# Optimal nonequilibrium thermometry in Markovian environments

Pavel Sekatski  Department of Applied Physics, University of Geneva, 1211
Geneva, Switzerland  Martí Perarnau-Llobet  Department of Applied Physics,
University of Geneva, 1211 Geneva, Switzerland

###### Abstract

What is the minimum time required to take a temperature? In this paper, we
solve this question for a large class of processes where temperature is
inferred by measuring a probe (the thermometer) weakly coupled to the sample
of interest, so that the probe’s evolution is well described by a quantum
Markovian master equation. Considering the most general control strategy on
the probe (adaptive measurements, arbitrary control on the probe’s state and
Hamiltonian), we provide bounds on the achievable measurement precision in a
finite amount of time, and show that in many scenarios these fundamental
limits can be saturated with a relatively simple experiment. We find that for
a general class of sample-probe interactions the scaling of the measurement
uncertainty is inversely proportional to the time of the process, a shot-noise
like behaviour that arises due to the dissipative nature of thermometry. As a
side result, we show that the Lamb shift induced by the probe-sample
interaction can play a relevant role in thermometry, allowing for finite
measurement resolution in the low-temperature regime. More precisely, the
measurement uncertainty decays polynomially with the temperature as
$T\rightarrow 0$, in contrast to the usual exponential decay with $T^{-1}$. We
illustrate these general results for (i) a qubit probe interacting with a
bosonic sample, where the role of the Lamb shift is highlighted, and (ii) a
collective superradiant coupling between a $N$-qubit probe and a sample, which
enables a quadratic decay with $N$ of the measurement uncertainty.

##  1 Introduction

Temperature is a physical quantity important in most natural sciences and some
aspects of daily life. The task of thermometry – measuring the temperature –
is essential in physics, chemistry, medicine, and cooking, to name a few. A
good thermometer should be accurate (on average it provides the right
temperature) and precise (fluctuations around the true value are small). The
former requires an accurate design of the thermometer, whereas the latter
demands collecting enough statistics: either by using a macroscopic
thermometer where each subsystem equilibrates with the sample, or by repeating
the experiment enough times when dealing with thermometers at the quantum
scale [1, 2, 3, 4, 5]. Such processes take time, and in fact we also wish a
good thermometer to be fast. Motivated by this practical limitation, in this
paper we derive fundamental limits on finite-time thermometry, and discover
that in some situations these limits can be saturated with a relatively simple
experiment.

Concretely, we consider a probe (acting as a thermometer) in contact with a
sample – a reservoir at temperature $T$. Their interaction leaks information
on the sample’s temperature (or another physical property) to the state of the
probe. After some time, the latter is measured in order to estimate the
parameter $T$. To benchmark the quality of such estimation schemes we consider
the quantum Fisher information (QFI) of the final state of the probe
$\mathcal{F}$ [6].

The most natural starting point is equilibrium thermometry, where the probe is
given sufficient time to thermalize to its Gibbs state, $\rho=e^{-\beta H}/\tr
e^{-\beta H}$ with $\beta=1/k_{\rm B}T$ and $k_{\rm B}$ the Boltzmann constant
($k_{\rm B}=1$ in what follows) [7, 8, 9, 10, 11, 12]. Remarkably, the
(quantum) Fisher information of the Gibbs state is proportional to the heat
capacity of the system, because the average energy turns out to be the optimal
estimator for temperature as it saturates the Cramer-Rao bound [13, 14, 7, 8].
This provides a practical recipe to design good equilibrium thermometers: the
probe’s Hamiltonian $H$ needs to be engineered to maximise the heat capacity
of the probe around the estimated temperature $T$. For example, this can be
achieved by preparing the probe close to a critical point, see some examples
in [3, 15]. The fundamental limits of equilibrium thermometry where derived by
Correa _et al_ in [8] by maximising the QFI with respect to probe’s
Hamiltonian, showing in particular that it can scale at most quadratically
with the number of subsystems in the probe [16, 12].

Nonequilibrium thermometry offers an alternative framework to enhance the QFI
of the probe. In this case, the probe is measured before reaching equilibrium,
so that extra information can be potentially achieved. This requires knowledge
of the dissipative dynamics and precise timing for the measurement, but can
lead to substantially higher QFI than standard equilibrium thermometry [17,
18, 19, 9, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33]. The
maximum achievable QFI in any such scenario has been recently derived in [32]
(note that, in contrast to the present work, the sensing time is not taken to
be a resource in [32]).

These results are promising for the design of quantum thermometers. However,
in most cases they disregard the importance of the sensing time, i.e. how long
it takes to reach the desired state. A clear example is equilibrium
thermometry: whereas critical systems have a higher heat capacity (i.e. QFI),
they also require longer thermalisation times [34]. There is hence an
interesting tradeoff between the probe’s QFI and the required time of the
estimation process. In fact, in analogy with frequency estimation [35, 36], it
is natural to consider both the number of probes and the sensing time are
resources for thermometry.

To take time into account, in this article we consider the QFI rate as the
figure of merit – the QFI of the final state divided by the total interaction
time [8, 3]. Considering the most general control strategy on the probe
(adaptive measurements, control on the probe’s state and Hamiltonian), we
derive upper bounds on the QFI rate whenever the thermalisation process is
well described by a general class of Markovian master equations by exploiting
techniques derived in [37, 38, 39] in the context of frequency estimation.
Using Markovian master equations for non-equilibrium thermometry is
ubiquiquituous in theoretical works (see e.g. the review [3]), but also of
crucial experimental relevance: recent experiments in non-equilibrium
thermometry involving single-atom probes for quantum gases [31] can be well
described by this framework [40].

Our main results and their implications can be summarised as follows:

  * •

We show that, for a large class of Markovian dynamics, the QFI grows at most
linearly with the total sensing time $\tau$, a shot-noise like behaviour
common of dissipative processes. More precisely, the QFI $\mathcal{F}$ is
upper bounded by an inequality of the form $\mathcal{F}\leq
f(\mathcal{L})\tau$, see Eq. (17) and (30), where $f(\mathcal{L})$ is a
function that depends on the specific Linblad evolution. The bounds can be
easily computed from the jump operators and the corresponding rates without
the need to solve the dynamics.

  * •

We apply these bounds to Lindblad master equations derived from a probe-sample
weak interaction of the form $H_{I}=A\otimes B$, with $A$ ($B$) acting on the
probe (sample). Then, we show that for arbitraty probe Hamiltonian $H$ our
bounds can be put in a very intuitive form (see Eq. (28)):

$\displaystyle\mathcal{F}\leq\tau\tilde{f}(B)(\Delta A)^{2}$ (1)

where $\Delta(A)=\lambda_{\mathrm{max}}-\lambda_{\mathrm{min}}$ is the
spectral gap of the operator $A$ (the difference between its maximal and
minimal eigenvalues), and $\tilde{f}(B)$ is a function that depends on the
spectral density of the sample. This can be understood as a speed limit
relating the amount of information that the probe can acquire about the sample
with their coupling given by $H_{I}=A\otimes B$. We also show how to engineer
the probe Hamiltonian $H$ in order to attain the bound.

  * •

In absence of temperature dependent Lamb shift, these upper bounds on the QFI
can be saturated by a fast measure-and-prepare strategy. The optimality of
these strategies were already noted numerically for qubit probes [8, 3], and
our results generalise these considerations to arbitrary Markovian evolutions.
We also develop an autonomous scheme where the fast measure-and-prepare
strategy is achieved through coupling the probe to an external zero-
temperature bath, and monitoring the energy flux from the sample to the bath.

  * •

The fast measure-and-prepare strategy can in principle be improved in the
presence of a temperature-dependent Lamb shift. Whereas this contribution is
usually negligible compared to that obtained from the temperature-dependent
decay rates, we argue that it plays a dominant role at low temperatures. By
focusing on a qubit probe with frequency $w$ interacting with a bosonic
sample, we show that the QFI can scale as $\mathcal{F}\propto T^{1+\alpha}$ as
$T\rightarrow 0$ ($\alpha$ is the Ohmicity of the sample), in contrast to the
standard exponential $\mathcal{F}\propto e^{-w/T}$ obtained when the Lamb
shift is neglected [7, 41]. Similar polynomial decays had been obtained in the
steady state of strongly coupled sample-probes states [41, 42, 43, 27]. To
exploit this advantage, we show that the probe has to be left to evolve
coherently for a finite time analogously to frequency estimation scenarios
(see in this context [25, 33]), which is markedly different from the fast
measure-and-prepare strategy.

