# Approaching Heisenberg-scalable thermometry with built-in robustness against
noise

Da-Jian Zhang<EMAIL_ADDRESS>Department of Physics, Shandong University,
Jinan 250100, China  D. M. Tong<EMAIL_ADDRESS>Department of Physics,
Shandong University, Jinan 250100, China

###### Abstract

It is a major goal in quantum thermometry to reach a $1/N$ scaling of
thermometric precision known as Heisenberg scaling but is still in its infancy
to date. The main obstacle is that the resources typically required are highly
entangled states, which are very difficult to produce and extremely vulnerable
to noises. Here, we propose an entanglement-free scheme of thermometry to
approach Heisenberg scaling for a wide range of $N$, which has built-in
robustness irrespective of the type of noise in question. Our scheme is
amenable to a variety of experimental setups. Moreover, it can be used as a
basic building block for promoting previous proposals of thermometry to reach
Heisenberg scaling, and its applications are not limited to thermometry but
can be straightforwardly extended to other metrological tasks.

##  I Introduction

Accurately measuring temperature is of universal importance, underpinning many
fascinating applications in material science Aigouy  _et al._ (2005); Linden
_et al._ (2010), medicine and biology Klinkert and Narberhaus (2009);
Schirhagl  _et al._ (2014), and quantum thermodynamics Gemmer  _et al._
(2004). The advent of quantum technologies has opened up exciting
possibilities of exploiting quantum effects to yield the quantum enhancement
of thermometric precision that cannot otherwise be obtained using classical
methods A. De Pasquale and T. M. Stace (2018); Mehboudi  _et al._ (2019). The
emerging field, known as quantum thermometry nowadays, could have a large
impact on quantum platforms demanding precise temperature control, such as
cold atoms, trapped ions, and superconducting circuits. This has motivated a
vibrant activity on quantum thermometry over the past decade Stace (2010);
Sabín  _et al._ (2014); Correa  _et al._ (2015); A. De Pasquale  _et al._
(2016); Seah  _et al._ (2019); Kulikov  _et al._ (2020); Mitchison  _et al._
(2020); Latune  _et al._ (2020); Xie  _et al._ (2020a); Rubio  _et al._
(2021); Zhang  _et al._ (2022).

A major goal in quantum thermometry is to reach a $1/N$ scaling of precision
known as Heisenberg scaling (HS), which represents an important quantum
advantage of central interest in quantum metrology Demkowicz-Dobrzański  _et
al._ (2015); Degen  _et al._ (2017); et al. (2018a). Here, $N$ stands for the
amount of physical resources which usually refers to the number of probes
employed. Conventionally, HS is achieved by exploiting highly entangled states
Stace (2010); Sabín  _et al._ (2014); Latune  _et al._ (2020); Xie  _et al._
(2020a). However, along this line, the HS permitted in theory is typically
elusive in reality, due to the vulnerability of highly entangled states to
noises as well as the difficulty in producing these states. This point has
been theoretically shown in Refs. Ji  _et al._ (2008); Escher  _et al._
(2011); Demkowicz-Dobrzański and Maccone (2014); Alipour  _et al._ (2014) and
is also reflected in the fact that less experimental progress has been made in
implementing Heisenberg-scalable thermometry to date. Indeed, although the
issue of how to attain HS has received much attention since the early days of
quantum thermometry Stace (2010), the first experiment demonstrating HS was
carried out only recently Uhlig  _et al._ (2019). This experiment explored
NOON states to reach HS and observed that the decoherence effect becomes
increasingly severe as the order of NOON states increases. As such, HS was
only demonstrated for small $N$, i.e., $N\leq N_{\textrm{max}}=9$, for which
the scaling advantage is far from allowing one to beat the best possible
classical methods. Besides, several no-go theorems Fujiwara and Imai (2008);
Demkowicz-Dobrzański  _et al._ (2012, 2017); Zhou and Jiang (2021) show that
HS is forbidden to reach in the asymptotic limit $N\rightarrow\infty$ in the
presence of most types of noises. For these reasons, it is crucial to find a
noise-robust scheme capable of reaching HS in the regime of large $N$ et al.
which is of interest from a practical standpoint and allowed by these
theorems.

Figure 1: Schematic of our scheme. (a) Configuration of the setup implementing
our scheme. To estimate the temperature of a given sample, we put $N$ probes
in contact with the sample and weakly couple all of them to a same external
ancilla. The continuous interplay between the strong Markovian thermalization
of the probes and the relatively weak coupling to the ancilla ensures
information transmission from the probes to the ancilla. This leads to the
coherent accumulation of a temperature-dependent phase, and measuring the
phase yields an estimate of the temperature. (b) Illustration of one possible
implementation of our scheme using magnetic resonance force microscopy. The
spins immersed in the sample serve as the probes and the magnetic tip serves
as the ancilla. The force produced by the spins on the magnetic tip affects
the mechanical vibrations of the cantilever, which can be detected by optical
methods.

Note that a number of noise-robust schemes have been proposed for other
metrological tasks Chaves  _et al._ (2013); Kessler  _et al._ (2014); Dür  _et
al._ (2014); Arrad  _et al._ (2014); Lu  _et al._ (2015); et al. (2016);
Reiter  _et al._ (2017); Matsuzaki  _et al._ (2018); Zhou  _et al._ (2018);
Layden  _et al._ (2019); Bai  _et al._ (2019). However, there has not been a
noise-robust scheme in quantum thermometry so far. In this work, we fill the
gap. We find that a temperature-dependent phase can be accumulated coherently
through the continuous interplay between the strong Markovian thermalization
of $N$ probes and the relatively weak coupling of the $N$ probes to a same
external ancilla (see Fig. 1). This mechanism enables us to propose a scheme
of thermometry to approach HS for a wide range of $N$, with robustness
irrespective of the type of noise in question but without complicated error
correction techniques.

A salient feature of our scheme is that the whole estimation procedure is free
of entanglement, unlike in previous works et al. (2018a), where highly
entangled states are either introduced in the state-preparation stage or
generated via some interactions in the interrogation stage. Another feature of
our scheme is that the probing time is not increasingly long with $N$. These
two features distinguish our scheme from previous schemes like the parallel
scheme and sequential scheme Giovannetti  _et al._ (2006) on a fundamental
level. As detailed below, our scheme is amenable to a variety of experimental
setups. Moreover, it can be used as a basic building block for promoting
previous proposals of thermometry to reach HS, and its applications are not
limited to thermometry but can be straightforwardly extended to other
metrological tasks.

##  II Results

The basic dynamical equation. Suppose that we are given a sample of
temperature $T$, with which a probe $\mathpzc{S}$ keeps in contact. The
thermalization process of $\mathpzc{S}$ can be described by a Markovian master
equation $\partial_{t}\rho{S}(t)=\mathcal{L}{S}\rho{S}(t)$ A. De Pasquale and
T. M. Stace (2018); Mehboudi  _et al._ (2019). Here, $\rho{S}(t)$ is the
evolving state of $\mathpzc{S}$. $\mathcal{L}{S}$ is a Liouville superoperator
assumed to have the following two properties: (i) it admits a temperature-
dependent state $\rho_{{}_{T}}$ as the unique steady state; and (ii) the
nonzero eigenvalues $\lambda_{\mu}$ of $\mathcal{L}{S}$ have negative real
parts, where $\mu$ is an index labeling the eigenvalues. Property (i) implies
that $\mathcal{L}{S}\rho_{{}_{T}}=0$, and property (ii) means that there is a
dissipative gap $\lambda:={\min_{\lambda_{\mu}\neq
0}\absolutevalue{\real(\lambda_{\mu})}}$ in the Liouvillian spectrum Zhang
_et al._ (2016a, b); Zhang and Gong (2020). We do not impose any restriction
on the explicit forms of $\mathcal{L}{S}$ and $\rho_{{}_{T}}$, so that the
scheme to be presented is applicable to a variety of physical models. In
particular, $\rho_{{}_{T}}$ may or may not be the Gibbs state, depending on
the specific physical model in question. Starting from an arbitrary initial
state, $\rho{S}(t)$ undergoing the thermalization process automatically
evolves towards $\rho_{{}_{T}}$ at an exponentially fast rate Zhang  _et al._
(2016a, b); Zhang and Gong (2020). Therefore, we may assume that $\mathpzc{S}$
(approximately) reaches $\rho_{{}_{T}}$ at a certain time $t_{0}$.

We weakly couple $\mathpzc{S}$ to an ancilla $\mathpzc{A}$ after the time
$t_{0}$. The free Hamiltonian of $\mathpzc{A}$ is denoted by $H{A}$. The
interaction between $\mathpzc{S}$ and $\mathpzc{A}$ is assumed to be of the
form $H_{I}=\hbar gS\otimes A$ with $[A,H{A}]=0$, where $g$ is a positive
number describing the coupling strength, and $S$ and $A$ are two Hermitian
operators of $\mathpzc{S}$ and $\mathpzc{A}$, respectively. In the frame
rotating with $H{A}$, the dynamics of $\mathpzc{S}\mathpzc{A}$ can be
described by

$\displaystyle\partial_{t}\rho(t)=\mathcal{L}{S}\rho(t)-\textrm{i}g[S\otimes
A,\rho(t)]=:\mathcal{L}\rho(t).$ (1)

Note that we temporarily do not take noise into account, as our purpose here
is to show how the continuous interplay between the thermalization
$\mathcal{L}{S}$ and the interaction $H_{I}$ leads to the coherent
accumulation of a temperature-dependent phase.

Let us figure out the reduced dynamics of $\mathpzc{A}$. Inspired by the
Nakajima-Zwanzig projection operator technique H.-P. Breuer and F. Petruccione
(2007), we introduce a superoperator $\mathcal{P}$, defined as
$\mathcal{P}X=\rho_{{}_{T}}\otimes\tr{S}X$, for an operator $X$ acting on the
joint Hilbert space of $\mathpzc{SA}$. Evidently,
$\mathcal{P}\rho(t)=\rho_{{}_{T}}\otimes\rho{A}(t)$, where $\rho{A}(t)$
denotes the evolving state of $\mathpzc{A}$. The superoperator complementary
to $\mathcal{P}$, denoted by $\mathcal{Q}$, is defined as
$\mathcal{Q}X=X-\mathcal{P}X$.

Applying $\mathcal{P}$, $\mathcal{Q}$ to Eq. (1) separately and invoking
$\mathcal{P}+\mathcal{Q}=1$, where $1$ stands for the identity map, we have

$\displaystyle\partial_{t}\mathcal{P}\rho(t)=\mathcal{P}\mathcal{L}\mathcal{P}\rho(t)+\mathcal{P}\mathcal{L}\mathcal{Q}\rho(t),$
(2)
$\displaystyle\partial_{t}\mathcal{Q}\rho(t)=\mathcal{Q}\mathcal{L}\mathcal{P}\rho(t)+\mathcal{Q}\mathcal{L}\mathcal{Q}\rho(t).$
(3)

The solution of Eq. (3) can be formally expressed as H.-P. Breuer and F.
Petruccione (2007)

$\displaystyle\mathcal{Q}\rho(t)=\mathcal{G}(t,t_{0})\mathcal{Q}\rho(t_{0})+\int_{t_{0}}^{t}ds\mathcal{G}(t,s)\mathcal{Q}\mathcal{L}\mathcal{P}\rho(s),$
(4)

where $\mathcal{G}(t,s):=e^{\mathcal{Q}\mathcal{L}(t-s)}$. This point can be
verified by inserting Eq. (4) into Eq. (3). Substituting Eq. (4) into Eq. (2)
and noting that $\rho(t_{0})=\rho_{{}_{T}}\otimes\rho{A}(t_{0})$ and therefore
$\mathcal{Q}\rho(t_{0})=0$, we obtain an exact equation of motion for
$\mathpzc{A}$:

$\displaystyle\partial_{t}\mathcal{P}\rho(t)=\mathcal{P}\mathcal{L}\mathcal{P}\rho(t)+\int_{t_{0}}^{t}ds\mathcal{P}\mathcal{L}\mathcal{G}(t,s)\mathcal{Q}\mathcal{L}\mathcal{P}\rho(s).$
(5)

Here, the second term on the right-hand side represents some memory effect, as
it depends on the past history of $\mathcal{P}\rho(s)$.

To figure out the explicit expression for the first term on the right-hand
side of Eq. (5), we deduce from property (i) that
$\mathcal{L}{S}\rho_{{}_{T}}=0$ and hence
$\mathcal{LP}\rho(t)=-\textrm{i}g[S\otimes A,\rho_{{}_{T}}\otimes\rho{A}(t)]$.
Then, from the defining property of $\mathcal{P}$, it follows that

$\displaystyle\mathcal{P}\mathcal{L}\mathcal{P}\rho(t)=\rho_{{}_{T}}\otimes\left(-\textrm{i}g[\varphi_{{}_{T}}A,\rho{A}(t)]\right),$
(6)

where $\varphi_{{}_{T}}:=\tr(S\rho_{{}_{T}})$. On the other hand, using
property (ii), the norm of the second term on the right-hand side of Eq. (5)
can be bounded as

$\displaystyle\norm{\int_{t_{0}}^{t}ds\mathcal{P}\mathcal{L}\mathcal{G}(t,s)\mathcal{Q}\mathcal{L}\mathcal{P}\rho(s)}\leq\frac{\varepsilon
g\norm{\mathcal{P}}^{2}\norm{\mathcal{K}}^{2}}{(\lambda/g)-\varepsilon\norm{\mathcal{K}}},$
(7)

where $\mathcal{K}$ is the superoperator defined as
$\mathcal{K}X:=-\textrm{i}[S\otimes A,X]$, and $\varepsilon$ is a
dimensionless constant determined by the damping basis of $\mathcal{L}{S}$
Zhang  _et al._ (2016a, b). The proof of the bound is given in the Method
section.

