# Uncertain Quantum Critical Metrology: From Single to Multi Parameter Sensing

George Mihailescu  0000-0002-0048-9622<EMAIL_ADDRESS>School
of Physics, University College Dublin, Belfield, Dublin 4, Ireland  Centre for
Quantum Engineering, Science, and Technology, University College Dublin,
Dublin 4, Ireland  Steve Campbell  0000-0002-3427-9113<EMAIL_ADDRESS>School of Physics, University College Dublin, Belfield, Dublin 4, Ireland
Centre for Quantum Engineering, Science, and Technology, University College
Dublin, Dublin 4, Ireland  Dahlem Center for Complex Quantum Systems, Freie
Universität Berlin, 14195 Berlin, Germany  Karol Gietka  0000-0001-7700-3208
<EMAIL_ADDRESS>Institut für Theoretische Physik, Universität
Innsbruck, Technikerstraße 21a, A-6020 Innsbruck, Austria

###### Abstract

Critical quantum metrology relies on the extreme sensitivity of a system’s
eigenstates near the critical point of a quantum phase transition to
Hamiltonian perturbations. This means that these eigenstates are extremely
sensitive to _all_ the parameters of the Hamiltonian. In practical settings,
there always exists a degree of experimental uncertainty in the control
parameters—which are approximately known quantities. Despite such
uncertainties representing the most relevant source of noise in critical
metrology, their impact on the attainable precision has been largely
overlooked. In this work we present a general framework, interpolating between
the single and multi-parameter estimation settings, allowing for the proper
bookkeeping of relevant errors. We apply this framework to the paradigmatic
transverse field Ising and Lipkin-Meshkov-Glick models, explicitly showing how
uncertainty in control parameters impacts the sensitivity of critical sensors.
For finite-size systems, we establish that there exists a trade-off between
the amount of uncertainty a many-body probe can withstand while still
maintaining a quantum advantage in parameter estimation.

Introduction.—Critical quantum metrology [1, 2, 3] aims to leverage the
increased sensitivity to parameter variations for a system close to a quantum
phase transition (QPT) in order to achieve high precision measurements [4, 5,
6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24]. This
can be quantified by directly translating the sensitivity of Hamiltonian
parameters to be estimated through the Cramér-Rao bound, which relates the
quantum Fisher information (QFI) with the estimation uncertainty [25]. When
the parameter of interest can be related to the relevant order parameter of
the QPT, such increased sensitivity can be achieved. However, an important
caveat is that all other parameters of the problem must be precisely known for
the framework of single parameter estimation theory to be applicable. In
general, this assumption is rather stringent. Typically, some degree of
experimental uncertainty or noise is present [26, 27, 28]. In the context of
quantum critical systems, while high precision experiments are possible [29],
as complexity and system size increase—a requisite for critical quantum many-
body systems—the resources required to perfectly control and characterize
these systems, such as time, also increase [4, 5, 30]. Furthermore,
universality is a hallmark of systems approaching their critical point [31,
32], making the system dependent on many, if not all, Hamiltonian parameters
precisely where the expected critical enhancement can be achieved. It is
therefore intuitive that any uncertainty in at least one of these parameters
is expected to negatively affect the QFI and thus deteriorate any potential
quantum advantage. Although the effect of noise on performance of critical
sensing has been studied in the literature [5, 19, 21], the impact of the most
crucial one—the uncertainty of control parameters—has thus far been largely
overlooked.

A more appropriate treatment of critical metrology requires a methodology that
relaxes the core constraint of the single parameter framework, i.e., that all
parameters are assumed to be perfectly known [33, 34, 35, 36]. In addition to
providing a more realistic description of the actual achievable precision,
this approach also allows for mitigating the necessity of complex and
resource-intensive experimental control, ultimately enabling the
identification of regions where a true quantum advantage is achievable. A
candidate framework for such a purpose is multiparameter quantum metrology
[37, 38, 33, 34, 39, 40], where the aim is typically to estimate multiple
parameters, in principle simultaneously. Clearly, within this framework, we
can still assume there is a central parameter we wish to infer, and
additionally, there are other relevant parameters about which we have no a
priori information. Contextually, in terms of critical metrology, the QPT is
driven by a single, experimentally controllable system parameter $\beta$, and
our objective is to infer the desired system parameter $\alpha$. The
multiparameter paradigm represents the situation in which the driving
(control) parameter $\beta$ is unknown. However, typically the driving
parameter is an experimentally controllable knob and therefore is known within
some finite window of resolution. It should be clear then that the
multiparameter paradigm provides us with a worst-case scenario for the
estimation of a central desired quantity, $\alpha$, whereby we have a complete
lack of knowledge regarding the value of the driving parameter, $\beta$.

In this work, we present a general approach to quantum parameter estimation,
interpolating between the two extreme scenarios of single and multi-parameter
estimation, allowing for the proper bookkeeping of relevant errors [8, 41,
35]. We apply this formalism to quantum critical metrology and investigate how
uncertainties in the driving parameter influence the ability to infer an
unknown parameter and whether a relevant advantage is maintained. We consider
the Ising and Lipkin-Meshkov-Glick models as two paradigmatic critical
systems, showing that the multiparameter estimation framework predicts a
catastrophic failure in the sensing capabilities of the probe in these models.
By allowing for a degree of statistical uncertainty in the driving parameter,
our framework demonstrates that a quantum advantage can still be achieved with
critical probes even in the presence of such uncertainties.