  * •

To illustrate the general applicability of our framework, we also develop
optimal thermometric protocols for more complex probes, consisting of a $N$
qubits coupled to the sample through a collective coupling [44, 45]. In this
case, we show that the QFI can scale as $N^{2}$, and characterise the
experimental requirements to obtain this enhanced scaling.

It is also important to emphasise that the bounds and measurement schemes we
derive are not limited to the task of thermometry. They apply to any scenario
where the probe’s interaction with the sample is well described by a Markovian
master equation for the state of the probe, with the jump rates and the Lamb
shift depending on some physical property of the sample. Our bounds only rely
on the form of this master equation, and hence set a general (and in many
cases attainable) limit on how well a physical property of the sample can be
estimated through the probe in a finite time. As an illustration we consider
the estimation of the Ohmicity of the bath in sec. 4.1.3.

The paper is structured as follows. In Sec. 2, we introduce the different
techniques and quantities of interest. The main results, consisting of the
differents bounds on QFI as well as optimal measurement strategies, are then
presented in Sec. 3. In Sec. 4, we apply such general considerations to a
qubit interacting with a bosonic sample, and to a collective coupling between
the sample and a $N$-qubit probe. We close in Sec. 5 with a short summary and
a discussion of future endeavours.

Figure 1: The general sensing scheme we consider. The temperature $T$ (or
another parameter) of a large sample at thermal equilibrium is estimated by
coupling it to a probe for a finite time $\tau$. We assume that the probe’s
evolution is well modeled by Markovian semi-group dynamics. In addition, we
allow the possibility that probe’s dynamics is assisted with general fast
quantum control. In particular, this may include continuous measurements,
adaptive schemes with feedback, and entangling gates with auxiliary quantum
systems.

##  2 Framework

In this section, we start by reviewing the basics of parameter estimation,
with emphasis on thermometry. We then introduce Lindblad master equations,
which are used to describe the thermalisation of the probe when put in contact
with the sample, as well as the general framework to describe arbitrary
quantum control on the probe (including measurements and adaptive strategies),
see Fig. 1. Finally, we describe the results of [37, 38, 39], which are the
main technical tools used to find optimal finite-time protocols for
thermometry.

###  2.1 Parameter estimation and Quantum Fisher Information

Consider a random variable $X$ collecting the results of all measurements
performed during an experiment. To simplify the following discussion we assume
that $X$ is discrete. For thermometry (or any parameter estimation task) the
description of the experiment is fully captured by the dependence of $X$ on
the parameter $T$, as given by the family of probability distributions
$p(x|T)$ modelling $X$. The Fisher information
$\mathcal{I}(T)=\mathds{E}\left[\left(\frac{\partial}{\partial T}\log
p(X|T)\right)^{2}\Big{|}T\right]$, associated to $X$ at a given $T$,
quantifies how much information the experiment reveal about the parameter
(note that the expected value $H(X)=\mathds{E}\left[\log
p(X|T)\big{|}T\right]$ is the entropy of $X$). Denoting $p_{x}=p(x|T)$ the
Fisher information can be easily expressed as

$\mathcal{I}(T)=\sum_{x}\frac{(\dot{p}_{x})^{2}}{p_{x}}$ (2)

under some regularity conditions111The expression is not well defined if some
of the probabilities $p_{x}$ are strictly zero.. Here and in the rest of the
paper we use the dot to denote the derivative with respect to the estimated
parameter $\dot{p}_{x}=\frac{\partial}{\partial T}p_{x}$. One verifies from
Eq. (2) that $\mathcal{I}(T)$ can also be understood geometrically as the
temperature susceptibility of the probability measure
$\bm{p}(T)=(p_{1},p_{2},\dots)$ with respect to Kullback-Leibler divergence
(also Hellinger or Bhattacharyya distances).

Fisher information plays a central role in statistics. In particular, via the
famous Cramér-Rao bound it sets a limit on how well the parameter can be
estimated with an unbiased estimator $\hat{T}(X)$

$\mathds{E}\left[(\hat{T}-T)^{2}\right]\geq\frac{1}{\mathcal{I}(T)}.$ (3)

This bound can also be adjusted to the context of Bayesian inference [46],
where the prior knowledge of the parameter $T$ is explicitly taken into
account. Futhermore, for a fixed prior Bernstein–von Mises theorem suggests
(under some regularity conditions) that if the experiment is long enough and
done well the posterior distribution is fully determined by $\mathcal{I}(T)$.

In a quantum experiment the final state of the system (that can include
classical registers) is not described by a probability measure $\bm{p}(T)$ but
by a density matrix $\rho(T)$. Fixing the measurement performed on the final
state relates $\rho(T)$ to a random variable $X$ and its Fisher information
$\mathcal{I}(T)$. It it then natural to define the quantum Fisher information
$\mathcal{F}(T)$ (QFI) of the state $\rho(T)$ as the maximal Fisher
information for all possible measurements. With this definition it is clear
that the role that $\mathcal{I}(T)$ plays for parameter estimation is endorsed
by QFI in quantum experiments. Remarkably, $\mathcal{F}(T)$ can be written in
a closed form

$\mathcal{F}_{\rho(T)}=\tr L\dot{\rho}\\\ $ (4)

with the symmetric logarithmic derivative operator $L$ given by the equation
$\frac{1}{2}\\{L,\rho\\}=\dot{\rho}$ [6, 47, 48]. Analogously to the classical
case the QFI can be understood geometrically as temperature susceptibility of
the state $\rho(T)$ with respect to Bures distance (or Uhlmann’s fidelity).

The final state of the system depends on how it was prepared at the beginning
of the experiment. Just like the final measurement the preparation procedure
is in principle under control. It is thus more natural to describe the
experiment by a quantum channel $\mathcal{E}_{T}$ mapping the initial state of
the system $\varrho_{0}$ to its final state
$\rho(T)=\mathcal{E}_{T}(\varrho_{0})$, and define the QFI of the channel by
optimizing over the preparation

$\mathcal{F}_{\mathcal{E}_{T}}=\max_{\varrho_{0}}\mathcal{F}_{\rho(T)}.$ (5)

As for the states the QFI of a channel can be expressed in a simple form, if
one allows trivial extensions of the channel
$\mathcal{E}_{T}\otimes\mathrm{id}$ [49].

Finally, in practice it is important to consider the duration of the
experiment $\tau$ as a resource for parameter estimation. To benchmark
experiments we analyse the QFI as a function of time
$\mathcal{F}_{\mathcal{E}_{|T\tau}}$. Intuitively, the longer the experiment
the better can one estimate the parameter. For example, because the (quantum)
Fisher information is additive for independent random variables
$\mathcal{I}_{(X,Y)}(T)=\mathcal{I}_{X}(T)+\mathcal{I}_{Y}(T)$ and product
states
$\mathcal{F}_{\rho(T)\otimes\sigma(T)}=\mathcal{F}_{\rho(T)}+\mathcal{F}_{\sigma(T)}$,
it is doubled if an experiment is repeated twice. This observation gives rise
to the so called diffusive (or standard quantum limit) scaling
$\mathcal{F}_{\mathcal{E}_{\tau|T}}\propto\tau$. Among others it describes
experiments where the same preparation and measurement is repeated many times.
The best scaling, called ballistic (or Heisenberg), possible in quantum
experiments is $\mathcal{F}_{\mathcal{E}_{\tau|T}}\propto\tau^{2}$ [50]. This
is for example the case when the dynamics of the system is unitary
$\mathcal{E}_{\tau|T}[\rho_{0}]=e^{-\mathrm{i}\,\tau
T\,H}\rho_{0}\,e^{\mathrm{i}\,\tau T\,H}$. This gives rise to a faster scaling
of precision with which the parameter is estimated with respect to the sensing
time.