Note that $\lambda$ characterizes the strength of the thermalization
$\mathcal{L}{S}$; that is, the larger $\lambda$ is, the stronger the
thermalization is. Likewise, $g$ characterizes the strength of the interaction
$H_{I}$. Under the condition that the thermalization is strong whereas the
interaction is relatively weak, i.e., $\lambda/g\gg 1$, we deduce from Eq. (7)
that the second term on the right-hand side of Eq. (5), i.e., the memory
effect, is negligible. Upon neglecting this term and inserting Eq. (6) into
Eq. (5), we reach the basic dynamical equation:

$\displaystyle\partial_{t}\rho{A}(t)=-\textrm{i}g[\varphi_{{}_{T}}A,\rho{A}(t)],$
(8)

representing a unitary dynamics able to coherently accumulate
$\varphi_{{}_{T}}$ in the state of $\mathpzc{A}$ superposed by eigenstates of
$A$. Hereafter, in line with the studies on quantum phase estimation
Giovannetti  _et al._ (2006), we refer to $\varphi_{{}_{T}}$ as a temperature-
dependent phase.

The scheme of thermometry. To estimate the unknown temperature $T$ of a given
sample, we put $N$ probes, $\mathpzc{S}_{1},\cdots,\mathpzc{S}_{N}$, in
contact with the sample. Upon preparing each probe in $\rho_{{}_{T}}$ at time
$t_{0}$, we couple all of the probes to a same ancilla $\mathpzc{A}$. The
above configuration is schematically shown in Fig. 1a. A natural platform for
implementing it could be the magnetic resonance force microscopy Berman  _et
al._ (2006) (see Fig. 1b), where the spins immersed in the sample and the
magnetic tip serve as the probes and the ancilla, respectively. The dynamical
equation for the $N$ probes and the ancilla in the rotating frame reads

$\displaystyle\partial_{t}\rho(t)=\left(\mathcal{L}_{\mathpzc{S}_{1}\mathpzc{A}}+\cdots+\mathcal{L}_{\mathpzc{S}_{N}\mathpzc{A}}\right)\rho(t),$
(9)

with
$\mathcal{L}_{\mathpzc{S}_{n}\mathpzc{A}}\rho=\mathcal{L}_{\mathpzc{S}_{n}}\rho-\textrm{i}g_{n}[S_{n}\otimes
A,\rho]$. Here, $\mathcal{L}_{\mathpzc{S}_{n}}$ denotes the Liouville
superoperator for $\mathpzc{S}_{n}$, $g_{n}$ represents the coupling strength
between $\mathpzc{S}_{n}$ and $\mathpzc{A}$, and $S_{n}$ is a Hermitian
operator acting on $\mathpzc{S}_{n}$. For simplicity, we assume that
$\mathcal{L}_{\mathpzc{S}_{n}}=\mathcal{L}_{\mathpzc{S}}$, $g_{n}=g$, and
$S_{n}=S$, for all $n$. Our following discussion can be straightforwardly
extended to the general case.

Let $\mathcal{E}_{N}(t_{1},t_{0})$ denote the dynamical map associated with
Eq. (9),

$\displaystyle\mathcal{E}_{N}(t_{1},t_{0})=\exp\left[\left(\mathcal{L}_{\mathpzc{S}_{1}\mathpzc{A}}+\cdots+\mathcal{L}_{\mathpzc{S}_{N}\mathpzc{A}}\right)(t_{1}-t_{0})\right],$
(10)

transforming the state $\rho_{{}_{T}}^{\otimes N}\otimes\rho{A}(t_{0})$ at
time $t_{0}$ into the state
$\mathcal{E}_{N}(t_{1},t_{0})\rho_{{}_{T}}^{\otimes N}\otimes\rho{A}(t_{0})$
at time $t_{1}$. Then, the state of $\mathpzc{A}$ at time $t_{1}$ reads

$\displaystyle\rho{A}(t_{1})=\tr_{\mathpzc{S}_{1}\cdots\mathpzc{S}_{N}}\left[\mathcal{E}_{N}(t_{1},t_{0})\rho_{{}_{T}}^{\otimes
N}\otimes\rho{A}(t_{0})\right].$ (11)

To find the explicit expression for $\rho{A}(t_{1})$, we resort to the fact
that $\mathcal{L}_{\mathpzc{S}_{n}\mathpzc{A}}$’s commute with each other and
rewrite Eq. (10) as

$\displaystyle\mathcal{E}_{N}(t_{1},t_{0})=\mathcal{E}_{\mathpzc{S}_{1}\mathpzc{A}}(t_{1},t_{0})\cdots\mathcal{E}_{\mathpzc{S}_{N}\mathpzc{A}}(t_{1},t_{0}),$
(12)

with
$\mathcal{E}_{\mathpzc{S}_{n}\mathpzc{A}}(t_{1},t_{0})=\exp[\mathcal{L}_{\mathpzc{S}_{n}\mathpzc{A}}(t_{1}-t_{0})]$.
Besides, according to the basic dynamical equation (8),

$\displaystyle\tr_{\mathpzc{S}_{n}}\left[\mathcal{E}_{\mathpzc{S}_{n}\mathpzc{A}}(t_{1},t_{0})\rho_{{}_{T}}\otimes\rho{A}(t_{0})\right]=\exp[-\textrm{i}g\varphi_{{}_{T}}A(t_{1}-t_{0})]\times$
$\displaystyle\rho{A}(t_{0})\exp[\textrm{i}g\varphi_{{}_{T}}A(t_{1}-t_{0})].$
(13)

Substituting Eqs. (12) and (II) into Eq. (11), we have

$\displaystyle\rho{A}(t_{1})=e^{-\textrm{i}gN\varphi_{{}_{T}}A(t_{1}-t_{0})}\rho{A}(t_{0})e^{\textrm{i}gN\varphi_{{}_{T}}A(t_{1}-t_{0})},$
(14)

implying that $\varphi_{{}_{T}}$ is intensified $N$ times due to the use of
$N$ probes.

In deriving Eq. (II), we have neglected the memory effect. Indeed, the memory
effect can be made arbitrarily weak by choosing either a large $\lambda$ or a
small $g$ so that $\lambda/g$ is large enough. Here, $N$ should be confined to
a certain appropriate range, as detailed below. It is worth noting that the
past two decades have witnessed enormous experimental progresses in
engineering strongly dissipative processes (see the review article Müller  _et
al._ (2012) and references therein). Nowadays, experimentalists are able to
realize arbitrary Markovian processes for a number of experimentally mature
platforms such as trapped ions et al. (2013a) and Rydberg atoms Weimer  _et
al._ (2010). These developments imply that it is experimentally feasible to
obtain a large $\lambda$. Besides, the value of $g$ is often fixed once the
platform adopted has been calibrated, whereas the value of $\lambda$ may be
still allowed to be tuned. For example, in the magnetic resonance force
microscopy, the value of $g$ is determined by the distance between the spins
and the magnetic tip Berman  _et al._ (2006). So, $g$ is fixed once the
distance has been fixed. However, $\lambda$ is still tunable through varying
the intensity and frequency of an external magnetic field Berman  _et al._
(2006). For the above reasons, we consider the scenario that $\lambda$ is
large whereas $g$ is fixed.

Without loss of generality, we set the probing time $t_{1}-t_{0}$ to be a
“unit” of time $1/g$ for simplicity. It follows from Eq. (14) that

$\displaystyle\rho{A}(t_{1})=\exp[-\textrm{i}N\varphi_{{}_{T}}A]\rho{A}(t_{0})\exp[\textrm{i}N\varphi_{{}_{T}}A].$
(15)

Then, an estimate of $T$ can be obtained by measuring $\rho{A}(t_{1})$. It
remains to find the optimal state $\rho{A}(t_{0})$ and observable $O$. This
can be carried out with the aid of the quantum Cramér-Rao theorem Braunstein
and Caves (1994), stating that the estimation error $\delta T$ from measuring
any observable is bounded by the inequality $\delta T\geq
1/\sqrt{\nu\mathcal{F}_{N}(T)}$, where $\nu$ denotes the number of
measurements, and $\mathcal{F}_{N}(T)$ is the quantum Fisher information (QFI)
for $\rho{A}(t_{1})$. Therefore, the optimal $\rho{A}(t_{0})$ is the state
maximizing the QFI, and the optimal $O$ is the observable saturating the
inequality. It is well-known Giovannetti  _et al._ (2006) that such
$\rho{A}(t_{0})$ and $O$ are respectively
$\left(\ket{a_{M}}+\ket{a_{m}}\right)/\sqrt{2}$ and
$\ket{a_{M}}\bra{a_{m}}+\ket{a_{m}}\bra{a_{M}}$. Here, $\ket{a_{M}}$ and
$\ket{a_{m}}$ denote the eigenvectors of $A$ corresponding to the maximum
eigenvalue $a_{M}$ and minimum eigenvalue $a_{m}$, respectively.

In light of the above analysis, we may now specify our scheme as follows: (i)
Put $N$ probes $\mathpzc{S}_{1},\cdots,\mathpzc{S}_{N}$ in contact with the
sample and prepare each of them in $\rho_{{}_{T}}$ at time $t_{0}$. (ii)
Initialize the state of ancilla $\mathpzc{A}$ to be
$\left(\ket{a_{M}}+\ket{a_{m}}\right)/\sqrt{2}$ and couple all of the probes
to $\mathpzc{A}$ for the time $1/g$. (iii) Perform the measurement associated
with $O=\ket{a_{M}}\bra{a_{m}}+\ket{a_{m}}\bra{a_{M}}$ on $\mathpzc{A}$ at
time $t_{1}=t_{0}+1/g$. (iv) Repeat steps (ii) and (iii) $\nu$ times to build
a sufficient statistics for determining $T$. As the QFI for $\rho{A}(t_{1})$
reads

$\displaystyle\mathcal{F}_{N}(T)=N^{2}(a_{M}-a_{m})^{2}\absolutevalue{\partial_{{}_{T}}\varphi_{{}_{T}}}^{2},$
(16)

the precision attained in our scheme approaches the HS

$\displaystyle\delta
T={1}/\left[{\sqrt{\nu}N(a_{M}-a_{m})\absolutevalue{\partial_{{}_{T}}\varphi_{{}_{T}}}}\right].$
(17)

Notably, the evolving state of the $N$ probes and the ancilla is approximately
$\rho_{{}_{T}}^{\otimes N}\otimes\rho{A}(t)$. Thus, no highly entangled state
is involved in the whole estimation procedure of our scheme. Moreover, the
probing time in our scheme is a single “unit” $1/g$ and does not increase as
$N$ increases. In passing, we point out that the precision of the form (17) is
termed as the weak Heisenberg limit in Ref. Luis (2017).

It may be instructive to give a physical picture for comprehending how our
scheme works. To gain some insight into the physical origin of Eq. (8), we may
interpret the probe employed in our scheme as an effective transducer, which
helps to extract the information about temperature from the sample and
transmit it to the ancilla under the condition that the Markovian
thermalization is strong enough so that the memory effect is negligible. The
ancilla, on the other hand, may be interpreted as an information storage and
is responsible for storing the information about temperature, with its initial
state determining the storage capacity. Such a continuous interplay between
the probe and the ancilla leads to the basic dynamical equation (8) permitting
the coherent accumulation of the temperature-dependent phase
$\varphi_{{}_{T}}$ in the state of the ancilla. The accumulating rate is
described by $g$, which is reminiscent of the transduction parameter in
electrical or magnetic field sensing Degen  _et al._ (2017). When $N$ probes
are coupled to the same ancilla, the accumulating rate is increased from $g$
to $Ng$ as indicated in Eq. (14), since there are now $N$ effective
transducers transmitting the information about temperature to the ancilla. We
emphasize that the working principle of our scheme is built upon the very
nature of open dynamics, which distinguishes our scheme from previous schemes
involving Ramsey and Mech-Zehnder interferometers where the dynamics is
unitary.

Built-in robustness of our scheme against noises. Taking noises into account,
we shall rewrite Eq. (9) as

$\displaystyle\partial_{t}\rho(t)=$
$\displaystyle[(\mathcal{L}_{\mathpzc{S}_{1}\mathpzc{A}}+\mathcal{L}_{\mathpzc{S}_{1}}^{\textrm{noise}})+\cdots+(\mathcal{L}_{\mathpzc{S}_{N}\mathpzc{A}}+\mathcal{L}_{\mathpzc{S}_{N}}^{\textrm{noise}})+$
(18) $\displaystyle\mathcal{L}_{\mathpzc{A}}^{\textrm{noise}}]\rho(t),$

with $\mathcal{L}_{\mathpzc{S}_{n}}^{\textrm{noise}}$ and
$\mathcal{L}_{\mathpzc{A}}^{\textrm{noise}}$ denoting the Liouville
superoperators describing the noises acting on $\mathpzc{S}_{n}$ and
$\mathpzc{A}$, respectively. Here, no restriction is imposed on the explicit
forms of $\mathcal{L}_{\mathpzc{S}_{n}}^{\textrm{noise}}$ and
$\mathcal{L}_{\mathpzc{A}}^{\textrm{noise}}$. Under the condition that the
Markovian thermalization $\mathcal{L}_{\mathpzc{S}_{n}}$ is strong, the QFI of
our scheme in the presence of the noises can be evaluated as

$\displaystyle\mathcal{F}_{N}^{\textrm{noise}}(T)=2N^{2}\absolutevalue{\partial_{{}_{T}}\varphi_{{}_{T}}}^{2}\sum_{k\neq
l}p_{k,l}\absolutevalue{\bra{\phi_{k}}A\ket{\phi_{l}}}^{2},$ (19)

with coefficients

$\displaystyle p_{k,l}=\begin{cases}0,&\mbox{if }p_{k}=p_{l}=0,\\\
\frac{(p_{k}-p_{l})^{2}}{p_{k}+p_{l}},&\mbox{otherwise}.\end{cases}$ (20)

Here, $p_{k}$ and $\ket{\phi_{k}}$ are the eigenvalue and associated
eigenvector of a density operator $\delta$ which is determined by
$\mathcal{L}{A}^{\textrm{noise}}$ and $\rho{A}(t_{0})$. The proof of Eq. (19)
is presented in Supplementary Note 1, where the expression of $\delta$ is
given. Using Eqs. (16) and (19), we arrive at the result that the ratio
${\mathcal{F}_{N}^{\textrm{noise}}(T)}/{\mathcal{F}_{N}(T)}$ is independent of
$N$. This result means that the detrimental influence of noises on our scheme
does not become increasingly severe as $N$ increases.