Parameter estimation in the presence of uncertainties.—In the most general
quantum sensing scenario, there are a set of unknown parameters $\vec{x}$ to
be estimated through suitable measurements of a quantum probe [42, 43, 44].
Information regarding this set of parameters is encoded in the state of the
probe, given by the density matrix
$\hat{\varrho}\left(\vec{x}\right)=\sum_{i}\lambda_{i}\ket{\lambda_{i}}\bra{\lambda_{i}}$,
with $\lambda_{i}$ being the probability of the probe to occupy state
$|\lambda_{i}\rangle$ which depends on $\vec{x}$. This information is
determined by performing a large number of measurements. The precision of
parameter estimation is given by the quantum Cramér-Rao bound [25] (hat
notation indicates a matrix):

$\displaystyle\mathrm{Cov}\left[\vec{x}\right]\geq{\hat{\mathcal{I}}}^{-1},$
(1)

which is an element-wise matrix inequality, lower bounding the precision of
parameter estimation through the covariance matrix. The elements are
$\text{Cov}(\alpha,\beta)=\langle(\alpha-\langle\alpha\rangle)(\beta-\langle\beta\rangle)\rangle$,
where $\alpha$ and $\beta$ label parameters from $\vec{x}$. Entries of the
quantum Fisher information matrix (QFIM) are given by:

$\displaystyle{\mathcal{I}}_{\alpha\beta}=\sum_{i,j}\frac{2\real\left[\langle\lambda_{i}|\partial_{\alpha}\hat{\varrho}|\lambda_{j}\rangle\langle\lambda_{j}|\partial_{\beta}\hat{\varrho}|\lambda_{i}\rangle\right]}{\lambda_{i}+\lambda_{j}},$
(2)

where diagonal elements relate to the single parameter precision and are
obtained by suitably maximizing over all possible positive operator-valued
measures. The off-diagonal elements relate to the correlation between
parameters. For pure states, the QFIM may be simplified to:

$\displaystyle\mathcal{I}_{\alpha\beta}=4\real\left[\langle\partial_{\alpha}\psi|\partial_{\beta}\psi\rangle-\langle\partial_{\alpha}\psi|\psi\rangle\langle\psi|\partial_{\beta}\psi\rangle\right].$
(3)

It is informative to consider the simple case where we have two parameters to
be estimated, although the results readily extend to an arbitrary number of
variables. From Eq. (1), the attainable precision with respect to parameter
$\alpha$ is given by:

$\displaystyle\Delta\alpha\geq\frac{{\mathcal{I}}_{\beta\beta}}{{\mathcal{I}}_{\alpha\alpha}{\mathcal{I}}_{\beta\beta}-{\mathcal{I}}_{\alpha\beta}{\mathcal{I}}_{\beta\alpha}}.$
(4)

To contrast this, in the single parameter setting where $\beta$ is assumed to
be known precisely (equivalently
$\mathcal{I}_{\beta\alpha}=\mathcal{I}_{\alpha\beta}=0$), Eq. (4) reduces to
$\Delta\alpha\geq 1/{\mathcal{I}}_{\alpha\alpha}$. As the QFIM is a positive
semi-definite matrix, it follows from Eq. (4) that the precision in single
parameter estimation is lower bounded by the multi-parameter counterpart [34].
Thus, while the single-parameter estimation framework gives an optimistic
lower bound on the precision, the multi-parameter case gives a pessimistic
one. This becomes particularly evident when considering cases where the
parameters of interest are related to one another. From Eq. (4), it is clear
that in order for the multiparameter problem to be well posed, the QFIM must
be an invertible quantity [45, 46], requiring that:

$\displaystyle\text{det}\,{\hat{\mathcal{I}}}={\mathcal{I}}_{\alpha\alpha}{\mathcal{I}}_{\beta\beta}-{\mathcal{I}}_{\alpha\beta}^{2}\neq
0.$ (5)

Thus, when the QFIM is singular, i.e., $\text{det}\,{\hat{\mathcal{I}}}=0$,
the quantum Cramér-Rao bound is not well defined, essentially implying that
the lowest possible uncertainty for estimating a parameter is infinite. Often,
the presence of the singularity can be interpreted as indicating that we are
not inferring distinct parameters $\alpha$ and $\beta$, but rather an
effective ratio, $\alpha/\beta$ [46]. In critical metrology, this becomes
particularly relevant since the aim is to leverage the sudden change in
Hamiltonian parameters near the critical point. However, at the critical
point, the ground state depends on many, if not all, system parameters, making
it apparent that the lack of knowledge about any system parameters other than
the central parameter of interest may lead to a complete loss of metrological
utility.

In most relevant settings, control parameters are implemented with a given
tolerance, making it more meaningful to consider bounding the attainable
precision on $\alpha$ when $\beta$ is known within some finite interval of
resolution. In full generality, we may model the sensitivity to parameter
$\alpha$ given some uncertainty in the driving parameter $\beta$ by
introducing a probability distribution, $p(\beta)$, obtained from sampling the
generated states for a given value of $\beta$. Formally, this can be achieved
by considering the probe state as an infinite mixture of states, each
corresponding to a different value of $\beta$. The corresponding sensitivity
is then given by evaluating the effective single parameter QFI for this family
of states, thereby explicitly accounting for imprecision or fluctuations in
$\beta$.