###  2.2 Probe-Sample interaction as a Lindblad master equation

In the context of this paper the sample is a large reservoir at thermal
equilibrium. Ideally, the state of the reservoir is unaffected by the
interaction with a small probe, so that the reservoir can be virtually
replaced with a fresh one all the time. In consequence the probe-sample
interaction can be naturally described by a time-independent Markovian semi-
group evolution of the probe system alone. Thus we assume the dynamics of the
probe to be given by a master equation of the form

$\begin{split}\frac{\differential}{\differential
t}\rho=\mathcal{L}(\rho)&=-\mathrm{i}[H+H_{{LS}},\rho]+\mathcal{D}(\rho),\\\
\mathrm{with}\qquad\mathcal{D}(\rho)&=\sum_{\omega}\gamma_{\omega}\left(A_{\omega}\rho
A_{\omega}^{\dagger}-\frac{1}{2}\\{A_{\omega}^{\dagger}A_{\omega},\rho\\}\right).\end{split}$
(6)

We assume, that the dissipator $\mathcal{D}(\rho)$ is of the Lindblad form
with respect to the jump operators $A_{\omega}$ that do not depend on bath
temperatures (but the rates $\gamma_{\omega}$ do). Furthermore, the jump
operators satisfy

$A_{\omega}=\sum_{(\varepsilon^{\prime},\varepsilon)\in
P_{\omega}}\Pi_{\varepsilon}A_{\omega}\Pi_{\varepsilon^{\prime}}$ (7)

for a family of orthogonal projectors $\\{\Pi_{\varepsilon}\\}_{\varepsilon}$,
where each pair $(\varepsilon,\varepsilon^{\prime})$ with
$\varepsilon\neq\varepsilon^{\prime}$ can only appear once (for one $\omega$).
In words, the operators $A_{\omega}$ describe jumps between subspaces labeled
by $\varepsilon^{\prime}$ and $\varepsilon$ appearing in $P_{\omega}$. A
simple example, is where $\Pi_{\epsilon}$ defines the subspace associate to
energy $\epsilon$ of the probe’s Hamiltonian $H$ [51]. Then, $P_{\omega}$
might collect all pairs of energies $(\varepsilon,\varepsilon^{\prime})$ such
that $\varepsilon^{\prime}-\varepsilon=\omega$. We chose to write the jump
operators in a more abstract form so that the master equation (6) can be
applied very generally, in particular to the local master equation [52, 53,
54], partial secular approximations [55, 56, 57, 58, 59, 60], as well as
reaction-coordinate mappings to deal with strong coupling [61].

The Hamiltonian $H_{{LS}}$ is the Lamb shift correction to the energy of the
probe system induced by the interaction with the sample. $H_{LS}$ is typically
small as compared to the energy of the probe and is very often neglected in
the thermometry literature (see Ref. [62] for an exception). Nevertheless, it
is generally a _temperature dependent_ correction $\dot{H}_{LS}\neq 0$ and may
play an important role in low-temperature thermometry as we shall see.

###  2.3 The master equation from a microscopic model

To vehicle some physical insight into Eq. (6) let us give explicit expressions
for the microscopic derivation of the so-called global master equation [51].
As the starting point consider the probe-sample coupling given by the global
Hamiltonian in the form

$H_{\rm tot}=H+H_{B}+A\otimes B,$ (8)

where $H(H_{B})$ is the Hamiltonian of the probe (bath or sample), and the
interaction Hamiltonin has a form $H_{I}=A\otimes B$ with Hermitian operators
$A$ and $B$ acting on the probe and sample respectively. It is convenient to
diagonalize the probe Hamiltonian

$H=\sum_{\varepsilon\in\mathrm{Sp}(H)}\varepsilon\,\Pi_{\varepsilon},$ (9)

where $\mathrm{Sp}(H)$ is the spectrum of $H$, defining the energy subspaces
given by the set of orthogonal projectors $\\{\Pi_{\varepsilon}\\}$.

Assuming that the coupling between the probe and the sample is weak and using
the Born-Markov and secular approximations222The secular approximation
requires that any non-zero frequency $w=\varepsilon^{\prime}-\varepsilon$ is
larger than the interaction strength, see e.g. [55, 56, 57, 58, 59, 60]., one
can derive the master equation for the probe with of the form of Eq. (6) [51]:

$\begin{split}\mathcal{D}(\rho)&=\sum_{\omega}\gamma_{\omega}\left(A_{\omega}\rho
A_{\omega}^{\dagger}-\frac{1}{2}\\{A_{\omega}^{\dagger}A_{\omega},\rho\\}\right)\\\
H_{{LS}}&=\sum_{\omega}s_{\omega}\,A_{\omega}^{\dagger}A_{\omega}\\\
A_{\omega}&=\sum_{\varepsilon^{\prime}-\varepsilon=\omega}\Pi_{\varepsilon}A\,\Pi_{\varepsilon^{\prime}}\end{split}$
(10)

The coefficeints $\gamma_{\omega}$ and $s_{\omega}$ are real and imaginaly
parts of the correlation function of the reservoir
$\Gamma_{\omega}=\frac{1}{2}\gamma_{\omega}+\mathrm{i}s_{\omega}=\int_{0}^{\infty}e^{\mathrm{i}\omega
t}\langle{B(0)B(t)}\rangle\differential t$ where $B(t)\equiv
e^{-iH_{B}t}Be^{-iH_{B}t}$ are operators of the sample in the interaction
picture. One notes that Eq. (10) is a particular case of Eqs. (6-7).

In the following we consider the possibility of engineering the probe
Hamiltonian $H$. In contrast, we treat the operator $A$ describing the
coupling of the probe to the sample as being fixed.

###  2.4 Fast quantum control

On top of the thermalisation process, we assume that the probe and an
additional auxilliary system of arbitrary dimension (not interacting with the
sample) can be externally measured and controlled at will. The control
capabilities are introduced in two steps. First, the Hamiltonian $H$ (acting
on the probe and the ancillae) can be engineered. In principle, we allow it to
vary in time as long as instantaneous thermalization at $t$ is well described
by Eq. (6) for some fixed $H(t)$ 333This is well justified when the driving is
sufficiently slow, for the so called adiabatic master equation [63, 64], or if
the driving is very fast, e.g. quenches, so that the bath does not have time
to react. However, we will see that all protocols considered here, including
optimal ones, do not require driving of $H$ when the QFI is the figure of
merit. It is enough to optimise $H$ at the beginning of the process. . Second,
the dissipative evolution is being interjected with arbitrary control
operations and measurements on the probe and the ancillae. Following [37] this
can be formally described by dividing the total time $\tau$ in $m$ steps,
$\differential t=\tau/m$ (in principle $m$ can be arbitrary large). The
systems evolve accordingly to the master equation during each step
$e^{\differential t\mathcal{L}}$ but in-between any two steps an arbitrary
gate, measurement or initialization can be applied. In fact because the
dimension of the auxiliary system is kept free, all control operations can be
implemented as unitary gates $U_{k}$. In particular, the information acquired
with a measurement along the process can mapped coherently onto some ancilla
and used in the next steps, thus allowing for adaptive schemes. It is
important to stress that, as above, we assume that the control operations do
not hinder the approximations and time-scale separation used to derive the
master equation (6) in the first place.

At any time $\tau$ the state of the probe and ancillae is given by a density
matrix $\rho(\tau|T)$. The QFI of this state with respect to the temperature
sets a fundamental limit on the precision of the whole scheme seen as a
thermometer. Our goal is then to develop finite-time strategies that maximise
the QFI of the channel

$\mathcal{E}_{\tau|T}=e^{\differential t\mathcal{L}}\cdot U_{m}\dots\circ
e^{\differential t\mathcal{L}}\circ U_{1}.$ (11)

Crucially, the control and presence of ancillae can not alter the coupling
$H_{I}=A\otimes B$ between the probe and the sample. The intuition is that
$H_{I}$ gives the bottleneck for how fast a change of temperature in the
sample causes an effect on the probe, setting a limit on thermometry even in
presence of arbitrary quantum control.

