 1318 (2012).
  * (18) A. M. Kaufman, M. E. Tai, A. Lukin, M. Rispoli, R. Schittko, P. M. Preiss, and M. Greiner, Science 353, 794 (2016).
  * (19) J. Smith, A. Lee, P. Richerme, B. Neyenhuis, P. W. Hess, P. Hauke, M. Heyl, D. A. Huse, and C. Monroe, Nat. Phys. 12, 907 (2016).
  * (20) G. Clos, D. Porras, U. Warring, and T. Schaetz, Phys. Rev. Lett. 117, 1 (2016).
  * (21) F. Kranzl, A. Lasek, M. K. Joshi, A. Kalev, R. Blatt, C. F. Roos, and N. Y. Halpern, Phys. Rev. X Quantum 4, 020318 (2023).
  * (22) C. Neill _et al_., Nat. Phys. 12, 1037 (2016).
  * (23) B. Swingle. Nature Phys. 14, 988 (2018).
  * (24) M. A. Rampp, R. Moessner, and P. W. Claeys Phys. Rev. Lett. 130, 130402 (2023).
  * (25) W. Xiao-Qian, M. Jian, Z. Xi-He, and W. Xiao-Guang, Chin. Phys. B 20, 050510 (2011).
  * (26) L. J. Song, J. Ma, D. Yan, and X. G. Wang, Eur. Phys. J. D. 66, 201 (2012).
  * (27) L. J. Fiderer and D. Braun, Nature Communications 9, 1351 (2018).
  * (28) T. Gorin, T. Prosen, T. H. Seligman, and M. Znidaric, Phys. Rep. 435, 33 (2006).
  * (29) M. Schmitt, D. Sels, S. Kehrein, and A. Polkovnikov, Phys. Rev. B 99, 134301 (2019).
  * (30) A. V. Kirkova, D. Porras, and P. A. Ivanov, Phys. Rev. A 105, 032444 (2022).
  * (31) M. G. A. Paris, Int. J. Quantum Inf. 7, 125 (2009).
  * (32) L. Pezze and A. Smerzi, arXiv:1411.5164.
  * (33) J. Liu, H. Yuan, X.-M. Lu, and X. Wang, J. Phys. A: Math. Theor. 53, 023001 (2020).
  * (34) V. Giovannetti, S. Lloyd, and L. Maccone, Nat. Photonics 5, 222 (2011).
  * (35) L. Pezze and A. Smerzi, Phys. Rev. Lett. 102, 100401 (2009).
  * (36) G. Toth, Phys. Rev. A 85, 022322 (2012).
  * (37) M. Brenes, S. Pappalardi, J. Goold, and A. Silva, Phys. Rev. Lett. 124, 040605 (2020).
  * (38) V. P. Pavlov, D. Porras, and P. A. Ivanov, Phys. Scr. 98, 095103 (2023).
  * (39) C. Nation and D. Porras, New J. Phys. 20, 103003 (2018).
  * (40) ”See Supplemental Material at [URL will be inserted by publisher]”.
  * (41) C. Nation and D. Porras, Quantum 3, 207 (2019).
  * (42) L. Dabelow and P. Reimann, Phys. Rev. Lett. 124, 120602 (2020).
  * (43) C. Nation and D. Porras, Phys. Rev. E 99, 052139 (2019).
  * (44) C Nation, D Porras Phys. Rev. E 102, 042115 (2020).
  * (45) C Nation, PhD thesis, U. of Sussex, (2020).
  * (46) N. Linden, S. Popescu, A. J. Short, and A. Winter, Phys. Rev. E 79, 061103 (2009).
  * (47) M. Schiulaz, E. J. Torres-Herrera and L. F. Santos, Phys. Rev. B 99, 174313 (2019).
  * (48) Y. Y. Atas and E. Bogomolny, J. Phys. A: Math. Theor. 50 385102 (2017).
  * (49) E. J. Torres-Herrera and L. F. Santos, Phys. Rev. A 89, 043620 (2014).
  * (50) E. J. Torres-Herrera, A. M. Garcia-Garcia, and L. F. Santos, Phys. Rev. B 97, 060303(R) (2018).

Supplemental Material: Random Matrix Theory Approach to Quantum Fisher
Information in Quantum Many-Body Systems

Venelin P. Pavlov1, Yoana R. Chorbadzhiyska1, Charlie Nation2, Diego Porras3,
and Peter A. Ivanov1

1 _Center for Quantum Technologies, Department of Physics, St. Kliment
Ohridski University of Sofia, James Bourchier 5 blvd, 1164 Sofia, Bulgaria_

2 _Department of Physics and Astronomy, University College London, London WC1E
6BT, United Kingdom_

3 _Institute of Fundamental Physics IFF-CSIC, Calle Serrano 113b, 28006
Madrid, Spain_

##  I Quantum Fisher Information in the many-body interacting basis

The quantum Fisher information for a pure state is given by

$F_{Q}(\lambda)=4\\{\langle\partial_{\lambda}\psi|\partial_{\lambda}\psi\rangle-\langle\psi|\partial_{\lambda}\psi\rangle\langle\partial_{\lambda}\psi|\psi\rangle\\},$
(S1)

where the state vector is $|\psi\rangle=e^{-i\hat{H}t}|\Psi_{0}\rangle$ and
$|\Psi_{0}\rangle$ is the initial state which is independent on parameter
$\lambda$. Therefore, we have
$|\partial_{\lambda}\psi\rangle=(\partial_{\lambda}e^{-i\hat{H}t})|\Psi_{0}\rangle$.
The partial derivative can be written as

$\partial_{\lambda}e^{-i\hat{H}t}=-it\int_{0}^{1}dse^{-i\hat{H}t}e^{i\hat{H}ts}(\partial_{\lambda}\hat{H})e^{-i\hat{H}ts}.$
(S2)

The quantum Fisher information can be rewritten as

$F_{Q}(\lambda)=4\\{\langle\Psi_{0}|(\partial_{\lambda}e^{i\hat{H}t})(\partial_{\lambda}e^{-i\hat{H}t})|\Psi_{0}\rangle-|\langle\Psi_{0}|e^{i\hat{H}t}(\partial_{\lambda}e^{-i\hat{H}t})|\Psi_{0}\rangle|^{2}\\}.$
(S3)

Now, let’s consider the first term. We have

$\langle\Psi_{0}|(\partial_{\lambda}e^{i\hat{H}t})(\partial_{\lambda}e^{-i\hat{H}t})|\Psi_{0}\rangle=\sum_{\mu\nu\rho}a^{*}_{\mu}a_{\nu}\langle\psi_{\mu}|\partial_{\lambda}e^{i\hat{H}t}e^{-i\hat{H}t}|\psi_{\rho}\rangle\langle\psi_{\rho}|e^{i\hat{H}t}\partial_{\lambda}e^{-i\hat{H}t}|\psi_{\nu}\rangle,$
(S4)

where we use that $|\Psi_{0}\rangle=\sum_{\mu}a_{\mu}|\psi_{\mu}\rangle$ and
$\sum_{\rho}|\psi_{\rho}\rangle\langle\psi_{\rho}|=\textbf{1}$. Using (S2) we
obtain

$\displaystyle\langle\psi_{\rho}|e^{i\hat{H}t}\partial_{\lambda}e^{-i\hat{H}t}|\psi_{\nu}\rangle$
$\displaystyle=$ $\displaystyle-
it\int_{0}^{1}ds\langle\psi_{\rho}|e^{is\hat{H}t}\partial_{\lambda}\hat{H}e^{-is\hat{H}t}|\psi_{\nu}\rangle=-it\langle\psi_{\rho}|\partial_{\lambda}\hat{H}|\psi_{\nu}\rangle\int_{0}^{1}dse^{i(E_{\rho}-E_{\nu})st}$
(S5)
$\displaystyle=-it\langle\psi_{\rho}|\partial_{\lambda}\hat{H}|\psi_{\nu}\rangle
e^{i\theta_{\rho\nu}t}{\rm sinc}(\theta_{\rho\nu}t),$

Here we have defined $\theta_{\mu\nu}=\frac{E_{\mu}-E_{\nu}}{2}$. Therefore,
we get

$\langle\Psi_{0}|(\partial_{\lambda}e^{i\hat{H}t})(\partial_{\lambda}e^{-i\hat{H}t})|\Psi_{0}\rangle=t^{2}\sum_{\mu\nu\rho}a^{*}_{\mu}a_{\nu}\langle\psi_{\mu}|\partial_{\lambda}\hat{H}|\psi_{\rho}\rangle\langle\psi_{\rho}|\partial_{\lambda}\hat{H}|\psi_{\nu}\rangle
e^{i\theta_{\mu\nu}}{\rm sinc}(\theta_{\mu\rho}t){\rm
sinc}(\theta_{\rho\nu}t).$ (S6)

Similarly for the second term we obtain

$\displaystyle\langle\Psi_{0}|e^{i\hat{H}t}(\partial_{\lambda}e^{-i\hat{H}t})|\Psi_{0}\rangle=-it\sum_{\mu\nu}a^{*}_{\mu}a_{\nu}\int_{0}^{1}ds\langle\psi_{\mu}|e^{is\hat{H}t}\partial_{\lambda}\hat{H}e^{-is\hat{H}t}|\psi_{\nu}\rangle=-it\sum_{\mu\nu}a^{*}_{\mu}a_{\nu}e^{i\theta_{\mu\nu}t}\langle\psi_{\mu}|\partial_{\lambda}\hat{H}|\psi_{\nu}\rangle{\rm
sinc}(\theta_{\mu\nu}t).$ (S7)

Using (S1), (S6) and (S7) we obtain

$\displaystyle
F_{Q}(\lambda)=4t^{2}\left\\{\sum_{\mu\nu\rho}a^{*}_{\mu}a_{\nu}(\partial_{\lambda}\hat{H}_{0})_{\mu\rho}(\partial_{\lambda}\hat{H}_{0})_{\rho\nu}e^{i\theta_{\mu\nu}t}{\rm
sinc}(\theta_{\mu\rho}t){\rm
sinc}(\theta_{\rho\nu}t)-|\sum_{\mu\nu}a^{*}_{\mu}a_{\nu}e^{i\theta_{\mu\nu}t}(\partial_{\lambda}\hat{H}_{0})_{\mu\nu}{\rm
sinc}(\theta_{\mu\nu}t)|^{2}\right\\}.$ (S8)

##  II Correlation Functions

In this section we outline the core RMT approach to eigenstate correlations
formulated in Ref. a . We can calculate arbitrary correlation functions of the
random wave functions $c_{\mu}(\alpha)$ by defining the respective generating
function. For $\mu=\nu$ it reads

$G_{\mu\mu}(\vec{\xi}_{\mu})\propto
e^{\frac{1}{2}\sum_{\alpha}\xi^{2}_{\mu,\alpha}\Lambda(\mu,\alpha)}.$ (S9)

where $\vec{\xi}_{\mu}=(\xi_{\mu,1},\xi_{\mu,2}\ldots\xi_{\mu,N})$ are
ancillary fields. An arbitrary correlation function of the random wave
functions can be obtained via

$\langle c_{\mu}(\alpha)c_{\mu}(\alpha^{\prime})\ldots
c_{\mu}(\beta)c_{\mu}(\beta^{\prime})\rangle_{V}=\frac{1}{G_{\mu\mu}}\partial_{\xi_{\mu,\alpha}}\partial_{\xi_{\mu,\alpha^{\prime}}}\ldots\partial_{\xi_{\mu,\beta}}\partial_{\xi_{\mu,\beta^{\prime}}}\left.G_{\mu\mu}\right|_{\xi_{\mu,\alpha}=0}.$
(S10)

Similarly for $\mu\neq\nu$ the generating function is

$G_{\mu\nu}(\vec{\xi}_{\mu},\vec{\xi}_{\nu})\propto
e^{\frac{1}{2}\sum_{\alpha}\xi^{2}_{\mu,\alpha}\Lambda(\mu,\alpha)+\frac{1}{2}\sum_{\alpha}\xi^{2}_{\nu,\alpha}\Lambda(\nu,\alpha)-\frac{1}{2}\sum_{\alpha\beta}\xi_{\mu,\alpha}\xi_{\mu,\beta}\xi_{\nu,\alpha}\xi_{\nu,\beta}\frac{\Lambda(\mu,\alpha)\Lambda(\mu,\beta)\Lambda(\nu,\alpha)\Lambda(\nu,\beta)}{\sum_{\gamma}\Lambda(\mu,\gamma)\Lambda(\nu,\gamma)}}.$
(S11)

and the correlation function becomes

$\langle c_{\mu}(\alpha)c_{\nu}(\alpha^{\prime})\ldots
c_{\mu}(\beta)c_{\nu}(\beta^{\prime})\rangle_{V}=\frac{1}{G_{\mu\nu}}\partial_{\xi_{\mu,\alpha}}\partial_{\xi_{\nu,\alpha^{\prime}}}\ldots\partial_{\xi_{\mu,\beta}}\partial_{\xi_{\nu,\beta^{\prime}}}\left.G_{\mu\nu}\right|_{\xi_{\mu,\alpha}=0,\xi_{\nu,\alpha}=0}.$
(S12)

In order to evaluate the QFI we focus on two sets of four-point correlation
functions of interest: $\langle
c_{\mu}(\alpha)c_{\nu}(\beta)c_{\mu}(\alpha^{\prime})c_{\nu}(\beta^{\prime})\rangle_{V}$
for $\mu=\nu$ and $\mu\neq\nu$.

For $\mu=\nu$ the random wave functions can be treated as an independent
random variables, namely

$\langle
c_{\mu}(\alpha)c_{\mu}(\beta)c_{\mu}(\alpha^{\prime})c_{\mu}(\beta^{\prime})\rangle_{V}=\Lambda(\mu,\alpha)\Lambda(\mu,\beta)\delta_{\alpha\alpha^{\prime}}\delta_{\beta\beta^{\prime}}+\Lambda(\mu,\alpha)\Lambda(\mu,\alpha^{\prime})(\delta_{\alpha^{\prime}\beta^{\prime}}\delta_{\alpha\beta}+\delta_{\alpha\beta^{\prime}}\delta_{\alpha^{\prime}\beta}).$
(S13)

For $\mu\neq\nu$ we have

$\langle
c_{\mu}(\alpha)c_{\nu}(\beta)c_{\mu}(\alpha^{\prime})c_{\nu}(\beta^{\prime})\rangle_{V}=\Lambda(\mu,\alpha)\Lambda(\nu,\beta)\delta_{\alpha\alpha^{\prime}}\delta_{\beta\beta^{\prime}}-\frac{\Lambda(\mu,\alpha)\Lambda(\nu,\beta)\Lambda(\mu,\alpha^{\prime})\Lambda(\nu,\beta^{\prime})}{\sum_{\gamma}\Lambda(\mu,\gamma)\Lambda(\nu,\gamma)}(\delta_{\alpha\beta}\delta_{\alpha^{\prime}\beta^{\prime}}+\delta_{\alpha\beta^{\prime}}\delta_{\alpha^{\prime}\beta}).$
(S14)

The first term in (S14) describes the four-point correlation function as an
independent random Gaussian variables, while the last two terms correspond to
the non-Gaussian correction which arises as a result of the orthogonality
condition.

In fact we may express graphically the correlation functions as a sum of
products of two-point correlation functions. Consider first $\mu=\nu$. Then we
have

$\langle
c_{\mu}(\alpha)c_{\mu}(\beta)c_{\mu}(\alpha^{\prime})c_{\mu}(\beta^{\prime})\rangle_{V}=\langle
c_{\mu}(\overbracket{\alpha)c_{\mu}(\beta})c_{\mu}(\overbracket{\alpha^{\prime})c_{\mu}(\beta^{\prime}})\rangle_{V}+\langle
c_{\mu}(\overbracket{\alpha)c_{\mu}(\underbracket{\beta)c_{\mu}(\alpha^{\prime}})c_{\mu}(\beta^{\prime}})\rangle_{V}+\langle
c_{\mu}(\hbox
to0.0pt{$\displaystyle\overbracket{\phantom{\alpha)c_{\mu}(\beta)c_{\mu}(\alpha^{\prime}}}$\hss}\alpha)c_{\mu}(\underbracket{\beta)c_{\mu}(\alpha^{\prime})c_{\mu}(\beta^{\prime}})\rangle_{V}.$
(S15)

Each of the terms can be written as a product of two-point correlation
functions. For example

$\langle
c_{\mu}(\overbracket{\alpha)c_{\mu}(\beta})c_{\mu}(\overbracket{\alpha^{\prime})c_{\mu}(\beta^{\prime}})\rangle_{V}=\langle
c_{\mu}(\alpha)c_{\mu}(\beta)\rangle_{V}\langle
c_{\mu}(\alpha^{\prime})c_{\mu}(\beta^{\prime})\rangle_{V}=\Lambda(\mu,\alpha)\delta_{\alpha\beta}\Lambda(\mu,\alpha^{\prime})\delta_{\alpha^{\prime}\beta^{\prime}}.$
(S16)

The case for $\mu\neq\nu$ we need also to include the non-Gaussian
corrections. We have

$\langle
c_{\mu}(\alpha)c_{\nu}(\beta)c_{\mu}(\alpha^{\prime})c_{\nu}(\beta^{\prime})\rangle_{V}=\langle
c_{\mu}(\hbox
to0.0pt{$\displaystyle\overbracket{\phantom{\alpha)c_{\nu}(\beta)c_{\mu}(\alpha^{\prime}}}$\hss}\alpha)c_{\nu}(\underbracket{\beta)c_{\mu}(\alpha^{\prime})c_{\nu}(\beta^{\prime}})\rangle_{V}+\langle
c_{\mu}(\overbracket{\alpha)c_{\nu}(\beta})c_{\mu}(\overbracket{\alpha^{\prime})c_{\nu}(\beta^{\prime}})\rangle_{V}+\langle
c_{\mu}(\overbracket{\alpha)c_{\nu}(\underbracket{\beta)c_{\mu}(\alpha^{\prime}})c_{\nu}(\beta^{\prime}})\rangle_{V}.$
(S17)

The last two terms in (S17) arise as a result of the orthogonality condition
between the many-body eigenstates. For example the first non-Gaussian term is

$\langle
c_{\mu}(\overbracket{\alpha)c_{\nu}(\beta})c_{\mu}(\overbracket{\alpha^{\prime})c_{\nu}(\beta^{\prime}})\rangle_{V}=-\frac{\Lambda(\mu,\alpha)\Lambda(\nu,\beta)\Lambda(\mu,\alpha^{\prime})\Lambda(\nu,\beta^{\prime})}{\sum_{\gamma}\Lambda(\mu,\gamma)\Lambda(\nu,\gamma)}\delta_{\alpha\beta}\delta_{\alpha^{\prime}\beta^{\prime}}$
(S18)

and similarly for the second one.

##  III Calculation of the QFI using Random Matrix Approach

Here we provide the method which we use to evaluate the QFI (3). We set
$\partial_{\lambda}\hat{H}_{0}=\hat{H}_{0}^{\prime}$ and assume that
$\hat{H}^{\prime}_{0}$ is a diagonal matrix in the non-interacting basis.
Therefore, the QFI is

$\displaystyle
F_{Q}(\lambda)=4t^{2}\left\\{\sum_{\mu\nu\rho}a^{*}_{\mu}a_{\nu}(\hat{H}^{\prime}_{0})_{\mu\rho}(\hat{H}^{\prime}_{0})_{\rho\nu}e^{i\theta_{\mu\nu}t}{\rm
sinc}(\theta_{\mu\rho}t){\rm
sinc}(\theta_{\rho\nu}t)-|\sum_{\mu\nu}a^{*}_{\mu}a_{\nu}e^{i\theta_{\mu\nu}t}(\hat{H}^{\prime}_{0})_{\mu\nu}{\rm
sinc}(\theta_{\mu\nu}t)|^{2}\right\\}.$ (S19)

Let us now consider separately the first term in (S19), namely

$\displaystyle\sum_{\mu\nu\rho}a^{*}_{\mu}a_{\nu}\langle\psi_{\mu}|\hat{H}^{\prime}_{0}|\psi_{\rho}\rangle\langle\psi_{\rho}|\hat{H}^{\prime}_{0}|\psi_{\nu}\rangle
e^{i\theta_{\mu\nu}t}{\rm sinc}(\theta_{\mu\rho}t){\rm
sinc}(\theta_{\rho\nu}t)=\sum_{\mu}|a_{\mu}|^{2}|\langle\psi_{\mu}|\hat{H}^{\prime}_{0}|\psi_{\mu}\rangle|^{2}+\sum_{\mu\nu\atop\mu\neq\nu}|a_{\mu}|^{2}|\langle\psi_{\mu}|\hat{H}^{\prime}_{0}|\psi_{\nu}\rangle|^{2}{\rm
sinc}^{2}(\theta_{\mu\nu}t)$
$\displaystyle+\sum_{\mu\nu\atop\mu\neq\nu}a^{*}_{\mu}a_{\nu}\langle\psi_{\mu}|\hat{H}^{\prime}_{0}|\psi_{\nu}\rangle\langle\psi_{\nu}|\hat{H}^{\prime}_{0}|\psi_{\nu}\rangle
e^{i\theta_{\mu\nu}t}{\rm
sinc}(\theta_{\mu\nu}t)+\sum_{\mu\nu\atop\mu\neq\nu}a^{*}_{\mu}a_{\nu}\langle\psi_{\mu}|\hat{H}^{\prime}_{0}|\psi_{\mu}\rangle\langle\psi_{\mu}|\hat{H}^{\prime}_{0}|\psi_{\nu}\rangle
e^{i\theta_{\mu\nu}t}{\rm sinc}(\theta_{\mu\nu}t)$
$\displaystyle+\sum_{\mu\nu\rho\atop\mu\neq\nu\neq\rho}a^{*}_{\mu}a_{\nu}\langle\psi_{\mu}|\hat{H}^{\prime}_{0}|\psi_{\rho}\rangle\langle\psi_{\rho}|\hat{H}^{\prime}_{0}|\psi_{\nu}\rangle
e^{i\theta_{\mu\nu}t}{\rm sinc}(\theta_{\mu\rho}t){\rm
sinc}(\theta_{\rho\nu}t).$ (S20)

Now we apply the self-averaging condition for each of the terms in (S20). For
the first one we have

$\sum_{\mu}|a_{\mu}|^{2}|\langle\psi_{\mu}|\hat{H}^{\prime}_{0}|\psi_{\mu}\rangle|^{2}=\sum_{\mu}\langle|a_{\mu}|^{2}|\langle\psi_{\mu}|\hat{H}^{\prime}_{0}|\psi_{\mu}\rangle|^{2}\rangle_{V}.$
(S21)

In order to evaluate (S21) we may further decouple the coefficients $a_{\mu}$
describing the initial state part and observable in the sense that (see
Section ”Corrections due to self-averaging decoupling” for more details)

$\sum_{\mu}\langle|a_{\mu}|^{2}|\langle\psi_{\mu}|\hat{H}^{\prime}_{0}|\psi_{\mu}\rangle|^{2}\rangle_{V}=\sum_{\mu}\langle|a_{\mu}|^{2}\rangle_{V}\langle|\langle\psi_{\mu}|\hat{H}^{\prime}_{0}|\psi_{\mu}\rangle|^{2}\rangle_{V}.$
(S22)

Therefore, we have

$\displaystyle\langle|\langle\psi_{\mu}|\hat{H}^{\prime}_{0}|\psi_{\mu}\rangle|^{2}\rangle_{V}$
$\displaystyle=$ $\displaystyle\sum_{\alpha\beta}\langle
c_{\mu}(\alpha)c_{\mu}(\alpha)c_{\mu}(\beta)c_{\mu}(\beta)\rangle_{V}(\hat{H}^{\prime}_{0})_{\alpha\alpha}(\hat{H}^{\prime}_{0})_{\beta\beta}$
(S23)
$\displaystyle=\sum_{\alpha\beta}\left(2\Lambda^{2}(\mu,\alpha)\delta_{\alpha\beta}(\hat{H}^{\prime}_{0})_{\alpha\alpha}(\hat{H}^{\prime}_{0})_{\beta\beta}+\Lambda(\mu,\alpha)(\hat{H}^{\prime}_{0})_{\alpha\alpha}\Lambda(\mu,\beta)(\hat{H}^{\prime}_{0})_{\beta\beta}\right).$

Figure S1: Relative error $d=|1-(F_{Q}(\omega))_{\rm RMT}/(F_{Q}(\omega))|$
between results for the QFI derived from Eqs. (3) and (1).

We define the average
$[(\hat{H}_{0}^{\prime})_{\alpha\alpha}]_{\mu}=\sum_{\alpha}\Lambda(\mu,\alpha)(\hat{H}^{\prime}_{0})_{\alpha\alpha}$,
which is essentially a microcanonical average centered on the energy
$E_{\mu}$. We also apply the smoothness condition which implies that the
variation of $[(\hat{H}_{0}^{\prime})_{\alpha\alpha}]_{\mu}$ as a function of
$E_{\mu}$ can be neglected. Using this, we obtain

$\langle|\langle\psi_{\mu}|\hat{H}^{\prime}_{0}|\psi_{\mu}\rangle|^{2}\rangle_{V}\approx
2[(\hat{H}^{\prime}_{0})^{2}_{\alpha\alpha}]_{\mu}\sum_{\alpha}\Lambda^{2}(\mu,\alpha)+[(\hat{H}^{\prime}_{0})_{\alpha\alpha}]^{2}_{\mu}.$
(S24)

Further, we take the continuum limit, substituting
$\sum_{\alpha}\rightarrow\int_{-\infty}^{\infty}\frac{dE_{\alpha}}{\omega}$,
and thereby obtain

$\sum_{\alpha}\Lambda^{2}(\mu,\alpha)=\frac{1}{\omega}\int_{-\infty}^{\infty}\Lambda^{2}(\mu,\alpha)dE_{\alpha}=\frac{\omega}{2\pi\Gamma}.$
(S25)

Substituting in Eq. (S21) we have

$\sum_{\mu}|a_{\mu}|^{2}|\langle\psi_{\mu}|\hat{H}^{\prime}_{0}|\psi_{\mu}\rangle|^{2}=\sum_{\mu}\langle|a_{\mu}|^{2}\rangle_{V}\left(\frac{\omega}{\pi\Gamma}[(\hat{H}^{\prime}_{0})^{2}_{\alpha\alpha}]_{\mu}+[(\hat{H}^{\prime}_{0})_{\alpha\alpha}]^{2}_{\mu}\right).$
(S26)

As long as $[(\hat{H}^{\prime}_{0})^{2}_{\alpha\alpha}]_{\mu}$ and
$[(\hat{H}^{\prime}_{0})_{\alpha\alpha}]^{2}_{\mu}$ are smooth function of the
energy $E_{\mu}$ and the probabilities $|a_{\mu}|^{2}$ take non-vanishing
value close to the mean energy $E_{0}=\langle\Psi_{0}|\hat{H}|\Psi_{0}\rangle$
with $|\Psi_{0}\rangle$ being the initial state, the ETH ensures that Eq.
(S26) is equivalent to a microcanonical average,

$\sum_{\mu}|a_{\mu}|^{2}|\langle\psi_{\mu}|\hat{H}^{\prime}_{0}|\psi_{\mu}\rangle|^{2}=\frac{\omega}{\pi\Gamma}(\hat{H}^{\prime
2}_{0})_{\rm mc}+(\hat{H}^{\prime}_{0})_{\rm mc}^{2}.$ (S27)

Consider the second term in (S20)

$\sum_{\mu\nu\atop\mu\neq\nu}|a_{\mu}|^{2}|\langle\psi_{\mu}|\hat{H}^{\prime}_{0}|\psi_{\nu}\rangle|^{2}{\rm
sinc}^{2}(\theta_{\mu\nu}t)=\sum_{\mu\nu\atop\mu\neq\nu}\langle|a_{\mu}|^{2}\rangle_{V}\langle|\langle\psi_{\mu}|\hat{H}^{\prime}_{0}|\psi_{\nu}\rangle|^{2}\rangle_{V}{\rm
sinc}^{2}(\theta_{\mu\nu}t).$ (S28)

For the matrix element we have

$\langle|\langle\psi_{\mu}|\hat{H}^{\prime}_{0}|\psi_{\nu}\rangle|^{2}\rangle_{V}=\sum_{\alpha\beta}\langle
c_{\mu}(\alpha)c_{\nu}(\alpha)c_{\mu}(\beta)c_{\nu}(\beta)\rangle_{V}(\hat{H}^{\prime}_{0})_{\alpha\alpha}(\hat{H}^{\prime}_{0})_{\beta\beta}.$
(S29)

Now, using (S14) we obtain

$\langle
c_{\mu}(\alpha)c_{\nu}(\alpha)c_{\mu}(\beta)c_{\nu}(\beta)\rangle_{V}=\Lambda(\mu,\alpha)\Lambda(\nu,\alpha)\delta_{\alpha\beta}-\frac{\Lambda(\mu,\alpha)\Lambda(\nu,\alpha)\Lambda(\mu,\beta)\Lambda(\nu,\beta)}{\sum_{\gamma}\Lambda(\mu,\gamma)\Lambda(\nu,\gamma)}-\frac{\Lambda^{2}(\mu,\alpha)\Lambda^{2}(\nu,\alpha)}{\sum_{\gamma}\Lambda(\mu,\gamma)\Lambda(\nu,\gamma)}\delta_{\alpha\beta}$
(S30)

and the matrix element becomes

$\displaystyle\langle|\langle\psi_{\mu}|\hat{H}^{\prime}_{0}|\psi_{\nu}\rangle|^{2}\rangle_{V}$
$\displaystyle=$
$\displaystyle\sum_{\alpha\beta}\left(\Lambda(\mu,\alpha)\Lambda(\nu,\alpha)\delta_{\alpha\beta}-\frac{\Lambda(\mu,\alpha)\Lambda(\nu,\alpha)\Lambda(\mu,\beta)\Lambda(\nu,\beta)}{\sum_{\gamma}\Lambda(\mu,\gamma)\Lambda(\nu,\gamma)}-\frac{\Lambda^{2}(\mu,\alpha)\Lambda^{2}(\nu,\alpha)}{\sum_{\gamma}\Lambda(\mu,\gamma)\Lambda(\nu,\gamma)}\delta_{\alpha\beta}\right)(\hat{H}^{\prime}_{0})_{\alpha\alpha}(\hat{H}^{\prime}_{0})_{\beta\beta}$
(S31)
$\displaystyle\approx[(\Delta\hat{H}^{\prime}_{0})_{\alpha\alpha}]^{2}_{\bar{\mu}}\sum_{\alpha}\Lambda(\mu,\alpha)\Lambda(\nu,\alpha),$

where $[(\Delta\hat{H}^{\prime}_{0})_{\alpha\alpha}]^{2}_{\bar{\mu}}$ is the
variance, $\bar{\mu}=\frac{\mu+\nu}{2}$. Going in the continuum limit we get

$\sum_{\alpha}\Lambda(\mu,\alpha)\Lambda(\nu,\alpha)=\frac{2\omega\Gamma}{\pi}\frac{1}{(E_{\mu}-E_{\nu})^{2}+4\Gamma^{2}}.$
(S32)

Therefore, the second term becomes

$\displaystyle\sum_{\mu\nu\atop\mu\neq\nu}\langle|a_{\mu}|^{2}\rangle_{V}\langle|\langle\psi_{\mu}|\hat{H}^{\prime}_{0}|\psi_{\nu}\rangle|^{2}\rangle_{V}{\rm
sinc}^{2}(\theta_{\mu\nu}t)=\frac{2\omega\Gamma}{\pi}\sum_{\mu\nu\atop\mu\neq\nu}\langle|a_{\mu}|^{2}\rangle_{V}[(\Delta\hat{H}^{\prime}_{0})_{\alpha\alpha}]^{2}_{\bar{\mu}}\frac{{\rm
sinc}^{2}(\theta_{\mu\nu}t)}{(E_{\mu}-E_{\nu})^{2}+4\Gamma^{2}}.$ (S33)

We now replace the sum over the index $\nu$ with integration, namely

$\sum_{\nu}\frac{{\rm
sinc}^{2}(\theta_{\mu\nu}t)}{(E_{\mu}-E_{\nu})^{2}+4\Gamma^{2}}\rightarrow\frac{1}{\omega}\int_{-\infty}^{\infty}\frac{{\rm
sinc}^{2}(\theta_{\mu\nu}t)}{(E_{\mu}-E_{\nu})^{2}+4\Gamma^{2}}dE_{\nu}=\frac{t}{2\omega}\int_{-\infty}^{\infty}\frac{{\rm
sinc}^{2}(x)}{x^{2}+(\Gamma t)^{2}}dx=\frac{\pi t}{4\omega(\Gamma
t)^{3}}(e^{-2\Gamma t}-1+2\Gamma t).$ (S34)

Finally, we have

$\sum_{\mu\nu\atop\mu\neq\nu}|a_{\mu}|^{2}|\langle\psi_{\mu}|\hat{H}^{\prime}_{0}|\psi_{\nu}\rangle|^{2}{\rm
sinc}^{2}(\theta_{\mu\nu}t)=\frac{(\Delta\hat{H}^{\prime 2}_{0})_{\rm
mc}}{2(\Gamma t)^{2}}(e^{-2\Gamma t}-1+2\Gamma t).$ (S35)

We note that in Eq. (S31) we have neglected the contribution from the third
term which is of order of $(\omega/\Gamma)^{2}$. Indeed, we have

$\displaystyle\sum_{\alpha}\Lambda^{2}(\mu,\alpha)\Lambda^{2}(\nu,\alpha)\rightarrow\left(\frac{\omega\Gamma}{\pi}\right)^{4}\frac{1}{\omega}\int_{-\infty}^{\infty}\frac{dE_{\alpha}}{((E_{\mu}-E_{\alpha})^{2}+\Gamma^{2})^{2}((E_{\nu}-E_{\alpha})^{2}+\Gamma^{2})^{2}}$
$\displaystyle=\left(\frac{\omega\Gamma}{\pi}\right)^{4}\frac{\pi}{\omega\Gamma^{3}}\frac{(E_{\mu}-E_{\nu})^{2}+20\Gamma^{2}}{((E_{\mu}-E_{\nu})^{2}+4\Gamma^{2})^{3}}.$
(S36)

Therefore, using (S32) we conclude that the third term in (S31) is of order of
$(\omega/\Gamma)^{2}$. We also note that the contribution for $\mu=\nu$ in Eq.
(S33) can be neglected.

Consider the third term in (S20). We have

$\sum_{\mu\nu\atop\mu\neq\nu}a^{*}_{\mu}a_{\nu}\langle\psi_{\mu}|\hat{H}^{\prime}_{0}|\psi_{\nu}\rangle\langle\psi_{\nu}|\hat{H}^{\prime}_{0}|\psi_{\nu}\rangle
e^{i\theta_{\mu\nu}t}{\rm
sinc}(\theta_{\mu\nu}t)=\sum_{\mu\nu\atop\mu\neq\nu}\langle
a^{*}_{\mu}a_{\nu}\rangle_{V}\langle\langle\psi_{\mu}|\hat{H}^{\prime}_{0}|\psi_{\nu}\rangle\langle\psi_{\nu}|\hat{H}^{\prime}_{0}|\psi_{\nu}\rangle\rangle_{V}e^{i\theta_{\mu\nu}t}{\rm
sinc}(\theta_{\mu\nu}t).$ (S37)

The matrix elements express in the non-interacting basis are

$\langle\langle\psi_{\mu}|\hat{H}^{\prime}_{0}|\psi_{\nu}\rangle\langle\psi_{\nu}|\hat{H}^{\prime}_{0}|\psi_{\nu}\rangle\rangle_{V}=\sum_{\alpha\beta}\langle
c_{\mu}(\alpha)c_{\nu}(\alpha)c_{\nu}(\beta)c_{\nu}(\beta)\rangle_{V}(\hat{H}^{\prime}_{0})_{\alpha\alpha}(\hat{H}^{\prime}_{0})_{\beta\beta}.$
(S38)

Using the generating function Eq. (S11) the four-point correlation function is

$\langle
c_{\mu}(\alpha)c_{\nu}(\alpha)c_{\nu}(\beta)c_{\nu}(\beta)\rangle_{V}=0.$
(S39)

Similarly, for the last term in (S20) we have

$\langle\langle\psi_{\mu}|\hat{H}^{\prime}_{0}|\psi_{\rho}\rangle\langle\psi_{\rho}|\hat{H}^{\prime}_{0}|\psi_{\nu}\rangle\rangle_{V}=\sum_{\alpha\beta}\langle
c_{\mu}(\alpha)c_{\rho}(\alpha)c_{\rho}(\beta)c_{\nu}(\beta)\rangle_{V}(\hat{H}^{\prime}_{0})_{\alpha\alpha}(\hat{H}^{\prime}_{0})_{\beta\beta}.$
(S40)

Now, the four-point correlation function contains three different indices. In
order to evaluate it we need to introduce three auxiliary fields. Because the
indexes $\mu$ and $\nu$ repeat only once we have

$\langle
c_{\mu}(\alpha)c_{\rho}(\alpha)c_{\rho}(\beta)c_{\nu}(\beta)\rangle_{V}=0.$
(S41)

Combining all averages in (S20) we obtain

$\displaystyle\sum_{\mu\nu\rho}a^{*}_{\mu}a_{\nu}\langle\psi_{\mu}|\hat{H}^{\prime}_{0}|\psi_{\rho}\rangle\langle\psi_{\rho}|\hat{H}^{\prime}_{0}|\psi_{\nu}\rangle
e^{i\theta_{\mu\nu}t}{\rm sinc}(\theta_{\mu\rho}t){\rm
sinc}(\theta_{\rho\nu}t)$ $\displaystyle=$
$\displaystyle\frac{\omega}{\pi\Gamma}(\hat{H}^{\prime 2}_{0})_{\rm
mc}+(\hat{H}^{\prime}_{0})_{\rm mc}^{2}+\frac{(\Delta\hat{H}^{\prime
2}_{0})_{\rm mc}}{2(\Gamma t)^{2}}(e^{-2\Gamma t}-1+2\Gamma t).$ (S42)

Let us now consider separately the second term in (S19), namely

$\displaystyle\sum_{\mu\nu}\sum_{\mu^{\prime}\nu^{\prime}}a^{*}_{\mu}a_{\nu}a_{\mu^{\prime}}a^{*}_{\nu^{\prime}}\langle\psi_{\mu}|\hat{H}^{\prime}_{0}|\psi_{\nu}\rangle\langle\psi_{\nu^{\prime}}|\hat{H}^{\prime}_{0}|\psi_{\mu^{\prime}}\rangle
e^{i\theta_{\mu\nu}t}e^{-i\theta_{\mu^{\prime}\nu^{\prime}}t}{\rm
sinc}(\theta_{\mu\nu}t){\rm
sinc}(\theta_{\mu^{\prime}\nu^{\prime}}t)=\sum_{\mu}|a_{\mu}|^{4}|\langle\psi_{\mu}|\hat{H}^{\prime}_{0}|\psi_{\mu}\rangle|^{2}$
$\displaystyle+\sum_{\mu\nu\atop\mu\neq\nu}|a_{\mu}|^{2}|a_{\nu}|^{2}\langle\psi_{\mu}|\hat{H}^{\prime}_{0}|\psi_{\mu}\rangle\langle\psi_{\nu}|\hat{H}^{\prime}_{0}|\psi_{\nu}\rangle+\sum_{\mu\nu\atop\mu\neq\nu}|a_{\mu}|^{2}|a_{\nu}|^{2}\langle\psi_{\mu}|\hat{H}^{\prime}_{0}|\psi_{\nu}\rangle\langle\psi_{\nu}|\hat{H}^{\prime}_{0}|\psi_{\mu}\rangle{\rm
sinc}^{2}(\theta_{\mu\nu}t)$
$\displaystyle+\sum_{\mu\nu\atop\mu\neq\nu}a_{\mu}^{*2}a_{\nu}^{2}\langle\psi_{\mu}|\hat{H}^{\prime}_{0}|\psi_{\nu}\rangle\langle\psi_{\mu}|\hat{H}^{\prime}_{0}|\psi_{\nu}\rangle
e^{2i\theta_{\mu\nu}t}{\rm sinc}^{2}(\theta_{\mu\nu}t)+\ldots$ (S43)

We have

$\sum_{\mu}|a_{\mu}|^{4}|\langle\psi_{\mu}|\hat{H}^{\prime}_{0}|\psi_{\mu}\rangle|^{2}=\sum_{\mu}\langle|a_{\mu}|^{4}\rangle_{V}\left(\frac{\omega}{\pi\Gamma}[(\hat{H}^{\prime}_{0})^{2}_{\alpha\alpha}]_{\mu}+[(\hat{H}^{\prime}_{0})_{\alpha\alpha}]^{2}_{\mu}\right)=\left(\frac{\omega}{\pi\Gamma}(\hat{H}^{\prime
2}_{0})_{\rm mc}+(\hat{H}^{\prime}_{0})^{2}_{\rm
mc}\right)\sum_{\mu}\langle|a_{\mu}|^{4}\rangle_{V}.$ (S44)

Note that $\left(\frac{\omega}{\pi\Gamma}\right)(\hat{H}^{\prime 2}_{0})_{\rm
mc}\langle\sum_{\mu}|a_{\mu}|^{4}\rangle_{V}$ is of order of
$(\omega/\Gamma)^{2}$ and thereby we neglect. The second term in (S43) is

$\sum_{\mu\nu\atop\mu\neq\nu}|a_{\mu}|^{2}|a_{\nu}|^{2}\langle\psi_{\mu}|\hat{H}^{\prime}_{0}|\psi_{\mu}\rangle\langle\psi_{\nu}|\hat{H}^{\prime}_{0}|\psi_{\nu}\rangle=\sum_{\mu\nu\atop\mu\neq\nu}\langle|a_{\mu}|^{2}|a_{\nu}|^{2}\rangle_{V}\langle\langle\psi_{\mu}|\hat{H}^{\prime}_{0}|\psi_{\mu}\rangle\langle\psi_{\nu}|\hat{H}^{\prime}_{0}|\psi_{\nu}\rangle\rangle_{V}.$
(S45)

The matrix elements are

$\displaystyle\langle\langle\psi_{\mu}|\hat{H}^{\prime}_{0}|\psi_{\mu}\rangle\langle\psi_{\nu}|\hat{H}^{\prime}_{0}|\psi_{\nu}\rangle\rangle_{V}$
$\displaystyle=$ $\displaystyle\sum_{\alpha\beta}\langle
c_{\mu}(\alpha)c_{\mu}(\alpha)c_{\nu}(\beta)c_{\nu}(\beta)\rangle_{V}(\hat{H}_{0}^{\prime})_{\alpha\alpha}(\hat{H}_{0}^{\prime})_{\beta\beta}$
(S46)
$\displaystyle=\sum_{\alpha\beta}\left(\Lambda(\mu,\alpha)\Lambda(\nu,\beta)-2\frac{\Lambda^{2}(\mu,\alpha)\Lambda^{2}(\nu,\beta)}{\sum_{\gamma}\Lambda(\mu,\gamma)\Lambda(\nu,\gamma)}\delta_{\alpha\beta}\right)(\hat{H}_{0}^{\prime})_{\alpha\alpha}(\hat{H}_{0}^{\prime})_{\beta\beta}$
$\displaystyle\approx[(\hat{H}^{\prime}_{0})_{\alpha\alpha}]^{2}_{\bar{\mu}},$

where we neglect the second term which is of order of $(\omega/\Gamma)^{2}$.
Therefore, we obtain

$\sum_{\mu\nu\atop\mu\neq\nu}\langle|a_{\mu}|^{2}|a_{\nu}|^{2}\rangle_{V}[(\hat{H}^{\prime}_{0})_{\alpha\alpha}]^{2}_{\bar{\mu}}\approx(\hat{H}^{\prime}_{0})_{\rm
mc}^{2}(\sum_{\mu\nu}\langle|a_{\mu}|^{2}|a_{\nu}|^{2}\rangle_{V}-\sum_{\mu}\langle|a_{\mu}|^{4}\rangle_{V})=(\hat{H}^{\prime}_{0})_{\rm
mc}^{2}(1-\sum_{\mu}\langle|a_{\mu}|^{4}\rangle_{V}).$ (S47)

Consider the term

$\sum_{\mu\nu\atop\mu\neq\nu}|a_{\mu}|^{2}|a_{\nu}|^{2}\langle|\langle\psi_{\mu}|\hat{H}^{\prime}_{0}|\psi_{\nu}\rangle|^{2}\rangle_{V}{\rm
sinc}^{2}(\theta_{\mu\nu}t)=\sum_{\mu\nu\atop\mu\neq\nu}\langle|a_{\mu}|^{2}|a_{\nu}|^{2}\rangle_{V}\langle|\langle\psi_{\mu}|\hat{H}^{\prime}_{0}|\psi_{\nu}\rangle|^{2}\rangle_{V}{\rm
sinc}^{2}(\theta_{\mu\nu}t).$ (S48)

Using (S31) we obtain

$\sum_{\mu\nu\atop\mu\neq\nu}\langle|a_{\mu}|^{2}|a_{\nu}|^{2}\rangle_{V}\langle|\langle\psi_{\mu}|\hat{H}^{\prime}_{0}|\psi_{\nu}\rangle|^{2}\rangle_{V}{\rm
sinc}^{2}(\theta_{\mu\nu}t)\approx(\Delta\hat{H}^{\prime 2}_{0})_{\rm
mc}\frac{2\Gamma\omega}{\pi}\sum_{\mu\nu\atop\mu\neq\nu}\langle|a_{\mu}|^{2}|a_{\nu}|^{2}\rangle_{V}\frac{{\rm
sinc}^{2}(\theta_{\mu\nu}t)}{(E_{\mu}-E_{\nu})^{2}+4\Gamma^{2}}.$ (S49)

Let us now assume that the initial state is an eigenstate of the non-
interaction Hamiltonian $\hat{H}_{0}$, namely
$|\psi(0)\rangle=|\varphi_{\alpha_{0}}\rangle$. Then, we have

$\langle|a_{\mu}|^{2}|a_{\nu}|^{2}\rangle_{V}=\Lambda(\mu,\alpha_{0})\Lambda(\nu,\alpha_{0})$
(S50)

and we get

$(\Delta\hat{H}^{\prime 2}_{0})_{\rm
mc}\frac{2\Gamma\omega}{\pi}\sum_{\mu\nu\atop\mu\neq\nu}\langle|a_{\mu}|^{2}|a_{\nu}|^{2}\rangle_{V}\frac{{\rm
sinc}^{2}(\theta_{\mu\nu}t)}{(E_{\mu}-E_{\nu})^{2}+4\Gamma^{2}}=(\Delta\hat{H}^{\prime
2}_{0})_{\rm
mc}\frac{2\Gamma\omega}{\pi}\sum_{\mu\nu\atop\mu\neq\nu}\Lambda(\mu,\alpha_{0})\Lambda(\nu,\alpha_{0})\frac{{\rm
sinc}^{2}(\theta_{\mu\nu}t)}{(E_{\mu}-E_{\nu})^{2}+4\Gamma^{2}}.$ (S51)

We replace the sum with the integration, such that we have

$\frac{2\Gamma\omega}{\pi}\frac{1}{\omega}\int_{-\infty}^{\infty}\Lambda(\mu,\alpha_{0})\frac{{\rm
sinc}^{2}(\theta_{\mu\nu}t)}{(E_{\mu}-E_{\nu})^{2}+4\Gamma^{2}}dE_{\mu}\leq\frac{2\Gamma}{\pi}\frac{\omega}{\pi\Gamma}\int_{-\infty}^{\infty}\frac{{\rm
sinc}^{2}(\theta_{\mu\nu}t)}{(E_{\mu}-E_{\nu})^{2}+4\Gamma^{2}}dE_{\mu}=\frac{\omega}{\pi\Gamma}\frac{1}{2(\Gamma
t)^{2}}(e^{-2\Gamma t}-1+2\Gamma t).$ (S52)

In the above equation we have used that for any two functions $f(x)>0$ and
$g(x)>0$, which obey $f(x)g(x)\leq f_{\rm max}g(x)$ it follow that
$\int_{-\infty}^{\infty}f(x)g(x)dx\leq f_{\rm
max}\int_{-\infty}^{\infty}g(x)dx$. Therefore we obtain

$\sum_{\mu\nu\atop\mu\neq\nu}\langle|a_{\mu}|^{2}|a_{\nu}|^{2}\rangle_{V}\langle|\langle\psi_{\mu}|\hat{H}^{\prime}_{0}|\psi_{\nu}\rangle|^{2}\rangle_{V}{\rm
sinc}^{2}(\theta_{\mu\nu}t)\leq(\Delta\hat{H}^{\prime 2}_{0})_{\rm
mc}\frac{\omega}{\pi\Gamma}\frac{1}{2(\Gamma t)^{2}}(e^{-2\Gamma t}-1+2\Gamma
t).$ (S53)

As long as $\Gamma\gg\omega$ we neglect this term.

Similarly, we have

$2\sum_{\mu\nu\atop\mu\neq\nu}a_{\mu}^{*2}a_{\nu}^{2}|\langle\psi_{\mu}|\hat{H}^{\prime}_{0}|\psi_{\nu}\rangle|^{2}\cos(2\theta_{\mu\nu}t){\rm
sinc}^{2}(\theta_{\mu\nu}t)\approx 2(\Delta\hat{H}^{\prime 2}_{0})_{\rm
mc}\frac{2\Gamma\omega}{\pi}\sum_{\mu\nu\atop\mu\neq\nu}\Lambda(\mu,\alpha_{0})\Lambda(\nu,\alpha_{0})\frac{\cos(2\theta_{\mu\nu}t){\rm
sinc}^{2}(\theta_{\mu\nu}t)}{(E_{\mu}-E_{\nu})^{2}+4\Gamma^{2}}.$ (S54)

Replacing the sum with integration we get

$\frac{2\Gamma\omega}{\pi}\int_{-\infty}^{\infty}\Lambda(\mu,\alpha_{0})\frac{\cos(2\theta_{\mu\nu}t){\rm
sinc}^{2}(\theta_{\mu\nu}t)}{(E_{\mu}-E_{\nu})^{2}+4\Gamma^{2}}dE_{\mu}\leq\frac{2\Gamma}{\pi}\frac{\omega}{\pi\Gamma}\int_{-\infty}^{\infty}\frac{\cos(2\theta_{\mu\nu}t){\rm
sinc}^{2}(\theta_{\mu\nu}t)}{(E_{\mu}-E_{\nu})^{2}+4\Gamma^{2}}dE_{\mu}.$
(S55)

The integral is given by

$\frac{2\Gamma}{\pi}\int_{-\infty}^{\infty}\frac{\cos(2\theta_{\mu\nu}t){\rm
sinc}^{2}(\theta_{\mu\nu}t)}{(E_{\mu}-E_{\nu})^{2}+4\Gamma^{2}}dE_{\mu}=\frac{(e^{-2\Gamma
t}-1)^{2}}{4(\Gamma t)^{2}}.$ (S56)

Therefore we obtain

$2\sum_{\mu\nu\atop\mu\neq\nu}a_{\mu}^{*2}a_{\nu}^{2}\langle\psi_{\mu}|\hat{H}^{\prime}_{0}|\psi_{\nu}\rangle\langle\psi_{\mu}|\hat{H}^{\prime}_{0}|\psi_{\nu}\rangle\cos(2\theta_{\mu\nu}t){\rm
sinc}^{2}(\theta_{\mu\nu}t)\leq(\Delta\hat{H}^{\prime 2}_{0})_{\rm
mc}\left(\frac{\omega}{\pi\Gamma}\right)\frac{(e^{-2\Gamma t}-1)^{2}}{4(\Gamma
t)^{2}},$ (S57)

which we neglect in the limit $\Gamma\gg\omega$. All other terms in (S43)
contain matrix elements with two and three equal indexes and respectively four
different indexes and their ensemble average is zero.

Combining all averages in the second term (S43) we get

$\sum_{\mu\nu}\sum_{\mu^{\prime}\nu^{\prime}}a^{*}_{\mu}a_{\nu}a_{\mu^{\prime}}a^{*}_{\nu^{\prime}}\langle\psi_{\mu}|\hat{H}^{\prime}_{0}|\psi_{\nu}\rangle\langle\psi_{\nu^{\prime}}|\hat{H}^{\prime}_{0}|\psi_{\mu^{\prime}}\rangle
e^{i\theta_{\mu\nu}t}e^{-i\theta_{\mu^{\prime}\nu^{\prime}}t}{\rm
sinc}(\theta_{\mu\nu}t){\rm
sinc}(\theta_{\mu^{\prime}\nu^{\prime}}t)\approx(\hat{H}^{\prime}_{0})^{2}_{\rm
mc}.$ (S58)

Finally, using Eq. (S19), (S42), and (S58) we obtain

$F_{Q}(\lambda)=4t^{2}\left\\{\frac{\omega}{\pi\Gamma}(\hat{H}^{\prime
2}_{0})_{\rm mc}+\frac{(\Delta\hat{H}_{0}^{\prime 2})_{\rm mc}}{2(\Gamma
t)^{2}}(e^{-2\Gamma t}-1+2\Gamma t)\right\\}.$ (S59)

In Fig. (S1) we show the relative error between the exact result derived from
Eq. (S1) and the analytical formula Eq. (S59) for various $g$ and $N$.

We see that increasing time the scaling of the QFI passes from linear to
quadratic. In fact we can obtain the long time scaling of QFI using that
$\lim_{t\rightarrow\infty}{\rm sinc}(x)=0$. Using (S19) we obtain

$\displaystyle\lim_{t\rightarrow\infty}\frac{F_{Q}(\lambda)}{t^{2}}$
$\displaystyle=$ $\displaystyle
4\\{\sum_{\mu}|a_{\mu}|^{2}|\langle\psi_{\mu}|\hat{H}^{\prime}_{0}|\psi_{\mu}\rangle|^{2}-\sum_{\mu\nu}|a_{\mu}|^{2}|a_{\nu}|^{2}\langle\psi_{\mu}|\hat{H}^{\prime}_{0}|\psi_{\mu}\rangle\langle\psi_{\nu}|\hat{H}^{\prime}_{0}|\psi_{\nu}\rangle\\}$
(S60)
$\displaystyle=4\\{\sum_{\mu}|a_{\mu}|^{2}|\langle\psi_{\mu}|\hat{H}^{\prime}_{0}|\psi_{\mu}\rangle|^{2}-\sum_{\mu}|a_{\mu}|^{4}|\langle\psi_{\mu}|\hat{H}^{\prime}_{0}|\psi_{\mu}\rangle|^{2}-\sum_{\mu\nu\atop\mu\neq\nu}|a_{\mu}|^{2}|a_{\nu}|^{2}\langle\psi_{\mu}|\hat{H}^{\prime}_{0}|\psi_{\mu}\rangle\langle\psi_{\nu}|\hat{H}^{\prime}_{0}|\psi_{\nu}\rangle\\}$
$\displaystyle=4\\{\frac{\omega}{\pi\Gamma}(\hat{H}^{\prime 2}_{0})_{\rm
mc}+(\hat{H}^{\prime}_{0})_{\rm mc}^{2}-(\hat{H}^{\prime}_{0})^{2}_{\rm
mc}\sum_{\mu}\langle|a_{\mu}|^{4}\rangle_{V}-(\hat{H}^{\prime}_{0})_{\rm
mc}^{2}(1-\sum_{\mu}\langle|a_{\mu}|^{4}\rangle_{V})\\}.$

Therefore, neglecting terms of order of $(\omega/\Gamma)^{2}$ we obtain the
long time limit of QFI as $F_{Q}(\lambda)\approx(\hat{H}^{\prime 2}_{0})_{\rm
mc}(4\omega/\pi\Gamma)t^{2}$. Similarly, we may consider the short-time limit
of the QFI by using that $\lim_{t\rightarrow 0}{\rm sinc}(x)=1$. Then we have

$\displaystyle\lim_{t\rightarrow 0}\frac{F_{Q}(\lambda)}{t^{2}}$
$\displaystyle=$ $\displaystyle
4\\{\sum_{\mu}|a_{\mu}|^{2}|\langle\psi_{\mu}|\hat{H}^{\prime}_{0}|\psi_{\mu}\rangle|^{2}+\sum_{\mu\nu\atop\mu\neq\nu}|a_{\mu}|^{2}|\langle\psi_{\mu}|\hat{H}^{\prime}_{0}|\psi_{\nu}\rangle|^{2}-\sum_{\mu}|a_{\mu}|^{4}|\langle\psi_{\mu}|\hat{H}^{\prime}_{0}|\psi_{\mu}\rangle|^{2}$
(S61)
$\displaystyle-\sum_{\mu\nu\atop\mu\neq\nu}|a_{\mu}|^{2}|a_{\nu}|^{2}\langle\psi_{\mu}|\hat{H}^{\prime}_{0}|\psi_{\mu}\rangle\langle\psi_{\nu}|\hat{H}^{\prime}_{0}|\psi_{\nu}\rangle\\}=4\\{\frac{\omega}{\pi\Gamma}(\hat{H}^{\prime
2}_{0})_{\rm mc}+\frac{2\omega\Gamma}{\pi}(\Delta\hat{H}_{0}^{\prime 2})_{\rm
mc}\sum_{\mu\nu\atop\mu\neq\nu}\frac{|a_{\mu}|^{2}}{(E_{\mu}-E_{\nu})^{2}+4\Gamma^{2}}\\}.$

Furthermore, we replace the sum with integration such that we have,

$\sum_{\nu}\frac{1}{(E_{\mu}-E_{\nu})^{2}+4\Gamma^{2}}\rightarrow\frac{1}{\omega}\int_{-\infty}^{\infty}\frac{dE_{\nu}}{(E_{\mu}-E_{\nu})^{2}+4\Gamma^{2}}=\frac{\pi}{2\omega\Gamma}.$
(S62)

Therefore, neglecting the terms of order of $\omega/\Gamma$, the short-time
scaling of the QFI is $F_{Q}(\lambda)\approx 4t^{2}(\Delta\hat{H}_{0}^{\prime
2})_{\rm mc}$.

##  IV Corrections due to self-averaging decoupling

Let us assume that the initial state is an eigenstate of non-interaction
Hamiltonian $\hat{H}_{0}$, namely
$|\Psi_{0}\rangle=|\varphi_{\alpha_{0}}\rangle$. Therefore, we have

$\sum_{\mu}\langle|a_{\mu}|^{2}|\langle\psi_{\mu}|\hat{H}^{\prime}_{0}|\psi_{\mu}\rangle|^{2}\rangle_{V}=\sum_{\mu}\sum_{\alpha\beta}\langle
c_{\mu}(\alpha_{0})c_{\mu}(\alpha_{0})c_{\mu}(\alpha)c_{\mu}(\alpha)c_{\mu}(\beta)c_{\mu}(\beta)\rangle_{V}(\hat{H}^{\prime}_{0})_{\alpha\alpha}(\hat{H}^{\prime}_{0})_{\beta\beta}.$
(S63)

In order to evaluate the average we use the generating function (S9). There
are in total $15$ terms. Consider the term

$\langle c_{\mu}(\overbracket{\alpha_{0})c_{\mu}(\alpha_{0}})c_{\mu}(\hbox
to0.0pt{$\displaystyle\overbracket{\phantom{\alpha)c_{\mu}(\alpha)c_{\mu}(\alpha}}$\hss}\alpha)c_{\mu}(\underbracket{\alpha)c_{\mu}(\beta)c_{\mu}(\beta})\rangle_{V}=\Lambda^{2}(\mu,\alpha)\Lambda(\mu,\alpha_{0})\delta_{\alpha\beta}.$
(S64)

Hence we get

$\displaystyle\sum_{\mu}\sum_{\alpha\beta}\langle
c_{\mu}(\overbracket{\alpha_{0})c_{\mu}(\alpha_{0}})c_{\mu}(\hbox
to0.0pt{$\displaystyle\overbracket{\phantom{\alpha)c_{\mu}(\alpha)c_{\mu}(\alpha}}$\hss}\alpha)c_{\mu}(\underbracket{\alpha)c_{\mu}(\beta)c_{\mu}(\beta})\rangle_{V}(\hat{H}^{\prime}_{0})_{\alpha\alpha}(\hat{H}^{\prime}_{0})_{\beta\beta}=\sum_{\mu}(\sum_{\alpha}\Lambda^{2}(\mu,\alpha)(\hat{H}^{\prime}_{0})^{2}_{\alpha\alpha})\Lambda(\mu,\alpha_{0})$
$\displaystyle\approx[(\hat{H}^{\prime}_{0})^{2}_{\alpha\alpha}]_{\mu}\sum_{\mu}\sum_{\alpha}\Lambda^{2}(\mu,\alpha)\Lambda(\mu,\alpha_{0})=\frac{\omega}{2\pi\Gamma}[(\hat{H}^{\prime}_{0})^{2}_{\alpha\alpha}]_{\mu}.$
(S65)

Similarly we have

$\sum_{\mu}\sum_{\alpha\beta}\langle
c_{\mu}(\overbracket{\alpha_{0})c_{\mu}(\alpha_{0}})c_{\mu}(\overbracket{\alpha)c_{\mu}(\underbracket{\alpha)c_{\mu}(\beta})c_{\mu}(\beta})\rangle_{V}(\hat{H}^{\prime}_{0})_{\alpha\alpha}(\hat{H}^{\prime}_{0})_{\beta\beta}\approx\frac{\omega}{2\pi\Gamma}[(\hat{H}^{\prime}_{0})^{2}_{\alpha\alpha}]_{\mu}.$
(S66)

Other six-point correlation term is

$\displaystyle\sum_{\mu}\sum_{\alpha\beta}\langle
c_{\mu}(\overbracket{\alpha_{0})c_{\mu}(\alpha_{0}})c_{\mu}(\overbracket{\alpha)c_{\mu}(\alpha})c_{\mu}(\overbracket{\beta)c_{\mu}(\beta})\rangle_{V}(\hat{H}^{\prime}_{0})_{\alpha\alpha}(\hat{H}^{\prime}_{0})_{\beta\beta}=\sum_{\mu}\sum_{\alpha\beta}\Lambda(\mu,\alpha)\Lambda(\mu,\beta)\Lambda(\mu,\alpha_{0})(\hat{H}^{\prime}_{0})_{\alpha\alpha}(\hat{H}^{\prime}_{0})_{\beta\beta}$
$\displaystyle\approx[(\hat{H}^{\prime}_{0})_{\alpha\alpha}]^{2}_{\mu}\sum_{\mu}\sum_{\alpha\beta}\Lambda(\mu,\alpha)\Lambda(\mu,\beta)\Lambda(\mu,\alpha_{0})=[(\hat{H}^{\prime}_{0})_{\alpha\alpha}]^{2}_{\mu}.$
(S67)

Combining Eqs. (S65), (S66), and (S67) we obtain Eq. (S27). Now, let’s
consider the corrections. We have

$\langle
c_{\mu}(\underbracket{\alpha_{0})c_{\mu}(\overbracket{\alpha_{0})c_{\mu}(\alpha})c_{\mu}(\overbracket{\alpha)c_{\mu}(\beta})c_{\mu}(\beta})\rangle_{V}=\Lambda(\mu,\beta)\delta_{\alpha_{0},\beta}\Lambda(\mu,\alpha_{0})\delta_{\alpha,\alpha_{0}}\Lambda(\mu,\alpha)\delta_{\alpha\beta}.$
(S68)

Therefore, we obtain

$\sum_{\mu}\sum_{\alpha\beta}\langle
c_{\mu}(\underbracket{\alpha_{0})c_{\mu}(\overbracket{\alpha_{0})c_{\mu}(\alpha})c_{\mu}(\overbracket{\alpha)c_{\mu}(\beta})c_{\mu}(\beta})\rangle_{V}(\hat{H}^{\prime}_{0})_{\alpha\alpha}(\hat{H}^{\prime}_{0})_{\beta\beta}=(\hat{H}^{\prime}_{0})^{2}_{\alpha_{0}\alpha_{0}}\sum_{\mu}\Lambda^{3}(\mu,\alpha_{0}).$
(S69)

Such a term gives correction of order of $(\omega/\Gamma)^{2}$. In fact there
are in total $8$ terms which give correction of such order.

Consider now the term

$\langle
c_{\mu}(\overbracket{\alpha_{0})c_{\mu}(\underbracket{\alpha_{0})c_{\mu}(\alpha})c_{\mu}(\alpha})c_{\mu}(\overbracket{\beta)c_{\mu}(\beta})\rangle_{V}=\Lambda^{2}(\mu,\alpha)\delta_{\alpha,\alpha_{0}}\Lambda(\mu,\beta).$
(S70)

Hence we get

$\displaystyle\sum_{\mu}\sum_{\alpha\beta}\langle
c_{\mu}(\overbracket{\alpha_{0})c_{\mu}(\underbracket{\alpha_{0})c_{\mu}(\alpha})c_{\mu}(\alpha})c_{\mu}(\overbracket{\beta)c_{\mu}(\beta})\rangle_{V}(\hat{H}^{\prime}_{0})_{\alpha\alpha}(\hat{H}^{\prime}_{0})_{\beta\beta}=\sum_{\mu}\sum_{\beta}\Lambda(\mu,\beta)(\hat{H}^{\prime}_{0})_{\beta\beta}\Lambda^{2}(\mu,\alpha_{0})(\hat{H}^{\prime}_{0})_{\alpha_{0}\alpha_{0}}$
$\displaystyle\approx[(\hat{H}^{\prime}_{0})_{\alpha\alpha}]_{\mu}(\hat{H}^{\prime}_{0})_{\alpha_{0}\alpha_{0}}\sum_{\mu}\sum_{\beta}\Lambda(\mu,\beta)\Lambda^{2}(\mu,\alpha_{0})=\frac{\omega}{2\pi\Gamma}[(\hat{H}^{\prime}_{0})_{\alpha\alpha}]_{\mu}(\hat{H}^{\prime}_{0})_{\alpha_{0}\alpha_{0}}.$
(S71)

There are in total $4$ terms with the same contribution. Note that for spin
chain that we consider the microcanonical average of the spin observable is
zero and thus these terms can be neglected.

##  V Spin-Spin Correlations

Figure S2: Exact time evolution of the ratio $F_{Q}^{(1)}/F_{Q}^{(\rm SQL)}$
for a spin chain consisting of two system spins. The initial state is
$|\Psi_{0}\rangle=\left|\uparrow\uparrow\right\rangle_{S}\left|\uparrow\downarrow\uparrow...\right\rangle_{B}$
for $N=15$. (Case 1) The system spins interact with different bath spins
($r_{1}=5$ and $r_{2}=8$) with QFI $F_{Q}^{(\rm SQL)}$. (Case 2) Both systems
spins interact with the same bath spin ($r_{1}=5$) with corresponding QFI
$F_{Q}^{(1)}$. Time evolution of the correlation
$\langle\sigma^{z}_{1}\sigma^{z}_{2}\rangle$ for cases 1 (inset (a)) and 2
(inset (b)).

Here we extend the discussion by consider a spin system Hamiltonian consisting
of two spins

$\hat{H}_{\rm S}=B(\sigma^{z}_{1}+\sigma^{z}_{2})$ (S72)

and spin-bath interaction

$\displaystyle\hat{H}_{\rm SB}=J_{z}^{(\rm
SB)}\sigma^{z}_{1}\sigma^{z}_{r_{1}}+J_{x}^{(\rm
SB)}(\sigma^{+}_{1}\sigma^{-}_{r_{1}}+\sigma^{-}_{1}\sigma^{+}_{r_{1}})+J_{z}^{(\rm
SB)}\sigma^{z}_{2}\sigma^{z}_{r_{2}}+J_{x}^{(\rm
SB)}(\sigma^{+}_{2}\sigma^{-}_{r_{2}}+\sigma^{-}_{2}\sigma^{+}_{r_{2}}),$
(S73)

where $r_{k}$ ($k=1,2$) denotes the position of the bath spins. For two spins
coupled to different bath spins $r_{1}\neq r_{2}$ we find that no correlation
is created between the system spins in the sense that
$(\sigma^{z}_{1}\sigma^{z}_{2})_{\rm mc}=(\sigma^{z}_{1})_{\rm
mc}(\sigma^{z}_{2})_{\rm mc}\approx 0$, see Fig. (S2) (inset (a)). In that
case, the QFI is twice the QFI for a single system spin, $F^{(\rm
SQL)}_{Q}(B)=2F_{Q}(B)$, which corresponds to the standard quantum limit
(SQL). Let us now consider $r_{1}=r_{2}$, namely, the two system spins are
coupled to the single bath spin. Then the spin-bath interaction creates a
correlation between the two system spins in a sense that
$(\sigma^{z}_{1}\sigma^{z}_{2})_{\rm mc}\neq 0$, see Fig. (S2) (inset (b)).
Using Eq. (3) with $\hat{H}^{\prime}_{0}=\sigma^{z}_{1}+\sigma^{z}_{2}$ we
obtain

$\displaystyle F^{(1)}_{Q}(B)=F^{(\rm
SQL)}_{Q}(B)+4t^{2}(\sigma^{z}_{1}\sigma^{z}_{2})_{\rm mc}\left\\{\frac{2}{\pi
D(E_{0})\Gamma}+\frac{1}{(\Gamma t)^{2}}(e^{-2\Gamma t}-1+2\Gamma
t)\right\\}.$ (S74)

As long as $(\sigma^{z}_{1}\sigma^{z}_{2})_{\rm mc}>0$ we have
$F^{(1)}_{Q}(B)>F^{(\rm SQL)}_{Q}(B)$ and thus one can overcome the SQL. We
plot in Fig. S2 the exact time evolution of the ratio $F^{(1)}_{Q}/F^{(\rm
SQL)}_{Q}$. We see that the positive quantum correlation between the system
spins leads to enhancement of the QFI compare with the $F^{(\rm SQL)}_{Q}(B)$.

##  VI Defining a local observable in RMT

In the main we have analysed the QFI for a _local_ observable of the system
interacting with a bath. In this section we will see that such an approach can
be formalised within our RMT approach, and indeed gives way to a crucial
condition - observable sparsity - of the application of RMT. We will see that
an additional condition is required on the system and bath parts of the total
system in order for RMT to apply to such local observables, namely, that the
system energy is much smaller than that of the bath.

We begin by separating the system into system and bath components via
$\hat{H}_{0}=\hat{H}_{S}(\lambda)\otimes\mathbf{1}_{B}+\mathbf{1}_{S}\otimes\hat{H}_{B}$,
with $\mathbf{1}_{S(B)}$ the identity on the system (bath) Hilbert space.
Crucially, here the system part of the Hamiltonian is assumed to depend on
some parameter $\lambda$. The eigenstates of $\hat{H}_{0}$ are then

$\displaystyle|\phi_{\alpha}\rangle=|s(\alpha)\rangle_{S}\otimes|\phi_{\alpha_{B}(\alpha)}^{(B)}\rangle_{B},$
(S75)

with energies

$\displaystyle E_{\alpha}$ $\displaystyle={}_{S}\langle
s(\alpha)|{}_{B}\langle\phi_{\alpha_{B}(\alpha)}^{(B)}|\hat{H}_{S}(\lambda)+\hat{H}_{B}|s(\alpha)\rangle_{S}\otimes|\phi_{\alpha_{B}(\alpha)}^{(B)}\rangle_{B}$
$\displaystyle=\epsilon_{s(\alpha)}(\lambda)+E^{(B)}_{\alpha_{B}(\alpha)},$
(S76)

where we have denoted eigenenergies of the system and bath Hamiltonians by
$\epsilon_{s(\alpha)}(\lambda)$ and $E^{(B)}_{\alpha_{B}(\alpha)}$
respectively.

Relevant observables in our approach act on the system Hilbert space as
$\hat{O}=\hat{O}_{S}\otimes\mathbf{1}_{B}$, which have matrix elements

$\displaystyle O_{\alpha\beta}$ $\displaystyle={}_{S}\langle
s(\alpha)|{}_{B}\langle\phi_{\alpha_{B}(\alpha)}^{(B)}|\hat{O}|s(\beta)\rangle_{S}\otimes|\phi_{\alpha_{B}(\beta)}^{(B)}\rangle_{B}$
$\displaystyle=(O_{S})_{s(\alpha)s(\beta)}\delta_{\alpha_{B}(\alpha)\alpha_{B}(\beta)},$
(S77)

where $(O_{S})_{s(\alpha)s(\beta)}={}_{S}\langle
s(\alpha)|\hat{O}_{S}|s(\beta)\rangle_{S}$. We see here that the local
observable $\hat{O}$ is guaranteed to be sparse if the dimension of the system
Hilbert space $d_{S}$ is much lower than that of the bath, $d_{B}$, as there
are a maximum of $d_{S}(d_{S}-1)$ independent off diagonal matrix elements of
$O_{\alpha\beta}|_{\alpha\neq\beta}$, corresponding to the possible system
state transitions, plus $d_{S}$ possible diagonal matrix elements.

These possible transitions that the local operator $\hat{O}$ may induce must
obey

$\displaystyle
E_{\alpha}-E_{\beta}=\epsilon_{s(\alpha)}(\lambda)-\epsilon_{s(\beta)}(\lambda):=\Delta_{\alpha\beta}^{(S)}(\lambda),$
(S78)

and more generally, we have,

$\displaystyle
E_{\alpha}-E_{\beta}=\Delta_{\alpha\beta}^{(S)}(\lambda)+E^{(B)}_{\alpha_{B}(\alpha)}-E^{(B)}_{\alpha_{B}(\beta)}.$
(S79)

For the random matrix model, we have $E_{\alpha}=\alpha\omega$, so we require
for the RMT to hold that $E_{\alpha}-E_{\beta}\approx(\alpha-\beta)\omega$ can
be approximated by an equidistant spacing of energies that does not depend on
$\lambda$. This is understood to hold if $\hat{H}_{B}$ is itself a non-
integrable Hamiltonian, and if $\Delta_{\alpha\beta}^{(S)}(\lambda)\ll
E^{(B)}_{\alpha_{B}(\alpha)}-E^{(B)}_{\alpha_{B}(\beta)}$, indicating that the
possible transitions induced by the local observable are negligible in energy
in comparison to the bath energy for the state $|\phi_{\alpha}\rangle$.

## References

  * (1) C. Nation and D. Porras, New J. Phys. 20, 103003 (2018).