Critical metrology achieves increased sensitivity in parameter estimation by
exploiting properties of eigen states near the critical point of QPTs, and the
optimal sensitivity is quantified using Eq. (3). However, as we are dealing
with a mixture of states—specifically a mixture of ground states—we require
the ground state density matrix for a given choice of $\beta$, defined as
$\hat{\varrho}\left(\alpha,\beta\right)=\ket{\psi_{GS}}\bra{\psi_{GS}}$. To
sample the corresponding quantum state for a given choice of $\beta$, we then
weigh the ground state density matrix according to the probability
distribution $p(\beta)$. A natural choice is a Gaussian distribution, such
that
$p\left(\beta\right)=\tfrac{1}{\sigma\sqrt{2\pi}}\exp{{-\tfrac{1}{2}\left(\frac{\beta-\bar{\beta}}{\sigma}\right)^{2}}}$,
yielding on average a final state of the form:

$\displaystyle\hat{{\varrho}}\left(\alpha,\sigma\right)=\int_{-\infty}^{\infty}d\beta\frac{\exp{-\frac{1}{2}\left(\frac{\beta-\bar{\beta}}{\sigma}\right)^{2}}}{\sigma\sqrt{2\pi}}\hat{\varrho}\left(\alpha,\beta\right),$
(6)

with $\bar{\beta}$ the average value of parameter $\beta$, and $\sigma$ the
variance quantifying the (im)precision. The corresponding sensitivity is then
obtained from the QFI of the averaged state using Eq. (2), which we denote as
${\bar{\mathcal{I}}}_{\alpha\alpha}$. This scenario represents an
interpolation between the various relevant settings. For $\sigma\to 0$, we
recover the single parameter estimation case where $\beta$ is assumed to be
precisely known; while $\sigma\to\infty$ corresponds to the multi-parameter
setting where $\beta$ is unknown and, depending on the specific setting at
hand, the QFIM may become singular.

Figure 1: Left panel: Ising model. Right panel: LMG model. Sensitivity to the
bare frequency $\omega$ with uncertainty in the coupling $g$ quantified by
$\sigma/\omega$. In (a) and (c) as well as (e) and (g) the attainable
sensitivity to the frequency $\omega$ is shown for various degrees of
uncertainty for $N=20$ spins. In (a) and (e), the dotted red line shows the
ideal sensitivity $\mathcal{I}_{\omega\omega}^{SP}$ of the single parameter
case when $g$ is perfectly known. Solid lines show attainable sensitivity for
a given precision in $g$. (c) and (g) depict the full phase diagram for
parameter sensitivity. (b) and (f) show scaling of the peak QFI with a given
uncertainty, ${\bar{\mathcal{I}}}_{\omega\omega}$, in the coupling $g$ as a
function of system size $N$. (d) and (g) depict the maximal uncertainty at
which the QFI _feels_ the uncertainty of the control parameter $g$ as a
function of system size $N$.

Applications.—We apply our framework to the general class of Hamiltonians
given by:

$\displaystyle\hat{H}=\omega\sum_{i}^{N}\hat{\sigma}_{i}^{z}-\sum_{i\neq
j=1}^{N}g_{ij}\hat{\sigma}_{i}^{x}\hat{\sigma}_{j}^{x}$ (7)

with a transverse field $\omega$ that we wish to estimate and $g_{ij}$ the
interaction strength between the $i^{\text{th}}$ and $j^{\text{th}}$ spin,
which we assume is controllable. We will consider two paradigmatic examples of
the above: the transverse field Ising model (TFIM) when $g_{ij}\\!=\\!g$ is
non-zero only for the nearest neighbors [47, 48], and the Lipkin-Meshkov-Glick
(LMG) model when $g_{ij}\\!\\!=\\!\\!g/N$ [49, 50, 51, 52, 53] (see also
Appendix A for a toy-model Landau-Zener Hamiltonian). In both cases, it should
be immediately evident that one can perform a simple rescaling of the
Hamiltonian, and therefore the ground state will be dependent on the
parameters’ ratio, immediately implying that the QFIM is singular [34].

Explicitly, for the Ising limit assuming periodic boundary conditions, by
utilizing a Jordan-Wigner mapping and transforming to momentum space, we can
obtain an exact expression for the ground state of the many-body system,
$\ket{\psi_{GS}}_{N}=\bigotimes_{k>0}\ket{\psi_{GS}}_{k}$, where
$\ket{\psi_{GS}}_{k}=\cos{\theta_{k}/2}\ket{0}_{k}+\sin{\theta_{k}/2}\ket{1}_{k}$
(see Appendix B for details). Exploiting this and leveraging the fact that the
QFI is additive under a tensor product, such that ${\hat{\mathcal{I}}}_{\omega
g}\left[\ket{\psi_{GS}}_{N}\right]={\hat{\mathcal{I}}}_{\omega
g}\left[\ket{\psi_{GS}}_{k1}\right]+{\hat{\mathcal{I}}}_{\omega
g}\left[\ket{\psi_{GS}}_{k2}\right]+\ldots+{\hat{\mathcal{I}}}_{\omega
g}\left[\ket{\psi_{GS}}_{kN}\right]$, we obtain a closed-form expression for
the QFIM