###  2.5 QFI bounds for control-assisted Markovian semi-group dynamics

It is straightforward to verify that a thermalization of an infinitesimal
duration $\differential t$ is given by a quantum channel $e^{\differential
t\mathcal{L}}$ which admits a Kraus representation $\bm{K}=(K_{0}\dots
K_{\omega}\dots)$ composed of Kraus operators

$\begin{split}K_{0}&=\mathds{1}-\differential
t\frac{1}{2}\sum_{\omega}\gamma_{\omega}A_{\omega}^{\dagger}A_{\omega}-\mathrm{i}\,\differential
t\,(H+H_{LS})\\\ K_{\omega}&=\sqrt{\differential
t\gamma_{\omega}}A_{\omega}\qquad\forall\,\omega,\end{split}$ (12)

where the terms of order $\differential t^{3/2}$ and higher are ignored. A
Kraus representation of a quantum channel is of course not unique, any vector
of operators $\tilde{\bm{K}}=u\bm{K}$ with a unitary gauge matrix $u$
describes the same channel444Note that the gauge matrix can explicitly depend
on the parameter. It has been shown in [37, 38] that if there exist a
Hermitian matrix $h=-\mathrm{i}u^{\dagger}\dot{u}$, such that

$\qquad\|\bm{K}^{\dagger}(\dot{\bm{K}}+\mathrm{i}h\bm{K})\|=O(\differential
t^{3/2}),$ (13)

then the QFI of any channel $\mathcal{E}_{\tau|T}$ of form of Eq. (11) is
upper bounded by

$\mathcal{F}_{\mathcal{E}_{\tau|T}}\leq 4\frac{\tau}{\differential
t}\|(\dot{\bm{K}}^{\dagger}-\mathrm{i}\bm{K}^{\dagger}h)(\dot{\bm{K}}+\mathrm{i}h\bm{K})\|,$
(14)

with $\|...\|$ the operator norm, which is given by
$\|A\|\equiv\max_{j}\lambda_{j}$ where $\lambda_{j}$ are the eigenvalues of
the Hermitian operator $A$. Note that if the master equation has an explicit
time dependence, but Eq. (13) is fulfilled at all times for some $h_{t}$, one
can simply replace the bound with $\mathcal{F}_{\mathcal{E}_{\tau|T}}\leq
4\int_{0}^{\tau}\|(\dot{\bm{K}}^{\dagger}_{t}-\mathrm{i}\bm{K}_{t}^{\dagger}h_{t})(\dot{\bm{K}}_{t}+\mathrm{i}h_{t}\bm{K}_{t})\|$,
this is analogous to the phase-estimation scenario with time-dependent noise
discussed in the supplementary material of [38].

##  3 Results

We are now ready to present the main results of this article. We start by
showing that the diffusive scaling
$\mathcal{F}_{\mathcal{E}_{\tau|T}}\propto\tau$ induced by Eq. (14) applies to
a wide class of probe-sample interactions. We then derive upper bounds on
$\mathcal{F}_{\mathcal{E}_{\tau|T}}$ for both fixed and engineered probe’s
Hamiltonian $H$, and show how to saturate them by simple measurement
strategies. Finally, we discuss the potential effect of the Lamb shift
Hamiltonian in thermometry.

###  3.1 The diffusive scaling of the QFI in thermometry

The first question to ask is whether ballistic scaling of QFI is possible in
thermometry, i.e. whether there exists a $h$ satisfying Eq. (13) as we just
discussed. Following the argumentation of [38], we derive the general
condition for diffusive scaling of the QFI that can be found in the Appendix
6. A simpler but slightly more restrictive version of it reads

$\dot{H}_{LS}\in\,\mathrm{span}\left\\{\mathds{1},\gamma_{\omega}A_{\omega}^{\dagger}A_{\omega}\right\\},$
(15)

see Appendix 6. In words, if the derivative of the Lamb shift Hamiltonian
$\dot{H}_{LS}$ is linearly dependent of identity $\mathds{1}$ and the
operators $\gamma_{\omega}A_{\omega}^{\dagger}A_{\omega}$ for all $\omega$,
then $\mathcal{F}_{\mathcal{E}_{\tau|T}}\propto\tau$. It is worth noting that
the condition (15) is also sufficient to enforce a diffusive scaling of the
QFI if the Lamb shift Hamiltonian is block-digonal
$H_{LS}=\sum_{\varepsilon}\Pi_{\varepsilon}H_{LS}\Pi_{\varepsilon}$, c.f.
Appendix 6.

In the case of our microscopic model introcuced in section 2.3 the Lamb shift
Hamiltonian takes the form
$H_{LS}=\sum_{\omega}s_{\omega}A^{\dagger}_{\omega}A_{\omega}$. Clearly, a
sufficient condition for (15) is that all the jump rates $\gamma_{\omega}$ are
non-vanishing whenever the jump operator $A_{\omega}\neq 0$ and the derivative
of the Lamb shift contribution $\dot{s}_{\omega}\neq$ are non zero. We argue
that this is usually the case. It is possible that some frequencies $\omega$
are missing in the spectrum of the bath, however in this case both
$\gamma_{\omega}$ and $s_{\omega}$ are identically zero. If the frequency is
present it typically gives rise to the dissipative term $\gamma_{\omega}$, and
not just a coherent temperature dependent Lamb shift $\dot{s}_{\omega}$. In
such cases we can conclude that the QFI is bound to a diffusive scaling

$\mathcal{F}_{\mathcal{E}_{\tau|T}}\leq\mathrm{const}\,\tau.$ (16)

Intuitively speaking, this scaling is a consequence of the inevitable presence
of noise/dissipation in the process of temperature estimation, which prevents
the ballistic scaling $\mathcal{F}_{\mathcal{E}_{\tau|T}}\propto\tau^{2}$
commonly obtained in quantum-coherent evolutions (e.g. in noiseless phase
interferometry [65]).

On the other hand, it is interesting to ask the question whether one can
engineer a more exotic reservoir in such a way that the condition for
diffusive scaling of the QFI is not fulfilled. For example, if the interaction
in (8) is replaced by the more general form $\sum_{i}A_{i}\otimes B_{i}$, then
the diffusive scaling of the QFI is not guaranteed by the argument above. In
this case, ballistic scaling of the QFI could be achieved, possibly requiring
active control [39].

###  3.2 Upper bound on QFI for a fixed $H$ (no Lamb shift)

For clarity of the exposition, we start by ignoring the Lamb shift $H_{LS}$
term in the master equation in the following sections. More precisely, we
assume that the Lamb shift is constant around the true value of the bath
temperature ($\dot{s}_{\omega}=0$ if $A_{\omega}^{\dagger}A_{\omega}\neq 0$),
so that it can be ignored. Indeed, any parameter independent term in the
Hamiltonian can be cancelled via appropriate control operations. We will come
back to the the effect of $H_{LS}$ later.

We start by considering a fixed probe Hamiltonian $H$, and discuss its
engineering in the following section. In absence of Lamb shift Eq. (13) can be
satisfyed by setting $h=0$. This gives a particularly simple upper-bound on
the QFI for thermalizaion. Straightforward application of Eq. (14) gives a
very intuitive expression

$\mathcal{F}_{\mathcal{E}_{\tau|T}}\leq\tau\left\|\sum_{w}\frac{\dot{\gamma}_{w}^{2}}{\gamma_{w}}\,A_{w}^{\dagger}A_{w}\right\|,$
(17)

see Appendix 7. Note that the bound only depends on the jump operators
$A_{\omega}$ and the rates $\gamma_{\omega}$, and neither requires solving the
master equation nor considering explicit control strategies. Remarkably, this
bound can be saturated with a simple strategy as we now show.