To see the significance of the above result, we may recall the reason why the
HS permitted in many previous schemes is extremely vulnerable to noises. So
far, a lot of effort has been devoted to exploring the possibilities of
encoding the parameter of interest into the relative phase of a quantum system
for reaching HS, which has become one main stream of research on Heisenberg-
scalable metrology nowadays. Generally speaking, along this line of
development, previous schemes require either highly entangled states or
increasingly long probing time. A well-known example is the parallel scheme
Giovannetti  _et al._ (2006), in which $N$ probes are employed and a relative
phase $N\varphi$ is obtained by preparing the initial state of the probes to
be an $N$-body maximally entangled state. Evidently, the $N$-body maximally
entangled state is increasingly vulnerable to noises as $N$ increases. Hence,
in the presence of noises, the parallel scheme usually suffers from an
exponential drop in performance as $N$ increases Degen  _et al._ (2017); et
al. (2018a). Another well-known example is the sequential scheme Giovannetti
_et al._ (2006), in which a single probe is employed and a relative phase
$N\varphi$ is obtained with an $N$ times long probing time. Since the
decoherence effect is accumulated with time, the longer the probing time is,
the severer the detrimental influence of noises on the sequential scheme
becomes. Likewise, the performance of the sequential scheme usually drops
exponentially as $N$ increases in the presence of noises Degen  _et al._
(2017); et al. (2018a). As a matter of fact, most of the experiments relying
on highly entangled states or increasingly long probing time are confined to
the regime of very small $N$ Degen  _et al._ (2017); et al. (2018a).

Our scheme provides a different means of encoding the parameter of interest
into the relative phase of a quantum system without appealing to highly
entangled states and increasingly long probing time. Here, by saying
“different means,” we mean that the phase $N\varphi_{{}_{T}}$ stems from the
information transmission from the $N$ probes to the same ancilla, which are
permitted by the very nature of open dynamics. The key advantage of exploring
this means for reaching HS is that the detrimental influence of noises is no
longer increasingly severe as $N$ increases. This means that our scheme is
robust against noises, which, as demonstrated below, allows for approaching HS
for a wide range of $N$. Notably, the robustness of our scheme is intrinsic,
since no error correction technique is involved in our scheme. Moreover, it is
worth noting that the robustness of our scheme is irrespective of the specific
type of noises in question, since Eq. (19) is derived without restricting the
explicit forms of $\mathcal{L}_{\mathpzc{S}_{n}}^{\textrm{noise}}$ and
$\mathcal{L}_{\mathpzc{A}}^{\textrm{noise}}$. In passing, we would like to
point out that it has been shown in several works Stace (2010); Sabín  _et
al._ (2014) that the problem of temperature estimation can be mapped into the
problem of phase estimation. Analogous to the parallel scheme, the schemes in
these works are based on highly entangled states like NOON states and have
been experimentally shown to be very vulnerable to noises Uhlig  _et al._
(2019).

Example. Let us furnish an analytically solvable model to demonstrate the
usefulness of our scheme. Consider the physical model of $N$ qubits
independently contacting with a Bosonic thermal reservoir of temperature $T$
Seah  _et al._ (2019). We take the qubits to be
$\mathpzc{S}_{1},\cdots,\mathpzc{S}_{N}$. The thermalization process of
$\mathpzc{S}_{n}$ is governed by the Liouvillian

$\displaystyle\mathcal{L}_{\mathpzc{S}_{n}}\rho=-\frac{\textrm{i}}{\hbar}[H_{\mathpzc{S}_{n}},\rho]+\gamma(\overline{n}+1)\mathcal{D}[\sigma_{-}^{\mathpzc{S}_{n}}]\rho+\gamma\overline{n}\mathcal{D}[\sigma_{+}^{\mathpzc{S}_{n}}]\rho.$

Here, $H_{\mathpzc{S}_{n}}=\hbar\Omega\sigma_{z}^{\mathpzc{S}_{n}}/2$ is the
free Hamiltonian of $\mathpzc{S}_{n}$, with $\sigma_{i}^{\mathpzc{S}_{n}}$,
$i=x,y,z$, denoting the Pauli matrices acting on ${\mathpzc{S}_{n}}$. The last
two terms stand for the process of energy exchange with the reservoir, with
$\gamma>0$ describing the exchange rate. Note that the dissipative gap
$\lambda$ is proportional to $\gamma$, i.e., $\lambda\propto\gamma$.
$\overline{n}=[\exp\left(\hbar\Omega/k_{B}T\right)-1]^{-1}$ and
$\mathcal{D}[\sigma_{\pm}^{\mathpzc{S}_{n}}]\rho=\sigma_{\pm}^{\mathpzc{S}_{n}}\rho\sigma_{\mp}^{\mathpzc{S}_{n}}-\\{\sigma_{\mp}^{\mathpzc{S}_{n}}\sigma_{\pm}^{\mathpzc{S}_{n}},\rho\\}/2$,
with $\sigma_{\pm}^{\mathpzc{S}_{n}}=(\sigma_{x}^{\mathpzc{S}_{n}}\mp
i\sigma_{y}^{\mathpzc{S}_{n}})/2$. Note that the unique steady state of
$\mathpzc{S}_{n}$ happens to be the Gibbs state
$\rho_{{}_{T}}=\exp(-H_{\mathpzc{S}_{n}}/k_{B}T)/Z_{T}$, where $k_{B}$ is the
Boltzmann’s constant and $Z_{T}=\tr\exp(-H_{\mathpzc{S}_{n}}/k_{B}T)$ the
partition function. $\mathpzc{A}$ is also taken to be a qubit. The interaction
between $\mathpzc{S}_{n}$ and $\mathpzc{A}$ is set to be $\hbar
g\sigma_{z}^{\mathpzc{S}_{n}}\otimes\sigma_{z}{A}$. So,

$\displaystyle\mathcal{L}_{\mathpzc{S}_{n}\mathpzc{A}}\rho=\mathcal{L}_{\mathpzc{S}_{n}}\rho-\textrm{i}g[\sigma_{z}^{\mathpzc{S}_{n}}\otimes\sigma_{z}{A},\rho].$
(22)

We examine a major source of noises, dephasing. That is, the Liouville
superoperator $\mathcal{L}_{\mathpzc{S}_{n}/\mathpzc{A}}^{\textrm{noise}}$
describing the noise acting on $\mathpzc{S}_{n}/\mathpzc{A}$ reads

$\displaystyle\mathcal{L}_{\mathpzc{S}_{n}/\mathpzc{A}}^{\textrm{noise}}\rho=\kappa_{\mathpzc{S}_{n}/\mathpzc{A}}\left(\sigma_{z}^{\mathpzc{S}_{n}/\mathpzc{A}}\rho\sigma_{z}^{\mathpzc{S}_{n}/\mathpzc{A}}-\rho\right),$
(23)

where $\kappa_{\mathpzc{S}_{n}/\mathpzc{A}}$ denotes the dephasing rate. The
initial state of the probes and the ancilla is set to be
$\rho_{{}_{T}}^{\otimes N}\otimes\rho_{\mathpzc{A}}(t_{0})$ with
$\rho_{\mathpzc{A}}(t_{0})=(\ket{0}+\ket{1})(\bra{0}+\bra{1})/2$. So far, we
have specified all the Liouville superoperators entering the full dynamical
equation (18) as well as the initial state assumed in our scheme.

Now, let us figure out the performance of our scheme in the presence of the
noises. To this end, we need to find out the output state of $\mathpzc{A}$ at
time $t_{1}=t_{0}+1/g$, which is given by

$\displaystyle\rho{A}(t_{1})=\frac{1}{2}\begin{pmatrix}1&\Gamma^{N}e^{-2\eta}\\\
\Gamma^{*N}e^{-2\eta}&1\end{pmatrix}.$ (24)

Here, $1/g$ is the probing time which is independent of $N$, and

$\displaystyle\Gamma=$
$\displaystyle\frac{e^{\omega_{+}(\xi)}}{2}\left[1+\frac{(2\overline{n}+1)\xi}{\sqrt{\Delta(\xi)}}-\frac{4\textrm{i}}{(2\overline{n}+1)\sqrt{\Delta(\xi)}}\right]+$
(25)
$\displaystyle\frac{e^{\omega_{-}(\xi)}}{2}\left[1-\frac{(2\overline{n}+1)\xi}{\sqrt{\Delta(\xi)}}+\frac{4\textrm{i}}{(2\overline{n}+1)\sqrt{\Delta(\xi)}}\right],$

where $\Delta(\xi)=(2\overline{n}+1)^{2}\xi^{2}-8\textrm{i}\xi-16$ and
$\omega_{\pm}(\xi)=[-(2\overline{n}+1)\xi\pm\sqrt{\Delta(\xi)}]/2$, with
$\xi=\gamma/g$ and $\eta=\kappa{A}/g$ characterizing (in units of $g$) the
strength of the Markovian thermalization and that of the dephasing noise
acting on the ancilla $\mathpzc{A}$, respectively. The proof of Eq. (24) is
presented in Supplementary Note 2. It is interesting to note that the noises
acting on the probes $\mathpzc{S}_{1},\cdots,\mathpzc{S}_{N}$ do not affect
the output state $\rho{A}(t_{1})$. The QFI for $\rho{A}(t_{1})$ reads

$\displaystyle\mathcal{F}_{N}^{\textrm{noise}}(T)=$ $\displaystyle
N^{2}\absolutevalue{\Gamma}^{2N-2}\absolutevalue{\partial_{{}_{T}}\Gamma}^{2}e^{-4\eta}+N^{2}\absolutevalue{\Gamma}^{4N-2}\times$
(26)
$\displaystyle\left(\partial_{{}_{T}}\absolutevalue{\Gamma}\right)^{2}e^{-8\eta}/\left(1-\absolutevalue{\Gamma}^{2N}e^{-4\eta}\right).$

Equations (24) and (26) are analytical results without numerical
approximation.

We first examine the limiting case of $\xi\rightarrow\infty$, which
corresponds to the ideal situation that the Markovian thermalization is
infinitely strong and there is no memory effect. Using Eq. (26) and noting
that $\lim_{\xi\rightarrow\infty}\Gamma=\exp(-2\textrm{i}\varphi_{{}_{T}})$,
where $\varphi_{{}_{T}}:=\tr(\sigma_{z}\rho_{{}_{T}})=1/(2\overline{n}+1)$, we
have that

$\displaystyle\mathcal{F}_{N}^{\textrm{noise}}(T)=4N^{2}\absolutevalue{\partial_{{}_{T}}\varphi_{{}_{T}}}^{2}e^{-4\eta}.$
(27)

Equation (27) represents a HS of precision. Since

$\displaystyle\rho{A}(t_{1})=\frac{1}{2}\begin{pmatrix}1&e^{-2\textrm{i}N\varphi_{{}_{T}}}e^{-2\eta}\\\
e^{2\textrm{i}N\varphi_{{}_{T}}}e^{-2\eta}&1\end{pmatrix},$ (28)

we see that the HS in Eq. (27) stems from the relative phase
$N\varphi_{{}_{T}}$ acquired in $\rho{A}(t_{1})$. Notably, this relative phase
is obtained without appealing to highly entangled states and increasingly long
probing time. A consequence of this fact is that the detrimental influence of
noises does not lead to an exponential drop of the HS. Indeed, using Eq. (27)
and noting that
$\mathcal{F}_{N}(T)=4N^{2}\absolutevalue{\partial_{{}_{T}}\varphi_{{}_{T}}}^{2}$,
we have that

$\displaystyle\mathcal{F}_{N}^{\textrm{noise}}(T)/\mathcal{F}_{N}(T)=e^{-4\eta},$
(29)

which is consistent with our general analysis that
$\mathcal{F}_{N}^{\textrm{noise}}(T)/\mathcal{F}_{N}(T)$ is independent of
$N$.

We now examine the situation that the strength of the Markovian thermalization
is large and within current experimental reach. To do this, we numerically
compute $\mathcal{F}_{N}^{\textrm{noise}}(T)$ in units of
$\mathcal{F}_{\textrm{th}}(T)$ for different $\xi$. Here,
$\mathcal{F}_{\textrm{th}}(T)=(\hbar\Omega/2k_{B}T^{2})^{2}\sech^{2}(\hbar\Omega/2k_{B}T)$
is the QFI for the Gibbs state
$\rho_{{}_{T}}=\exp(-H_{\mathpzc{S}_{n}}/k_{B}T)/Z_{T}$ Seah  _et al._ (2019).
We choose four values of $\xi$, namely, $100$, $200$, $300$, and $400$. To
reach these values, we can set, for instance, $g=100$ kHz and $\gamma=10$,
$20$, $30$, $40$ MHz, which are experimentally feasible for a number of
quantum platforms such as Rydberg atoms and trapped ions Müller  _et al._
(2012). We examine an exemplary temperature satisfying $k_{B}T/\hbar\Omega=2$,
which is of interest in quantum thermometry Seah  _et al._ (2019). The
numerical results are shown in Fig. 2, where $\eta$ is set to be $1/10$ in
view of the convention that the lifetime of an ancilla is usually assumed to
be long in quantum metrology (see, e.g., Refs. Kessler  _et al._ (2014); Dür
_et al._ (2014); Arrad  _et al._ (2014); et al. (2016); Zhou  _et al._
(2018)). Here, we also show the QFI given by Eq. (27) (see the black solid
line), which serves as a benchmark for the QFIs associated with the four
values of $\xi$.

Figure 2: Log-log plot of the quantum Fisher information
$\mathcal{F}_{N}^{\textrm{noise}}(T)$ as a function of $N$ in units of
$\mathcal{F}_{\textrm{th}}(T)$ for different $\xi$. The black solid line
corresponds to the quantum Fisher information in Eq. (27). Parameters used are
$k_{B}T/\hbar\Omega=2$ and $\eta=1/10$.

The four dashed curves in Fig. 2 correspond to the QFIs associated with the
four values of $\xi$ (see the figure legend). As can be seen from this figure,
these dashed curves approach the black solid line when $N$ is no larger than a
certain $N_{\textrm{max}}$. This means that the HS described by Eq. (27) is
attained in the regime of $N\leq N_{\textrm{max}}$. It is easy to see that
$N_{\textrm{max}}$ depends on $\xi$; more precisely, the larger $\xi$ is, the
greater $N_{\textrm{max}}$ can be. We numerically find that $N_{\textrm{max}}$
may be respectively taken to be $109$, $217$, $326$, $435$ for the four values
of $\xi$. Note that the HS achieved in current experiments is usually confined
to the regime of $N\leq N_{\textrm{max}}=10$ et al. due to the detrimental
effect of noises. The above numerical results demonstrate that our scheme
allows for approaching HS in the presence of noises for a wide range of $N$.
Moreover, it is worth noting that strongly dissipative processes have been
experimentally realized in a number of quantum platforms (see the review
article Müller  _et al._ (2012)). This means that the HS permitted in the
above wide ranges of $N$ may be achieved using a variety of experimental
setups (to be listed in the Discussion section), which makes our scheme
appealing from a practical perspective.

Figure 3: Log-log plot of $\mathcal{F}_{N}^{\textrm{noise}}(T)$ as a function
of $N$ in units of $\mathcal{F}_{\textrm{th}}(T)$ for different $\eta$.
Parameters used are $k_{B}T/\hbar\Omega=2$ and $\xi=400$.

Figure 3 depicts $\mathcal{F}_{N}^{\textrm{noise}}(T)$ as a function of $N$ in
units of $\mathcal{F}_{\textrm{th}}(T)$ for different noise strength $\eta$.
Here, $k_{B}T/\hbar\Omega=2$, $\xi=400$, and four values of $\eta$ are chosen,
namely, $\eta=0.1,0.2,0.3,0.4$. As can be seen from Fig. 3,
$\mathcal{F}_{N}^{\textrm{noise}}(T)$ decreases as $\eta$ increases, which is
expected from a physical point of view. It is interesting to note that
$N_{\textrm{max}}$ does not change much in the course of varying $\eta$,
indicating that the range in which the HS is attained is irrespective of the
value of $\eta$. Moreover, in Supplementary Note 3, we examine the influence
of the initial state of the $N$ probes and the ancilla on
$\mathcal{F}_{N}^{\textrm{noise}}(T)$. We consider the initial state of the
form $\rho^{\otimes N}\otimes\sigma$, where $\rho$ and $\sigma$ are
arbitrarily given $2\times 2$ density matrices. Both analytical and numerical
results show that $\mathcal{F}_{N}^{\textrm{noise}}(T)$ does not depend much
on $\rho$ but heavily relies on $\sigma$. More precisely,
$\mathcal{F}_{N}^{\textrm{noise}}(T)$ grows quadratically as a function of the
$l_{1}$ norm of coherence of $\sigma$ Zhang  _et al._ (2018).

We remark that the HS given by Eq. (27) is lost when $N$ is too large. The
underlying reason is that there is some memory effect for each
$\mathcal{E}_{\mathpzc{S}_{n}\mathpzc{A}}(t_{1},t_{0})$ appearing in
decomposition (12) of $\mathcal{E}_{N}(t_{1},t_{0})$ and these memory effects
may add linearly so that the total memory effect for
$\mathcal{E}_{N}(t_{1},t_{0})$ becomes non-negligible for a too large $N$.
Nevertheless, as long as the Markovian thermalization is strong enough, these
memory effects can be made very weak and any large value of $N_{\textrm{max}}$
can be obtained in our scheme. By the way, considering that a stronger
thermalization allows for producing a larger number of Gibbs states per unit
time, a direct approach of taking advantage of strong thermalization for
thermometry might be to repeatedly produce and measure Gibbs states with
individual probes. Yet, this approach is ineffective in practice. Indeed,
real-word quantum measurements suffer from correlated background noises if the
measurement time is shorter than the noise correlation time Feizpour  _et al._
(2011); Chu  _et al._ (2020). As these noises cannot be averaged out by
repetitive measurements Feizpour  _et al._ (2011); Chu  _et al._ (2020), the
direct approach may not even be able to beat the standard quantum limit in the
presence of these noises.

##  III Discussion

The key to the implementation of our scheme is to realize the weak coupling of
$N$ probes to a same ancilla. Apart from the magnetic resonance force
microscopy, this kind of coupling is achievable in a variety of experimental
setups, including superconducting qubits coupled to a microwave resonator et
al. (2013b), Rydberg atoms coupled to a microwave cavity Garcia  _et al._
(2019), trapped ions coupled to a common motional mode Pedernales  _et al._
(2015) or an optical cavity mode et al. (2019a), and nitrogen-vacancy (NV)
centers in diamond coupled to a microwave mode in a superconducting
transmission line cavity et al. (2017).

Thanks to the freedom in choosing explicit forms of $\mathcal{L}{S}$ and
$\rho_{{}_{T}}$, various kinds of probes can be used in our scheme.
Particularly, the probes in previous proposals of thermometry may be used in
our scheme, and our scheme may be exploited to promote these proposals to
reach HS. To this end, one only needs to find an ancilla that is able to
couple with the $N$ probes given in a previous proposal. The ancilla can be a
NV center if hybridization proposals et al. (2018b) are under consideration,
or a measuring apparatus when quantum nondemolition measurements are used et
al. (2019b); Pati  _et al._ (2020), or even an environment to which we have
access Giovannetti  _et al._ (2011); Degen  _et al._ (2017).

The applications of our scheme are not limited to thermometry but can be
straightforwardly extended to other metrological tasks. Indeed, our scheme is
applicable to any metrological task in which the parameter of interest can be
encoded in the unique steady state of a Markovian system. The metrological
tasks of this kind are far more than quantum thermometry and are frequently
encountered in quantum sensing, especially in noisy quantum metrology
Giovannetti  _et al._ (2011). Two examples are the detection of rotating
fields Chen  _et al._ and the sensing of low-frequency signals Xie  _et al._
(2020b).

In summary, we have proposed a scheme of thermometry to approach HS for a wide
range of $N$, based on the finding that an $N$-fold increase of the
temperature-dependent phase can be obtained from the continuous interplay
between the strong thermalization of $N$ probes and the weak coupling to the
same external ancilla. As opposed to conventional ones, our scheme gets rid of
a number of experimentally demanding requirements, e.g., the preparation of
highly entangled states, and is implementable in a variety of experimental
setups. More importantly, it offers the key advantage of robustness
irrespective of the type of noise in question, without resorting to
complicated error correction techniques. Therefore, our scheme provides a
feasible and robust pathway to the HS, with the entanglement-free feature
which is noteworthy in view of previous proposals of Heisenberg-scalable
thermometry. Two directions for future work are to exploit our scheme to
promote previous proposals of thermometry to reach HS and to apply our scheme
to other metrological tasks. Besides, throughout this paper, we have assumed
the Markovian dissipative process described by the Lindblad equation, which is
an approximative description relying on a number of simplifications. A more
detailed analysis of our scheme starting from a fully microscopic description,
while going beyond the scope of the present work, would be an interesting
topic for future studies.

##  IV Methods

To obtain Eq. (7), it is convenient to express $\mathcal{L}$ as
$\mathcal{L}=\mathcal{L}{S}+g\mathcal{K}$ with
$\mathcal{K}X:=-\textrm{i}[S\otimes A,X]$. Using the equalities
$\mathcal{L}{S}\mathcal{P}=\mathcal{P}\mathcal{L}{S}=0$ Zhang  _et al._
(2016a, b), we have

$\displaystyle\mathcal{PL}=g\mathcal{PK},~{}~{}~{}~{}\mathcal{LP}=g\mathcal{KP}.$
(30)

Note that
$\norm{\mathcal{E}_{1}\mathcal{E}_{2}}\leq\norm{\mathcal{E}_{1}}\norm{\mathcal{E}_{2}}$
for two superoperators $\mathcal{E}_{1}$ and $\mathcal{E}_{2}$. Here and
throughout this paper, the Hilbert-Schmidt norm is adopted. That is, for an
operator $X$, the norm reads $\norm{X}:=\sqrt{\tr(X^{\dagger}X)}$, and for a
superoperator $\mathcal{E}$, it is the induced norm defined as
$\norm{\mathcal{E}}:=\sup_{\norm{X}\leq 1}\norm{\mathcal{E}(X)}$. It follows
from Eq. (30) that $\norm{\mathcal{P}\mathcal{L}}\leq
g\norm{\mathcal{P}}\norm{\mathcal{K}}$ and $\norm{\mathcal{L}\mathcal{P}}\leq
g\norm{\mathcal{K}}\norm{\mathcal{P}}$. Using these two inequalities and
noting that $\norm{\rho(s)}=\sqrt{\tr\rho^{2}(s)}\leq 1$, we have

$\displaystyle\norm{\int_{t_{0}}^{t}ds\mathcal{P}\mathcal{L}\mathcal{G}(t,s)\mathcal{Q}\mathcal{L}\mathcal{P}\rho(s)}\leq\int_{t_{0}}^{t}ds\norm{\mathcal{P}\mathcal{L}\mathcal{G}(t,s)\mathcal{Q}\mathcal{L}\mathcal{P}\rho(s)}$
$\displaystyle\leq\norm{\mathcal{P}\mathcal{L}}\left(\int_{t_{0}}^{t}ds\norm{\mathcal{G}(t,s)\mathcal{Q}}\right)\norm{\mathcal{L}\mathcal{P}}\leq\alpha
g^{2},$ (31)

where
$\alpha:=\norm{\mathcal{P}}^{2}\norm{\mathcal{K}}^{2}\int_{t_{0}}^{t}ds\norm{\mathcal{G}(t,s)\mathcal{Q}}$.
It can be shown that

$\displaystyle\norm{\mathcal{G}(t,s)\mathcal{Q}}\leq\varepsilon\exp[\left(\varepsilon
g\norm{\mathcal{K}}-\lambda\right)(t-s)].$ (32)

The proof of Eq. (32) is very technical and presented in Supplementary Note 4,
where the expression for $\varepsilon$ is given. Under the condition

$\displaystyle\lambda/g>{\varepsilon\norm{\mathcal{K}}},$ (33)

$\norm{\mathcal{G}(t,s)\mathcal{Q}}$ decreases exponentially as $t-s$
increases. Then,

$\displaystyle\alpha\leq\norm{\mathcal{P}}^{2}\norm{\mathcal{K}}^{2}\left[\varepsilon/(\lambda-\varepsilon
g\norm{\mathcal{K}})\right].$ (34)

Inserting Eq. (34) into Eq. (IV), we can immediately obtain Eq. (7).

Data availability
Numerical data from the plots presented are available from D.-J.Z. upon
request.

Code availability
The codes used for this study are available from D.-J.Z. upon request.

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Acknowledgements
This work was supported by the National Natural Science Foundation of China
through Grant Nos. 11775129 and 12174224.

Author contributions
All the authors contribute equally to this work.

Competing interests
The authors declare no competing interests.

SUPPLEMENTAL INFORMATION

## Supplementary Note 1: Proof of Eq. (19) in the main text

Here, we explain why the detrimental influence of noises on our scheme does
not become increasingly severe as $N$ increases, irrespective of the type of
noises in question. By taking into account noises, the full dynamical equation
of the $N$ probes and the ancilla reads

$\displaystyle\partial_{t}\rho(t)=(\mathcal{L}_{\mathpzc{S}_{1}}+\mathcal{L}_{\mathpzc{S}_{1}}^{\textrm{noise}})\rho(t)-\textrm{i}g[S_{1}\otimes
A,\rho(t)]+\cdots+(\mathcal{L}_{\mathpzc{S}_{N}}+\mathcal{L}_{\mathpzc{S}_{N}}^{\textrm{noise}})\rho(t)-\textrm{i}g[S_{N}\otimes
A,\rho(t)]+\mathcal{L}{A}^{\textrm{noise}}\rho(t):=\mathcal{L}\rho(t).$

Here, $\mathcal{L}_{\mathpzc{S}_{n}}^{\textrm{noise}}$ and
$\mathcal{L}_{\mathpzc{A}}^{\textrm{noise}}$ denote the noises acting on the
$n$th probe $\mathpzc{S}_{n}$ and the ancilla $\mathpzc{A}$, respectively.
$S_{n}$ is a Hermitian operator acting on $\mathpzc{S}_{n}$. We assume for
simplicity that $\mathcal{L}_{\mathpzc{S}_{n}}^{\textrm{noise}}$’s are
identical with each other and $S_{n}$’s are identical with each other. Our
following discussion can be straightforwardly extended to the scenario that
$\mathcal{L}_{\mathpzc{S}_{n}}^{\textrm{noise}}$’s are different and $S_{n}$’s
are different.

Let $\rho_{{}_{T}}^{\prime}$ be the steady state associated with the
Liouvillian
$\mathcal{L}_{\mathpzc{S}_{n}}+\mathcal{L}_{\mathpzc{S}_{n}}^{\textrm{noise}}$;
that is,
$(\mathcal{L}_{\mathpzc{S}_{n}}+\mathcal{L}_{\mathpzc{S}_{n}}^{\textrm{noise}})\rho_{{}_{T}}^{\prime}=0$.
Define $\mathcal{P}$ as $\mathcal{P}X=\rho_{{}_{T}}^{\prime\otimes
N}\otimes\tr_{\mathpzc{S}_{1}\cdots\mathpzc{S}_{N}}X$ and $\mathcal{Q}$ as
$\mathcal{Q}X=X-\mathcal{P}X$. Using the same reasoning as in deriving Eq. (5)
in the main text, we have

$\displaystyle\partial_{t}\mathcal{P}\rho(t)=\mathcal{P}\mathcal{L}\mathcal{P}\rho(t)+\int_{t_{0}}^{t}ds\mathcal{P}\mathcal{L}\mathcal{G}(t,s)\mathcal{Q}\mathcal{L}\mathcal{P}\rho(s),$
(IV.0S.2)

with $\mathcal{L}$ given by Eq. (Supplementary Note 1: Proof of Eq. (19) in
the main text). As in the main text, the second term on the right-hand side of
Eq. (IV.0S.2) represents the total memory effect and is of the order
$\mathcal{O}(g^{2})$. Under the assumption that the Markovian thermalization
of each probe ${\mathpzc{S}_{n}}$ is strong so that the total memory effect is
negligible, we can neglect the second term on the right-hand side of Eq.
(IV.0S.2). Then, we have

$\displaystyle\partial_{t}\mathcal{P}\rho(t)=\mathcal{P}\mathcal{L}\mathcal{P}\rho(t).$
(IV.0S.3)

Clearly,

$\displaystyle\mathcal{P}\rho(t)=\rho_{{}_{T}}^{\prime\otimes
N}\otimes\rho{A}(t).$ (IV.0S.4)

A direct calculation shows that

$\displaystyle\mathcal{P}\mathcal{L}\mathcal{P}\rho(t)=\rho_{{}_{T}}^{\prime\otimes
N}\otimes\left\\{-\textrm{i}Ng[\varphi_{{}_{T}}^{\prime}A,\rho{A}(t)]+\mathcal{L}{A}^{\textrm{noise}}\rho{A}(t)\right\\},$
(IV.0S.5)

where $\varphi_{{}_{T}}^{\prime}=\tr(S_{i}\rho_{{}_{T}}^{\prime})$.
Substituting Eqs. (IV.0S.4) and (IV.0S.5) into Eq. (IV.0S.3) yields

$\displaystyle\partial_{t}\rho{A}(t)=-\textrm{i}Ng[\varphi_{{}_{T}}^{\prime}A,\rho{A}(t)]+\mathcal{L}{A}^{\textrm{noise}}\rho{A}(t).$
(IV.0S.6)

Note that $\mathcal{L}_{\mathpzc{S}_{n}}$ is assumed to be strong and,
therefore, the steady state associated with
$\mathcal{L}_{\mathpzc{S}_{n}}+\mathcal{L}_{\mathpzc{S}_{n}}^{\textrm{noise}}$
is mainly determined by $\mathcal{L}_{\mathpzc{S}_{n}}$. So,

$\displaystyle\rho_{{}_{T}}^{\prime}\approx\rho_{{}_{T}},$ (IV.0S.7)

and hence,

$\displaystyle\varphi_{{}_{T}}^{\prime}\approx\varphi_{{}_{T}}.$ (IV.0S.8)

That is, the effect of $\mathcal{L}_{\mathpzc{S}_{n}}^{\textrm{noise}}$ is to
slightly alter the steady state of $\mathpzc{S}_{n}$, which does not affect
much the performance of our scheme. Then, we can rewrite Eq. (IV.0S.6) as

$\displaystyle\partial_{t}\rho{A}(t)=-\textrm{i}Ng[\varphi_{{}_{T}}A,\rho{A}(t)]+\mathcal{L}{A}^{\textrm{noise}}\rho{A}(t).$
(IV.0S.9)

$\mathcal{L}{A}^{\textrm{noise}}$ on the right-hand side of Eq. (IV.0S.9) is
independent of $N$. Also, no restriction is imposed on the form of
$\mathcal{L}{A}^{\textrm{noise}}$. These facts indicate that the detrimental
influence of noises on our scheme does not become increasingly severe as $N$
increases, irrespective of the type of noises in question.

To confirm the above point, let us figure out the QFI associated with the
output state of $\mathpzc{A}$. Without loss of generality, we can express
$\mathcal{L}{A}^{\textrm{noise}}$ as

$\displaystyle\mathcal{L}{A}^{\textrm{noise}}\rho=\sum_{l}\kappa_{l}\left(K_{l}\rho
K_{l}^{\dagger}-\frac{1}{2}\\{K_{l}^{\dagger}K_{l},\rho\\}\right),$ (IV.0S.10)

where $\kappa_{l}$ is the decaying rate and $K_{l}$ denotes the jump operator.
Define

$\displaystyle\widetilde{{\rho}}{A}(t):=U^{\dagger}(t,t_{0}){\rho}{A}(t)U(t,t_{0}),$
(IV.0S.11)

where $U(t,t_{0})=e^{-\textrm{i}Ng\varphi_{{}_{T}}A(t-t_{0})}$. Using Eqs.
(IV.0S.9), (IV.0S.10), and (IV.0S.11), we have that the dynamical equation for
$\widetilde{\rho}{A}(t)$ reads

$\displaystyle\partial_{t}\widetilde{\rho}{A}(t)=\mathcal{L}{A}^{\textrm{noise}}(t,t_{0})\widetilde{\rho}{A}(t),$
(IV.0S.12)

where

$\displaystyle\mathcal{L}{A}^{\textrm{noise}}(t,t_{0})\rho=\sum_{l}\kappa_{l}\left[{K}_{l}(t,t_{0})\rho{K}_{l}^{\dagger}(t,t_{0})-\frac{1}{2}\left\\{{K}_{l}^{\dagger}(t,t_{0}){K}_{l}(t,t_{0}),\rho\right\\}\right],$
(IV.0S.13)

with

$\displaystyle{K}_{l}(t,t_{0})=U^{\dagger}(t,t_{0})K_{l}U(t,t_{0}).$
(IV.0S.14)

Noting that

$\displaystyle
U(t,t_{0})=\sum_{\mu}e^{-\textrm{i}Ng\varphi_{{}_{T}}a_{\mu}(t-t_{0})}\ket{a_{\mu}}\bra{a_{\mu}},$
(IV.0S.15)

where $a_{\mu}$ and $\ket{a_{\mu}}$ denote the eigenvalue and the associated
eigenstate of $A$, we have that

$\displaystyle{K}_{l}(t,t_{0})=\sum_{\mu,\nu}e^{\textrm{i}Ng\varphi_{{}_{T}}(a_{\mu}-a_{\nu})(t-t_{0})}K_{l\mu\nu}\ket{a_{\mu}}\bra{a_{\nu}},$
(IV.0S.16)

with

$\displaystyle K_{l\mu\nu}=\bra{a_{\mu}}K_{l}\ket{a_{\nu}}$ (IV.0S.17)

Substituting Eq. (IV.0S.16) into Eq. (IV.0S.13) yields

$\displaystyle\mathcal{L}{A}^{\textrm{noise}}(t,t_{0})\rho=$
$\displaystyle\sum_{l}\kappa_{l}\sum_{\mu,\nu,\mu^{\prime},\nu^{\prime}}e^{\textrm{i}Ng\varphi_{{}_{T}}(a_{\mu}-a_{\nu}-a_{\mu}^{\prime}+a_{\nu}^{\prime})(t-t_{0})}K_{l\mu\nu}K_{l\mu^{\prime}\nu^{\prime}}^{*}\left(\ket{a_{\mu}}\bra{a_{\nu}}\rho\ket{a_{\nu}^{\prime}}\bra{a_{\mu}^{\prime}}-\frac{1}{2}\left\\{\ket{a_{\nu}^{\prime}}\bra{a_{\mu}^{\prime}}\ket{a_{\mu}}\bra{a_{\nu}},\rho\right\\}\right).$
(IV.0S.18)

Note that, for a large $N$, the complex exponentials
$e^{\textrm{i}Ng\varphi_{{}_{T}}(a_{\mu}-a_{\nu}-a_{\mu}^{\prime}+a_{\nu}^{\prime})(t-t_{0})}$
are rapidly oscillating when $a_{\mu}-a_{\nu}\neq
a_{\mu}^{\prime}-a_{\nu}^{\prime}$. Hence, on an appreciable time scale, the
oscillations will quickly average to zero. Using the rotating wave
approximation, we can neglect the terms in Eq. (Supplementary Note 1: Proof of
Eq. (19) in the main text) for $a_{\mu}-a_{\nu}\neq
a_{\mu}^{\prime}-a_{\nu}^{\prime}$ and, therefore, obtain the following
effective Liouvillian,

$\displaystyle\widetilde{\mathcal{L}}_{\mathpzc{A}}^{\textrm{noise}}\rho=\sum_{l}\kappa_{l}\sum_{a_{\mu}-a_{\nu}=a_{\mu}^{\prime}-a_{\nu}^{\prime}}K_{l\mu\nu}K_{l\mu^{\prime}\nu^{\prime}}^{*}\left(\ket{a_{\mu}}\bra{a_{\nu}}\rho\ket{a_{\nu}^{\prime}}\bra{a_{\mu}^{\prime}}-\frac{1}{2}\left\\{\ket{a_{\nu}^{\prime}}\bra{a_{\mu}^{\prime}}\ket{a_{\mu}}\bra{a_{\nu}},\rho\right\\}\right).$
(IV.0S.19)

Let $I:=\\{a_{\mu}-a_{\nu}|\textrm{$a_{\mu}$ and $a_{\nu}$ are the eigenvalues
of $A$}\\}$ be the set comprised of the differences among the eigenvalues of
$A$. We can rewrite Eq. (IV.0S.19) as

$\displaystyle\widetilde{\mathcal{L}}_{\mathpzc{A}}^{\textrm{noise}}\rho=\sum_{l}\sum_{\Delta\in
I}\kappa_{l}\left(K_{l,\Delta}\rho
K_{l,\Delta}^{\dagger}-\frac{1}{2}\left\\{K_{l,\Delta}^{\dagger}K_{l,\Delta},\rho\right\\}\right),$
(IV.0S.20)

with

$\displaystyle
K_{l,\Delta}:=\sum_{a_{\mu}-a_{\nu}=\Delta}K_{l\mu\nu}\ket{a_{\mu}}\bra{a_{\nu}}=\sum_{a_{\mu}-a_{\nu}=\Delta}\ket{a_{\mu}}\bra{a_{\mu}}K_{l}\ket{a_{\nu}}\bra{a_{\nu}}.$
(IV.0S.21)

Hence,

$\displaystyle\widetilde{\rho}{A}(t)=e^{\widetilde{\mathcal{L}}_{\mathpzc{A}}^{\textrm{noise}}(t-t_{0})}\widetilde{\rho}{A}(t_{0}).$
(IV.0S.22)

Noting that $\rho{A}(t)=U(t,t_{0})\widetilde{\rho}{A}(t)U^{\dagger}(t,t_{0})$
and
$\widetilde{\rho}{A}(t_{0})={\rho{A}}(t_{0})=\frac{1}{2}(\ket{a_{M}}+\ket{a_{m}})(\bra{a_{M}}+\bra{a_{m}})$,
we have that the output state of the ancilla reads

$\displaystyle\rho{A}(t_{0}+1/g)=U^{-\textrm{i}N\varphi_{{}_{T}}A}\delta
U^{\textrm{i}N\varphi_{{}_{T}}A}.$ (IV.0S.23)

Here $\delta$ is defined to be the state

$\displaystyle\delta=e^{\widetilde{\mathcal{L}}_{\mathpzc{A}}^{\textrm{noise}}/g}\left[\frac{1}{2}(\ket{a_{M}}+\ket{a_{m}})(\bra{a_{M}}+\bra{a_{m}})\right],$
(IV.0S.24)

whose eigendecomposition can be expressed as

$\displaystyle\delta=\sum_{k}p_{k}\ket{\phi_{k}}\bra{\phi_{k}}.$ (IV.0S.25)

Then, the QFI associated with the output state reads

$\displaystyle\mathcal{F}_{N}^{\textrm{noise}}(T)=\left(2\sum_{k\neq
l}p_{k,l}\absolutevalue{\bra{\phi_{k}}A\ket{\phi_{l}}}^{2}\right)N^{2}\absolutevalue{\partial_{{}_{T}}\varphi_{{}_{T}}}^{2},$
(IV.0S.26)

with coefficients

$\displaystyle p_{k,l}=\begin{cases}0,&\mbox{if }p_{k}=p_{l}=0\\\
\frac{(p_{k}-p_{l})^{2}}{p_{k}+p_{l}},&\mbox{otherwise}.\end{cases}$
(IV.0S.27)

## Supplementary Note 2: Proof of Eq. (24) in the main text

Here, we figure out the expression of $\rho{A}(t_{1})$ for the model under
consideration. Upon taking into account the dephasing noise, the full dynamics
of the $N$ probes and the ancilla reads

$\displaystyle\partial_{t}\rho(t)=\left[(\mathcal{L}_{\mathpzc{S}_{1}\mathpzc{A}}+\mathcal{L}_{\mathpzc{S}_{1}}^{\textrm{noise}})+\cdots+(\mathcal{L}_{\mathpzc{S}_{N}\mathpzc{A}}+\mathcal{L}_{\mathpzc{S}_{N}}^{\textrm{noise}})+\mathcal{L}{A}^{\textrm{noise}}\right]\rho(t),$
(IV.0S.28)

where the expressions for $\mathcal{L}_{\mathpzc{S}_{n}\mathpzc{A}}$,
$\mathcal{L}_{\mathpzc{S}_{n}}^{\textrm{noise}}$, and
$\mathcal{L}{A}^{\textrm{noise}}$ are given in the main text. It is worth
noting that the Liouville superoperators
$\mathcal{L}_{\mathpzc{S}_{n}\mathpzc{A}}+\mathcal{L}_{\mathpzc{S}_{n}}^{\textrm{noise}}$
commute with each other and, moreover, they commute with the Liouville
superoperator $\mathcal{L}{A}^{\textrm{noise}}$,

$\displaystyle[\mathcal{L}_{\mathpzc{S}_{n}\mathpzc{A}}+\mathcal{L}_{\mathpzc{S}_{n}}^{\textrm{noise}},\mathcal{L}_{\mathpzc{S}_{m}\mathpzc{A}}+\mathcal{L}_{\mathpzc{S}_{m}}^{\textrm{noise}}]=0,~{}~{}~{}~{}[\mathcal{L}_{\mathpzc{S}_{n}\mathpzc{A}}+\mathcal{L}_{\mathpzc{S}_{n}}^{\textrm{noise}},\mathcal{L}_{\mathpzc{A}}^{\textrm{noise}}]=0.$
(IV.0S.29)

Hence, the dynamical map
$\mathcal{E}_{N}(t_{1},t_{0})=\exp\left[\left(\mathcal{L}_{\mathpzc{S}_{1}\mathpzc{A}}+\mathcal{L}_{\mathpzc{S}_{1}}^{\textrm{noise}}+\cdots+\mathcal{L}_{\mathpzc{S}_{N}\mathpzc{A}}+\mathcal{L}_{\mathpzc{S}_{N}}^{\textrm{noise}}+\mathcal{L}{A}^{\textrm{noise}}\right)(t_{1}-t_{0})\right]$
can be decomposed as

$\displaystyle\mathcal{E}_{N}(t_{1},t_{0})=\mathcal{E}_{\mathpzc{S}_{1}\mathpzc{A}}(t_{1},t_{0})\cdots\mathcal{E}_{\mathpzc{S}_{N}\mathpzc{A}}(t_{1},t_{0})\mathcal{E}_{\mathpzc{A}}(t_{1},t_{0}),$
(IV.0S.30)

where

$\displaystyle\mathcal{E}_{\mathpzc{S}_{n}\mathpzc{A}}(t_{1},t_{0})=\exp[\left(\mathcal{L}_{\mathpzc{S}_{n}\mathpzc{A}}+\mathcal{L}_{\mathpzc{S}_{n}}^{\textrm{noise}}\right)(t_{1}-t_{0})],~{}~{}~{}~{}\mathcal{E}_{\mathpzc{A}}(t_{1},t_{0})=\exp[\mathcal{L}{A}^{\textrm{noise}}(t_{1}-t_{0})],$
(IV.0S.31)

with $t_{1}=t_{0}+1/g$. Noting that
$\rho_{\mathpzc{A}}(t_{0})=(\ket{0}+\ket{1})(\bra{0}+\bra{1})/2$ and
$\rho{A}(t_{1})=\tr_{\mathpzc{S}_{1}\cdots\mathpzc{S}_{N}}\left[\mathcal{E}_{N}(t_{1},t_{0})\rho_{{}_{T}}^{\otimes
N}\otimes\rho{A}(t_{0})\right]$, we have

$\displaystyle\rho{A}(t_{1})$ $\displaystyle=$
$\displaystyle\frac{1}{2}\tr_{\mathpzc{S}_{1}\cdots\mathpzc{S}_{N}}\left[\mathcal{E}_{N}(t_{1},t_{0})\rho_{{}_{T}}^{\otimes
N}\otimes\ket{0}\bra{0}\right]+\frac{1}{2}\tr_{\mathpzc{S}_{1}\cdots\mathpzc{S}_{N}}\left[\mathcal{E}_{N}(t_{1},t_{0})\rho_{{}_{T}}^{\otimes
N}\otimes\ket{1}\bra{1}\right]$ (IV.0S.32) $\displaystyle+$
$\displaystyle\frac{1}{2}\tr_{\mathpzc{S}_{1}\cdots\mathpzc{S}_{N}}\left[\mathcal{E}_{N}(t_{1},t_{0})\rho_{{}_{T}}^{\otimes
N}\otimes\ket{0}\bra{1}\right]+\frac{1}{2}\tr_{\mathpzc{S}_{1}\cdots\mathpzc{S}_{N}}\left[\mathcal{E}_{N}(t_{1},t_{0})\rho_{{}_{T}}^{\otimes
N}\otimes\ket{1}\bra{0}\right].$

In the following, we make use of the decomposition in Eq. (IV.0S.30) to figure
out all of the four terms on the right-hand side of Eq. (IV.0S.32).

First, we figure out the term
$\frac{1}{2}\tr_{\mathpzc{S}_{1}\cdots\mathpzc{S}_{N}}\left[\mathcal{E}_{N}(t_{1},t_{0})\rho_{{}_{T}}^{\otimes
N}\otimes\ket{0}\bra{0}\right]$. It is easy to see that
$\mathcal{E}_{\mathpzc{A}}(t_{1},t_{0})$ maps the state
$\rho_{{}_{T}}^{\otimes N}\otimes\ket{0}\bra{0}$ into itself,

$\displaystyle\rho_{{}_{T}}^{\otimes
N}\otimes\ket{0}\bra{0}\xrightarrow{\mathcal{E}_{\mathpzc{A}}(t_{1},t_{0})}\rho_{{}_{T}}^{\otimes
N}\otimes\ket{0}\bra{0}.$ (IV.0S.33)

Hence,

$\displaystyle\mathcal{E}_{N}(t_{1},t_{0})\rho_{{}_{T}}^{\otimes
N}\otimes\ket{0}\bra{0}=\mathcal{E}_{\mathpzc{S}_{1}\mathpzc{A}}(t_{1},t_{0})\cdots\mathcal{E}_{\mathpzc{S}_{N}\mathpzc{A}}(t_{1},t_{0})\rho_{{}_{T}}^{\otimes
N}\otimes\ket{0}\bra{0}.$ (IV.0S.34)

Noting that

$\displaystyle\left(\mathcal{L}_{\mathpzc{S}_{n}\mathpzc{A}}+\mathcal{L}_{\mathpzc{S}_{n}}^{\textrm{noise}}\right)\rho\otimes\ket{0}\bra{0}=\left(\mathcal{L}_{\mathpzc{S}_{n}}\rho-\textrm{i}g[\sigma_{z}^{\mathpzc{S}_{n}},\rho]+\mathcal{L}_{\mathpzc{S}_{n}}^{\textrm{noise}}\rho\right)\otimes\ket{0}\bra{0}=:\mathcal{L}_{\mathpzc{S}_{n}\mathpzc{A}}^{(00)}\rho\otimes\ket{0}\bra{0},$
(IV.0S.35)

where
$\mathcal{L}_{\mathpzc{S}_{n}\mathpzc{A}}^{(00)}\rho=\mathcal{L}_{\mathpzc{S}_{n}}\rho-\textrm{i}g[\sigma_{z}^{\mathpzc{S}_{n}},\rho]+\mathcal{L}_{\mathpzc{S}_{n}}^{\textrm{noise}}\rho$
is a Liouville superoperator acting on $\mathpzc{S}_{n}$, we have

$\displaystyle\mathcal{E}_{\mathpzc{S}_{n}\mathpzc{A}}(t_{1},t_{0})\rho_{{}_{T}}\otimes\ket{0}\bra{0}=\exp[\mathcal{L}_{\mathpzc{S}_{n}\mathpzc{A}}^{(00)}(t_{1}-t_{0})]\rho_{{}_{T}}\otimes\ket{0}\bra{0}.$
(IV.0S.36)

Since $\mathcal{L}_{\mathpzc{S}_{n}\mathpzc{A}}^{(00)}$ is in the Lindblad
form, $\exp[\mathcal{L}_{\mathpzc{S}_{n}\mathpzc{A}}^{(00)}(t_{1}-t_{0})]$ is
a completely positive and trace-preserving (CPTP) map. Then, we have

$\displaystyle\tr_{\mathpzc{S}_{n}}\left[\mathcal{E}_{\mathpzc{S}_{n}\mathpzc{A}}(t_{1},t_{0})\rho_{{}_{T}}\otimes\ket{0}\bra{0}\right]=\tr_{\mathpzc{S}_{n}}\left[e^{\mathcal{L}_{\mathpzc{S}_{n}\mathpzc{A}}^{(00)}(t_{1}-t_{0})}\rho_{{}_{T}}\right]\ket{0}\bra{0}=\ket{0}\bra{0}.$
(IV.0S.37)

Using Eqs. (IV.0S.34) and (IV.0S.37), we have

$\displaystyle\frac{1}{2}\tr_{\mathpzc{S}_{1}\cdots\mathpzc{S}_{N}}\left[\mathcal{E}_{N}(t_{1},t_{0})\rho_{{}_{T}}^{\otimes
N}\otimes\ket{0}\bra{0}\right]=\frac{1}{2}\ket{0}\bra{0}.$ (IV.0S.38)

Second, we figure out the term
$\frac{1}{2}\tr_{\mathpzc{S}_{1}\cdots\mathpzc{S}_{N}}\left[\mathcal{E}_{N}(t_{1},t_{0})\rho_{{}_{T}}^{\otimes
N}\otimes\ket{1}\bra{1}\right]$. Likewise,
$\mathcal{E}_{\mathpzc{A}}(t_{1},t_{0})$ maps the state
$\rho_{{}_{T}}^{\otimes N}\otimes\ket{1}\bra{1}$ into itself,

$\displaystyle\rho_{{}_{T}}^{\otimes
N}\otimes\ket{1}\bra{1}\xrightarrow{\mathcal{E}_{\mathpzc{A}}(t_{1},t_{0})}\rho_{{}_{T}}^{\otimes
N}\otimes\ket{1}\bra{1}.$ (IV.0S.39)

Hence,

$\displaystyle\mathcal{E}_{N}(t_{1},t_{0})\rho_{{}_{T}}^{\otimes
N}\otimes\ket{1}\bra{1}=\mathcal{E}_{\mathpzc{S}_{1}\mathpzc{A}}(t_{1},t_{0})\cdots\mathcal{E}_{\mathpzc{S}_{N}\mathpzc{A}}(t_{1},t_{0})\rho_{{}_{T}}^{\otimes
N}\otimes\ket{1}\bra{1}.$ (IV.0S.40)

Noting that

$\displaystyle\left(\mathcal{L}_{\mathpzc{S}_{n}\mathpzc{A}}+\mathcal{L}_{\mathpzc{S}_{n}}^{\textrm{noise}}\right)\rho\otimes\ket{1}\bra{1}=\left(\mathcal{L}_{\mathpzc{S}_{n}}\rho+\textrm{i}g[\sigma_{z}^{\mathpzc{S}_{n}},\rho]+\mathcal{L}_{\mathpzc{S}_{n}}^{\textrm{noise}}\rho\right)\otimes\ket{1}\bra{1}=:\mathcal{L}_{\mathpzc{S}_{n}\mathpzc{A}}^{(11)}\rho\otimes\ket{1}\bra{1},$
(IV.0S.41)

where
$\mathcal{L}_{\mathpzc{S}_{n}\mathpzc{A}}^{(11)}\rho=\mathcal{L}_{\mathpzc{S}_{n}}\rho+\textrm{i}g[\sigma_{z}^{\mathpzc{S}_{n}},\rho]+\mathcal{L}_{\mathpzc{S}_{n}}^{\textrm{noise}}\rho$,
we have

$\displaystyle\mathcal{E}_{\mathpzc{S}_{n}\mathpzc{A}}(t_{1},t_{0})\rho_{{}_{T}}\otimes\ket{1}\bra{1}=\exp[\mathcal{L}_{\mathpzc{S}_{n}\mathpzc{A}}^{(11)}(t_{1}-t_{0})]\rho_{{}_{T}}\otimes\ket{1}\bra{1}.$
(IV.0S.42)

Noting that $\mathcal{L}_{\mathpzc{S}_{n}\mathpzc{A}}^{(11)}$ is in the
Lindblad form and
$\exp[\mathcal{L}_{\mathpzc{S}_{n}\mathpzc{A}}^{(11)}(t_{1}-t_{0})]$ is a CPTP
map, we have

$\displaystyle\tr_{\mathpzc{S}_{n}}\left[\mathcal{E}_{\mathpzc{S}_{n}\mathpzc{A}}(t_{1},t_{0})\rho_{{}_{T}}\otimes\ket{1}\bra{1}\right]=\tr_{\mathpzc{S}_{n}}\left[e^{\mathcal{L}_{\mathpzc{S}_{n}\mathpzc{A}}^{(11)}(t_{1}-t_{0})}\rho_{{}_{T}}\right]\ket{1}\bra{1}=\ket{1}\bra{1}.$
(IV.0S.43)

Using Eqs. (IV.0S.40) and (IV.0S.43), we have

$\displaystyle\frac{1}{2}\tr_{\mathpzc{S}_{1}\cdots\mathpzc{S}_{N}}\left[\mathcal{E}_{N}(t_{1},t_{0})\rho_{{}_{T}}^{\otimes
N}\otimes\ket{1}\bra{1}\right]=\frac{1}{2}\ket{1}\bra{1}.$ (IV.0S.44)

Third, we figure out the term
$\frac{1}{2}\tr_{\mathpzc{S}_{1}\cdots\mathpzc{S}_{N}}\left[\mathcal{E}_{N}(t_{1},t_{0})\rho_{{}_{T}}^{\otimes
N}\otimes\ket{0}\bra{1}\right]$. It is easy to see that

$\displaystyle\rho_{{}_{T}}^{\otimes
N}\otimes\ket{0}\bra{1}\xrightarrow{\mathcal{E}_{\mathpzc{A}}(t_{1},t_{0})}e^{-2\eta}\rho_{{}_{T}}^{\otimes
N}\otimes\ket{0}\bra{1},$ (IV.0S.45)

where $\eta=\kappa{A}/g$. Hence,

$\displaystyle\mathcal{E}_{N}(t_{1},t_{0})\rho_{{}_{T}}^{\otimes
N}\otimes\ket{0}\bra{1}=e^{-2\eta}\mathcal{E}_{\mathpzc{S}_{1}\mathpzc{A}}(t_{1},t_{0})\cdots\mathcal{E}_{\mathpzc{S}_{N}\mathpzc{A}}(t_{1},t_{0})\rho_{{}_{T}}^{\otimes
N}\otimes\ket{0}\bra{1}.$ (IV.0S.46)

Noting that

$\displaystyle\left(\mathcal{L}_{\mathpzc{S}_{n}\mathpzc{A}}+\mathcal{L}_{\mathpzc{S}_{n}}^{\textrm{noise}}\right)\rho\otimes\ket{0}\bra{1}=\left(\mathcal{L}_{\mathpzc{S}_{n}}\rho-\textrm{i}g\\{\sigma_{z}^{\mathpzc{S}_{n}},\rho\\}+\mathcal{L}_{\mathpzc{S}_{n}}^{\textrm{noise}}\rho\right)\otimes\ket{0}\bra{1}=:\mathcal{L}_{\mathpzc{S}_{n}\mathpzc{A}}^{(01)}\rho\otimes\ket{0}\bra{1},$
(IV.0S.47)

where
$\mathcal{L}_{\mathpzc{S}_{n}\mathpzc{A}}^{(01)}\rho=\mathcal{L}_{\mathpzc{S}_{n}}\rho-\textrm{i}g\\{\sigma_{z}^{\mathpzc{S}_{n}},\rho\\}+\mathcal{L}_{\mathpzc{S}_{n}}^{\textrm{noise}}\rho$,
we have

$\displaystyle\mathcal{E}_{\mathpzc{S}_{n}\mathpzc{A}}(t_{1},t_{0})\rho_{{}_{T}}\otimes\ket{0}\bra{1}=\exp[\mathcal{L}_{\mathpzc{S}_{n}\mathpzc{A}}^{(01)}(t_{1}-t_{0})]\rho_{{}_{T}}\otimes\ket{0}\bra{1}.$
(IV.0S.48)

Then, we have

$\displaystyle\tr_{\mathpzc{S}_{n}}\left[\mathcal{E}_{\mathpzc{S}_{n}\mathpzc{A}}(t_{1},t_{0})\rho_{{}_{T}}\otimes\ket{0}\bra{1}\right]=\tr_{\mathpzc{S}_{n}}\left[e^{\mathcal{L}_{\mathpzc{S}_{n}\mathpzc{A}}^{(01)}(t_{1}-t_{0})}\rho_{{}_{T}}\right]\ket{0}\bra{1}=\Gamma\ket{0}\bra{1},$
(IV.0S.49)

where
$\Gamma=\tr_{\mathpzc{S}_{n}}\left[e^{\mathcal{L}_{\mathpzc{S}_{n}\mathpzc{A}}^{(01)}(t_{1}-t_{0})}\rho_{{}_{T}}\right]$.
Using MATHEMATICA, we can work out the expression for $\Gamma$, given by

$\displaystyle\Gamma=\frac{e^{\omega_{+}(\xi)}}{2}\left[1+\frac{(2\overline{n}+1)\xi}{\sqrt{\Delta(\xi)}}-\frac{4\textrm{i}}{(2\overline{n}+1)\sqrt{\Delta(\xi)}}\right]+\frac{e^{\omega_{-}(\xi)}}{2}\left[1-\frac{(2\overline{n}+1)\xi}{\sqrt{\Delta(\xi)}}+\frac{4\textrm{i}}{(2\overline{n}+1)\sqrt{\Delta(\xi)}}\right],$
(IV.0S.50)

where

$\displaystyle\Delta(\xi)=(2\overline{n}+1)^{2}\xi^{2}-8\textrm{i}\xi-16,$
(IV.0S.51)

and

$\displaystyle\omega_{\pm}(\xi)=[-(2\overline{n}+1)\xi\pm\sqrt{\Delta(\xi)}]/2$
(IV.0S.52)

with $\xi=\gamma/g$. Using Eqs. (IV.0S.46) and (IV.0S.49), we have

$\displaystyle\frac{1}{2}\tr_{\mathpzc{S}_{1}\cdots\mathpzc{S}_{N}}\left[\mathcal{E}_{N}(t_{1},t_{0})\rho_{{}_{T}}^{\otimes
N}\otimes\ket{0}\bra{1}\right]=\frac{1}{2}\Gamma^{N}e^{-2\eta}\ket{0}\bra{1}.$
(IV.0S.53)

Now, substituting Eqs. (IV.0S.38), (IV.0S.44), and (IV.0S.53) into Eq.
(IV.0S.32), we have

$\displaystyle\rho{A}(t_{1})=\frac{1}{2}\begin{pmatrix}1&\Gamma^{N}e^{-2\eta}\\\
\Gamma^{*N}e^{-2\eta}&1\end{pmatrix},$ (IV.0S.54)

where we have used the fact that the term
$\frac{1}{2}\tr_{\mathpzc{S}_{1}\cdots\mathpzc{S}_{N}}\left[\mathcal{E}_{N}(t_{1},t_{0})\rho_{{}_{T}}^{\otimes
N}\otimes\ket{0}\bra{1}\right]$ is complex conjugate to the term
$\frac{1}{2}\tr_{\mathpzc{S}_{1}\cdots\mathpzc{S}_{N}}\left[\mathcal{E}_{N}(t_{1},t_{0})\rho_{{}_{T}}^{\otimes
N}\otimes\ket{1}\bra{0}\right]$. This completes the proof of Eq. (24).

## Supplementary Note 3: Influence of the initial state on the quantum Fisher
information

Here, focusing on the model in the main text, we examine the influence of the
initial state of the $N$ probes and the ancilla on the quantum Fisher
information (QFI) for the output state of the ancilla $\rho{A}(t_{1})$. We
consider the initial state of the form

$\displaystyle\rho^{\otimes N}\otimes\sigma,$ (IV.0S.55)

where $\rho$ and $\sigma$ are arbitrarily given $2\times 2$ density matrices.
In the following, we figure out the analytical expression of the QFI for the
output state of the ancilla and then examine the influence of the initial
state given by Eq. (IV.0S.55) on the QFI.

Writing $\sigma$ as
$\sigma=\sigma_{00}\ket{0}\bra{0}+\sigma_{11}\ket{1}\bra{1}+\sigma_{01}\ket{0}\bra{1}+\sigma_{01}^{*}\ket{1}\bra{0}$,
we have

$\displaystyle\rho{A}(t_{1})$ $\displaystyle=$
$\displaystyle\sigma_{00}\tr_{\mathpzc{S}_{1}\cdots\mathpzc{S}_{N}}\left[\mathcal{E}_{N}(t_{1},t_{0})\rho^{\otimes
N}\otimes\ket{0}\bra{0}\right]+\sigma_{11}\tr_{\mathpzc{S}_{1}\cdots\mathpzc{S}_{N}}\left[\mathcal{E}_{N}(t_{1},t_{0})\rho^{\otimes
N}\otimes\ket{1}\bra{1}\right]$ (IV.0S.56) $\displaystyle+$
$\displaystyle\sigma_{01}\tr_{\mathpzc{S}_{1}\cdots\mathpzc{S}_{N}}\left[\mathcal{E}_{N}(t_{1},t_{0})\rho^{\otimes
N}\otimes\ket{0}\bra{1}\right]+\sigma_{01}^{*}\tr_{\mathpzc{S}_{1}\cdots\mathpzc{S}_{N}}\left[\mathcal{E}_{N}(t_{1},t_{0})\rho^{\otimes
N}\otimes\ket{1}\bra{0}\right].$

Using the same reasoning as in deriving Eqs. (IV.0S.38), (IV.0S.44), and
(IV.0S.53), we have

$\displaystyle\tr_{\mathpzc{S}_{1}\cdots\mathpzc{S}_{N}}\left[\mathcal{E}_{N}(t_{1},t_{0})\rho^{\otimes
N}\otimes\ket{0}\bra{0}\right]=1,$ (IV.0S.57)
$\displaystyle\tr_{\mathpzc{S}_{1}\cdots\mathpzc{S}_{N}}\left[\mathcal{E}_{N}(t_{1},t_{0})\rho^{\otimes
N}\otimes\ket{1}\bra{1}\right]=1,$ (IV.0S.58)
$\displaystyle\tr_{\mathpzc{S}_{1}\cdots\mathpzc{S}_{N}}\left[\mathcal{E}_{N}(t_{1},t_{0})\rho^{\otimes
N}\otimes\ket{0}\bra{1}\right]=\Gamma^{N}e^{-2\eta},$ (IV.0S.59)

where

$\displaystyle\Gamma=\tr_{\mathpzc{S}_{n}}\left[e^{\mathcal{L}_{\mathpzc{S}_{n}\mathpzc{A}}^{(01)}(t_{1}-t_{0})}\rho\right],$
(IV.0S.60)

with
$\mathcal{L}_{\mathpzc{S}_{n}\mathpzc{A}}^{(01)}\rho=\mathcal{L}_{\mathpzc{S}_{n}}\rho-\textrm{i}g\\{\sigma_{z}^{\mathpzc{S}_{n}},\rho\\}+\mathcal{L}_{\mathpzc{S}_{n}}^{\textrm{noise}}\rho$.
Using MATHEMATICA, we can work out the expression for $\Gamma$, given by

$\displaystyle\Gamma=\frac{e^{\omega_{+}(\xi)}}{2}\left[1+\frac{(2\overline{n}+1)\xi}{\sqrt{\Delta(\xi)}}-\frac{4\textrm{i}}{\sqrt{\Delta(\xi)}}(\rho_{00}-\rho_{11})\right]+\frac{e^{\omega_{-}(\xi)}}{2}\left[1-\frac{(2\overline{n}+1)\xi}{\sqrt{\Delta(\xi)}}+\frac{4\textrm{i}}{\sqrt{\Delta(\xi)}}(\rho_{00}-\rho_{11})\right],$
(IV.0S.61)

where we have expressed $\rho$ as
$\rho=\rho_{00}\ket{0}\bra{0}+\rho_{11}\ket{1}\bra{1}+\rho_{01}\ket{0}\bra{1}+\rho_{01}^{*}\ket{1}\bra{0}$
and $\Delta(\xi)$ and $\omega_{\pm}(\xi)$ are given by Eqs. (IV.0S.51) and
(IV.0S.52), respectively. Substituting Eqs. (IV.0S.57), (IV.0S.58), and
(IV.0S.59) into Eq. (IV.0S.56), we have the output state of the ancilla

$\displaystyle\rho{A}(t_{1})=\begin{pmatrix}\sigma_{00}&\sigma_{01}\Gamma^{N}e^{-2\eta}\\\
\sigma_{01}^{*}\Gamma^{*N}e^{-2\eta}&\sigma_{11}\end{pmatrix}.$ (IV.0S.62)

Then, the QFI associated with $\rho{A}(t_{1})$ reads

$\displaystyle\mathcal{F}_{N}^{\textrm{noise}}(T)=4N^{2}\absolutevalue{\sigma_{01}}^{2}\absolutevalue{\Gamma}^{2N-2}\absolutevalue{\partial_{{}_{T}}\Gamma}^{2}e^{-4\eta}+4N^{2}\absolutevalue{\sigma_{01}}^{4}\absolutevalue{\Gamma}^{4N-2}\left(\partial_{{}_{T}}\absolutevalue{\Gamma}\right)^{2}e^{-8\eta}/\left(\sigma_{00}\sigma_{11}-\absolutevalue{\sigma_{01}}^{2}\absolutevalue{\Gamma}^{2N}e^{-4\eta}\right).$
(IV.0S.63)

Notably, $\mathcal{F}_{N}^{\textrm{noise}}(T)$ depends on $\rho$ through
$\Gamma$ which, as can be seen from Eq. (IV.0S.61), is dependent on
$\rho_{00}$ and $\rho_{11}$ and is independent of $\rho_{01}$.

We first examine the limiting case of $\xi\rightarrow\infty$. Using Eq.
(IV.0S.61), we have that
$\lim_{\xi\rightarrow\infty}\Gamma=\exp(-2i\varphi_{{}_{T}})$, which further
leads to

$\displaystyle\lim_{\xi\rightarrow\infty}\mathcal{F}_{N}^{\textrm{noise}}(T)=16N^{2}\absolutevalue{\sigma_{01}}^{2}\absolutevalue{\partial_{{}_{T}}\varphi_{{}_{T}}}^{2}e^{-4\eta}.$
(IV.0S.64)

With Eq. (IV.0S.64), we reach the following observations: (i)
$\lim_{\xi\rightarrow\infty}\mathcal{F}_{N}^{\textrm{noise}}(T)$ is
irrespective of the state $\rho^{\otimes N}$ of the $N$ probes, and (ii)
$\lim_{\xi\rightarrow\infty}\mathcal{F}_{N}^{\textrm{noise}}(T)$ is a
quadratic function of the absolute value $\absolutevalue{\sigma_{01}}$ of the
off-diagonal element of the state $\sigma$ of the ancilla. Note that
$\absolutevalue{\sigma_{01}}$ is proportional to the coherence of $\sigma$
quantified via the so-called $l_{1}$ norm of coherence Zhang  _et al._ (2018),

$\displaystyle C_{l_{1}}(\sigma)=\sum_{i\neq
j}\absolutevalue{\sigma_{ij}}=2\absolutevalue{\sigma_{01}}.$ (IV.0S.65)

So, observation (ii) amounts to the statement that
$\lim_{\xi\rightarrow\infty}\mathcal{F}_{N}^{\textrm{noise}}(T)$ is a
quadratic function of the $l_{1}$ norm of coherence of $\sigma$. We expect
that the above two observations approximately hold for a large $\xi$. To
confirm this point, we numerically compute
$\mathcal{F}_{N}^{\textrm{noise}}(T)$ in units of
$\mathcal{F}_{\textrm{th}}(T)$ for $\xi=400$. The numerical results are
presented in Supplementary Fig. 4. In Supplementary Fig. 4a, $\rho$ is
randomly chosen whereas $\sigma$ is set to
$(\ket{0}+\ket{1})(\bra{0}+\bra{1})/2$. As can be seen from this subfigure,
$\mathcal{F}_{N}^{\textrm{noise}}(T)$ does not change much in the course of
varying $\rho_{00}$, which confirms that observation (i) approximately hold.
In Supplementary Fig. 4b, $\sigma$ is randomly chosen whereas $\rho$ is set to
the steady state $\rho_{{}_{T}}$. As can be seen from this subfigure,
$\mathcal{F}_{N}^{\textrm{noise}}(T)$ grows quadratically as a function of
$\absolutevalue{\sigma_{01}}$, which confirms that observation (ii)
approximately hold.

Supplementary Figure 4: Numerical results of
$\mathcal{F}_{N}^{\textrm{noise}}(T)$ in units of
$\mathcal{F}_{\textrm{th}}(T)$ for different initial states of the $N$ probes
and the ancilla. Here, the form of the initial state is assumed to be
$\rho^{\otimes N}\otimes\sigma$. (a) $\rho$ is randomly chosen whereas
$\sigma$ is set to $(\ket{0}+\ket{1})(\bra{0}+\bra{1})/2$. (b) $\sigma$ is
randomly chosen whereas $\rho$ is set to the steady state $\rho_{{}_{T}}$.
Parameters used are $k_{B}T/\hbar\Omega=2$, $N=100$, $\xi=400$, and
$\eta=1/10$.

## Supplementary Note 4: Proof of Eq. (32) in the main text

We first collect some useful facts:

(i) $\mathcal{L}{S}\mathcal{P}=\mathcal{P}\mathcal{L}{S}=0$. This follows from
the facts $\mathcal{L}{S}\rho_{{}_{T}}=0$ and
$[\mathcal{L}{S},\mathcal{P}]=0$.

(ii) $\mathcal{L}{S}\mathcal{Q}=\mathcal{Q}\mathcal{L}{S}=\mathcal{L}{S}$.
This follows immediately from fact (i).

(iii) For any complex number $c$, operators $X$, $Y$, and superoperator
$\mathcal{E}$, $\norm{cX}=\absolutevalue{c}\norm{X}$,
$\norm{X+Y}\leq\norm{X}+\norm{Y}$, $\norm{XY}\leq\norm{X}\norm{Y}$, and
$\norm{\mathcal{E}(X)}\leq\norm{\mathcal{E}}\norm{X}$.

(iv) $\mathcal{L}{S}$ has a non-degenerate zero eigenvalue, and all its
nonvanishing eigenvalues have negative real parts Zhang  _et al._ (2016a, b).

Let $\lambda_{\mu}$, $\mu=0,\cdots,M$, denote the eigenvalues of
$\mathcal{L}{S}$. We arrange them in ascending order of the absolute values of
their real parts; that is, when $\mu<\nu$, there is
$\absolutevalue{\real(\lambda_{\mu})}\leq\absolutevalue{\real(\lambda_{\nu})}$.
So, $\lambda_{0}=0$, and $\absolutevalue{\real(\lambda_{\mu})}>0$, for
$\mu=1,\cdots,M$.
$\lambda:={\min_{\mu>0}\absolutevalue{\real(\lambda_{\mu})}}={\absolutevalue{\real(\lambda_{1})}}$
is known as the dissipative gap. In the following, we prove the inequality
(32) in the main text,

$\displaystyle\norm{\mathcal{G}(t,s)\mathcal{Q}}\leq\varepsilon\exp[\left(\varepsilon
g\norm{\mathcal{K}}-\lambda\right)(t-s)],$ (IV.0S.66)

where $\varepsilon$ is a dimensionless constant determined by the damping
basis of $\mathcal{L}{S}$ [see Eq. (IV.0S.77)], and $\mathcal{K}$ is the
superoperator defined as $\mathcal{K}X:=-\textrm{i}[S\otimes A,X]$.

First, we derive a convenient formula for $\mathcal{G}(t,s)\mathcal{Q}$. Using
fact (ii) and noting that $\mathcal{L}=\mathcal{L}{S}+g\mathcal{K}$, we have
$\mathcal{QL}=\mathcal{L}{S}+g\mathcal{Q}\mathcal{K}$. It then follows that

$\displaystyle\partial_{t}\mathcal{G}(t,s)=(\mathcal{L}{S}+g\mathcal{Q}\mathcal{K})\mathcal{G}(t,s).$
(IV.0S.67)

It is not difficult to see that Eq. (IV.0S.67) can be formally solved as

$\displaystyle\mathcal{G}(t,s)=e^{\mathcal{L}{S}(t-s)}+g\int_{s}^{t}dt_{1}e^{\mathcal{L}{S}(t-t_{1})}\mathcal{QK}\mathcal{G}(t_{1},s).$
(IV.0S.68)

Using Eq. (IV.0S.68) iteratively, we obtain

$\displaystyle\mathcal{G}(t,s)=e^{\mathcal{L}{S}(t-s)}+g\int_{s}^{t}dt_{1}e^{\mathcal{L}{S}(t-t_{1})}\mathcal{QK}e^{\mathcal{L}{S}(t_{1}-s)}+g^{2}\int_{s}^{t}dt_{1}\int_{s}^{t_{1}}dt_{2}e^{\mathcal{L}{S}(t-t_{1})}\mathcal{QK}e^{\mathcal{L}{S}(t_{1}-t_{2})}\mathcal{QK}e^{\mathcal{L}{S}(t_{2}-s)}+\cdots.$

As an immediate consequence, we arrive at the desired formula:

$\displaystyle\mathcal{G}(t,s)\mathcal{Q}$ $\displaystyle=$ $\displaystyle
e^{\mathcal{L}{S}(t-s)}\mathcal{Q}+g\int_{s}^{t}dt_{1}e^{\mathcal{L}{S}(t-t_{1})}\mathcal{QK}e^{\mathcal{L}{S}(t_{1}-s)}\mathcal{Q}$
(IV.0S.70) $\displaystyle+$ $\displaystyle
g^{2}\int_{s}^{t}dt_{1}\int_{s}^{t_{1}}dt_{2}e^{\mathcal{L}{S}(t-t_{1})}\mathcal{QK}e^{\mathcal{L}{S}(t_{1}-t_{2})}\mathcal{QK}e^{\mathcal{L}{S}(t_{2}-s)}\mathcal{Q}+\cdots.$

Second, we show that there exists a constant $\varepsilon$ such that

$\displaystyle\norm{e^{\mathcal{L}{S}(t-s)}\mathcal{Q}}\leq\varepsilon
e^{-\lambda(t-s)},$ (IV.0S.71)

i.e., $\norm{e^{\mathcal{L}{S}(t-s)}\mathcal{Q}X}\leq\varepsilon
e^{-\lambda(t-s)}$, for any operator $X$ with $\norm{X}\leq 1$. An easy way to
see this might be to employ the notion of the damping basis introduced in
Supplementary Ref. 1993Briegel3311. A damping basis of $\mathcal{L}{S}$ is a
set of operators, $\\{R_{\mu},\mu=0,\cdots,M\\}$, such that

$\displaystyle\mathcal{L}{S}X=\sum_{\mu=0}^{M}\lambda_{\mu}R_{\mu}\tr(L_{\mu}^{\dagger}X).$
(IV.0S.72)

Here, $L_{\mu}$, $\mu=0,\cdots,M$, constitute a dual basis satisfying
$\tr(L_{\mu}^{\dagger}R_{\nu})=\delta_{\mu\nu}$. Evidently,
$\mathcal{L}{S}R_{\mu}=\lambda_{\mu}R_{\mu}$, i.e., $R_{\mu}$ is the
eigenvector of $\mathcal{L}{S}$ corresponding to the eigenvalue
$\lambda_{\mu}$. Particularly, $R_{0}$ is the eigenvector associated with
$\lambda_{0}=0$, i.e., $R_{0}=c\rho_{{}_{T}}$ for a coefficient $c$. We then
have $QR_{0}=0$. Using fact (ii), we have
$\mathcal{L}{S}\mathcal{Q}R_{\mu}=\mathcal{Q}\mathcal{L}{S}R_{\mu}=\lambda_{\mu}\mathcal{Q}R_{\mu}$
and
$\mathcal{L}{S}\mathcal{Q}R_{\mu}=\mathcal{L}{S}R_{\mu}=\lambda_{\mu}R_{\mu}$.
So, $\mathcal{Q}R_{\mu}=R_{\mu}$, for $\mu=1,\cdots,M$. On the other hand,
from Eq. (IV.0S.72), it follows that

$\displaystyle
e^{\mathcal{L}{S}(t-s)}X=\sum_{\mu=0}^{M}e^{\lambda_{\mu}(t-s)}R_{\mu}\tr(L_{\mu}^{\dagger}X).$
(IV.0S.73)

Using fact (ii), Eq. (IV.0S.73), $QR_{0}=0$, and $QR_{\mu}=R_{\mu}$, for
$\mu=1,\cdots,M$, we have

$\displaystyle
e^{\mathcal{L}{S}(t-s)}\mathcal{Q}X=\mathcal{Q}e^{\mathcal{L}{S}(t-s)}X=\mathcal{Q}\sum_{\mu=0}^{M}e^{\lambda_{\mu}(t-s)}R_{\mu}\tr(L_{\mu}^{\dagger}X)=\sum_{\mu=1}^{M}e^{\lambda_{\mu}(t-s)}R_{\mu}\tr(L_{\mu}^{\dagger}X).$
(IV.0S.74)

So,

$\displaystyle\norm{e^{\mathcal{L}{S}(t-s)}\mathcal{Q}X}=\absolutevalue{e^{\lambda_{1}(t-s)}}\norm{\sum_{\mu=1}^{M}e^{(\lambda_{\mu}-\lambda_{1})(t-s)}R_{\mu}\tr(L_{\mu}^{\dagger}X)}=e^{-\lambda(t-s)}\norm{\sum_{\mu=1}^{M}e^{(\lambda_{\mu}-\lambda_{1})(t-s)}R_{\mu}\tr(L_{\mu}^{\dagger}X)}.$
(IV.0S.75)

Using fact (iii) and the Cauchy-Schwarz inequality
$\absolutevalue{\tr(L_{\mu}^{\dagger}X)}\leq\norm{L_{\mu}}\norm{X}$ and noting
that $\norm{X}\leq 1$ and
$\absolutevalue{e^{(\lambda_{\mu}-\lambda_{1})(t-s)}}\leq 1$, we have

$\displaystyle\norm{\sum_{\mu=1}^{M}e^{(\lambda_{\mu}-\lambda_{1})(t-s)}R_{\mu}\tr(L_{\mu}^{\dagger}X)}\leq\sum_{\mu=1}^{M}\absolutevalue{e^{(\lambda_{\mu}-\lambda_{1})(t-s)}}\absolutevalue{\tr(L_{\mu}^{\dagger}X)}\norm{R_{\mu}}\leq\sum_{\mu=1}^{M}\norm{L_{\mu}}\norm{R_{\mu}}.$
(IV.0S.76)

Combining Eqs. (IV.0S.75) and (IV.0S.76), we obtain Eq. (IV.0S.71) with

$\displaystyle\varepsilon=\sum_{\mu=1}^{M}\norm{L_{\mu}}\norm{R_{\mu}}.$
(IV.0S.77)

Third, with the aid of Eqs. (IV.0S.70) and (IV.0S.71), we evaluate
$\norm{\mathcal{G}(t,s)\mathcal{Q}}$. Using fact (iii) and Eqs. (IV.0S.70) and
(IV.0S.71), we have

$\displaystyle\norm{\mathcal{G}(t,s)\mathcal{Q}}$ $\displaystyle\leq$
$\displaystyle\norm{e^{\mathcal{L}{S}(t-s)}\mathcal{Q}}+g\int_{s}^{t}dt_{1}\norm{e^{\mathcal{L}{S}(t-t_{1})}\mathcal{Q}}\norm{\mathcal{K}}\norm{e^{\mathcal{L}{S}(t_{1}-s)}\mathcal{Q}}$
(IV.0S.78) $\displaystyle+$ $\displaystyle
g^{2}\int_{s}^{t}dt_{1}\int_{s}^{t_{1}}dt_{2}\norm{e^{\mathcal{L}{S}(t-t_{1})}\mathcal{Q}}\norm{\mathcal{K}}\norm{e^{\mathcal{L}{S}(t_{1}-t_{2})}\mathcal{Q}}\norm{\mathcal{K}}\norm{e^{\mathcal{L}{S}(t_{2}-s)}\mathcal{Q}}+\cdots$
$\displaystyle\leq$ $\displaystyle\varepsilon
e^{-\lambda(t-s)}+\varepsilon^{2}g\norm{\mathcal{K}}(t-s)e^{-\lambda(t-s)}+\frac{1}{2!}\varepsilon^{3}g^{2}\norm{\mathcal{K}}^{2}(t-s)^{2}e^{-\lambda(t-s)}+\cdots$
$\displaystyle=$ $\displaystyle\varepsilon\left[1+\varepsilon
g\norm{\mathcal{K}}(t-s)+\frac{\varepsilon^{2}g^{2}\norm{\mathcal{K}}^{2}(t-s)^{2}}{2!}+\cdots\right]e^{-\lambda(t-s)}$
$\displaystyle=$ $\displaystyle\varepsilon\exp[\left(\varepsilon
g\norm{\mathcal{K}}-\lambda\right)(t-s)].$

This completes the proof of Eq. (32) in the main text.