$\displaystyle{\hat{\mathcal{I}}}_{\text{Ising}}=\sum_{k}\begin{pmatrix}\frac{g^{2}\sin^{2}{k}}{\left(g^{2}+\omega^{2}-2g\omega\cos{k}\right)^{2}}&-\frac{g\omega\sin^{2}{k}}{\left(g^{2}+\omega^{2}-2g\omega\cos{k}\right)^{2}}\\\
-\frac{g\omega\sin^{2}{k}}{\left(g^{2}+\omega^{2}-2g\omega\cos{k}\right)^{2}}&\frac{\omega^{2}\sin^{2}{k}}{\left(g^{2}+\omega^{2}-2g\omega\cos{k}\right)^{2}}\end{pmatrix},$
(8)

with $k=\pi(2n+1)/N$ and $n=0,1,2,\ldots,N/2-1$, which is clearly singular for
all $N$, irrespective of the distance to the critical point. Consequently, the
estimation of the bare frequency $\omega$ is strictly prohibited when the
coupling $g$ is unknown. The single-parameter sensitivity to the bare
frequency $\omega$ corresponds to the first element of Eq. (8).

Allowing for uncertainty in the coupling, we construct a mixture of ground
states sampled for a given value of $g$ according to Eq. (6). In Fig. 1, we
show the QFI for this averaged state, denoted
${\bar{\mathcal{I}}}_{\omega\omega}$, for the TFIM. Moreover, in Fig. 1(a), we
explicitly compare the ideal single-parameter sensitivity,
${\mathcal{I}}_{\omega\omega}$ shown in the dotted red line, with attainable
sensitivity for a given uncertainty, in the range
$\sigma/\omega\\!=\\![0,100]$ from solid light blue to solid dark blue for
$N=20$ spins. Crucially, we note that for small uncertainty in $g$, the QFI
retains the peak of the single-parameter case, and this peak sensitivity
decreases as our uncertainty in $g$ increases. Notably, in the limit
$\sigma\\!\rightarrow\\!\infty$, uncertainty in the coupling completely
deteriorates sensitivity to the bare frequency $\omega$, recovering the
expected multi-parameter result when the QFIM is singular. Fig. 1(c) shows the
full phase diagram of the QFI for $N=20$ spins as a function of the variance,
$\sigma/\omega$, where we clearly see that introducing uncertainty in our
knowledge of the coupling strength reduces and smooths out the peak
sensitivity.

In the best-case scenario when the coupling $g$ is perfectly known, the peak
sensitivity at the critical point $g=\omega$ as a function of system size
scales as $\left(N^{2}+N\right)/8\omega^{2}$. Herein lies the power of
critical quantum metrology, whereby treating the system size $N$ as a resource
for sensing, the QFI scales as $N^{\gamma}$ with an exponent which is
typically $\gamma>1$ with respect to parameter(s) which drive the phase
transition. Such scaling represents an advantage over the independent (non-
interacting) case where $\gamma=1$. An important consideration is therefore if
such an advantage is indeed attainable in the presence of relevant
experimental imprecision. In Fig. 1(b), we plot the scaling of the QFI in the
presence of uncertainty, ${\bar{\mathcal{I}}}_{\omega\omega}$ as a function of
system size $N$ for various degrees of imprecision in the control parameter
$g$. The red region indicates scaling that is worse than linear ($\gamma<1$),
the yellow region indicates scaling better than linear but worse than
quadratic ($2>\gamma>1$), and the green region indicates better than quadratic
scaling ($\gamma>2$). The red crosses indicate the point at which the QFI is
affected by the imprecision: specifically, when the QFI no longer scales as in
the ideal scenario when $g$ is known with absolute certainty,
$\left(N^{2}+N\right)/8\omega^{2}$. Importantly, we note from the red crosses
that larger system sizes are less robust to relevant uncertainties—indicating
an important experimental trade-off between attainable sensitivity endowed by
larger system sizes and their robustness to relevant perturbations. This
provides a crucial practical consideration in the design of criticality-
enhanced sensors, whereby there is an optimal trade-off between the
sensitivity offered by larger systems and their robustness to uncertainties.
Furthermore, treating the definition of criticality to hold strictly for an
infinite system, we note that in the asymptotic limit of system size any
finite perturbation or uncertainty would necessarily prohibit scaling larger
than $\gamma=1$, leading to a constant factor improvement, consistent with
Ref. [54]. Fig. 1(d) depicts the maximal uncertainty at which the QFI _feels_
the uncertainty of the control parameter $g$—the red crosses from (b), denoted
$\sigma_{F}/\omega$, as a function of system size $N$ [for clarity we do not
show all the generated data in (b), hence there are more red crosses in (d)
than in (b). For more details, see Appendix C]. From here we can see that
smaller system sizes are more robust to the uncertainty than bigger systems.
According to a numerical fit, $\sigma_{F}/\omega\approx 0.32\times N^{-0.97}$
in the TFIM, indicating how precisely the control parameter $g$ has to be
known as a function of $N$ to retain the same sensitivity as in the single-
parameter paradigm.

A qualitatively similar behavior can be demonstrated for the LMG model. The
isometric all-to-all coupling means it is convenient to recast Eq. (7) in
terms of collective spin operators,
$\hat{S}_{i}\equiv\sum_{n}\hat{\sigma}_{i}^{n}/2$ with $i\in\\{x,y,z\\}$ [41],

$\displaystyle\hat{H}=\omega\hat{S}_{z}-\frac{g}{N}\hat{S}_{x}^{2}.$ (9)

The LMG Hamiltonian experiences a second order QPT in the thermodynamic limit
at $g=g_{c}=\omega$ [55]. This can be explicitly seen by using the Holstein-
Primakoff transformation and applying the $N\rightarrow\infty$ approximation,

$\displaystyle\hat{H}=\omega\hat{a}^{\dagger}\hat{a}-\frac{g}{4}\left(\hat{a}^{\dagger}+\hat{a}\right)^{2}.$
(10)

In the thermodynamic limit, the eigenstates are squeezed Fock states of the
non-interacting system $|n_{\xi}\rangle=\hat{S}(\xi)|n\rangle$, where
$\hat{S}(\xi)=\exp{\frac{1}{2}\left(\xi^{*}\hat{a}^{2}-\xi\hat{a}^{\dagger
2}\right)}$ is the squeezing operator and $\xi=\frac{1}{4}\ln\\{1-g/\omega\\}$
is the squeezing parameter. At the critical point, the eigenstates become
infinitely squeezed, which is responsible for the non-analytic behavior at the
critical point [56], while for finite-sized systems, this non-analytic
behavior is smoothed out as a function of $N$ [57].

Since in the thermodynamic limit the ground state is completely defined by the
squeezing parameter $\xi$, the QFIM can be simply expressed as,

$\displaystyle\begin{split}\hat{\mathcal{I}}_{\text{LMG}}&=2\begin{pmatrix}[\partial_{\omega}\xi]^{2}&[\partial_{\omega}\xi][\partial_{g}\xi]\\\
[\partial_{\omega}\xi][\partial_{g}\xi]&[\partial_{g}\xi]^{2}\end{pmatrix}\\\
&=\frac{1}{32}\begin{pmatrix}\frac{\omega^{2}}{(\omega-g)^{2}\omega^{2}}&\frac{g\omega}{(\omega-g)^{2}\omega^{2}}\\\
\frac{\omega
g}{(\omega-g)^{2}\omega^{2}}&\frac{g^{2}}{(\omega-g)^{2}\omega^{2}}\end{pmatrix},\end{split}$
(11)

which, again due to the fact that the ground state depends on the ratio of
parameters, is evidently singular regardless of the distance to the critical
point. In order to calculate the QFI matrix elements for a finite-size system
including uncertainties in the control parameter $g$, we resort to using
numerical calculations. The uncertainty of the control parameter $g$ is
incorporated in the same way as for the TFIM. The exemplary QFI with respect
to $\omega$ for $N=20$ spins is presented in Fig. 1(e) and the corresponding
full phase diagram in the presence of uncertainty in Fig. 1(g). We also plot
the maximal attainable QFI as a function of the system size for various levels
of uncertainty in the control parameter [see Fig. 1(f)]. Interestingly,
although the LMG model is an all-to-all interacting system, the QFI scales
less than quadratically as in the case of nearest-neighbor interacting TFIM,
$\gamma=4/3$. Similar to the TFIM, we also notice that for every $N$ there
exists a corresponding $\sigma_{F}/\omega$ at which the QFI stops following
the ideal case single-parameter scaling, denoted by the red crosses. The
numerical fit reveals that $\sigma_{F}/\omega\approx 0.2\times N^{-0.8}$,
shown in Fig. 1(h).

Conclusions.—In this work, we have developed a framework to characterize the
effectiveness of critical quantum probes, taking into account inherent
uncertainties in other parameters of the model which we expect to be the most
serious obstacle for achieving quantum enhancement in critical sensing. We
demonstrate that this framework interpolates between the best and worst-case
scenarios, as captured by single and multiparameter estimation settings. While
the power of critical quantum probes typically lies in the scaling of the
quantum Fisher information with respect to system size $N$, we reveal that
practical settings involving relevant uncertainties exhibit a fundamental
trade-off between the sensitivity offered by larger system sizes and their
robustness to imprecision. Furthermore, we show that in the asymptotic limit
of system size, any finite perturbation negates the advantage in quantum
Fisher information scaling.

We applied our framework to two paradigmatic spin systems: the Lipkin-Meshkov-
Glick model and the Transverse Field Ising Model. In doing so, we also
provided an analytical approach for determining the quantum Fisher information
matrix and its determinant for any system that can be mapped to free fermions.
Therefore, we anticipate that our results can be extended to a wide range of
other spin systems, such as $XY$, $XXZ$ models, and topological systems like
the Su-Schrieffer-Heeger and Chern models [58].

For the models explicitly considered, we have shown that for every system
size, there exists a range of uncertainties in the control parameters for
which the predicted critical enhancement of metrological utility in the
scaling of quantum Fisher information can be maintained. Our results
demonstrate that, in critical metrology, while uncertainty in the precise
value of other model parameters will ultimately limit the maximal precision
with which the parameter of interest can be determined, a useful quantum
advantage can still be achieved. This establishes that, even in the presence
of such uncertainties, critical systems remain viable probes for quantum-
enhanced sensing tasks.

###### Acknowledgements.

The authors are grateful to Victor Montenegro, Chiranjib Mukhopadhyay, and
Andrew Mitchell for fruitful discussions and thank the organizers of QUMINOS.
G.M. acknowledges support from Equal1 Laboratories Ireland Limited. SC
acknowledges support from the John Templeton Foundation Grant ID 62422 and the
Alexander von Humboldt Foundation. K.G. was supported by the Lise-Meitner
Fellowship M3304-N of the Austrian Science Fund (FWF).

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##  I Appendix A: quantum Fisher information matrix for Landau-Zener toy-model

The Landau-Zener model [59] describes a two-level system in a control field
$g$,

$\displaystyle\hat{H}_{\mathrm{LZ}}=\frac{\omega}{2}\hat{\sigma}_{z}-\frac{g}{2}\hat{\sigma}_{x}.$
(A1)

where $\omega$ is the level splitting for $g=0$, and $\hat{\sigma}_{i}$ is the
$i$th Pauli matrix. Being a single-particle system, the Landau-Zener model
does not exhibit a QPT. However, it features an avoided crossing at $g=0$ and
can therefore be considered as a toy model for criticality [60, 61, 30]. The
instantaneous ground state of the Landau-Zener model is given by,

$\displaystyle|\mathrm{\psi_{0}}\rangle=-\frac{g+\sqrt{\omega^{2}+g^{2}}}{\sqrt{2g\left(g+\sqrt{\omega^{2}+g^{2}}\right)+2\omega^{2}}}|\\!\downarrow\,\rangle$
(A2)
$\displaystyle+\frac{\omega}{\sqrt{2g\left(g+\sqrt{\omega^{2}+g^{2}}\right)+2\omega^{2}}}|\\!\uparrow\,\rangle.$
(A3)

Using Eq. (3) we can readily obtain the QFI matrix for the above ground state,

$\displaystyle{\hat{\mathcal{I}}}_{LZ}=\begin{pmatrix}\frac{g^{2}}{\left(g^{2}+\omega^{2}\right)^{2}}&-\frac{g\omega}{\left(g^{2}+\omega^{2}\right)^{2}}\\\
-\frac{\omega
g}{\left(g^{2}+\omega^{2}\right)^{2}}&\frac{\omega^{2}}{\left(g^{2}+\omega^{2}\right)^{2}}\end{pmatrix}.$
(A4)

The above QFI matrix is always singular regardless of the distance to the
avoided crossing, i.e.,
$\rm{\det}\left[\hat{\mathcal{I}}_{LZ}\right]={\mathcal{I}}_{\omega\omega}{\mathcal{I}}_{gg}-{\mathcal{I}}_{\omega
g}^{2}=0$. In Fig. A1(a) we show the elements of the QFIM as a function of
driving parameter $g$ when $\omega=1$, with ${\mathcal{I}}_{\omega\omega}$,
${\mathcal{I}}_{gg}$, and ${\mathcal{I}}_{\omega g}$ depicted in the solid
blue, dash-dotted red, and dashed grey lines respectively. We note that the
quantity $\omega^{2}{\mathcal{I}}_{\alpha\beta}$ is a universal scaling
function of the rescaled parameter $g/\omega$. In general, we may expect
accidental crossings of the elements of the QFIM, in (a) we highlight the
coalescence when $\omega=g=1$. We attribute the singularity of the QFIM in the
Landau-Zener model to the fact that we are not sensing separately parameters
$\omega$ and $g$, but rather, an effective ratio of $g/\omega$. As we are
sensing an effective ratio of parameters, precisely when $\omega=g$, the ratio
$g/\omega=1$, and elements of the QFIM attain the same value, and coalesce.

Figure A1: Landau-Zener model: (a) Elements of the QFIM for various $g$ when
the bare frequency $\omega=1$. (b) Sensitivity to the bare frequency $\omega$
with various degrees of uncertainty in the control parameter $g$. The dashed
red line shows the ideal sensitivity ${\mathcal{I}}_{\omega\omega}^{SP}$ of
the single parameter scenario when $g$ is known. The solid blue lines show
attainable QFI ${\bar{\mathcal{I}}}_{\omega\omega}$ with uncertainty in the
coupling $g$. (c) The full phase diagram in attainable QFI with varying degree
of precision quantified by $\sigma$.

The multiparameter case represents the scenario when neither the bare
frequency $\omega$ or the driving field $g$ are known. As in the main text, we
investigate the scenario in which we wish to estimate the bare frequency
$\omega$, but we have some finite window of resolution in the driving
parameter $g$. This finite window of resolution represents the degree of
uncertainty we have, quantified by the variance of a Gaussian distribution
$\sigma$. The corresponding sensitivity is found viz.,

$\displaystyle\hat{{\varrho}}\left(\omega,\sigma\right)=\int_{-\infty}^{\infty}dg\frac{\exp{-\frac{1}{2}\left(\frac{g-\bar{g}}{\sigma}\right)^{2}}}{\sqrt{2\pi\sigma^{2}}}\hat{\varrho}\left(\omega,g\right),$
(A5)

with $\hat{\varrho}\left(\omega,g\right)$ the ground state density matrix of
Eq. (A2), $\sigma$ is the variance, and $\bar{g}$ is the expected value of the
driving parameter $g$. In Fig. A1 (b) the ideal sensitivity when the control
field $g$ is perfectly known—i.e., the single parameter sensitivity,
${\mathcal{I}}_{\omega\omega}^{SP}$, is shown in the dashed red line. The
solid blue lines depict the attainable sensitivity
${\bar{\mathcal{I}}}_{\omega\omega}$ for various uncertainty quantified by
$\sigma$. We note that in the limit $\sigma\rightarrow\infty$, when the
driving parameter $g$ is essentially unknown, there is zero attainable
sensitivity to the bare frequency $\omega$. This is because the
$\sigma\rightarrow\infty$ limit is essentially the multiparameter scenario: as
the QFIM is singular, when the driving parameter $g$ is unknown, the frequency
$\omega$ cannot be inferred. In scenarios where the QFIM is non-singular, we
would expect the $\sigma\rightarrow\infty$ limit to yield some finite
sensitivity. In panel (c) we show the full phase diagram for the attainable
sensitivity when there is uncertainty in the driving parameter $g$.

## Appendix B: quantum Fisher information matrix for transverse field Ising
model

In this section, we provide a simple prescription for finding a closed form
solution for the QFIM and determinant of QFIM for any system that can be
mapped to free fermions. We present these results by considering the TFIM as a
paradigmatic example. The system can be diagonalized by first using the
Jordan-Wigner transformation and then going to momentum space to obtain,

$\displaystyle\hat{H}_{k}=2\sum_{k}\left(~{}\left[\omega-g\cos{k}\right]\left[\hat{c}_{k}^{\dagger}\hat{c}_{k}^{\phantom{\dagger}}-\hat{c}_{-k}^{\dagger}\hat{c}_{-k}^{\phantom{\dagger}}\right]-g\sin{k}\left[ie^{-2i\phi}\hat{c}_{k}^{\dagger}\hat{c}_{-k}^{\dagger}-ie^{2i\phi}\hat{c}_{k}^{\phantom{\dagger}}\hat{c}_{-k}^{\phantom{\dagger}}\right]~{}\right),$
(A6)

where the arbitrary overall phase, $\phi$, was introduced in writing the model
in the quasi-momentum $k$-space basis. The phase factor, $\phi$, simply
corresponds to changing the phase of pairs created in momentum space. All
Hamiltonian terms now come in momentum space pairs, $\left(k,-k\right)$, where
one can define the full Hamiltonian by considering only the positive momentum
$k$ values, corresponding to $k=\pi\left(2n+1\right)/N$ where
$n=0,1,2,...,N/2-1$. As noted in [62], the Hamiltonian’s $\hat{H}_{k}$ live in
a 4-dimensional space spanned by the states
$\\{\ket{0},\hat{c}_{k}^{\dagger}\ket{0},\hat{c}_{-k}^{\dagger}\ket{0},\hat{c}_{k}^{\dagger}\hat{c}_{-k}^{\dagger}\ket{0}\\}$,
which has a single non-trivial $2\times 2$ block. To deal with the necessary
combination of states living in these non-trivial blocks, spanned by
$\\{\ket{0},\\{\hat{c}_{k}^{\dagger}\hat{c}_{-k}^{\dagger}\ket{0}\\}$, we
introduce fermionic two-component spinors

$\hat{\Psi}_{k}^{\phantom{\dagger}}=\begin{pmatrix}\hat{c}_{k}^{\phantom{\dagger}}\\\
\hat{c}_{-k}^{\dagger}\end{pmatrix},\qquad\hat{\Psi}_{k}^{\dagger}=\begin{pmatrix}\hat{c}_{k}^{\dagger}&\hat{c}_{-k}^{\phantom{\dagger}}\end{pmatrix},$
(A7)

allowing us then to write the Hamiltonian of each momentum space block,
$\hat{H}_{k}$, in the form

$\hat{H}_{k}=2\Psi_{k}^{\dagger}\begin{pmatrix}\left(\omega-g\cos{k}\right)&-ie^{-2i\phi}g\sin{k}\\\
ie^{2i\phi}g\sin{k}&-\left(\omega-g\cos{k}\right)\end{pmatrix}\Psi_{k}^{\phantom{\dagger}},$
(A8)

where $\hat{H}=\sum_{k>0}\hat{H}_{k}$. Making the common choice of phase,
$\phi=-\pi/4$, corresponds to working in the $x$-$z$ pseudo-spin basis, allows
us to write the Hamiltonian in a more familiar manner viz. the pseudo-spin
Pauli matrices as,

$\hat{H}_{k}^{spin}=2\Psi_{k}^{\dagger}\left[\left(\omega-g\cos{k}\right)\sigma_{z}+\left(g\sin{k}\right)\sigma_{x}\right]\Psi_{k}^{\phantom{\dagger}}.$
(A9)

Solving the $2\times 2$ eigenvalue problem for the pseudo-spin Hamiltonian
provides us with the ground state energy of each momentum-space block,

$\epsilon_{k}=-2\sqrt{g^{2}+\omega^{2}-2g\omega\cos{k}},$ (A10)

with a corresponding (un-normalised) ground state solution of each block given
by