###  3.3 Saturation of the bound (17) with a continuous measure-and-prepare
scheme

To save some space let us denote the norm appearing in the rhs of Eq. (17) as
$\mathds{O}=\left\|\sum_{w}\frac{\dot{\gamma}_{w}^{2}}{\gamma_{w}}\,A_{w}^{\dagger}A_{w}\right\|$.
For simplicity we start with the expression of the the jump operators
$A_{\omega}$ in Eq. (10). It gives

$\begin{split}\mathds{O}&=\left\|\sum_{\omega}\frac{\dot{\gamma}_{\omega}^{2}}{\gamma_{\omega}}\sum_{\bar{\varepsilon}^{\prime}-\bar{\varepsilon}=\omega}\sum_{\varepsilon^{\prime}-\varepsilon=\omega}\Pi_{\bar{\varepsilon}^{\prime}}A\,\Pi_{\bar{\varepsilon}}\Pi_{\varepsilon}A\,\Pi_{\varepsilon^{\prime}}\right\|\\\
&=\left\|\sum_{\omega}\frac{\dot{\gamma}_{\omega}^{2}}{\gamma_{\omega}}\sum_{\varepsilon^{\prime}-\varepsilon=\omega}\Pi_{\varepsilon^{\prime}}A\,\Pi_{\varepsilon}A\,\Pi_{\varepsilon^{\prime}}\right\|.\end{split}$
(18)

Here, we used
$\Pi_{\bar{\varepsilon}}\Pi_{\varepsilon}=\delta_{\bar{\varepsilon}\varepsilon}\Pi_{\varepsilon}$
and the fact that for a fixed $\omega$ and $\varepsilon$ there can only be one
$\varepsilon^{\prime}$ fulfilling $\varepsilon^{\prime}-\varepsilon=\omega$.
Since all the operators
$\Pi_{\varepsilon^{\prime}}O\Pi_{\varepsilon^{\prime}}$ for different
$\varepsilon^{\prime}$ are orthogonal, we can simplify the operator norm to

$\mathds{O}=\max_{\varepsilon^{\prime}}\left\|\Pi_{\varepsilon^{\prime}}\left(\sum_{\varepsilon\neq\varepsilon^{\prime}}\frac{\dot{\gamma}_{\varepsilon^{\prime}-\varepsilon}^{2}}{\gamma_{\varepsilon^{\prime}-\varepsilon}}\,A\,\Pi_{\varepsilon}\,A\,\right)\Pi_{\varepsilon^{\prime}}\right\|.$
(19)

We denote the energy which attains the maximum by $\varepsilon_{*}$.

With the more general form of the jump operators
$A_{\omega}=\sum_{(\varepsilon^{\prime},\varepsilon)\in
P_{\omega}}\Pi_{\varepsilon}A_{\omega}\Pi_{\varepsilon^{\prime}}$ in Eq. (7)
the expression takes the form

$\mathds{O}=\max_{\varepsilon^{\prime}}\left\|\Pi_{\varepsilon^{\prime}}\left(\sum_{\begin{subarray}{c}\omega,\varepsilon\\\
(\varepsilon,\varepsilon^{\prime})\in
P_{\omega}\end{subarray}}\frac{\dot{\gamma}_{\omega}^{2}}{\gamma_{\omega}}\,A_{\omega}^{\dagger}\,\Pi_{\varepsilon}\,A_{\omega}\,\right)\Pi_{\varepsilon^{\prime}}\right\|,$
(20)

which follows from identical arguments, with the maximum attained for
$\varepsilon^{\prime}=\varepsilon_{*}$.

So far we have just rewritten Eq. (17). To show that it can be attained,
consider the following strategy. Prepare the probe in some state
$\ket{\Psi_{\varepsilon_{*}}}=\Pi_{\varepsilon_{*}}\ket{\Psi_{\varepsilon_{*}}}$
in the $\varepsilon_{*}$-subspace. Let it evolve for an infinitesimal time
$\differential t$ and measure in which subspace $\Pi_{\varepsilon}$ it is
(i.e. the energy of the probe). The probability that the probe jumps to
another energy subspace labeled by $\varepsilon\neq\varepsilon_{*}$ is given
by

$\begin{split}p_{\varepsilon}&=\tr\Pi_{\varepsilon}\,e^{\mathcal{L}\differential
t}\left(\mathinner{|{\Psi_{\varepsilon_{*}}}\rangle}\\!\\!\mathinner{\langle{\Psi_{\varepsilon_{*}}}|}\right)\\\
&=\differential
t\,\tr\Pi_{\varepsilon}\sum_{\omega}\gamma_{\omega}A_{\omega}\mathinner{|{\Psi_{\varepsilon_{*}}}\rangle}\\!\\!\mathinner{\langle{\Psi_{\varepsilon_{*}}}|}A_{\omega}^{\dagger}\\\
&=\differential
t\,\gamma_{\omega}\bra{\Psi_{\varepsilon_{*}}}A_{\omega}^{\dagger}\Pi_{\varepsilon}A_{\omega}\ket{\Psi_{\varepsilon_{*}}},\end{split}$
(21)

for $\omega=\omega(\varepsilon,\varepsilon_{*})$ such that
$(\varepsilon,\varepsilon_{*})\in P_{\omega}$. While the probability that it
remains at $\varepsilon_{*}$ is simply
$p_{\varepsilon_{*}}=1-\sum_{\varepsilon\neq\varepsilon_{*}}p_{\varepsilon}$.
Computing the Fisher information for this measurement after $\differential t$
we find that in the leading order

$\mathcal{I}_{\differential
t}=\sum_{\varepsilon}\frac{(\dot{p}_{\varepsilon})^{2}}{p_{\varepsilon}}=\differential
t\bra{\Psi_{\varepsilon_{*}}}\sum_{\begin{subarray}{c}\omega,\varepsilon\\\
(\varepsilon,\varepsilon_{*})\in
P_{\omega}\end{subarray}}\frac{\dot{\gamma}_{\omega}^{2}}{\gamma_{\omega}}\,A_{\omega}^{\dagger}\,\Pi_{\varepsilon}\,A_{\omega}\ket{\Psi_{\varepsilon_{*}}}$
(22)

By maximizing this expression over all states $\ket{\Psi_{\varepsilon_{*}}}$
we find the optimal Fisher information $\mathcal{I}_{\differential
t}=\differential
t\,\left\|\Pi_{\varepsilon_{*}}\sum_{\omega,\varepsilon}\frac{\dot{\gamma}_{\omega}^{2}}{\gamma_{\omega}}\,A_{\omega}^{\dagger}\,\Pi_{\varepsilon}\,A_{\omega}\Pi_{\varepsilon_{*}}\right\|$
by definition of the operator norm. After the measurement the probe is
reinitialized to the optimal state $\ket{\Psi_{\varepsilon_{*}}}$, and the
same procedure is repeated. By additivity, the overall Fisher information of
this scheme reads

$\mathcal{I}_{\tau}=\int_{0}^{\tau}\mathcal{I}_{\differential
t}=\tau\left\|\Pi_{\varepsilon_{*}}\\!\\!\\!\sum_{\begin{subarray}{c}\omega,\varepsilon\\\
(\varepsilon,\varepsilon_{*})\in
P_{\omega}\end{subarray}}\frac{\dot{\gamma}_{\omega}^{2}}{\gamma_{\omega}}\,A_{\omega}^{\dagger}\,\Pi_{\varepsilon}\,A_{\omega}\,\Pi_{\varepsilon_{*}}\right\|$
(23)

which coincides with the upper-bound as expressed in the form of Eq. (20) (and
also Eq. (19) as a particular case).

This shows that the upper-bound (17) is indeed saturated with a simple
continuous measure-and-prepare strategy, where the jumps (the energy) of the
probe are constantly monitored and the probe is actively reinitialized to the
initial state. The optimality of fast-measurements for qubit probes was
already noted numerically in Ref. [8] (see also [21, 3]), and we generalised
these results to general probes and arbitrary control-assisted/adaptive
schemes. In contrast to such previous results, it is worth stressing that our
approach neither requires solving the Lindblad dynamics nor considering
specific strategies, instead the optimal protocol can be inferred directly
from simple calculation of (17) and (28). This enables us to study optimal
protocols for more complex interactions and probes, as we will illustrate in
Sec. 4.

###  3.4 Engineering of the optimal spectrum for $H$

The bound (17) is saturable and holds independently of the unitary control
operations $U_{k}$ applied during the sensing process. Nevertheless, in the
context of the microscopic model in Sec. 2.3 it does depend on the probe
Hamiltonian $H$ via the jump operators $A_{\omega}$ and the corresponding
rates. The goal of this section is to find the Hamiltonian maximizing the QFI
of our thermometer, given the form of the observable $A$ coupling the probe to
the sample, its temperature $T$ and the spectral dependence of the rates
$\gamma_{\omega}$ and $\dot{\gamma}_{\omega}$. Recall that we allow the usage
of auxiliary systems of arbitrary dimension. Hence, we consider Hamiltonians
$H=\sum_{\varepsilon}\varepsilon\Pi_{\varepsilon}$ acting on the probe and the
auxiliary system, while only the probe couples to the sample
$A=A_{\mathrm{probe}}\otimes\mathds{1}_{\mathrm{ancillae}}$ (this is implicit
in the following discussion).

