# Criticality-amplified quantum probing of a spontaneous collapse model

G. Zicari  Centre for Quantum Materials and Technologies, School of
Mathematics and Physics, Queen’s University Belfast, BT7 1NN, United Kingdom
M. Carlesso  Department of Physics, University of Trieste, Strada Costiera 11,
34151 Trieste, Italy  Istituto Nazionale di Fisica Nucleare, Trieste Section,
Via Valerio 2, 34127 Trieste, Italy  A. Trombettoni  Department of Physics,
University of Trieste, Strada Costiera 11, 34151 Trieste, Italy  CNR-IOM
DEMOCRITOS Simulation Center, Via Bonomea 265, 34136 Trieste, Italy  M.
Paternostro  Quantum Theory group, Dipartimento di Fisica e Chimica - Emilio
Segrè, Università degli Studi di Palermo, via Archirafi 36, I-90123 Palermo,
Italy  Centre for Quantum Materials and Technologies, School of Mathematics
and Physics, Queen’s University Belfast, BT7 1NN, United Kingdom

###### Abstract

Spontaneous collapse models, which are phenomenological mechanisms introduced
and designed to account for dynamical wavepacket reduction, are attracting a
growing interest from the community interested in the characterisation of the
quantum-to-classical transition. Here, we introduce a quantum-probing approach
to the quest of deriving metrological upper bounds on the free parameters of
such empirical models. To illustrate our approach, we consider an extended
quantum Ising chain whose elements are – either individually or collectively –
affected by a mechanism responsible for spontaneous collapse. We explore
configurations involving out-of-equilibrium states of the chain, which allows
us to infer information about the collapse mechanism before it is completely
scrambled from the state of the system. Moreover, we investigate potential
amplification effects on the probing performance based on the exploitation of
quantum criticality.

##  I Introduction

The quantum measurement problem, which was made famous by the Schrödinger-cat
paradox, is one of the most controversial aspects in the current formulation
of quantum theory. In order to tackle it, models for the potential spontaneous
collapse of the wavefunction of a quantum system have been built, leveraging
on suitable modifications of the quantum axioms [1]. Specifically, the
Schrödinger equation describing the evolution of the state of a quantum system
is modified so as to include stochastic and non-linear terms.

Consequently, the dynamics for the density operator $\hat{\rho}$ includes a
term describing the spontaneous, fundamental decoherence-like mechanisms,
which suppresses the quantum coherence in a specific basis. Among the various
formulations proposed to account for this, the Continuous Spontaneous
Localisation (CSL) model [2] and the Diósi-Penrose (DP) one [3, 4] are two of
the most celebrated. While the latter attributes the origin of the collapse
mechanism to gravity, the former is a fully phenomenological model and it is
typically taken as the paradigm of collapse models. In the last decade, the
experimental verification of collapse models has attracted a strong interest
[5, 6]. Owing to such endeavours, some important quantitative bounds on the
various parameters characterizing such models have been established.
Importantly, the full coverage of the models’ parameter space is still in need
of significant advances, including the deployment of new and sophisticated
techniques for their extensive exploration [7, 8].

The framework of quantum probing makes use of simple quantum systems coupled
with complex environments to precisely extract information about environmental
parameters, such as its temperature or spectral density [9, 10, 11, 12]. The
expectation values of the system observables are indeed sensitive to changes
in those parameters [13], as one can assess even in the linear response regime
[14, 15]. Quantum probes are recognised for their capacity to achieve superior
precision compared to classical sensors using equivalent resources [16].
Significant advantages are known to stem from the preparation of the probes in
specific non-classical, yet fragile [17], states [18, 19]. A way around the
problem embodied by the fragility of such an approach is offered by the
exploitation of sensitivity-enhancing mechanisms based on many-body effects:
the strong, long-range correlations emerging in the state of a quantum many-
body system near its critical point is expected to generally enhance the
sensing capabilities and result, in turn, in a higher estimation precision
[20, 21, 22, 23, 24]. More recently, several protocols have been introduced to
show that such an enhancement is not only featured in the thermodynamic limit,
but also in finite-component quantum phase transitions [25, 26, 27].

In this paper, we leverage criticality to amplify the effects of the collapse
of a quantum many-body system near a critical point to demonstrate the
criticality-enhanced estimate of the corresponding collapse rate. We show the
effectiveness of such an approach using the paradigmatic case of a quantum
Ising chain, whose elements are individually affected by a collapse mechanism,
coupled to a simple two-level quantum probe. In order to investigate the
influences of criticality, we explore the metrological performance of the
probing system – quantitatively provided by the Quantum Fisher Information
(QFI) – across an ample region of the phase diagram of the chain, highlighting
the situations that correspond to an enhanced sensitivity of the QFI to the
collapse rate. Moreover, we show that possible spatial correlations across the
collapse mechanisms affecting the elements of the chain do not alter
significantly the picture gathered by assuming individual events, which can
thus be considered as representative of the typical performance of the
proposed probing scheme.

The remainder of this manuscript is organised as follows. In Sec. II we give a
short outline of the probing mechanism at hand, discussing in particular the
use of non-equilibrium states. Sec. III is dedicated to the description of the
physical model at hand, the transverse Ising model, which is then used to
identify working points where a criticality-enhanced estimate of the rate of a
spontaneous collapse model is possible [cf. Sec. IV]. In Sec. V we give an
account of the set of results stemming from our study, and present an
assessment of the effects that spatially correlated collapse mechanisms would
have on the performance of our quantum probing approach. Finally, while Sec.
VI is devoted to both our conclusions and the forward look stemming from our
analysis, a set of Appendices report some technical aspects of our study.

##  II Non-equilibrium probing

Let us consider a typical scenario for an estimation problem. Suppose that one
has a generic quantum system $S$, whose dynamics depends on a parameter
$\lambda$ to be estimated. One can follow two approaches, which we can dub as
_equilibrium_ or _non-equilibrium_ probing. The former corresponds to the
following scenario: the dynamics of the system $S$ converges to a steady state
that encodes information about $\lambda$, thus we use that state to estimate
the parameter itself. By contrast, the latter non-equilibrium approach
consists in letting the system $S$ interact with a second one, called _probe_
, that can be fully controlled. This allows us to dynamically infer
information about the parameter $\lambda$ as the evolution time goes by,
without directly interfere with $S$. These approaches are often used in
quantum estimation problems, where they are sometimes referred to as _static_
or _dynamical_ encoding, respectively [28]. For instance, they are used in the
context of quantum thermometry [29, 30], where the parameter to be estimated
is the system temperature [31, 32, 33, 34, 35].

In a similar spirit, in what follows we will choose to apply a non-equilibrium
probing scheme to a particular problem of fundamental physics. To this end, we
can choose a two-level system (qubit) to serve as a probe for the system $S$.
The non-equilibrium scheme is described by the following Hamiltonian:

$\hat{H}=\hat{H}_{S}\otimes\mathbbm{1}_{p}+\mathbbm{1}_{S}\otimes\hat{H}_{p}+\hat{H}_{I}\,,$
(1)

where $\hat{H}_{S}$ and $\hat{H}_{p}$ are the Hamiltonian operators of the
system and the probe, respectively. The Hamiltonian $\hat{H}_{I}$ accounts for
the interaction between them, while $\mathbbm{1}_{S}$ and $\mathbbm{1}_{p}$
are the identity operators defined over the Hilbert spaces associated with the
system and the probe, respectively.

This non-equilibrium scheme requires solving the dynamics of such composite
system. Let us assume that the dynamics is governed by the following dynamical
equation:

$\frac{\operatorname{d}\\!\hat{\rho}}{\operatorname{d}\\!t}=\mathcal{L}_{\lambda}\hat{\rho},$
(2)

where $\mathcal{L}_{\lambda}$ is the Liouvillian (super)-operator encompassing
both the unitary and non-unitary contributions. Note that
$\mathcal{L}_{\lambda}$ implicitly depends on $\lambda$, which is the
parameter we want to estimate.

In the following, we narrow down our selection of Liouvillians by considering
those of the standard form [36]

$\displaystyle\mathcal{L}_{\lambda}\hat{\rho}\equiv-i[\hat{H},\hat{\rho}]+D(\hat{\rho})\,,$
(3)

where $D(\hat{\rho})$ is the dissipator in the standard Lindblad form [37, 38,
39]. Note that we will work in units such that $\hbar=k_{\text{B}}=1$. In
principle, either the unitary or the non-unitary part can bear dependence on
$\lambda$. In our study, however, we consider the case in which $\lambda$ is a
parameter characterising the dissipator.

Given the initial state $\hat{\rho}(0)$, one can formally solve Eq. (2),
yielding $\hat{\rho}(t)=e^{t\mathcal{L}_{\lambda}}\hat{\rho}(0)$ at any time
$t$. One can thus trace out the degrees of freedom of the system $S$,
obtaining the state of the probe
$\hat{\rho}_{p}(t)=\operatorname{tr}_{S}\hat{\rho}(t)$, which – to ease the
notation – will be denoted as $\hat{\rho}_{\lambda}$, as the probe density
matrix implicitly depends on the parameter $\lambda$ to be estimated.

Our problem can be rephrased through the language of quantum metrology [40,
41, 42, 43], where the key quantity is represented by the Quantum Fisher
Information (QFI), defined as

$G(\lambda)\equiv\operatorname{Tr}\left[\hat{\varrho}_{\lambda}\hat{\Lambda}_{\lambda}^{2}\right]\,,$
(4)

where the symmetric logarithmic derivative $\hat{\Lambda}_{\lambda}$ (such
that $\hat{\Lambda}_{\lambda}^{\dagger}=\hat{\Lambda}_{\lambda}$) is
implicitly defined through the equation

${\partial_{\lambda}\hat{\varrho}_{\lambda}}=\frac{1}{2}(\hat{\Lambda}_{\lambda}\hat{\varrho}_{\lambda}+\hat{\varrho}_{\lambda}\hat{\Lambda}_{\lambda})\,.$
(5)

The QFI provides the ultimate limit to the precision of the estimate of the
unknown parameter $\lambda$, as it enters in the so-called _quantum Cramér-Rao
bound_ for the variance $\operatorname{Var}\left(\lambda\right)$ of the
unbiased estimator, i.e.

$\operatorname{Var}\left(\lambda\right)\geq[MG(\lambda)]^{-1}\,,$ (6)

where $M$ is the number of independent measurements performed on the quantum
system. Note that the QFI is independent of the particular measurement that
has been performed, thus it is an intrinsic feature of the family of the
probing states [40]. From Eq. (6), one can easily deduce that for a given $M$,
maximising the QFI $G(\lambda)$ leads to the mimimum lower bound for the
variance. Our aim is to investigate under which physical conditions one can
maximise $G(\lambda)$.

In order to calculate the QFI, we need a closed formula which specialises the
general definition in Eq. (4) to the case of a qubit. Let us suppose that we
can diagonalise the density matrix $\hat{\varrho}_{\lambda}$ as
$\hat{\varrho}_{\lambda}=\varrho_{+}\left|\psi_{+}\right\rangle\\!\left\langle\psi_{+}\right|+\varrho_{-}\left|\psi_{-}\right\rangle\\!\left\langle\psi_{-}\right|$,
which allows us to express the QFI as

$\displaystyle G(\lambda)$
$\displaystyle=\sum_{j=\pm}\frac{(\partial_{\lambda}\varrho_{j})^{2}}{\varrho_{j}}{+}2\kappa\left[|\langle\psi_{-}|\partial_{\lambda}\psi_{+}\rangle|^{2}{+}|\langle\psi_{+}|\partial_{\lambda}\psi_{-}\rangle|^{2}\right]\,,$
(7)

with

$\kappa=\frac{(\varrho_{+}-\varrho_{-})^{2}}{\varrho_{+}+\varrho_{-}}=\left(1-2\varrho_{+}\right)^{2}\,,$
(8)

where we used the normalization condition $\varrho_{+}+\varrho_{-}=1$. The
derivatives of the eigenvectors are decomposed over the basis
$\\{\left|0\right\rangle,\left|1\right\rangle\\}$ of eigenstates of the $z$
Pauli operator $\hat{\sigma}^{z}=\left|1\right\rangle\\!\left\langle
1\right|-\left|0\right\rangle\\!\left\langle 0\right|$, i.e.

$\displaystyle\left|\partial_{\lambda}\psi_{\pm}\right\rangle=\partial_{\lambda}{\langle
0|\psi_{\pm}\rangle}\left|0\right\rangle+\partial_{\lambda}{\langle
1|\psi_{\pm}\rangle}\left|1\right\rangle.$ (9)

##  III The model

To introduce the model, we first focus on the system $S$ alone. Let us
consider the latter being a single two-level system undergoing a spontaneous
collapse dynamics. The latter can be formally expressed via a stochastic and
non-linear Schrödinger equation [1], which can be recast in the form of a
master equation for the statistical operator $\hat{\rho}_{S}$, i.e.

$\frac{\operatorname{d}\\!\hat{\rho}_{S}}{\operatorname{d}\\!t}=-i[\hat{H}_{S},\hat{\rho}_{S}]+D(\hat{\rho}_{S})\,.$
(10)

The unitary part of the dynamics is controlled by the Hamiltonian
$\hat{H}_{S}=(\omega_{0}/2)\hat{\sigma}^{x}$, where $\omega_{0}$ is the energy
splitting between the two levels, and
$\hat{\sigma}^{x}=\left|1\right\rangle\\!\left\langle
0\right|+\left|0\right\rangle\\!\left\langle 1\right|$ is the $x$ Pauli
operator. The collapse mechanism is effectively seen as the effect of an
external bath with which the system interacts. The latter leads to a non-
unitary evolution, which can be expressed through the following dissipator in
the Lindblad form [44, 45]:

$D(\hat{\rho}_{S})=-\frac{\lambda}{2}\,[\hat{\sigma}^{z},[\hat{\sigma}^{z},\hat{\rho}_{S}]]\,,$
(11)

where $\lambda$ is the _collapse rate_ , which is the parameter we aim at
estimating. Note that free Hamiltonian $\hat{H}_{S}$ does not commute with the
operator $\hat{\sigma}^{z}$ entering in Eq. (11), which results in non-trivial
dynamics, bringing about both decoherence and dissipation. The dynamics
governed by Eq. (10) can be analytically solved, as shown in Ref. [44]. The
solution is found by vectorising Eq. (10), i.e. by following the prescriptions
given in Appendix A. By so doing, one can obtain the state of the system at
any time $t$, i.e. $\hat{\rho}_{S}=\hat{\rho}_{S}(t)$. We can recall that the
density matrix of a qubit can be written as

$\displaystyle\hat{\rho}_{S}=\frac{\mathbbm{1}+\bm{\sigma}\cdot\bm{\tau}}{2}\,,$
(12)

where $\bm{\sigma}\equiv(\hat{\sigma}^{x},\hat{\sigma}^{y},\hat{\sigma}^{z})$
is the Pauli vector, while $\bm{\tau}\equiv(\tau_{x},\tau_{y},\tau_{z})$ is
the Bloch vector, whose components are defined as
$\tau_{k}\equiv\operatorname{Tr}(\hat{\sigma}^{k}\hat{\rho}_{S})$, with
$k=x,y,z$.

In Fig. 1, we plot the components of the Bloch vector against time, i.e.
$\tau_{k}=\tau_{k}(t)$, together with the corresponding Bloch sphere
representation. Furthermore, it is worth mentioning that one can compute the
Liouvillian gap $\Delta_{\mathcal{L}}$, which identifies the rate of the
slowest relaxation mode [46, 47], thus correctly providing an estimate of the
typical system relaxation time. The details of the calculation are given in
Appendix B. One sees that the generic state at any time $t$, i.e.
$\hat{\rho}_{S}=\hat{\rho}_{S}(t)$, bears dependence on the collapse rate
$\lambda$. However, this dependence is completely washed out by the dynamical
process, as one can immediately deduce from the form of the steady state
$\hat{\rho}_{\rm ss}=\mathbbm{1}/2$. In other words, regardless of the choice
of the initial state, the system relaxes towards a unique maximally mixed
state [48, 44]. These feature have an important consequence for our estimation
problem: We cannot indeed infer any information about $\lambda$ by solely
looking at the steady state. Therefore, in order to estimate the collapse rate
$\lambda$, we need to devise a different strategy that is able to capture
information about the collapse rate before the system reaches its unique
steady state.

Figure 1: Dynamics of a qubit undergoing the collapse mechanism described by
the dissipator in Eq. (11). We simulate the system dynamics choosing
$\left|1\right\rangle$ as initial state, and $\ \lambda=0.1\omega_{0}$. On the
left, we show the plots of the Bloch vector components
$\tau_{k}=\langle\hat{\sigma}^{k}\rangle$ (with $k=x,y,x$), along with the
envelopes $e^{\pm t\Delta_{\mathcal{L}}}$, which captures the typical
relaxation time, with $\Delta_{\mathcal{L}}$ being the Liouvillian gap. On the
right, we display the corresponding trajectory in the Bloch sphere.

###  III.1 Non-equilibrium probing

To probe the collapse dynamics of a single qubit undergoing the spontaneous
collapse mechanism described by the dissipator in Eq. (10), one can follow the
non-equilibrium scheme described in Sec. II. We thus introduce a second qubit
that serves as a probe for the first one. The total dynamics is then given by
Eq. (1), where we take $\hat{H}_{p}=\omega_{p}\hat{\sigma}^{x}_{p}/2$ as the
Hamiltonian of the probe, while we engineer the interaction between the two
qubits to be along $z$, i.e.

$\hat{H}_{I}=g\,\hat{\sigma}^{z}_{S}\otimes\hat{\sigma}^{z}_{p}\,,$ (13)

where $g$ is the coupling constant. Our scheme is based on the assumption to
have full control on the probe, whereas the system is subject to the collapse
mechanism, as described by Eq. (10), where the Pauli operator need to be
suitably defined as
$\hat{\sigma}^{z}\to\hat{\sigma}^{z}_{S}\otimes\mathbbm{1}_{p}$. Similarly to
the previous case, we can solve the dynamics by vectorisation of the
corresponding master equation for a given initial state [cf. Appendix A].
Specifically, we assume that the system is initially prepared in the product
state

$\hat{\rho}(0)=\hat{\rho}_{S}(0)\otimes\hat{\rho}_{p}(0)\,,$ (14)

where the system is in a thermal Gibbs state
$\hat{\rho}_{S}(0)=\exp(-\beta\hat{H}_{S})/\mathcal{Z}_{S}$, with
$\mathcal{Z}_{S}=\operatorname{Tr}{[\exp(-\beta\hat{H}_{S})]}$, $\beta=1/T$
being the inverse temperature. The probe is prepared in a pure state in the
form
$\hat{\rho}_{p}=\left|\psi_{p}\right\rangle\\!\left\langle\psi_{p}\right|$,
where we write as usual
$\left|\psi_{p}\right\rangle=\cos(\theta/2)\left|1\right\rangle+e^{i\phi}\sin(\theta/2)\left|0\right\rangle$,
with the constraints $\theta\in[0,\pi]$ and $\phi\in[0,2\pi]$.

Figure 2: Sketch of the non-equilibrium probing scheme to estimate the
collapse rate $\lambda$. The system is made of a chain of qubits intearcting
in a Ising-like form: each of them is independently undergoing the collapse
mechanism. The chain is attached to a perfectly isolated qubit that serves as
a probe.

The explicit solution of the dynamics yields the density matrix
$\hat{\rho}=\hat{\rho}(t)$. Thus, by tracing out the system, we can obtain the
reduced state of the probe at any time $t$, i.e.
$\hat{\rho}_{p}(t)\equiv\operatorname{tr}_{S}\hat{\rho}(t)$. It is
straightforward to assess the qualitative effect of the collapse mechanism
onto the system dynamics: larger rates $\lambda$ results in damping the
oscillations over a shorter timescale. This complicates the detection of the
effects of collapse for small values of $\lambda$.

###  III.2 Numerical results

Since $\hat{\rho}_{p}(t)$ implicitly depends on the parameter $\lambda$, and
such a dependence can be used to compute the QFI via Eq. (7). It is not
redundant to stress that the QFI, apart from being a function of the collapse
rate $\lambda$, depends on several additional parameters: the evolution time
$t$ and the angles that specify the initial (pure) state of the probe, i.e.
$\theta$ and $\phi$. We verified that the function $G$ is independent of the
choice of the azimuthal angle $\phi$. Differently, the role played by the
parameters $t$ and $\theta$ is nontrivial. Specifically, the evolution time,
depending on the magnitude of the collapse rate $\lambda$, should satisfy two
physical constraints. On one hand, it should not be too small so that we
depart form fully coherent oscillations; on the other hand, it should not be
too large to guarantee that the amplitude of the oscillations is still non-
vanishing.

Therefore, before studying $G$ as a function of the parameter $\lambda$, we
perform an optimisation over the pair $(t,\theta)$ yielding $(t_{\rm
opt},\theta_{\rm opt})$ such that $G_{\rm max}=G(t_{\rm opt},\theta_{\rm
opt})$. For instance, we can consider the situation where
$\lambda=0.1\omega_{0}$, while we choose the remaining physical parameters as
$\omega_{p}=0.3\omega_{0}$, $g=0.2\omega_{0}$, $\beta=0.01/\omega_{0}$, and
$\phi=\pi/4$. Using the Nelder-Mead algorithm [49], we obtain
$\omega_{0}t_{\rm opt}\approx 56.82$ and $\theta_{\rm opt}\approx 3.13$. This
procedure allows us to determine an optimal working point.

Alternatively, one can choose to fix $t$ and $\theta$ as $t=\bar{t}$ and
$\theta=\bar{\theta}$, which essentially means to fix the maximum evolution
time and the initial state of the probe. We can then study how the QFI varies
as a function of the parameter we would like to estimate, i.e. $\lambda$.
Beside the QFI $G(\lambda)$, one can also look at the Quantum Signal-to-Noise
Ratio (QSNR), defined as $Q_{\lambda}\equiv\lambda^{2}G(\lambda)$, which
quantifies how the variance scales with the mean value [40]. In Fig. 3, we
show that the QFI $G(\lambda)$ decreases monotonically with the parameter
$\lambda$. The QSNR shows that there is a trade-off between the information we
can gather about the system and the noise.

Figure 3: Non-equilibrium probing of the collapse rate $\lambda$ appearing in
Eq. (11). We plot the QFI $G$ as a function of $\lambda$, while in the inset
we show the QSNR $Q_{\lambda}$. For the numerical simulations we chose
$\omega_{p}=0.3\omega_{0}$ and $g=0.2\omega_{0}$. The initial state of the
probe is fully specified by the angles $\theta=\phi=\pi/4$, while the system
is in a thermal Gibbs state such that $\beta=0.01/\omega_{0}$. The maximum
evolution time is given by $\omega_{0}\bar{t}=2\pi/g$.

##  IV Many-body approach to the estimation of the collapse rate

In this Section we introduce a many-body approach to provide the best estimate
of the collapse rate $\lambda$. To this end, motivated by the discussion of
the previous section, we introduce a non-equilibrium setup. We use a single
qubit to probe a many-body system affected by the collapse mechanism, as
sketched in Fig. 2. Our aim is essentially twofold. On one hand, we assess how
the estimability of the parameter $\lambda$ scales with the system size; on
the other hand, we investigate whether and under which conditions quantum
criticality [50, 51] is resourceful for our estimation problem. The former can
also be regarded as a way to assess the existence of cooperative effects in a
many-body system, which could enhance our ability to estimate a single
parameter. The latter, instead, hinges upon a key result of quantum metrology:
quantum criticality represents a resource for the estimation of Hamiltonian
parameters [20].

To address the aforementioned issues, we consider the prototypical example of
a quantum Ising chain. The Hamiltonian of the system reads as

$\hat{H}_{\rm
Ising}=-h\sum_{j=1}^{N}\hat{\sigma}_{j}^{x}-J\sum_{j=1}^{N-1}\hat{\sigma}_{j}^{z}\hat{\sigma}_{j+1}^{z}\,,$
(15)

where $h>0$ is the intensity of the local fields, $J>0$ is the interaction
strength between adjacent spins (sometimes dubbed as exchange constant) [50].
We assume, for the sake of simplicity, that $h$ and $J$ are the same across
all the $N$ spins of the chain. Note that $\hat{\sigma}_{j}^{k}$ (for
$k=x,y,z$) are the local Pauli operators acting on the $j$-th spin.

Eq. (15) exhibits a competition between the interaction among adjacent spins
in the $z$ direction and the external field in the $x$ direction. The external
field adds quantum fluctuations to the system, driving the system across a
quantum phase transition from an ordered to a disordered phase. This model can
be analytically solved via Jordan-Wigner transformation [52], showing that –
at zero temperature – the QPT occurs exactly when $h=J$ [53, 50, 54]. This
critical point separates the ordered ($h<J$) from the disordered ($h>J$)
phase, the latter emerging when the transverse field dominates.

Notwithstanding its numerous interesting features, here we consider the
quantum Ising model for its metrological properties. In Ref. [55] it has been
shown that the accuracy with which we are able to estimate the Hamiltonian
parameters $h$ and $J$ is greatly enhanced at critical points with respect to
non-critical ones.

Differently, here we aim at discussing the possible advantage offered by
quantum criticality in estimating the non-Hamiltonian parameter $\lambda$,
i.e. the collapse rate. To this end, we need to let each spin of the chain
interact with a local bath mimicking the collapse mechanism, as in Eq. (11),
whose many-body version reads

$D_{j}(\hat{\rho})=-\frac{\lambda}{2}\,[\hat{\sigma}^{z}_{j},[\hat{\sigma}^{z}_{j},\hat{\rho}]]\,,$
(16)

for each site $j=1,\ldots,N$.

Following the non-equilibrium approach put forth in Sec. II, we attach a probe
(i.e. a qubit) to the Ising chain, so that in the full Hamiltonian of Eq. (1)
we take $\hat{H}_{S}=\hat{H}_{\rm Ising}$,
$\hat{H}_{p}=-h_{p}\hat{\sigma}^{x}_{p}$, while the interaction Hamiltonian is
given by

$\hat{H}_{I}=-J_{p}\hat{\sigma}_{N}^{z}\otimes\hat{\sigma}_{p}^{z}\,,$ (17)

where $J_{p}$ is the coupling constant between the $N$-th spin of the chain
and the probe. In other terms, owing to this choice, the composite system
described by the Hamiltonian (1) can be regarded as a $(N+1)$-dimensional
Ising chain, where one site, serving as a probe, is not subject to the
collapse mechanism.

In order to study the estimability of $\lambda$ in relation to the critical
behaviour of the Ising model, we work in a situation close to the transition
point for the corresponding infinite-dimensional closed system, i.e. $h=J$. In
other words, since $\lambda$ is small compared to the other physical
quantities characterising the unitary dynamics, we can work in a low-
dissipation regime, where the dissipation induced by Eq. (16) acts _de facto_
as a perturbation [56].

##  V Analysis and Results

Given the many-body probing setup described in Sec. IV, we discuss in this
section the problem of estimating the collapse rate $\lambda$.

The first step is to solve the master equation for an open many-body system,
which can be – in general – a formidable task [57]. In our study we follow a
simple approach consisting in vectorising [cf. Appendix A] the Lindblad master
equation

$\frac{\operatorname{d}\\!\hat{\rho}}{\operatorname{d}\\!t}=-i[\hat{H},\hat{\rho}]+\sum_{j=1}^{N}D_{j}(\hat{\rho})\,,$
(18)

and numerically solving the corresponding set of ordinary differential
equations. We should mention that this method scales exponentially with the
system size $N$, as the Hilbert space scales as $2^{N+1}$. However, it allows
us to gain insight on the issue that we want to investigate, without resorting
to more sophisticated simulation techniques.

Specifically, we choose the following initial state
$\hat{\rho}(0)=\hat{\rho}_{\rm Ising}(0)\otimes\hat{\rho}_{p}(0)$, where the
Ising chain is in a thermal Gibbs state $\hat{\rho}_{\rm
Ising}(0)\equiv\exp(-\beta\hat{H}_{\rm Ising})/\mathcal{Z}_{\rm Ising}$, with
$\mathcal{Z}_{\rm Ising}=\operatorname{Tr}{[\exp(-\beta\hat{H}_{\rm
Ising})]}$, and $\beta=1/T$ being the inverse temperature. As in Eq. (14), the
probe is prepared in a pure state in the form
$\hat{\rho}_{p}=\left|\psi_{p}\right\rangle\\!\left\langle\psi_{p}\right|$.

As for the single-body case, since we need to compute the QFI at a given time,
we have to carefully choose the timescale so that, depending on the value of
$\lambda$, we can detect the effect of the collapse mechanism. On one hand, as
$\lambda$ gets smaller, it takes longer to see the probe’s dynamics departing
from the fully coherent oscillations as a result of the collapse. Moreover,
the size $N$ of the chain plays a crucial role on the dynamics of the probe.
By increasing $N$, the probe converges quicker to its steady state. A rough
estimate of the typical relaxation time is given by $\tau_{R}\approx
1/\Delta_{\mathcal{L}}$, where $\Delta_{\mathcal{L}}$ is the Liouvillian gap
[cf. Appendix A].

By tracing out the degrees of freedom associated with the Ising chain only, we
obtain the state of the probe at any time $t$, i.e.
$\hat{\rho}_{p}(t)=\operatorname{tr}_{S}{\hat{\rho}(t)}$. The latter is needed
to tackle our estimation problem, using it to calculate the QFI by resorting
to Eq. (7).

To study the behaviour of the QFI as a function of $\lambda$, we proceed as
follows. For a given set of physical parameters (including the value of the
collapse rate $\lambda$), we calculate the QFI as a function of time, i.e.
$G=G(t)$, determining the optimal time $t_{\rm opt}$ such that $G_{\rm
max}=G(t_{\rm opt})$. We then fix $t=t_{\rm opt}$ and we study the QFI as a
function of $\lambda$, i.e. $G=G(\lambda)$ over an interval of interest
$\mathcal{I}_{\lambda}=[\lambda_{\rm min},\lambda_{\rm max}]$.

To study the behaviour of the QFI as a function of the collapse parameter
$\lambda$, i.e. $G=G(\lambda)$, we consider four different intervals for
$\lambda$, namely $\mathcal{I}_{\lambda_{1}}=[0.1,0.5]$,
$\mathcal{I}_{\lambda_{2}}=[10^{-3},10^{-1}]$,
$\mathcal{I}_{\lambda_{3}}=[10^{-5},10^{-3}]$ and
$\mathcal{I}_{\lambda_{4}}=[10^{-7},10^{-5}]$. In each of these intervals we
determine $t=t_{\rm opt}$ corresponding to the lower bound of each interval,
i.e. $\lambda=\lambda_{\rm min}$.

In our numerical simulations, we test the behaviour of the QFI $G=G(\lambda)$
in three different scenarios: at the critical point (i.e. $h=J$), slightly
below or above it (i.e. $h=0.99J$ and $h=1.01J$, respectively). Furthermore,
we study how the QFI scales with the system size by taking $N=2,3,4$ and $5$.

According to our numerical simulations, in the interval
$\mathcal{I}_{\lambda_{1}}$ (where $\lambda$ is comparable to the relevant
energies of the system) there is no evidence that quantum criticality is
resourceful for estimating the collapse rate $\lambda$, as one can notice in
Fig. 4 (a). Similarly, our ability to discriminate $\lambda$ is not enhanced
by the system size: the QFI actually becomes smaller as the system size
increases [cf. Fig. 4 (b)]. This is likely due to the chosen values of
$\lambda$, for which the assumption of small collapse-induced perturbations
breaks down.

Figure 4: Non-equilibrium probing of a quantum Ising chain: plots of the QFI
$G$ of the probe as a function of the collapse rate $\lambda$, when
$\lambda\in\mathcal{I}_{\lambda_{1}}=[0.1,0.5]$. In Panel (a) we consider the
case $N=5$. We plot $G$ for three different values of $h$: at the critical
point (i.e. $h=J$, solid line), slightly below (i.e. $h=0.99J$, dash-dotted
line), and slightly above (i.e. $h=1.01J$, dashed line). In Panel (b), we plot
$G$ for different values of the system size $N$ when $h=J$. For the numerical
simulations, we chose $h_{p}=0.5J$, $J_{p}=0.5J$, while for the initial state,
we took $\beta=0.1/J$, $\theta=\pi$, $\phi=0$. The times at which the QFI is
measured, i.e. $t=t_{\rm opt}$ (in units of $J$), are displayed in the legend
of Panel (b), as reported in the Table (1) of the Appendix (C).

At variance, if we consider the parameter regime corresponding to the
intervals $\mathcal{I}_{\lambda_{j}}$ with $j=2,3$ and $4$, we gather
numerical evidence that, for a given $N$, quantum criticality helps when we
are to estimate $\lambda$. In Fig. 5, we show an explicit example for
$\mathcal{I}_{\lambda_{3}}$. This result is remarkable in that $h/J=1$ is the
critical point for infinite system ($N\to\infty$), but nevertheless we see a
marked difference between results for $\mid h/J-1\mid=0.01$ and for $\mid
h/J-1\mid=0$ already for the considered small values of $N$.

Figure 5: QFI as a function of the parameter $\lambda$ when we apply the non-
equilibrium probing scheme to an Ising chain, as described in Eq. (IV). We
plot $G$ for three different values of $h$, so that the critical, the ordered
vs disordered phases are explored. In these plots, we consider
$\lambda\in\mathcal{I}_{\lambda_{3}}=[10^{-5},10^{-3}]$, $N=4$, while all the
remaining physical parameters are $h_{p}=0.5J$, $J_{p}=0.5J$, $\beta=0.1/J$,
$\theta=\pi$, $\phi=0$. The QFI is evaluated at $t=t_{\rm opt}=6.84/J\cdot
10^{2}$. In the inset we show the corresponding QSNR $Q_{\lambda}$.

In the same parameter regimes, our numerical results show that there is no
clear monotonicity relationship between the QFI (or the QSNR) and the system
size, as one can deduce from the instances reproduced in Fig. 7. However, if
we consider the case corresponding to the interval $\mathcal{I}_{\lambda_{3}}$
[cf. Fig. 7 (e) and (f)], we can notice that there is a range of values for
which the system size plays a positive role in estimating the collapse rate
$\lambda$. Dealing with a larger system appears to enhance the ability to
estimate the collapse rate $\lambda$. Near the expected critical point (at
thermodynamic limit) the correlation length becomes comparable to the system
size, corroborating the expectation that, as $N$ grows to thermodynamic limit,
the entire system will contribute to amplify the signal. We stress that,
although we have used, as probing system, a spin at the edge of the quantum
spin chain, the previous argument about the diverging correlation length leads
us to conclude that effects qualitatively similar to the ones reported here
would hold if the probe is coupled to one spin (or possibly all of the spins)
in the bulk of the chain itself.

A further comment is related to the structure of the collapse dissipator in
Eq. (18), which is the incoherent sum of local channels. One can interpret
this as if the spatial correlation of the noise mechanisms acting on the
elements of the chainhas a typical scale $r_{\text{\tiny C}}$ smaller than the
sites interval distance $a$. We can go beyond this assumption by considering a
model where such a correlation is introduced. A possibility is to consider

$\frac{\operatorname{d}\\!\hat{\rho}}{\operatorname{d}\\!t}=-i[\hat{H},\hat{\rho}]+\sum_{i,j=1}^{N}\tilde{D}_{ij}(\hat{\rho})\,,$
(19)

where

$\tilde{D}_{ij}(\hat{\rho})=-\frac{\lambda}{2}f(i,j)\,[\hat{\sigma}^{z}_{i},[\hat{\sigma}^{z}_{j},\hat{\rho}]]$
(20)

with $f(i,j)$ describing the spatial correlation of the noise between the
sites $i$ and $j$. An example of this correlation function can be borrowed
from the CSL model [6], i.e.

$f(i,j)=e^{-a^{2}(i-j)^{2}/4r_{\text{\tiny C}}^{2}},$ (21)

which correlates sites $i$ and $j$ if their relative distance $a(i-j)$ is
smaller than the correlation length $r_{\text{\tiny C}}$. Conversely, if the
distance is larger than $r_{\text{\tiny C}}$, the noise acts incoherently as
described in Eq. (16). To quantify the effect of the correlated noise, we can
explicitly compute the relative difference $\delta
G=(G_{\text{corr}}-G_{\text{uncorr}})/G_{\text{corr}}$, where
$G_{\text{uncorr}}$ and $G_{\text{corr}}$ is the QFI computed when the the
dissipator entering in the Lindblad master equation is given by Eqs. (16) and
(20), respectively. In general, no significant changes are observed when one
considers the case of correlated noise, as emerges, for instance, in the case
plotted in Fig. 6.

Figure 6: Relative difference $\delta G$ of the QFI in presence of correlated
noise as described by Eqs. (20) and (21) against the case where the latter is
absent so that the dissipator reduces to the one in Eq. (16). In the numerical
simulations, we consider an Ising chain made of $N=4$ spins, while
$\lambda\in\mathcal{I}_{\lambda_{3}}=[10^{-5},10^{-3}]$. All other parmeters
are $h_{p}=0.5J$, $J_{p}=0.5J$, $\beta=0.1/J$, $\theta=\pi$, $\phi=0$,
$t=t_{\rm opt}=6.84/J\cdot 10^{2}$, and $r_{\text{\tiny C}}/a=2$.

Figure 7: Non-equilibrium probing of a quantum Ising chain: QFI (left column)
and QSNR (right column) as a function of $\lambda$, where
$\lambda\in\mathcal{I}_{\lambda_{j}}$, with $j=2$ (first raw), $3$ (second
raw), and $4$ (last raw) [Cf. main text] for different system sizes $N$. The
numerical simulations are carried out choosing $h_{p}=0.5J$, $J_{p}=0.5J$,
$\beta=0.1/J$, $\theta=\pi$, $\phi=0$. The optimal instant of time $t=t_{\rm
opt}$ (in units of $J$) for the cases considered above are given in the
legend, as well as in Eq. (1) of Eq. (C)

##  VI Conclusions

We have introduced a quantum probing approach to the estimation of the
collapse rate affecting a quantum critical system. The long-range correlations
established in proximity of the expected critical point can result in an
enhancement of the sensitivity of the QFI of a quantum probe, thus providing
an enhanced measurement precision. Our work paves the way to the further
exploration the exclusion plot for proposed collapse models through
criticality-assisted quantum metrology, and thus provides a new set of tools
in the quest for the falsification of quantum collapse models.

###### Acknowledgements.

We acknowledge support by the European Union’s Horizon Europe EIC-Pathfinder
project QuCoM (101046973), the Leverhulme Trust Research Project Grant
UltraQuTe (grant RPG-2018-266), the Royal Society Wolfson Fellowship
(RSWF/R3/183013), the UK EPSRC (EP/T028424/1), the Department for the Economy
Northern Ireland under the US-Ireland R&D Partnership Programme, the Italy-UK
CNR/RS Joint Project “Testing fundamental theories with ultracold atoms”, and
the PNRR PE National Quantum Science and Technology Institute (PE0000023).

##  Appendix A Vectorization method

In this Appendix, we summarise the so-called _vectorization_ operation that
maps a matrix $\mathbf{M}$ onto a vector
$\operatorname{vec}\left(\mathbf{M}\right)=\vec{M}$ in the following way:

$\operatorname{vec}\begin{pmatrix}a&b\\\ c&d\end{pmatrix}=\begin{pmatrix}a\\\
c\\\ b\\\ d\end{pmatrix}\,.$ (22)

It is useful to introduce the following property [58]:

$\operatorname{vec}\left(\mathbf{A}\mathbf{B}\mathbf{C}\right)=\left(\mathbf{C}^{T}\otimes\mathbf{A}\right)\operatorname{vec}\left(\mathbf{B}\right)\,,$
(23)

given three matrices $\mathbf{A},\mathbf{B},\mathbf{C}$.

This change of basis is particularly suitable for solving the dynamics of an
open quantum system, when it is governed by a master equation of the form [36,
59]

$\frac{\operatorname{d}\\!\hat{\rho}}{\operatorname{d}\\!t}=-i[\hat{H},\hat{\rho}]+D(\hat{\rho})\,,$
(24)

where the dissipator is in the GKSL form [37, 38, 39]

$D(\hat{\rho})=\sum_{k}\gamma_{k}\left(\hat{L}_{k}\hat{\rho}\,\hat{L}_{k}^{\dagger}-\frac{1}{2}\left\\{\hat{L}_{k}^{\dagger}\hat{L}_{k},\hat{\rho}\right\\}\right)\,,$
(25)

with $\gamma_{k}>0$ being the relaxation rates, and $\\{\hat{L}_{k}\\}_{k}$
the set of Lindblad operators.

The main idea behind vectorisation is to bring Eq. (24) into a first-order
ordinary differential equation in the matrix form
$\operatorname{d}\\!\mathbf{x}/\operatorname{d}\\!t=\mathbf{A}\mathbf{x}$.
Note that, if the system is a single qubit, the vectorisation of the density
matrix consist in mapping

$\hat{\rho}=\begin{pmatrix}\rho_{00}&\rho_{01}\\\
\rho_{10}&\rho_{11}\end{pmatrix}\longmapsto\left|\rho\right\rangle\rangle=\begin{pmatrix}\rho_{00}\\\
\rho_{10}\\\ \rho_{01}\\\ \rho_{11}\end{pmatrix}\,,$ (26)

which is a direct application of Eq. (22). In order to vectorise Eq. (24), we
can resort to the property (23). The right-hand side is readily vectorised by
inserting the identity operator $\mathbbm{1}$ where needed. For the unitary
part, we have

$\displaystyle-i$
$\displaystyle\left(\hat{H}\hat{\rho}\mathbbm{1}-\mathbbm{1}\hat{\rho}\hat{H}\right)\mapsto\vec{\mathcal{L}}_{U}\left|\rho\right\rangle\rangle\equiv-i\left(\mathbbm{1}\otimes\hat{H}-\hat{H}^{T}\otimes\mathbbm{1}\right)\left|\rho\right\rangle\rangle\,.$
(27)

Similarly, the dissipator can be vectorised as

$\displaystyle\sum_{k}{\gamma_{k}}\left(\hat{L}_{k}\hat{\rho}\,\hat{L}_{k}^{\dagger}-\frac{1}{2}\left(\hat{L}_{k}^{\dagger}\hat{L}_{k}\hat{\rho}\mathbbm{1}+\mathbbm{1}\hat{\rho}\hat{L}_{k}^{\dagger}\hat{L}_{k}\right)\right)\mapsto$
(28)
$\displaystyle\vec{\mathcal{L}}_{D}\left|\rho\right\rangle\rangle{\equiv}\sum_{k}\gamma_{k}\left[L_{k}^{*}{\otimes}L_{k}{-}\frac{1}{2}\left(\mathbbm{1}\otimes\hat{L}_{k}^{\dagger}\hat{L}_{k}{+}(\hat{L}_{k}^{\dagger}\hat{L}_{k})^{T}{\otimes}\mathbbm{1}\right)\right]\left|\rho\right\rangle\rangle\,.$

Therefore, Eq. (24) becomes

$\frac{\operatorname{d}\\!\left|\rho_{t}\right\rangle\rangle}{\operatorname{d}\\!t}=\vec{\mathcal{L}}\left|\rho_{t}\right\rangle\rangle\,,$
(29)

where $\vec{\mathcal{L}}\equiv\vec{\mathcal{L}}_{U}+\vec{\mathcal{L}}_{D}$.
The formal solution of Eq. (29) is obtained by exponentiation of the time-
independent Liouvillian $\vec{\mathcal{L}}$, i.e.

$\left|\rho_{t}\right\rangle\rangle=e^{t\vec{\mathcal{L}}}\left|\rho_{0}\right\rangle\rangle\,.$
(30)

Therefore, solving the dynamics of an open quantum system described by the Eq.
(24) is equivalent to exponentiating a complex, non-Hermitian matrix
$\vec{\mathcal{L}}$ [60, 61]. The vectorised Liouvillian $\vec{\mathcal{L}}$
possesses two distinct families of right and left eigenvectors, i.e.
$\\{\left|R_{\alpha}\right\rangle\rangle\\}_{\alpha}$ and
$\\{\left|L_{\alpha}\right\rangle\rangle\\}_{\alpha}$, respectively,
associated with the singular value $l_{\alpha}$, sometimes dubbed as
_rapidity_ , the latter being in general a complex number. The right and left
eigenvectors are defined by the equations
$\vec{\mathcal{L}}\left|R_{\alpha}\right\rangle\rangle=l_{\alpha}\left|R_{\alpha}\right\rangle\rangle$
and
$\langle\langle{L}_{\alpha}|\vec{\mathcal{L}}=l_{\alpha}\langle\langle{L}_{\alpha}|$,
respectively.

When vectorising a master equation, we are actually working in a Hilbert space
endowed with the scalar product
$\langle\langle\phi|\chi\rangle\rangle\equiv\operatorname{Tr}{[\hat{\phi}^{\dagger}\hat{\chi}]}$.
Note that the right and left eigenvectors are not normalised; they can easily
normalised by setting
$|\tilde{R}_{\alpha}\rangle\rangle=\left|R_{\alpha}\right\rangle\rangle/\sqrt{\langle\langle
L_{\alpha}|R_{\alpha}\rangle\rangle}$ and
$\langle\langle\tilde{L}_{\alpha}|=\langle\langle
L_{\alpha}|/\sqrt{\langle\langle L_{\alpha}|R_{\alpha}\rangle\rangle}$, so
that they automatically fulfil the orthornomality condition
$\langle\langle\tilde{L}_{\alpha}|\tilde{R}_{\alpha^{\prime}}\rangle\rangle=\delta_{\alpha\alpha^{\prime}}$.
We can decompose the full Liouvillian as

$\vec{\mathcal{L}}=\sum_{\alpha}l_{\alpha}|\tilde{R}_{\alpha}\rangle\rangle\langle\langle\tilde{L}_{\alpha}|\,.$
(31)

Starting from Eq. (31), one can expand the exponential and resort to the
condition
$\langle\langle\tilde{L}_{\alpha}|\tilde{R}_{\alpha^{\prime}}\rangle\rangle=\delta_{\alpha\alpha^{\prime}}$
to obtain

$e^{\hat{\mathcal{L}}t}=\sum_{\alpha}e^{l_{\alpha}t}|\tilde{R}_{\alpha}\rangle\rangle\langle\langle\tilde{L}_{\alpha}|\,.$
(32)

Combining Eq. (30) and (32), we get

$\left|\rho_{t}\right\rangle\rangle=\sum_{\alpha}c_{\alpha}e^{l_{\alpha}t}|\tilde{R}_{\alpha}\rangle\rangle\,,$
(33)

where
$c_{\alpha}\equiv\langle\langle\tilde{L}_{\alpha}|\rho_{0}\rangle\rangle$ are
determined by the initial conditions. Essentially, the evolved state
$\left|\rho_{t}\right\rangle\rangle$ is given a linear combination of the
right eigenvectors $|\tilde{R}_{\alpha}\rangle\rangle$ with certain
coefficients $c_{\alpha}$, while the time dependence is all contained in the
exponentials $e^{l_{\alpha}t}$.

The spectrum of the Liouvillians contains interesting information about the
system’s dynamics [62, 63]. Specifically, we can summarise some of the
properties of $\vec{\mathcal{L}}$ as follows [47]

  1. (i)

the spectrum $\\{l_{\alpha}\\}_{\alpha}$ is symmetric with respect to the real
line,

  2. (ii)

all eigenvalues $l_{\alpha}$ belong to the left half of the complex plane,
i.e. $\mathbb{C}_{-}=\\{z\in\mathbb{C}|\operatorname{Re}z\leq 0\\}$,

  3. (iii)

the right eigenvector corresponding to the eigenvalue $l_{0}=0$, namely
$|\tilde{R}_{0}\rangle\rangle$, defines the steady state, i.e.
$\vec{\mathcal{L}}\left|{\rho}_{\rm
ss}\right\rangle\rangle=\vec{\mathcal{L}}|\tilde{R}_{0}\rangle\rangle=0$.

One more property of the Liouvillian spectrum allows us to estimate how fast a
quantum system approaches the stationary state when is dissipatively coupled
to an external environment. To this end, it is worth mentioning that the
rapidities, being in general complex, can be denoted as
$l_{\alpha}=-i\omega_{\alpha}-\Gamma_{\alpha}$, with
$\omega_{\alpha}\equiv-\operatorname{Im}l_{\alpha}$ and
$\Gamma_{\alpha}\equiv-\operatorname{Re}l_{\alpha}$. The set
$\\{\Gamma_{\alpha}\\}_{\alpha}$ defines the relaxation rates. This is
justified by close inspection of Eq. (33), where the rapidities $l_{\alpha}$
enter: it is immediate to classify the decay mode $\alpha$ into oscillatory
[if $\Gamma_{\alpha}=0$], purely decaying [if $\omega_{\alpha}=0$], decaying
spirals [if $\Gamma_{\alpha}>0$ and $\omega_{\alpha}\neq 0$] [62]. In
particular, we can define the so-called _Liouvillian gap_ as
$\Delta_{\mathcal{L}}\equiv\min_{\alpha\neq 0}\Gamma_{\alpha}$, which
identifies the decay rate of the slowest relaxation mode [46]. It is
reasonable to give $1/\Delta_{\mathcal{L}}$ the meaning of an upper bound to
the relaxation time. While such interpretation will hold when considering
many-body systems [56, 51], it is worth mentioning that there are some
instances contradicting it [64, 65].

##  Appendix B Vectorisation of Eq. (10)

The results summarised in Appendix A can be applied to vectorise Eq. (10). We
have that Eq. (26) and (27) yield

$\displaystyle\vec{\mathcal{L}}_{U}$
$\displaystyle=-i\left(\mathbbm{1}_{2}\otimes\hat{H}-\hat{H}\otimes\mathbbm{1}_{2}\right)\,,$
(34) $\displaystyle\vec{\mathcal{L}}_{D}$
$\displaystyle=\lambda\left(\hat{\sigma}^{z}\otimes\hat{\sigma}^{z}-\mathbbm{1}_{4}\right)\,,$
(35)

where $\mathbbm{1}_{2}$ and $\mathbbm{1}_{4}$ are the identity operators
defined over the two- and four-dimensional Hilbert space, respectively. The
full Liouvillian
$\vec{\mathcal{L}}=\vec{\mathcal{L}}_{U}+\vec{\mathcal{L}}_{D}$ can be cast as
the non-Hermitian matrix

$\vec{\mathcal{L}}=\frac{1}{2}\begin{pmatrix}0&-i\omega_{0}&i\omega_{0}&0\\\
-i\omega_{0}&-4\lambda&0&i\omega_{0}\\\
i\omega_{0}&0&-4\lambda&-i\omega_{0}\\\
0&i\omega_{0}&-i\omega_{0}&0\end{pmatrix}\,,$ (36)

which can be diagonalised to get the following eigenvalues

$l_{0}=0,\quad l_{1}=-2\lambda,\quad l_{\pm}=\
\lambda\pm\sqrt{\lambda^{2}-\omega_{0}^{2}}\,,$ (37)

while the corresponding (unnormalised) right eigenvectors read as

$\displaystyle\left|R_{0}\right\rangle\rangle=$
$\displaystyle\begin{pmatrix}1\\\ 0\\\ 0\\\
1\end{pmatrix}\,,\quad\left|R_{1}\right\rangle\rangle=\begin{pmatrix}0\\\ 1\\\
1\\\ 0\end{pmatrix}\,,$ (38)
$\displaystyle\left|R_{\pm}\right\rangle\rangle=\frac{1}{\omega_{0}}$
$\displaystyle\begin{pmatrix}-\omega_{0}\\\
{i}\left(\lambda\mp\sqrt{-\lambda^{2}+\omega_{0}^{2}}\right)\\\
i\left(-\lambda\pm\sqrt{-\lambda^{2}+\omega_{0}^{2}}\right)\\\
\omega_{0}\end{pmatrix}.$

If $\omega_{0}\neq\lambda$, all the eigenvalues are simple and the
corresponding quantum dyanamical semigroup $\\{e^{t\mathcal{L}}\\}_{t\geq 0}$
is relaxing towards a unique maximally mixed state [66, 67, 47]. The steady
state $\hat{\rho}_{\rm ss}$ can be readily found considering the eigenvector
corresponding to the eigenvalue $l_{0}=0$, i.e.
$\left|R_{0}\right\rangle\rangle$. Upon de-vectorisation and normalisation, we
obtain the totally mixed state

$\displaystyle\hat{\rho}_{\rm ss}=\frac{1}{2}\mathbbm{1}_{2}.$ (39)

In general, the state at a time $t$ is given by

$\left|\rho_{t}\right\rangle\rangle=\sum_{\alpha=0}^{3}c_{\alpha}e^{l_{\alpha}t}\left|R_{\alpha}\right\rangle\rangle,~{}~{}\text{with}~{}~{}c_{\alpha}=\langle\langle
L_{\alpha}|\rho_{0}\rangle\rangle,$ (40)

together with Eqs. (37) and (38).

##  Appendix C Further data about numerical simulations

In this Appendix, we show some numerical data regarding the quantum Ising
chain discussed in Secs. IV and V. For studying the behaviour of the QFI
against the collapse rate $\lambda$, we consider the regime in which
$h_{p}=0.5J$, $J_{p}=0.5J$, $\beta=0.1/J$, $\theta=\pi$, $\phi=0$. In Table 1
we report the optimal time $t=t_{\rm opt}$ at which the QFI is determined in
the examples provided in Sec. V.

$J\lambda$ $N$ | $2$ | $3$ | $4$ | $5$
---|---|---|---|---
$10^{-1}$ | $7.19$ | $9.53$ | $8.19$ | $8.86$
$10^{-3}$ | $7.71\times 10^{2}$ | $5.24\times 10^{2}$ | $3.76\times 10^{2}$ | $2.41\times 10^{2}$
$10^{-5}$ | $9.51\times 10^{2}$ | $9.22\times 10^{2}$ | $6.84\times 10^{2}$ | $9.93\times 10^{2}$
$10^{-7}$ | $9.51\times 10^{2}$ | $7.46\times 10^{2}$ | $6.84\times 10^{2}$ | $9.93\times 10^{2}$
Table 1: Numerical values of $t_{\rm opt}/J$ for a given value of $\lambda$
and $N$.

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