$\ket{\psi_{GS}}_{k}=\cos{\frac{\theta_{k}}{2}}\ket{0}_{k}+\sin{\frac{\theta_{k}}{2}}\ket{1}_{k},$
(A11)

where the angle is defined as
$\tan{\theta_{k}}=\frac{\omega\sin{k}}{\omega\cos{k}-g}$. The ground state for
an $N$-site system is simply given as the tensor product
$\ket{\psi_{GS}}_{N}=\bigotimes_{k>0}\ket{\psi_{GS}}_{k}$ of the ground state
of each momentum space block $\hat{H}_{k}$. The QFI is additive under tensor
product and for a pure state of the form
$\ket{\Psi}=\ket{\psi}_{A}\otimes\ket{\psi}_{B}\otimes....,\ket{\psi}_{N}$ can
be decomposed as,

${\mathcal{I}}_{\alpha\beta}\left[\ket{\psi}_{A}\otimes\ket{\psi}_{B}\otimes...,\ket{\psi}_{N}\right]={\mathcal{I}}_{\alpha\beta}\left[\ket{\psi}_{A}\right]+{\mathcal{I}}_{\alpha\beta}\left[\ket{\psi}_{B}\right]+...,{\mathcal{I}}_{\alpha\beta}\left[\ket{\psi}_{N}\right]$
(A12)

with ${\mathcal{I}}_{\alpha\beta}\left[\ket{\psi}_{i}\right]$ defined in Eq.
(3). By exploiting the additive property of the QFI under tensor product and
the fact that the ground state of a free fermion system can be written as
$\ket{\psi_{GS}}_{N}=\bigotimes_{k>0}\ket{\psi_{GS}}_{k}$, elements of the
QFIM can readily be obtained

${\mathcal{I}}_{\alpha\beta}=4\sum_{k}\left[\langle\partial_{\alpha}\psi_{GS}|\partial_{\beta}\psi_{GS}\rangle_{k}-\langle\partial_{\alpha}\psi_{GS}|\psi_{GS}\rangle_{k}\langle\psi_{GS}|\partial_{\beta}\psi_{GS}\rangle_{k}\right],$
(A13)

as the sum of the QFI of each momentum space block. Alternatively, the angle
$\theta_{k}$ containing information regarding the unitary transformation
required to diagonalize the system can be used to compute elements of the QFIM
viz.,

${\mathcal{I}}_{\alpha\beta}=\sum_{k}\partial_{\alpha}\arctan{\theta_{k}}\times\partial_{\beta}\arctan{\theta_{k}}.$
(A14)

Equipped with this, the QFIM can readily be obtained for arbitrary system size
$N$ of the TFIM as,

$\displaystyle\hat{\mathcal{I}}=\sum_{k}\begin{pmatrix}\frac{g^{2}\sin^{2}{k}}{\left(g^{2}+\omega^{2}-2g\omega\cos{k}\right)^{2}}&-\frac{g\omega\sin^{2}{k}}{\left(g^{2}+\omega^{2}-2g\omega\cos{k}\right)^{2}}\\\
-\frac{g\omega\sin^{2}{k}}{\left(g^{2}+\omega^{2}-2g\omega\cos{k}\right)^{2}}&\frac{\omega^{2}\sin^{2}{k}}{\left(g^{2}+\omega^{2}-2g\omega\cos{k}\right)^{2}}\end{pmatrix},\qquad\text{with
$k=\pi(2n+1)/N$ and $n=0,1,2,\ldots,\frac{N}{2}-1$},$ (A15)

where at the critical point of the TFIM, $g=\omega$, elements of the above QFI
matrix follow a triangular sequence and coalesce to $(N^{2}+N)/8\omega^{2}$.
The determinant of the QFIM follows as determinant of the sum of the QFIM of
each momentum space block,

$\rm{det}\left[\hat{\mathcal{I}}\right]=\rm{\det}\left[\sum_{k}\begin{pmatrix}\frac{g^{2}\sin^{2}{k}}{\left(g^{2}+\omega^{2}-2g\omega\cos{k}\right)^{2}}&-\frac{g\omega\sin^{2}{k}}{\left(g^{2}+\omega^{2}-2g\omega\cos{k}\right)^{2}}\\\
-\frac{g\omega\sin^{2}{k}}{\left(g^{2}+\omega^{2}-2g\omega\cos{k}\right)^{2}}&\frac{\omega^{2}\sin^{2}{k}}{\left(g^{2}+\omega^{2}-2g\omega\cos{k}\right)^{2}}\end{pmatrix}\right],$
(A16)

which in the case of the QFIM is singular for all $N$, but in full generality
is not strictly singular for any free fermion system and set of parameters.

##  II Appendix C: Additional figures

For the sake of clarity, in Figs 1(b) and (f) from the main text, we presented
only part of the data used to generate Figs 1(d) and (h). Here we provide all
the relevant generated data and plot them in Fig. A2. Note that because of the
finite resolution in $\sigma$, the red crosses in Figs 1(d) and (h) appear
bumpy.

Figure A2: Scaling of the peak QFI ${\bar{\mathcal{I}}}_{\omega\omega}$, with
a given uncertainty $\sigma/\omega$ in the control parameter $g$ in shades of
blue, as a function of system size $N$ in the TFIM (a) and LMG (b). Red
crosses denote the value at which the QFI _feels_ the uncertainty of the
control parameter $g$ and shows the full compliment of data within the
relevant uncertainty range used to generate panels (d) of Fig. 1 and Fig. 1 of
the main text.