To optimize over all Hamiltonians $H$ one can vary its spectrum
$(\varepsilon_{1},\varepsilon_{2},\dots)$ and the subspaces
$\sum_{\varepsilon}\Pi_{\varepsilon}=\mathds{1}$ independently. To find the
best Hamiltonian we have to maximize the quantity
$\mathds{O}=\left\|\sum_{w}\frac{\dot{\gamma}_{w}^{2}}{\gamma_{w}}\,A_{w}^{\dagger}A_{w}\right\|$.
Starting with any Hamiltonain let us label the energy subspace that attains
the maximum in Eq. (19) with $\varepsilon_{*}$, without loss of generality we
can shift all the energies such that $\varepsilon_{*}=0$
($\Pi_{\varepsilon_{*}}\to\Pi_{0}$). The sum inside the norm in Eq. (19)
involves positive operators
$\Pi_{0}A\Pi_{\varepsilon}A\Pi_{0}=(\Pi_{0}A\Pi_{\varepsilon})(\Pi_{0}A\Pi_{\varepsilon})^{\dagger}$
and positive scalar coefficients
$(\dot{\gamma}_{-\varepsilon}^{2}/\gamma_{-\varepsilon})$. Therefore, for
fixed subspaces $\Pi_{\varepsilon}$ the expression $\mathds{O}$ is maximized
when all the scalar coefficients are maximized. But for any bath temperature
$T$ there is indeed an optimal energy difference $\delta_{T}$ maximizing the
coefficients

$f_{T}=\max_{\varepsilon}\frac{\dot{\gamma}_{-\varepsilon}^{2}}{\gamma_{-\varepsilon}},\qquad\delta_{T}=-\mathrm{argmax}_{\varepsilon}\frac{\dot{\gamma}_{-\varepsilon}^{2}}{\gamma_{-\varepsilon}}.$
(24)

Hence, for a fixed $\Pi_{0}$ it is optimal to set the energies of all the
other levels to the same value $\varepsilon=\delta_{T}$. Using
$\sum_{\varepsilon\neq 0}\Pi_{\varepsilon}=\mathds{1}-\Pi_{0}$ we get for the
optimal choice of energies

$\mathds{O}=f_{T}\,\|\Pi_{0}A(\mathds{1}-\Pi_{0})A\Pi_{0}\|.$ (25)

For a fixed $\Pi_{0}$ the norm is attained by some state
$\ket{\Psi}=\Pi_{0}\ket{\Psi}$. Denoting
$\Pi_{0\setminus\Psi}=\Pi_{0}-\mathinner{|{\Psi}\rangle}\\!\\!\mathinner{\langle{\Psi}|}$
we obtain

$\begin{split}\|\Pi_{0}A(\mathds{1}-\Pi_{0})A\Pi_{0}\|&=\bra{\Psi}A(\mathds{1}-\mathinner{|{\Psi}\rangle}\\!\\!\mathinner{\langle{\Psi}|}-\Pi_{0\setminus\Psi})A\ket{\Psi}\\\
&\leq\bra{\Psi}A(\mathds{1}-\mathinner{|{\Psi}\rangle}\\!\\!\mathinner{\langle{\Psi}|})A\ket{\Psi}.\end{split}$
(26)

But changing the Hamiltonian to one with
$\Pi_{0}=\mathinner{|{\Psi}\rangle}\\!\\!\mathinner{\langle{\Psi}|}$ saturates
this inequality. Thus, an optimal Hamiltonian is necessarily of this form
$H=\delta_{T}(\mathds{1}-\mathinner{|{\Psi}\rangle}\\!\\!\mathinner{\langle{\Psi}|})$,
which is fully degenerate except for one energy eigenstate (coincidentally,
this same structure is optimal for equilibrium thermometry [8]). It remains to
find the optimal state $\ket{\Psi}$. For a rank one projector
$\Pi_{0}=\mathinner{|{\Psi}\rangle}\\!\\!\mathinner{\langle{\Psi}|}$ the
expression
$\|\Pi_{0}A(\mathds{1}-\Pi_{0})A\Pi_{0}\|=\bra{\Psi}A^{2}\ket{\Psi}-\bra{\Psi}A\ket{\Psi}^{2}$
is simply the variance of $A$ with respect to the state $\ket{\Psi_{*}}$. The
variance of an operator is maximized on the state, which is an equal
superposition

$\ket{\Psi_{\Delta(A)}}=\frac{1}{\sqrt{2}}(\ket{a_{\mathrm{max}}}+\ket{a_{\mathrm{min}}})$
(27)

of two extremal eigenstates of
$A\ket{a_{\mathrm{max(min)}}}=\lambda_{\mathrm{max(min)}}\ket{a_{\mathrm{max(min)}}}$.
As a result we obtain the simple and general bound

$\mathcal{F}_{\mathcal{E}_{\tau|T}}\leq\tau\left(\max_{\omega}\frac{\dot{\gamma}_{\omega}^{2}}{\gamma_{\omega}}\right)\left(\frac{\Delta(A)}{2}\right)^{2},$
(28)

where $\Delta(A)=\lambda_{\mathrm{max}}-\lambda_{\mathrm{min}}$ is the
spectral gap of the operator $A$ (the difference between its maximal and
minimal eigenvalues).

In summary, we have shown that the QFI of our thermometer is maximized for the
almost degenerate Hamiltonian

$H=\delta_{T}(\mathds{1}-\mathinner{|{\Psi_{\Delta(A)}}\rangle}\\!\\!\mathinner{\langle{\Psi_{\Delta(A)}}|}),$
(29)

with the unique nondegenerate energy level $\ket{\Psi_{\Delta(A)}}$ of the
form in Eq. (27). This bound allows us to formalize the original intuition –
the form of the operator $A$ coupling the probe to the sample does set a
general “speed limit” on thermometry. Recall that the bound can be attained
with the continuous measure-and-prepare strategy, where the probe is
reinitialized to the state $\ket{\Psi_{\Delta(A)}}$. It is worth noting that
this strategy does not require an auxiliary system.

###  3.5 Autonomous implementation of the fast measure-and-prepare strategy

In order to exploit these ideas in practice, a possible autonomous
implementation is depicted in Fig. 2. We consider a scenario where the probe
is simultaneously coupled to the sample and to a zero temperature bath
($T=0$). If $\kappa$ and $\gamma$ are parameters describing the coupling
strength between the probe and each bath, $\kappa$ for the zero-temperature
bath and $\gamma$ for the sample, we consider the regime $\kappa\gg\gamma$.
This ensures that, every time the probe gains an excitation due to the
interaction with the sample, the cold bath quickly reinitialises the probe
back to its ground state. By monitoring the heat current between the probe and
the cold bath at the single-photon level, one is able to keep track of the
energy jumps induced on the probe by the sample (the same approach is used in
quantum thermodynamics to measure heat and work statistics, see e.g. [66,
67]). This provides a simple way of performing the fast measure-and-prepare
strategy introduced earlier, in which the cold bath simultaneously prepares
and measures the probe. It is worth mentioning that this type of environment
monitoring to measure single-photon emissions is already being implemented in
superconducting quantum circuits [66, 68, 69, 70].

To saturate the upper-bound (28) one needs to implement the optimal probe
Hamiltonian of the form given in Eq. (29). This might not be an easy task in
practice. Nevertheless, simpler Hamiltonians might already come close to the
upper-bound. This will be illustrated in Sec. 4.2 for collective two-body
Hamiltonians which are implemented in spin squeezing experiments [71].

Figure 2: A practical autonomous thermometer scheme. In the limit where the
interaction of the probe with the cold bath is much stronger then its
interaction with the sample the probe is constantly reset to its ground state.
Continuous measurements of the heat current between the probe and the cold
bath allow one to monitor the energy jumps in the probe induced by the sample.
Generally, this setup implements the continuous measure and prepare strategy,
where the probe is reset to its ground state. In particular, for collective
thermalisation (stemming from probe-sample interaction of the form
$H_{I}=J_{x}\otimes B$) and optimized two-body probe Hamiltonian of the form
$H=\frac{\omega}{2}J_{z}+bJ_{z}^{2}$ we show in Sec. 4.2 that the strategy
comes very close to saturating the upper-bound of Eq.(28).

###  3.6 Effect of the Lamb shift

We have derived very general and attainable bounds on the control-assisted
thermometry both in the case of a fixed probe Hamiltonian $H$ in (17) as well
as for an optimized $H$ in (28). In doing so we have however assumed that the
Lamb shift term in the master equation (6) can be ignored555This assumption is
common in the literature on quantum thermometry.. Let us now assume the
presence of a Lamb shift satisfying the condition (15) and include it in the
analysis.

As shown in Sec. 3.1, the time scaling of the Fisher information is still
diffusive $\mathcal{F}_{\mathcal{E}_{t}|T}\propto\tau$ in presence of a
temperature-dependent Lamb shift if the condition (15) holds. In this case,
the derivative of the Lamb shift Hamiltonian can be expressed as

$\dot{H}_{LS}=h_{\mathds{1}}\,\mathds{1}+\sum_{\omega}h_{\omega}A^{\dagger}_{\omega}A_{\omega},$
(30)

with some real coefficeints $h_{\mathds{1}}$ and $h_{\omega}$ such that
$h_{\omega}=0$ if $\gamma_{\omega}=0$. Then, we show in the appendix 7 that
the bounds for a fixed $H$ (17) and for the optimal $H$ (28) are modified to
the form

$\begin{split}\mathcal{F}_{\mathcal{E}_{\tau}|T}&\leq\tau\left\|\sum_{\omega}\frac{\dot{\gamma}_{\omega}^{2}+4h_{\omega}^{2}}{\gamma_{\omega}}\,A_{\omega}^{\dagger}A_{\omega}\right\|,\\\
\mathcal{F}_{\mathcal{E}_{\tau|T}}&\leq\tau\left(\max_{\omega}\sum_{\omega}\frac{\dot{\gamma}_{\omega}^{2}+4h_{\omega}^{2}}{\gamma_{\omega}}\right)\left(\frac{\Delta(A)}{2}\right)^{2}.\end{split}$
(31)

Here, to get the second inequality one simply repeats the derivation of
section 3.4 for the new scalar functions
$\left(\frac{\dot{\gamma}_{\omega}^{2}}{\gamma_{\omega}}\right)\mapsto\left(\frac{\dot{\gamma}_{\omega}^{2}+4h_{\omega}^{2}}{\gamma_{\omega}}\right)$.

In the case of the microscopic model, a particularly simple choice of the
coefficients $h_{\mathds{1}}$ and $h_{\omega}$ in Eq. (30) is offered by the
original expression for the Lamb shift

$\dot{H}_{LS}=\sum_{\omega}{\dot{s}}_{\omega}A^{\dagger}_{\omega}A_{\omega}$
(32)

assuming that $\gamma_{\omega}\neq 0$ if ${\dot{s}}_{\omega}\neq 0$. This
choice gives rise to simple upper bounds

$\begin{split}\mathcal{F}_{\mathcal{E}_{\tau}|T}&\leq\tau\left\|\sum_{\omega}\frac{\dot{\gamma}_{\omega}^{2}+4\dot{s}_{\omega}^{2}}{\gamma_{\omega}}\,A_{\omega}^{\dagger}A_{\omega}\right\|,\\\
\mathcal{F}_{\mathcal{E}_{\tau|T}}&\leq\tau\left(\max_{\omega}\sum_{\omega}\frac{\dot{\gamma}_{\omega}^{2}+4\dot{s}_{\omega}^{2}}{\gamma_{\omega}}\right)\left(\frac{\Delta(A)}{2}\right)^{2}.\end{split}$
(33)

that are directly expressible with quantities appearing in the master equation
(10). The disadvantage is that they might diverge in the low-temperature limit
where the jump rates $\gamma_{\omega}$ can become negligible compared to
$\dot{s}_{w}$, as we will see in the next section. In this case it is better
to use the more general bounds of Eqs. (30) and (31) with an appropriate
choice of $h_{w}$.

The main difference between Eq. (31) and (17) concerns the attainability of
these bounds. First, it is important to note that the Fisher information of
the fast measure-and-prepare strategy is not affected by a Lamb shift. Indeed,
due to the Zeno effect it does not affect the measurement probabilities after
an infinitesimal time $\differential t$. Therefore, the values on the rhs of
the inequalities (17) and (28) can still be attained in presence of a Lamb
shift. However, these values are not tight with the new upper-bounds.
Furthermore, the Zeno effect also implies that to approach the new bounds one
has to move away from continuous measurement schemes which are insensitive to
Lamb shift. Even if the bounds (31) and (33) are not necessarily attainable by
any measurement strategy, it is important to stress that the presence of the
Lamb shift can only improve the estimation precision. This will be illustrated
in the following example, where we will see that the presence of Lamb shift
can enable dramatic enhancements in the low-temperature regime.

##  4 Case studies

In this section we apply our general results to two pedagogical and
experimentally relevant examples. First, we consider a qubit probe with an
explicit model of the sample as a collection of bosonic modes. This allows us
to obtain explicit expressions for the jump rates in terms of sample
properties, and discuss different temperature regimes. Second, we consider the
case of an $N$-qubit probe with a two-body Hamiltonian and coupled
collectively to the sample. Here, we discuss the scaling of the thermometer
precision with $N$.

###  4.1 Qubit probe

We consider a qubit probe weakly interacting with a bosonic sample (bath).
This is a common scenario in thermometry experiments at the quantum scale,
such as in the temperature measurement of cold atomic ensembles [72, 73],
nanoresonators [17, 18], and black body radiation thermometry [74] through
two-level probes.

We start from the Hamiltonian

$\displaystyle H_{\rm tot}=$
$\displaystyle\frac{w}{2}\sigma_{z}+\sigma_{x}\int_{0}^{\Omega}\sqrt{J(x)}\left(b_{x}+b_{x}^{\dagger}\right){\rm
d}x$ $\displaystyle+\int_{0}^{\Omega}xb_{x}^{\dagger}b_{x}{\rm d}x,$ (34)

where $\omega$ is the frequency of the probe, $b_{x}$ ($b_{x}^{\dagger}$) are
annihilation (creation) operators of a bosonic mode at frequency $x$, $J(x)$
is the spectral density of the sample or bath, and $\Omega$ is a cut-off
frequency. This is a particular case of the model discussed in Sec. 2.3 with
$H=\frac{w}{2}\sigma_{z}$, $A=\sigma_{x}$,
$B=\int_{0}^{\Omega}\sqrt{J(x)}\left(b_{x}+b_{x}^{\dagger}\right){\rm d}x$ and
$H_{B}=\int_{0}^{\Omega}xb_{x}^{\dagger}b_{x}{\rm d}x$. Following the standard
derivation of the master equation in the weak coupling regime through the
Born-Markov and secular approximations [51], we obtain the Lindblad equation
(6) where

$A_{w}=\sigma_{-}\qquad A_{-w}=\sigma_{+}$ (35)

and the rates $\gamma_{w}=2\pi J(w)(1+N(w))$, $\gamma_{-w}=2\pi
J(w)N(w)<\gamma_{w}$, with $N(w)=(e^{\beta w}-1)^{-1}$ the Bose-Einstein
distribution. Furthermore, the Lamb shift takes the form
$H_{LS}=s_{w}A^{\dagger}_{w}A_{w}+s_{-w}A^{\dagger}_{-w}A_{-w},$ with
$s_{w}=-(\Delta_{T}+\Delta)$ , $s_{-w}=\Delta_{T}$ where $\Delta_{T},\Delta$
are given by (see e.g. [59]):

$\displaystyle\Delta=\mathbb{P}\left[\int_{0}^{\Omega}\frac{J(x)}{x-w}{\rm
d}x\right]$
$\displaystyle\Delta_{T}=\mathbb{P}\left[\int_{0}^{\Omega}\frac{J(x)N(x)}{x-w}{\rm
d}x\right].$ (36)

Here $\mathbb{P}$ denotes the Cauchy principle value of the integral. Note
that all the temperature dependence is encoded in $N(w)$, which enters into
$\gamma_{w}$, $\gamma_{-w}$ and $H_{LS}$. This implies
$\dot{s}_{-w}=-\dot{s}_{w}$ with

$\dot{H}_{LS}=-\dot{s}_{w}\,\sigma_{z},$ (37)

and $\dot{\gamma}_{-w}=\dot{\gamma}_{w}$. For concreteness, in what follows we
assume a spectral density of the form

$\displaystyle J(w)=gw^{\alpha},\hskip 22.76219ptw\leq\Omega$ (38)

where $\alpha$ determines the Ohmicity of the sample ($\alpha=1$ for an Ohmic
bath).

####  4.1.1 Upper bounds on the QFI

In order to compute Eq. (31), we first note that the condition (15,30)

$\dot{H}_{LS}=-\dot{s}_{w}\,\sigma_{z}=h_{\mathds{1}}\mathds{1}+h_{w}\mathinner{|{1}\rangle}\\!\\!\mathinner{\langle{1}|}+h_{-w}\mathinner{|{0}\rangle}\\!\\!\mathinner{\langle{0}|}$
(39)

implies

$\begin{cases}h_{w}=\dot{s}_{w}-x\\\ h_{-w}=-\dot{s}_{w}-x,\end{cases}$ (40)

leaving a free parameter $x=h_{\mathds{1}}$. The bound (31) then takes the
form

$\mathcal{F}_{\mathcal{E}_{\tau}|T}\leq\tau\min_{x}\left\|\begin{pmatrix}\frac{\dot{\gamma}_{w}^{2}+4(\dot{s}_{w}+x)^{2}}{\gamma_{-w}}&\\\
&\frac{\dot{\gamma}_{w}^{2}+4(\dot{s}_{w}-x)^{2}}{\gamma_{w}}\end{pmatrix}\right\|.$
(41)

In the appendix we derive the closed form expression of the bound optimized
with respect to $x$. The resulting expression is a little lengthy, but two
limiting regimes are simple to understand. In the low temperature limit $(T\to
0)$ the rate $\gamma_{-w}$ goes to zero faster than $\dot{s}_{w}^{2}$. To
cancel the diverging term $\frac{(\dot{s}_{w}+x)^{2}}{\gamma_{-w}}$ we then
choose $x=-\dot{s}_{w}$ so that that the bound reads

$\mathcal{F}_{\mathcal{E}_{\tau}|T}\leq\tau\frac{\dot{\gamma}_{w}^{2}+16\dot{s}_{w}^{2}}{\gamma_{w}}.$
(42)

In the high temperature limit $(T\to\infty)$ the values
$\gamma_{w}\approx\gamma_{-w}$ become relatively close. Then a good choice is
to set $x=0$, yielding the bound

$\mathcal{F}_{\mathcal{E}_{\tau}|T}\leq\tau\frac{\dot{\gamma}_{w}^{2}+4\dot{s}_{w}^{2}}{\gamma_{-w}}.$
(43)

Of course both bounds (42,43) remain valid for all $T$, but are the tightest
in the respective temperature limits.

In Fig. 3 we plot the upper bound (41) for an Ohmic sample (similar results
are obtained for different $\alpha$). In the temperature range of Fig. 3(a) we
observe that most of the contribution to (the upper bound of) $\mathcal{F}$
arises from the rates, whereas the Lamb shift provides a small but non-
negligible contribution. Interestingly, a completely different picture appears
in the low-temperature regime $T\ll w$ of Fig. 3(b). In this case, it is well
known that thermometry becomes exponentially hard: $\mathcal{F}\propto
e^{-w/T}$ [7, 41, 42], as it can be inferred from
$N(w)=\mathcal{O}(e^{-w/T})$. Remarkably, this is no longer the case if the
Lamb shift contribution is not neglected. We find that $\Delta_{T}$ at low
temperatures takes the form:

$\displaystyle\Delta_{T}\approx
J(w)b_{\alpha}\left(\frac{T}{w}\right)^{1+\alpha},\hskip 28.45274ptT\ll w,$
(44)

where we have defined $b_{\alpha}\equiv f_{\alpha}\Gamma(\alpha+1)$ with
$\Gamma(\alpha)$ the Gamma function and $f_{\alpha}=\mathcal{O}(1)$. From (42)
using $\gamma_{w}\approx 2\pi J(w)$ we obtain the upper bound

$\displaystyle\mathcal{F}_{\mathcal{E}_{\tau|T}}\leq\tau\frac{8J(w)(1+\alpha)^{2}b_{\alpha}^{2}}{\pi
w^{2}}\left(\frac{T}{w}\right)^{2\alpha}\quad\mathrm{for}\quad T\ll w.$ (45)

Notably, for a flat spectral density ($\alpha=0$), the QFI can in principle
remain constant with $T\rightarrow 0$, which is not in contradiction with the
fundamental bounds derived in Ref. [43, 27]. In Sec. 4.1.4 we will discuss the
saturability of this bound with explicit measurement schemes.

Figure 3: Plot of the upper bounds of $F$ in Eq. (41) (in green). The
different lines distinguish the contributions coming from either
$\dot{\gamma}_{w}$ (rates, in blue) or $\dot{s}_{w}$ (Lamb shift, in orange).
For both Figures, we take $w=1$ and cutoff frequency $\Omega=5$. Figure (a)
shows the regime $T/w=\mathcal{O}(1)$, where the contribution from the rates
is the most relevant one, whereas Fig. (b) focuses on the low-temperature
regime $T/w\ll 1$, where the Lamb shit contribution highly dominates.

####  4.1.2 Optimal finite-time thermometry through a qubit probe

We first consider the scenario where control over $w$ is possible, so that the
probe can be engineered to perform at the maximum QFI, as in Fig. 3 (a). In
this case, the Lamb shift plays a secondary role, and we neglect it in this
subsection666Note that accounting for the Lamb shift can only make the
protocol better. At the same time, the fast measure-and-prepare strategy is
completely insensitive to the presence of a Lamb shift. Hence, exploiting the
Lamb shift requires finite times between subsequent measurements of the
probe.. In this case, we can use the bound of Eq. (17), which becomes
$\mathcal{F}_{\mathcal{E}_{\tau|T}}\leq\tau\frac{\dot{\gamma}_{w}^{2}}{\gamma_{-w}}$
and takes the explicit form

$\displaystyle\mathcal{F}_{\mathcal{E}_{\tau|T}}\leq\tau\gamma
N(w)^{3}e^{2w\beta}w^{2+\alpha}\frac{1}{T^{4}}.$ (46)

As discussed above, it can be saturated with the fast measurement-and-prepare
strategy where the probe is continuously measured and reinitialized in the
ground state. The optimal frequency $w$, which depends on both $\beta$ and
$\alpha$, can be found numerically by maximising the right hand side of the
inequality (46), saturating (28) for this set-up. Setting $\alpha=3$ in (46),
we recover the results of Ref. [8], where the performance of the ground-state
was compared to other suboptimal states (see in particular Fig. 2 of [8]) and
to strategies where the probe’s state is not reinitialised [21, 3].

####  4.1.3 Optimal finite-time estimation of bath Ohmicity through a qubit
probe

One strength of our framework is that it allows one to obtain similar results
beyond temperature estimation. The determination of spectral properties of
mesoscopic environments is a highly non-trivial task, and realistic samples
can often become non-Ohmic [75, 76]. Inspired by such results, we consider the
estimation of the Ohmicity $\alpha\geq 0$ of a bosonic environment with rates
$\gamma_{w}=\gamma w^{\alpha}N(w)$, $\gamma_{-w}=\gamma w^{\alpha}(1+N(w))$
and $N(w)=(e^{\beta w}-1)^{-1}$. In this case, the bound of Eq. (17) takes the
simple form

$\displaystyle\mathcal{F}_{\mathcal{E}_{\tau|\alpha}}\leq\tau(\ln
w)^{2}\gamma_{-w}.$ (47)

Again, this bound can be saturated by the fast measurement-and-prepare
strategy.

####  4.1.4 Low temperature regime: Lamb shift thermometry

We have already noticed the potential importance of the Lamb shift
contribution to the bound in the low-temperature regime through the upper
bound shown in Fig. 3 (b). This is arguably the most challenging regime for
thermometry, and several strategies for equilibrium thermometry based on
strong sample-probe coupling have been suggested to avoid the exponential
decaying QFI [41, 42, 43, 27]. We now sketch a new non-equilibrium approach
that works in the weak coupling regime, and instead exploits the temperature
dependence of the Lamb shift via a coherent evolution of the probe.

We start by explicitly solving the qubit master equation. In the limit $T\ll
w$ the rate $\gamma_{-w}\to 0$, so that the jumps induced by the operator
$A_{-w}=\sigma_{+}$ can be neglected. