# Critical quantum thermometry and its feasibility in spin systems

Enes Aybar  ICFO - Institut de Ciencies Fotoniques, The Barcelona Institute of
Science and Technology, 08860 Castelldefels, Barcelona, Spain  Artur Niezgoda
ICFO - Institut de Ciencies Fotoniques, The Barcelona Institute of Science and
Technology, 08860 Castelldefels, Barcelona, Spain  Faculty of Physics,
University of Warsaw, ul. Pasteura 5, PL-02-093 Warsaw, Poland  Safoura S.
Mirkhalaf  Department of Physics, University of Tehran, P.O. Box 14395-547,
Tehran, Iran  School of Nano Science, Institute for Research in Fundamental
Sciences (IPM), P.O. Box 19395-5531, Tehran, Iran  Morgan W. Mitchell  ICFO -
Institut de Ciencies Fotoniques, The Barcelona Institute of Science and
Technology, 08860 Castelldefels, Barcelona, Spain  ICREA - Institució Catalana
de Recerca i Estudis Avançats, 08010 Barcelona, Spain  Daniel Benedicto Orenes
ICFO - Institut de Ciencies Fotoniques, The Barcelona Institute of Science and
Technology, 08860 Castelldefels, Barcelona, Spain  Emilia Witkowska  Institute
of Physics PAS, Aleja Lotnikow 32/46, 02-668 Warszawa, Poland

###### Abstract

In this work, we study temperature sensing with finite-sized strongly
correlated systems exhibiting quantum phase transitions. We use the quantum
Fisher information (QFI) approach to quantify the sensitivity in the
temperature estimation, and apply a finite-size scaling framework to link this
sensitivity to critical exponents of the system around critical points. We
numerically calculate the QFI around the critical points for two
experimentally-realizable systems: the spin-1 Bose-Einstein condensate and the
spin-chain Heisenberg XX model in the presence of an external magnetic field.
Our results confirm finite-size scaling properties of the QFI. Furthermore, we
discuss experimentally-accessible observables that (nearly) saturate the QFI
at the critical points for these two systems.

##  1 Introduction

Understanding and quantifying uncertainty in the measurement of a classical
parameter using quantum resources is a central goal of modern estimation
theory [1] with far-reaching implications both for fundamental studies and
quantum technology applications [2]. In the well-studied scenario of single-
parameter estimation, the geometry of quantum states [3], connects the
sensitivity of parameter estimation to the fidelity between neighboring
states. Concretely, if $\hat{\rho}(\lambda)$ is a parametrized state and
$\mathcal{F}_{\lambda,\lambda^{\prime}}$ is the fidelity between
$\hat{\rho}(\lambda)$ and $\hat{\rho}(\lambda^{\prime})$, the quantum Fisher
information (QFI) can be related to the second derivative
$\partial^{2}_{\lambda^{\prime}}\mathcal{F}_{\lambda,\lambda^{\prime}}|_{\lambda^{\prime}=\lambda}$
[3, 4, 5]; when this quantity is large, $\lambda$ can in principle be
estimated with high precision.

In order to enhance this sensitivity different resources have been studied,
including: entanglement [6], inter-particle interaction [7] and quantum phase
transitions (QPTs) [8]. Regarding the latter, prior work has studied
parametrized states, e.g., ground states of a parametrized Hamiltonian
$\hat{H}(\lambda)$. The rapid state changes at the QPT imply larger QFI for
$\lambda$, i.e., for the QPT’s control parameter [8, 9, 10, 11, 12, 13, 14,
15, 16, 17, 18]. A relation between sensitivity to $\lambda$ and universal
scaling behaviour near critical points has been established [19, 20, 21, 22].

Thermometry is another paradigmatic example of a quantum estimation problem;
temperature $T$ is not an observable in the quantum mechanical sense, but
rather a parameter that can be estimated from measurements of proper
observables. The study of fundamental limits to sensitivity in temperature
estimation has attracted attention in different areas: from thermalization
studies of open quantum systems [23] and characterization of trapped ions for
quantum computation [24] to temperature sensing by micro-organisms [25].

Here we study thermometry in quantum critical systems using the tools of
quantum parameter estimation. The parametrized state $\hat{\rho}(T,\lambda)$
of interest is the Gibbs state of a parametrized Hamiltonian
$\hat{H}(\lambda)$. We focus on systems that exhibit continuous QPTs at zero
temperature and in the thermodynamic limit. We explore their behaviour for
finite numbers of particles $N$ and finite temperature $T$. In addition, we
find relations between the sensitivity of temperature estimation and universal
scaling behaviours around the critical points. At zero temperature and in the
thermodynamic limit, a QPT shows power-law divergences of physical quantities,
described by critical exponents that quantify the speed of the divergence
[26]. When the system size is finite, the behaviour of physical quantities are
regular at critical points and can be described by analytic functions that are
subject to a universal finite-size scaling [26, 27, 28]. At small but finite
temperatures, Gibbs states can inherit signatures of quantum critical behavior
that, as in the zero-temperature case, enhance sensitivity in parameter
estimation around the QPT [29, 21, 30].

We quantify the sensitivity in temperature estimation using the quantum Fisher
information and relate it with the critical exponents at the QPT through a
universal scaling function that is directly proportional to the inverse of the
lowest energy gap squared. This relationship causes the abrupt growth of the
QFI in the critical region when temperature is lower than the energy gap. In
addition, we analyze the signal-to-noise ratio (SNR) showing its relation with
critical exponents and the associated scaling function. We find out that the
SNR does not exhibit the direct dependence on energy gap, i.e., diminishing
the energy gap in the critical region does not lead to a larger SNR.

Finally, we illustrate the resulting finite-size scaling properties of the
sensitivity using two interesting and experimentally-relevant examples: (i)
the ferromagnetic spin-1 Bose-Einstein condensate (BEC) and (ii) the
ferromagnetic XX spin-1/2 chain, both subject to the presence of an external
magnetic field. The two systems considered by us can be realized with current
experimental techniques using ultra-cold atoms [31, 32]. We discuss possible
experimental implementations, calculating the sensitivity of temperature
measurements in these systems for different experimentally-accessible
observables identifying the ones that would saturate the QFI. Our results
demonstrate the validity of the finite-size scaling analysis and show how
quantum thermometry in strongly correlated systems exhibiting continuous QPTs
can benefit from the criticality when comparing to their far-from-criticality
behavior. In this sense we refer to criticality-enhanced quantum thermometry.
This enhancement in the sensitivity can lead to the realization of useful
thermometers based on spin-measurements, i.e., not relying on external
ancillary probes [33, 34], further manipulation of the atomic sample [35] or
thermal fraction of the ultra-cold quantum gases [36].

##  2 Local Quantum Thermometry and Criticality

In this section we present the principles of temperature estimation and its
sensitivity bounds quantified by the QFI. We introduce the concept of a
finite-size scaling framework for systems exhibiting continuous QPTs. The
scaling approach is then applied to study the behavior of the QFI in the
vicinity of QPTs.

###  2.1 QFI approach for quantum thermometry

Thermometry is the classical parameter estimation problem conveniently
resolved using the parameter estimation protocols [37, 38]. More precisely,
the temperature of a system consisting of $N$ particles is inferred from
results of $m$ measurements $\\{\alpha_{i}\\}$ of an observable $\hat{A}$ of
the system under consideration through the estimator function $T_{\rm
est}(\\{\alpha_{i}\\})$ which we assume to be unbiased, i.e., $\langle T_{\rm
est}\rangle=T$ ($i$ labels independent measurements). We define the resulting
fluctuations of temperature in terms of the mean squared error of the
corresponding estimator $\delta^{2}T\equiv\langle(T_{\rm est}-T)^{2}\rangle$
which is subject to the Cramér-Rao lower bound [39]:

$\delta^{2}T\geq\frac{1}{mF_{Q}(T)},$ (1)

where $F_{Q}(T)$ is the QFI (we set $k_{\rm B}=1$). In the scenario of a
parameter dependent Hamiltonian $\hat{H}(\lambda)$ the quantum states are
given by a density matrix $\hat{\rho}(T,\lambda)$, and the QFI, with respect
to the temperature $T$ for a given value of the control parameter $\lambda$,
is defined as the maximization of the classical Fisher information (CFI)
$F_{c}(T,\hat{A})$ over all possible measurements [3]

$F_{Q}(T)=\underset{\hat{A}}{\rm max}\,F_{c}(T,\hat{A}),$ (2)

with $\hat{A}$ representing any possible observable of the state, and

$F_{c}(T,\hat{A})=\sum_{\alpha}\frac{1}{p(\alpha|T)}\left(\frac{\partial
p(\alpha|T)}{\partial(T)}\right)^{2},$ (3)

where $p(\alpha|T)=\langle\alpha|\hat{\rho}(T,\lambda)|\alpha\rangle$ (and
$\hat{A}|\alpha\rangle=\alpha|\alpha\rangle$) is a conditional probability
distribution of the measurement outcomes for $\hat{A}$, given a fixed value of
temperature $T$ [37, 38].

It has been proved that the optimal measurement (i.e., the measurement which
gives $F_{c}(T,\hat{A})=F_{Q}(T)$) is the energy of the system [37, 17], i.e.,
$\hat{A}=\hat{H}(\lambda)$ with eigenvalues
$\hat{H}(\lambda)|\psi_{n}\rangle=E_{n}(\lambda)|\psi_{n}\rangle$. Therefore
it is convenient to define the energy gap between $n-$th excited and ground
state as follows: $\Delta_{n}(\lambda)=E_{n}(\lambda)-E_{0}(\lambda)$.
Considering a state well described within the canonical Gibbs ensemble

$\hat{\rho}(T,\lambda)=\sum_{n=0}^{\mathcal{N}}\frac{e^{-\Delta_{n}(\lambda)/T}}{Z}|\psi_{n}\rangle\langle\psi_{n}|,$
(4)

where $\mathcal{N}$ is the total number of quantum states, $Z$ is the
partition function, and applying Eq. (3), one obtains

$F_{Q}(T,\lambda)=\frac{\Delta^{2}\hat{H}(T,\lambda)}{T^{4}},$ (5)

where $\Delta^{2}$ denotes the standard definition for the variance of an
operator,
$\Delta^{2}\hat{A}\equiv\langle\hat{A}^{2}\rangle-\langle\hat{A}\rangle^{2}$.
The relation (5) can also be obtained by using the fidelity susceptibility
approach [38] as conveniently used in the estimation of the parameter of the
Hamiltonian [29]. The value of the QFI depends on the structure of energy
levels and the temperature of a given quantum system. Note that in the
remainder of the work, unless required, we drop the explicit dependence of
$F_{Q}$ and $F_{c}$ with the observable $\hat{A}$ and the parameters $\lambda$
and $T$.

###  2.2 Finite-size scaling at continuous quantum phase transitions

We consider a $d$-dimensional quantum many-body system of size $L^{d}$
consisting of $N$ particles and described by the Hamiltonian
$\hat{H}(\lambda)=\hat{H}_{0}+\lambda\hat{V}$, where
$[\hat{H}_{0},\hat{V}]\neq 0$ and $\lambda$ is a dimensionless control
parameter. We assume the QPT takes place in the thermodynamic limit
($N\to\infty$, $L^{d}\to\infty$, $N/L^{d}={\rm const.}$) at zero temperature
and is driven by $\lambda$ with the critical point located at
$\lambda=\lambda_{c}$. We therefore define $\epsilon\equiv\lambda-\lambda_{c}$
as the distance from the critical point. The QPT is characterized by a
diverging power law behavior of a physical quantity $A\sim\epsilon^{a}$ given
by a critical exponent $a$ quantifying how rapidly $A$ changes at
$\lambda_{c}$, for example, if $A$ is the energy gap between the ground and
first excited energy states $\Delta_{g}\equiv\Delta_{1}(\epsilon)$ then
$\Delta_{g}\sim\epsilon^{-z\nu}$ [26], where $z$ is the dynamic critical
exponent and $\nu$ is the critical exponent associated with the divergence of
the correlation length.

The singular behavior of $A$ is observed in the thermodynamic limit. If the
size of the system ($L$ and $N$) is finite then the change of $A$ is an
analytic function of $\epsilon$. Moreover, around the transition point
physical quantities are subject to finite-size scaling which depends on the
general properties of the transition [27, 40, 28]. This was understood by the
finite-size scaling approach, which was also considered through the
renormalization group [27, 41, 42, 43, 28, 44]. In the finite-size scaling
framework, an existence of a regular function $f_{a}$ is postulated such that

$A\sim L^{-a/\nu}f_{a}(\epsilon L^{1/\nu})$ (6)

with the constraint $f_{a}(0)\neq 0$ [27, 28, 44, 45]. Note, the finite-size
behavior of $A$ is characterized by the scaling function and scaling laws,
with critical exponents determined by the universality class of the QPT.
Therefore, the behavior of $A$ in the critical region depends on global
properties, such as $L,\,N,\,d$ and the symmetry or nature of the interactions
[26, 44]. The finite-size scaling framework can also be considered in terms of
$N$ instead of $L$ as $L\propto N^{1/d}$. For example, if $A=\Delta_{g}$ and
$a=z\nu$, the finite-size scaling for the energy gap [28] then reads

$\Delta_{g}\sim N^{-z/d}f_{\Delta}(\epsilon N^{1/(\nu d)}).$ (7)

A non-zero value of temperature $T$ introduces an additional relevant
parameter, which, within the finite-size scaling framework, is taken into
account by adding a further dependence of the scaling functions on
$T/\Delta_{g}$ [26, 44].

####  2.2.1 Finite-size scaling of the QFI

We now apply the finite-size scaling approach to the thermometry problem at
the QPT in order to relate the behavior of the QFI with critical exponents and
relevant scaling functions. According to the discussion above we can expect
that the QFI is also subject to the finite-size scaling relation (6). However,
the value of the scaling exponent $a$ is not given a-priori. To analyze the
scaling of the QFI and obtain $a$, we first consider the dimensionless
quantity $F_{Q}T^{2}=\Delta^{2}\hat{H}/T^{2}$, which can be expressed as a
function of independent, dimensionless combinations of physical parameters of
the system, specifically, the product $\epsilon N^{1/(\nu d)}$ as for (7) and
the ratios $T/\Delta_{n}$. When the temperature is low, i.e,
$T\lesssim\Delta_{g}$, the main contribution to the quantity $F_{Q}T^{2}$
comes from the terms containing $T/\Delta_{g}$ while the contribution of
$T/\Delta_{n>1}$ is assumed to be negligible 111In fact, according to [28] not
only the lowest energy gap $\Delta_{g}$ but also other lower lying energy gaps
$\Delta_{n\neq 1}$ follows the scaling (7).. This approximation leads to the
following scaling form of the QFI near the critical point

$F_{Q}=\Delta_{g}^{-2}\tilde{g}\left(\epsilon N^{1/(\nu
d)},\frac{T}{\Delta_{g}}\right),$ (8)

where the low temperature approximate of the scaling function $\tilde{g}$ can
be expressed as $\tilde{g}\left(\epsilon N^{1/(\nu
d)},\frac{T}{\Delta_{g}}\right)\equiv\left(\Delta_{g}/T\right)^{2}\left(\Delta^{2}\hat{H}/T^{2}\right)$
(see Appendix B for details). Note, Eq. (8) shows the direct relation between
the QFI and the inverse of the energy gap squared. As a consequence, the QFI
grows at the critical point at a rate determined by the speed at which the
energy gap closes. Under this approximation, $\tilde{g}$ can be calculated
analytically as a function of the energy gap $\Delta_{g}$ (see Appendix B),
and it is

$\tilde{g}\left(\frac{T}{\Delta_{g}}\right)=\left(\frac{\Delta_{g}}{T}\right)^{4}\frac{1}{4\cosh^{2}{\left[\frac{\Delta_{g}}{2T}\right]}}.$
(9)

The scaling function can be seen as a single-parameter function of
$T/\Delta_{g}$, which tends to zero when temperature is much larger and
smaller than the energy gap. The condition in which $T\gg\Delta_{g}$ resembles
a classical situation in which the temperature is much larger than many
internal energy levels in addition to the lower-lying ones, thus frustrating
all the information enhancement provided by the closing of the energy gap,
making this limit uninteresting for our purposes. An analysis shows that the
maximal value of the QFI from (9) near the critical point is approximately
$F_{Q}^{\rm max}\approx 4.53\Delta_{g}^{-2}$ for $T/\Delta_{g}\approx 0.24$.
These values estimate well the position and maximum of the QFI for the
strongly correlated system we are considering in the next sections. Taking the
scaling of the energy gap (7) in (8), we obtain that the QFI around
criticality scales with the total number of particles as $N^{2z/d}$,

$F_{Q}\sim N^{2z/d}\frac{\tilde{g}\left(\epsilon N^{1/(\nu
d)},\frac{T}{\Delta_{g}}\right)}{f^{2}(\epsilon N^{1/(\nu d)})},$ (10)

when considered for fixed values of $\epsilon N^{1/(\nu d)}$ and
$T/\Delta_{g}$.

Until now, we have considered a general scenario for non-degenerate systems.
However, degeneracy might have important consequences for the scaling of the
QFI. Here we just note some simple observations. When the degeneracy is
independent of the number of particles $N$ it appears as a constant
multiplicative factor to the scaling function and can be neglected in the
analysis of the scaling exponents. However, a more complicated scenario
appears if the degeneracy of the energy levels is a function of $N$. In this
case the scaling exponents can differ from those considered in the non-
degenerate scenario, see Appendix B for more details. Indeed, the optimal
structure of energy levels, which leads to the highest value of the QFI,
involves a single ground state and all degenerated excited states with a well-
defined energy gap [46, 37]. The engineering of the optimal energy levels and
the resulting sensitivity are summarized in Appendix A.

We now discuss the SNR, defined as $T/\sqrt{\delta^{2}T}$, also called the
relative estimation precision. This quantity is dimensionless itself and it
can be related to the QFI using (1). This gives
$T\sqrt{\delta^{2}T}=\sqrt{F_{Q}}T$. From the scaling behavior of the QFI in
(8), we can see that the SNR can be expressed as

$\frac{T}{\sqrt{\delta^{2}T}}=\left(\frac{T}{\Delta_{g}}\right)\tilde{g}^{1/2}\left(\epsilon
N^{1/(\nu d)},\frac{T}{\Delta_{g}}\right).$ (11)

This implies that, as before, the SNR near the critical point is given by a
scaling function of two parameters $\epsilon N^{1/(\nu d)}$ and
$T/\Delta_{g}$. Unlike the QFI, the SNR does not exhibit the scaling with the
total number of particles for fixed values of $\epsilon N^{1/(\nu d)}$ and
$T/\Delta_{g}$.

$(a)$$(b)$$(c)$$(d)$$(e)$$(f)$ Figure 1: $(a)$ Energy levels of
$\hat{H}_{1}/c$ as a function of quadratic Zeeman energy $q=q_{z}/c$ for
$N=1000$. Contour plots of the quantum Fisher information $F_{Q}$ in units of
$c^{-2}$ $(b)$ and the signal-to-noise ratio $T/\sqrt{\delta^{2}T}$ $(c)$
versus $q$ and $T/\Delta_{\rm min}$ for $N=200$. Numerical demonstration of
finite-size scaling of $\Delta_{g}$ $(d)$, $F_{Q}$ $(e)$ and
$T/\sqrt{\delta^{2}T}$ $(f)$ versus the product $N^{2/3}\epsilon$ around the
left critical point when $T/\Delta_{\rm min}=0.25$ $(e)$, and $T/\Delta_{\rm
min}=0.17$ $(f)$, for various number of atoms $N$ as indicated in figure
legend. Here, $q_{c}$ is defined as a position of the minimum of $\Delta_{q}$
and depends on $N$. In Appendix E, we show the numerical results before
applying the scaling.

In this section, we have applied the finite-size scaling approach for critical
systems to the thermometry problem. We derived an expression for the scaling
function and scaling exponents of the sensitivity bound in temperature
estimation quantified by the QFI. In the following two sections, we
investigate the sensitivity of critical quantum thermometry and the scaling
properties around the critical region of two different systems. First, we
consider the spin-1 BEC belonging to the all-connected spins universality
class such as the Lipkin-Meshkov-Glick model [47]. Next, we do the same for
the spin-1/2 XX model with nearest neighbors interaction [13]. These systems
are different in the structure of their lowest energy levels. However, both of
them exhibit phase transition as a function of a Hamiltonian parameter and
display asymptotically closing energy gaps at the critical points.

##  3 Spin-1 Bose-Einstein Condensate

In this section we consider the finite size spin-1 BEC system in the $F=1$
hyperfine ground state manifold in the presence of an external homogeneous
magnetic field. Under the single mode approximation (SMA) [48, 49, 15], the
Hamiltonian for $N$ atoms reads

$\hat{H}_{1}=-\frac{c}{2N}\hat{J}^{2}-q_{\rm z}\hat{N}_{0}$ (12)

after dropping constant terms. Here,
$\hat{J}^{2}=\hat{J}_{x}^{2}+\hat{J}_{y}^{2}+\hat{J}_{z}^{2}$ where
$\hat{J}_{x,y,z}$ are the spin-1 operators, $\hat{N}_{m_{F}}$ is the number of
atoms in the ${m_{F}}=0,\pm 1$ Zeeman state, $q_{\rm z}$ is the quadratic
Zeeman energy shift which acts as the control parameter $\lambda$, and
${c}=N|c_{2}|\int d{\boldsymbol{r}}|\phi(\boldsymbol{r})|^{4}$, where
$\phi(\boldsymbol{r})$ is the spatial atomic wave function being a solution of
the Gross-Pitaevskii equation [48, 49] 222The explicit form of $c_{2}$ is
given by $c_{2}=4\pi\hbar^{2}(a_{0}-a_{2})/3m$, where $m$ is the mass of each
particle and $a_{0}(a_{2})$ is the s-wave scattering length for spin-1 atoms
colliding in symmetric channels of total spin $J=0\,(J=2)$. Note, $c$ is
proportional to the density $c\propto\rho=N/V$ for homogeneous systems.. In
the following we will work with the re-scaled Hamiltonian $\hat{H}_{1}/c$, and
we will refer to the re-scaled quadratic Zeeman energy as $q=q_{\rm z}/c$.

In general, the system can be viewed as $N$ interacting spin-1 particles
exposed to an external field where the interaction is not limited to the
nearest neighbors. The total magnetization $\mathcal{M}$ is defined as
$\mathcal{M}\equiv N_{1}-N_{-1}$. This quantity is a constant of motion due to
collisional symmetry [48]. Throughout the remainder of this work, we will only
consider the case in which $N_{1}=N_{-1}$, i.e., $\mathcal{M}=0$, and the
variance $\Delta^{2}\mathcal{M}=0$.

In the case of $\mathcal{M}=0$, the phase diagram of ground states contains
two critical points that separate the polar (P) phase (characterized by
$N_{\pm 1}=0$) from the broken axisymmetry (BA) phase ($N_{0},N_{\pm 1}>0$),
and the BA phase from the antiferromagnetic (AFM) phase ($N_{\pm 1}=N/2$)
[50]. In the thermodynamic limit, the two critical points for $\hat{H}_{1}/c$
are $q_{c}=2$ and $q_{c}=-2$, respectively. These critical points were shown
to provide an enhanced sensitivity in the estimation of the control parameter
$q$ [15].

In the following, we will explore the sensitivity bounds for quantum
thermometry with the spin-1 system by numerically evaluating the QFI (5) and
SNR given by $T/\sqrt{\delta^{2}T}=\sqrt{F_{Q}}T$ using the method described
in [14]. The results will be then interpreted within the finite-size scaling
approach described in section 2, confirming the validity of this description
and our derivation of the scaling behavior of the QFI and SNR.

Figure 1$(a)$ shows the eigenvalue spectrum $\Delta_{n}(q)$ (for the five
lowest energy levels) of $\hat{H}_{1}/c$ for $N=10^{3}$. This exhibits minima
in the gap $\Delta_{n}$ near $q=\pm 2$. Figure 1 $(b)$ and $(c)$ show by color
the values of the QFI and SNR, respectively, both in the $(q-T/\Delta_{\rm
min})$ parameter space. The QFI is largest around the critical points, and its
value peaks at $T/\Delta_{\rm min}\approx 0.26$. However, while the SNR also
increases around the critical points, it shows a different behavior, as
discussed in section 2.

To confirm the validity of the finite-size scaling approach, we have to
consider that the spin-1 BEC within the SMA is an example of the “infinite”
coordinate system, where the concept of correlation length and dimension is
lost. The scaling approach can be applied in this case with the exponents
modified by the critical dimension [43, 26]. This was discussed for the spin-1
system in [51, 15], where it was shown that $\Delta_{g}\sim N^{-1/3}f(\epsilon
N^{2/3})$. Having this into account, Figure 1$(d)$ shows the result of
applying the finite-size scaling to the lowest energy gap, as in Figure 1$(a)$
for different numbers of atoms. The scaling is confirmed with scaling
exponents $z\nu=1/2$, as in the Lipkin-Meshkov-Glick model and its
corresponding universality class [17, 52]. Figure 1 $(e)$ and $(f)$ show the
result of applying the scaling to the numerical results for $F_{Q}$ and
$T/\sqrt{\delta^{2}T}$, respectively, as a function of the product $\epsilon
N^{2/3}$ for fixed $T/\Delta_{g}=0.17$. The results evidence the scaling
$F_{Q}\sim\Delta_{g}^{-2}\tilde{g}(\epsilon N^{2/3})$ and
$T/\sqrt{\delta^{2}T}\sim(T/\Delta_{g})\tilde{g}^{1/2}(\epsilon N^{2/3})$ as
expected from the finite-size scaling approach. Directly, this demonstrates
that $F_{Q}\sim N^{2/3}$ when $T/\Delta_{g}$ is fixed, and the accuracy of the
general finite-size scaling arguments used in section 2 for the spin-1 BEC
system.

It is worth mentioning that in the antiferromagnetic condensate [53] the
energy gap follows the scaling law (7) for the zero magnetization case when
the system exhibits a first order phase transition [14]. The relevant scaling
exponent suggests $F_{Q}\sim N^{2}$ for fixed $T/\Delta_{g}$ around
criticality.

##  4 Spin-1/2 XXZ System

$(a)$$(b)$$(c)$$(d)$$(e)$$(f)$ Figure 2: $(a)$ Lowest energy level structure
of $\hat{H}_{1/2}/J$ versus transverse magnetic field $h=h_{x}/J$ for $M=8$.
Contour plot for the $F_{Q}$ in units of $J^{-2}$ $(b)$ and SNR $(c)$ as a
function of $h_{x}/J$ and $T/\Delta_{g}$ for $M=8$. $(d)$ Energy gap
$\Delta_{g}$ re-scaled according to (14). Panels $(e)$ and $(f)$ display
$\Delta_{g}^{2}F_{Q}$ and SNR respectively versus
$\rm{sgn}(\epsilon)M|\epsilon|^{4/7}$ when $T/\Delta_{\rm min}=0.17$ for the
total number of spins as indicated in the legend. We note that the different
lines in panels $(d),(e)$ and $(f)$ overlap. Thus, we show the direct
numerical results before applying the finite-size scaling in Appendix E.

In this section, we discuss the XXZ model of $M$ spins in the transverse field
with periodic boundary conditions described by the Hamiltonian [54]:

$\displaystyle\hat{H}_{1/2}=$
$\displaystyle-4J\sum_{j=1}^{M}\left(\hat{s}^{x}_{j}\hat{s}^{x}_{j+1}+\hat{s}^{y}_{j}\hat{s}^{y}_{j+1}+\zeta_{z}\hat{s}^{z}_{j}\hat{s}^{z}_{j+1}\right)$
$\displaystyle+2h_{x}\sum_{j=1}^{M}\hat{s}^{x}_{j},$ (13)

where $\hat{s}^{\sigma}_{j}$ with $\sigma=x,y,z$, are pseudo-spin-1/2
operators for $j$th site, $J$ is an exchange coupling in the XY plane,
$\zeta_{z}$ is the exchange coupling anisotropy in the $z$ direction, and
$h_{x}$ is the coupling strength between the system and an external,
transverse, magnetic field.

In this work, we set $\zeta_{z}=0$, and consider the re-scaled Hamiltonian
$\hat{H}_{1/2}/J$, for which the system exhibits a second-order phase
transition at the critical point $h_{x}/J=0$ [13, 55]. The values of the
relevant critical exponents are $z=1$ and $\nu=2/(4-\arccos(\zeta_{z}/J)/\pi$
[56]. In general, the model with $h_{x}\neq 0$ is not integrable due to the
transverse field considered. However, in the special case of $h_{x}=0$, the
Hamiltonian reduces to the XX model and can be solved analytically [57, 55,
58]. In what follows, we will explore the sensitivity bounds for quantum
thermometry in this system evaluating the QFI (5) and SNR near the phase
transition by performing an exact numerical diagonalization of the
Hamiltonian. Defining a local temperature in spin systems is not trivial, and
has led to much discussion [59, 60, 61, 62, 63, 64, 65], some of which has
been confirmed in different systems [63, 64, 65]. Here, in contrast, we are
considering global spin temperatures that describe the entire spin system. Our
results can nonetheless be expected to apply also locally when the boundary is
sufficiently large [64].

The structure of energy levels is shown in Figure 2$(a)$. Unlike the spin-1
BEC model discussed in the previous section, the spin-1/2 system features some
weak degeneracy of the low energy levels around $h_{x}=0$. We define the
distance from the critical point as $\epsilon=h_{x}/J$. According to (7), the
energy gap between the two lowest energy states is expected to scale as

$\Delta_{g}\sim M^{-1}f(\epsilon M^{7/4}),$ (14)

with the particular values of critical exponents $z\nu=4/7$. This is shown by
the numerical results presented in Figure 2$(d)$, where we plot the re-scaled
$\Delta_{g}$ according to (14) as a function of ${\rm
sgn}(\epsilon)|\epsilon|M^{7/4}$.

In Figure 2$(b)$ and $(c)$, we show by color $F_{Q}$ from (5) and
$T/\sqrt{\delta^{2}T}$ for $\hat{H}_{1/2}/J$, quantifying the sensitivity and
the SNR in the estimation of temperature in the $(h_{x}/J-T/\Delta_{\rm min})$
parameter space. The increase of the sensitivity at the critical region is
again clearly visible when $T\lesssim\Delta_{g}$ as expected by the finite-
size scaling approach discussed in section 2.

Finally, we examine the finite-size scaling of both, the QFI and SNR versus
${\rm sgn}(\epsilon)M|\epsilon|^{4/7}$ when $T/\Delta_{g}$ is fixed. As we
observe in Figure 2$(e)$ and Figure 2$(f)$ respectively, initially different
curves, that correspond to different total number of spins, collapse into each
other after the scaling is applied, when $T/\Delta_{g}=0.17$. Our results
clearly demonstrate that the sensitivity is subject to the finite-size
scaling, and $F_{Q}\sim\Delta_{g}^{-2}\tilde{g}(\epsilon M^{7/4})$ and
$T/\sqrt{\delta^{2}T}\sim(T/\Delta_{g})\tilde{g}^{1/2}(\epsilon M^{7/4})$
applies for the XXZ model even in the presence of a weak degeneracy, and that
the scaling of the QFI with the number of spins is $F_{Q}\sim M^{2}$ for fixed
$T/\Delta_{g}$ at the critical point.

##  5 Feasibility of Critical Quantum Thermometry in Spin Systems

In the previous sections, we discussed the critical quantum thermometry for
two examples of physical systems having different critical and scaling
exponents. We also demonstrated by means of exact numerical calculations, that
the finite-size scaling approach well captures universal scaling properties of
the sensitivity. In addition, both systems can be realized in ultra-cold
atomic experiments [32, 31]. In this section, we discuss the feasibility of
criticality-aided temperature sensing in such systems in reference to current
experimental capabilities. We assess this considering different observables of
the system, and quantifying the sensitivity that a possible experiment could
reach using (3) and the error propagation formula (EPF). The EPF is used in
the estimation of a parameter, in this case $T$, evaluated from accessible
experimental measurements of an observable $\hat{A}$, such that:

$\varDelta^{2}T=\frac{\Delta^{2}\hat{A}}{|\partial_{T}\langle\hat{A}\rangle|^{2}}.$
(15)

Let us start the analysis with the spin-1 system. We consider two possible
measurements that can be experimentally realized: _i_) non-destructive Faraday
rotation (FR) of the collective spin degree of freedom
$\hat{A}=\hat{J}_{\perp}^{2}\equiv\hat{J}^{2}-\hat{J}^{2}_{z}$ 333Note,
because $\hat{J}^{2}_{\perp}=\hat{J}^{2}-\hat{J}_{z}^{2}$ the equivalence of
${\langle\hat{J}^{2}_{\perp}\rangle}$ with ${\langle\hat{J}^{2}\rangle}$
arises due to the fact that we are considering
$\mathcal{M}=\langle\hat{J}_{z}\rangle=0$, and the assumption of fixed (non-
fluctuating) magnetization, i.e., $\Delta^{2}\mathcal{M}=0$. [66, 15, 67] and
_ii_) destructive absorption imaging of the $m_{F}=0$ state population
$\hat{A}=\hat{N}_{0}$. We note that the combination of such measurements is
possible within the same experimental run since FR measurements are non-
destructive and can be performed before a final destructive measurement of
$\hat{N}_{0}$.

$(a)$$(b)$$(c)$$(d)$ Figure 3: Numerical results showing the temperature
dependence of $c^{2}F_{Q}$ (red dashed line), $c^{2}F_{c}$ (green dot-dashed)
and $c^{2}/\varDelta^{2}T$ (orange solid) for $\hat{J}_{\perp}^{2}$ $(a)$ and
$(b)$, and $\hat{N}_{0}$ $(c)$ and $(d)$ with $N=200$ at the critical point
$q_{z}/c=-1.869$ $(a),(c)$, and a near-critical point $q_{z}/c=-1.8$
$(b),(d)$. The black-dashed lines indicate $\underset{T}{\rm
max}\,F^{\rm*}_{Q}(q_{z},T)$ to give a reference level.

Figure 3 shows the sensitivity in temperature estimation quantified by the
$F_{c}$ and $1/\varDelta^{2}T$ as in (3) and (15) for both
$\hat{A}=\\{\hat{J}_{\perp}^{2},\hat{N}_{0}\\}$ in units of the energy scale
$c$, versus $T/c$. As a reference, we also plot $F_{Q}$. Left panels $(a)$ and
$(c)$ show the sensitivities at the critical point $q_{z}/c=-1.869$ using
$\hat{J}_{\perp}^{2}$ $(a)$, and $\hat{N}_{0}$ $(c)$. Right panels $(b)$ and
$(d)$ show the sensitivities at $q_{z}/c=-1.8$. As expected, at the critical
point $F_{Q}$ shows the largest value. However, both $F_{c}$ and the
sensitivity derived from the EPF are larger at the slightly displaced value of
$q_{z}/c=-1.8$ than at the critical point itself. Therefore, although the
optimal (energy) measurement would reach a maximum sensitivity at the critical
point, the more experimentally accessible quantities show its maximum at a
slightly displaced position. The black-dashed lines indicate a baseline
sensitivity

$F^{\infty}_{Q}\equiv\underset{T}{\textrm{max}}\,F^{\rm*}_{Q}(T,q_{z})$ (16)

evaluated at the corresponding $q_{z}$ for the spin-1 system, where
$F^{\rm*}_{Q}(T,q_{z})$ corresponds to the QFI calculated for the Hamiltonian
in (12) when the interaction term is neglected. In the limit of large $N$ and
very low temperatures, analytic expression can be obtained giving
$F^{\rm*}_{Q}(T,q_{z})=(q_{z}/T)^{4}(q_{z})^{-2}\sinh^{-2}(q_{z}/T)$ for
$q_{z}\neq 0$, as discussed in Appendix C. This indicates that the critical
behavior of the system enhances the sensitivity in the estimation of
temperature. We can see that temperature estimation based in measurements of
$\hat{N}_{0},\,\hat{J}^{2}_{\perp}$ yield a similar sensitivity around the
critical point, and that they are suboptimal. We note that the measurement of
$\hat{J}_{\perp}^{2}$ gives the best sensitivity at $q_{z}=0$ since at this
particular value, the Hamiltonian reduces to
$\hat{H}_{1}/c=-\hat{J}_{\perp}^{2}/(2N)$. In the case of the spin-1 system,
the range of temperatures for which the system exhibit criticality-enhanced
sensitivity is found in the region of an experimentally accessible range of
temperatures $\sim\textrm{pK}-\textrm{nK}$ for $N=10^{3}$ and $|c|=h\times
17\textrm{Hz}$ [31].

$(a)$$(b)$$(c)$$(d)$ Figure 4: Numerical results showing $J^{2}F_{Q}$ (red
dashed line), $J^{2}F_{c}$ (green dot-dashed) and $J^{2}/\varDelta^{2}T$
(orange solid) with $M=4$ as a function of $T/J$ for $h_{x}/J=0$ $(a),(c)$,
and $h_{x}/J=0.5$ $(b),(d)$ when $\hat{A}=\hat{S}_{x}^{2}$ $(a),(b)$, and
$\hat{A}=\hat{S}_{z}^{2}$ $(c),(d)$.

The spin-1/2 XXZ model we are considering in this work has been recently
realized for the first time in an ultra-cold atomic experiment [32] using a
two-components ultra-cold bosonic gas in a 1D optical lattice. Each bosonic
component takes the role of the ground(excited) state in the equivalent
spin-$1/2$ model. In this case, the collective
$\hat{S}_{z}=\sum_{j}\hat{s}_{j}^{z}$ and its higher moments can be measured
through absorption imaging techniques, and the other collective spin
components could in principle be accessed after a suitable rotation
$\hat{S}_{x}\rightarrow\hat{S}_{z}$ and $\hat{S}_{y}\rightarrow\hat{S}_{z}$.
More sophisticated methods, like quantum non-demolition FR measurements are
also possible [68].

Figure 4 shows the numerical results of evaluating $F_{Q}$, $F_{c}$ and
$1/\varDelta^{2}T$ using (5), (3) and (15) respectively, for
$\hat{H}_{1/2}/J$, and the spin observables $\hat{A}=\hat{S}_{z}^{2}$ and
$\hat{A}=\hat{S}_{x}^{2}$, as a function of $T/J$. The optimality of measuring
collective quantum correlations for quantum thermometry was pointed out in
[58]. The results at the critical point $h_{x}=0$ are shown in Figure 4 in
panels $(a)$ and $(c)$. We observe that $F_{c}$ achieves maximum sensitivity
bounded by $F_{Q}$ using both observables in the low temperature regime. Away
from the critical point, at $h_{x}=0.5$ the two signals are sub-optimal, as
shown in panels $(b)$ and $(d)$. We also observe that at this value of $h_{x}$
and for $\hat{A}=\hat{S}_{x}^{2}$, the sensitivity obtained using the EPF
equals $F_{c}$.

Finally, let us briefly comment on the effect of imperfect detection.
Typically, there is a decrease in the sensitivity when the noise is of the
order of the parameter to be measured [69]. Consequently, in order to retain
an acceptable sensitivity in the estimation of $T$, the detection noise should
be smaller than the value of the energy gap $\Delta_{g}$. We briefly discuss
the effect of detection noise in a particular situation in Appendix D.

##  6 Conclusions

In this work, we have explored the critical properties of quantum thermometry
in systems exhibiting continuous QPT. Using the finite-size scaling approach
we linked the sensitivity in temperature estimation with universal scaling in
terms of critical exponents. We showed that in the critical region and for
fixed $T/\Delta_{g}$, the QFI scales with $N$ as $F_{Q}\sim N^{2z/d}$. This
general relation is confirmed by exact numerical calculation of two different
spin models with finite $N$: the spin-1 BEC and the Heisenberg XX spin chain
which are experimentally realizable. In addition, we discussed the feasibility
and optimal setup for realizing a critical quantum thermometer in such systems
using practical observables. We show that measurements accessible in current
experiments enable ultra-precise thermometry by exploiting critical quantum
resources. The method presented here allows for both fundamental studies of
thermalization processes and practical applications such as precise
characterization and preparation of systems for analog quantum simulation.

## Acknowledgements

The authors kindly thank B. Damski, J. Kołodyński and M. Mehboudi for fruitful
discussions. EW is supported by the Polish National Science Center Grants
DEC-2015/18/E/ST2/00760. AN is supported by Project no. 2017/25/Z/ST2/03039,
funded by the National Science Centre, Poland, under the QuantERA programme
and by MCIN/ AEI (project PID2020-115761RJ-I00). SM is supported by a grant
from Basic Sciences Research, Iran, Fund (No. BSRF-phys-399-05). MWM, DBO and
EA acknowledge support by the European Union from the European Metrology
Programme for Innovation and Research (EMPIR) project 17FUN03-USOQS. EMPIR
projects are co-funded by the European Union’s Horizon 2020 research and
innovation program and the EMPIR participating states. They also acknowledge
H2020 QuantERA ERA-NET Cofund Q-CLOCKS (PCI2018-092973 project funded by MCIN/
AEI /10.13039/501100011033/ FEDER “A way to make Europe”), and H2020 FET
Quantum Technologies Flagship project MACQSIMAL (Grant No. 820393), and the
Spanish Ministry of Science projects OCARINA (PGC2018-097056-B-I00 project
funded by MCIN/ AEI /10.13039/501100011033/ FEDER “A way to make Europe”) and
“Severo Ochoa” Center of Excellence CEX2019-000910-S. Generalitat de
Catalunya: CERCA, AGAUR Grant No. 2017-SGR-1354, Secretaria d’Universitats i
Recerca (project QuantumCat, ref. 001-P-001644). Fundació Privada Cellex, and
Fundació Mir-Puig.

##### Author contributions

AN, EA and DBO contributed equally to this work in the form of both analytic
and numerical calculations. EW provided the scaling relations in section 2.
MWM, EW and DBO conceived the idea. EW and DBO guided the research and wrote
the first draft. All the authors contributed to the discussion of the results
and the manuscript revision and preparation.

##  Appendix A Optimal Configuration for the Quantum Thermometry

The maximization of the Fisher information was already presented in the
literature, e.g., [46, 37], with the realistic proposal [70]. The general idea
is to calculate the classical Fisher information from (3), which leads to the
the expression

$\displaystyle F_{c}(T)=$
$\displaystyle\frac{1}{T^{2}}\frac{1}{Z^{2}}\sum_{\alpha=1}^{m}\tilde{\Delta}^{2}_{\alpha}e^{-\tilde{\Delta}_{\alpha}}$
$\displaystyle+\frac{1}{T^{2}}\frac{1}{Z^{2}}\sum_{\alpha,\alpha^{\prime}>\alpha}(\tilde{\Delta}_{\alpha}-\tilde{\Delta}_{\alpha^{\prime}})^{2}e^{-(\tilde{\Delta}_{\alpha}+\tilde{\Delta}_{\alpha^{\prime}})}$
$\displaystyle\equiv$
$\displaystyle\frac{g(\tilde{\Delta}_{\alpha})}{(T)^{2}},$ (17)

where we introduced $g(\tilde{\Delta}_{\alpha})$ to mark the whole sum in the
above expression which depends on $m$ re-scaled energy gaps
$\tilde{\Delta}_{\alpha}$. The minimum of $F_{c}^{-1}$ signals the lower bound
for the sensitivity in the temperature estimation. This idea is presented in
[37] and later generalized for all $m$ excited states in [46].

When taking into account the first excited state only, $m=1$ in (A), one deals
with the function $g(\tilde{\Delta}_{1})$ which after maximization determines
the bound for the Fisher information $F_{\rm max}(T)\approx 0.44/T^{2}$ (or
for the temperature fluctuations $\delta^{2}T_{\rm min}/T^{2}\geq 2.27$).

When taking into account two excited states, $m=2$ in (A), one can easily see
that the maximum for the CFI can be achieved when
$\tilde{\Delta}_{1}=\tilde{\Delta}_{2}$, meaning that degeneracy of the
excited states is required to obtain the maximal value. The maximal Fisher
information is then $F_{\rm max}(T)\approx 0.76/T^{2}$ (or for the temperature
fluctuations $\delta^{2}T_{\rm min}/T^{2}\geq 1.31$).

Finally, one can consider all the excited states and find that the maximum of
the FI is achieved when all excited states are degenerated. Then, the Fisher
information $F_{\rm max}(T)\approx({\rm log}\,m)^{2}/(2T)^{2}$ (or for the
temperature fluctuations $\delta^{2}T_{\rm min}/T^{2}\geq 4/({\rm
log\,}m)^{2}$ when $m\gg 1$). More precisely, the gap that maximizes the QFI
is given by:

$\frac{e^{x}(x-2)}{x+2}=m,\quad{\rm where}\,\,x=\Delta_{\rm max}/T$ (18)

and the maximal value of the QFI is

$F_{Q,\rm max}(T)=\frac{\mathcal{X}_{\rm max}}{T^{2}}$ (19)

with

$\mathcal{X}_{\rm max}=\frac{mx^{2}e^{-x}}{(1+me^{-x})^{2}}.$ (20)

##  Appendix B Scaling Function Around Criticality

In this section we present a simple derivation of the relation
$\Delta_{g}^{2}F_{Q}$ as a function of $x=\epsilon N^{1/(\nu d)}$ and
$y=T/\Delta_{g}$ as expressed in Equation 8.

To show this, we first express the temperature in units of $\Delta_{g}$,
namely $T=\Delta_{g}(T/\Delta_{g})=y\Delta_{g}$. Then
$F_{Q}=\Delta_{g}^{-2}(\Delta^{2}\hat{H})/(y^{4}\Delta_{g}^{2})$. Next, one
needs to consider the quantity $(\Delta^{2}\hat{H})/(\Delta_{g}^{2})$. The
variance of the energy is
$\Delta^{2}\hat{H}=\sum_{n}g_{n}E_{n}^{2}e^{-E_{n}/T}/Z-(\sum_{n}g_{n}E_{n}e^{-E_{n}/T}/Z)^{2}$,
where $g_{n}$ is degeneracy of the $n$th energy level. The ground and the
first excited states provide leading terms in the low temperature region
($T\lesssim\Delta_{g}$) [37, 29]. Under this approximation the variance of the
energy can be expressed as
$\Delta^{2}\hat{H}\approx(g_{0}E_{0}^{2}+g_{1}E_{1}^{2}e^{-\Delta_{g}/T})/Z-(g_{0}E_{0}+g_{1}E_{1}e^{-\Delta_{g}/T}/Z)^{2}$,
with $Z\approx g_{0}+g_{1}e^{-\Delta_{g}/T}$, which simplifies to
$\Delta^{2}\hat{H}\approx
g_{0}g_{1}\Delta_{g}^{2}/(g_{0}e^{\Delta_{g}/2T}+g_{1}e^{-\Delta_{g}/2T})^{2}$
after some algebra. The ratio of energy variance and the energy gap squared
then becomes $\Delta^{2}\hat{H}/\Delta_{g}^{2}\approx
g_{0}g_{1}/(g_{0}e^{\Delta_{g}/2T}+g_{1}e^{-\Delta_{g}/2T})^{2}\equiv\tilde{g}(g_{0},g_{1},y)$.
The dependence on $x$ arises indirectly through $y$ (since
$\Delta_{g}=\Delta_{g}(x)$). The QFI can be written as

$F_{Q}=g_{0}g_{1}\Delta_{g}^{-2}\left(\frac{\Delta_{g}}{T}\right)^{4}\left(g_{0}e^{\Delta_{g}/2T}+g_{1}e^{-\Delta_{g}/2T}\right)^{-2}$
(21)

Note here, that if degeneracy $g_{0/1}$ is independent of $N$ then it acts as
the numerical factor and can be neglected in scaling analysis. However, if
degeneracy is a function of $N$ then it affects the scaling properties of the
QFI and must be taken into account.

##  Appendix C Baseline Sensitivity Based on the Uncorrelated Counterpart of
the System

In this section we will consider the generic model Hamiltonian
$\hat{H}^{*}=g\hat{H}_{0}$, which effectively describes the two system models
considered in the main text after dropping the corresponding non-linear terms.
This corresponds to the situation at which the interaction term in the
Hamiltonian becomes negligible as compared to the linear term
$\hat{H}^{*}\propto g$, and takes place in the region far from the critical
point.We calculate the QFI for this generic model in its eigenbasis, assuming
$\hat{H}_{0}|n_{0}\rangle=E_{n_{0}}|n_{0}\rangle$ with $E_{n_{0}}=gn_{0}$,
$n_{0}=0,\cdots,\mathcal{N}$. Now, we can compute the QFI for $\hat{H}^{*}$
using (5) and obtain:

$\displaystyle F^{\rm*}_{Q}(T,g)=$ $\displaystyle
T^{-4}\sum_{n_{0}=0}^{\mathcal{N}}\frac{E_{n_{0}}^{2}}{Z}e^{-E_{n_{0}}/T}$
$\displaystyle-T^{-4}\left(\sum_{n_{0}=0}^{\mathcal{N}}\frac{E_{n_{0}}}{Z}e^{-E_{n_{0}}/T}\right)^{2}$
$\displaystyle\approx$
$\displaystyle\frac{1}{4g^{2}}\left(\frac{g}{2T}\right)^{4}\frac{1}{\sinh^{2}\left(g/2T\right)},$
(22)

for $g\neq 0$, $\mathcal{N}\to\infty$ and in the limit of low temperatures,
where $e^{-E_{n_{0}}/T}\to 0$ for large $n_{0}$. A careful analysis of (C)
shows that its maximal possible value as a function of temperature is

$F_{Q}^{\infty}\equiv\underset{T}{\rm max}\,F_{Q}^{\rm*}(T,g)\approx
4.88/g^{2}$ (23)

when the temperature that maximizes the above expression $T^{*}\approx
g/(3.83)$. This implies that the QFI tends to zero as $1/g^{2}$ when
$g\to\infty$. If one identifies the control parameter $g$ with the distance
from the critical point $\epsilon$, then one can conclude that the QFI tends
to zero as $1/\epsilon^{2}$.

For the spin-1 system, we can identify $g=-q_{z}$ and
$\hat{H}_{0}=\hat{N}_{0}$. The corresponding eigenbasis is given by the Fock
states $|n_{0}\rangle=|k,N-2k,k\rangle$ with $k=0,\cdots,N$ and
$E_{n_{0}}=-2qck$.

In particular, away from the critical point the analytic approximate
expression for the QFI in the large $\mathcal{N}$ limit is

$F^{\rm*}_{Q}(T,q_{z})=\frac{1}{q_{z}^{2}}\left(\frac{q_{z}}{T}\right)^{4}\frac{1}{\sinh^{2}\left(q_{z}/T\right)}.$
(24)

A similar expression can be obtained for the XXZ model. However, since in this
case the critical point is $h_{x}=0$, we do not use it to reference a baseline
for the sensitivity in the main text.

$F^{\rm*}_{Q}(T,h_{x})=\frac{4}{h_{x}^{2}}\left(\frac{h_{x}}{2T}\right)^{4}\frac{1}{\sinh^{2}\left(h_{x}/2T\right)}$
(25)

for the XXZ model.

Figure 5: The Fisher information with detection noise with respect to
unperturbed case for the optimal case with degenerate $m$ excited states when
$\Delta_{1}/T=\log(m)$.

##  Appendix D Effect of Detection Noise

Assuming the measurement of energy $\hat{A}=\hat{H}$, we show here the effect
of detection noise. Therefore, we consider the Fisher information as in (3)
where the probability distribution $p(E_{\alpha})=e^{-E_{\alpha}}/Z$ is
replaced with

$P_{dn}(E_{\alpha})=\sum_{E_{\alpha^{\prime}}}\frac{e^{-(E_{\alpha}-E_{\alpha^{\prime}})^{2}/2\sigma^{2}}}{\mathcal{N}_{\alpha^{\prime}}}P(E_{\alpha^{\prime}}),$
(26)

with

$\mathcal{N}_{\alpha^{\prime}}=\sum_{n}e^{-(E_{n}-E_{\alpha^{\prime}})^{2}/2\sigma^{2}}.$
(27)

In the optimal case, considered in Appendix A, with all degenerated $m$
excited states one can obtain an analytical formula for the classical Fisher
information

$F_{dn}=\frac{1}{(T)^{2}}\frac{m(1-y)^{2}\log^{2}m}{4(1+y+2my)(m+2y+my^{\log^{2}m})},$
(28)

where we have taken the optimal energy gap $\Delta_{g}/T=\log(m)$ and
introduced $\log y=(T\log m/\sigma)^{2}$. One can easily note that in this
optimal configuration the Fisher information is one parameter dependent
function of dimensionless parameter $\sigma/T$. In general, FI with detection
noise can be considered as unperturbed QFI multiplied by the function of the
only parameter $\sigma/T$. A similar conclusion was made in [69]. An example
of $F_{dn}$ is given in Figure 5.

##  Appendix E Numerical Results for Non-Scaled Quantities in Figure 1 and
Figure 2

$(a)$$(b)$$(c)$$(d)$$(e)$$(f)$ Figure 6: Upper row shows results for the
spin-1 systems. Lower row displays the corresponding quantities for the XXZ
model. $(a)$ Energy gap $\Delta_{g}$ of the BEC spin-1 system. Panels $(b)$
and $(c)$ display $\Delta_{g}^{2}F_{Q}$ and SNR respectively versus the
distance from the critical point $\epsilon$, around the left critical point
when $T/\Delta_{\rm min}=0.17$. Colors label different number of atoms $N$ as
indicated in the legend of Figure 1. $(d)$ Energy gap $\Delta_{g}$ of the XXZ
model. Panels $(e)$ and $(f)$ display $\Delta_{g}^{2}F_{Q}$ and SNR
respectively versus $\epsilon$ when $T/\Delta_{\rm min}=0.17$. The colors
indicate the total number of spins as shown in the legend of Figure 2.

Figure 6 shows the numerical results for the same quantities shown in panels
$(d),(e)$ and $(f)$ for both Figure 1 and Figure 2 before applying the
corresponding scaling.

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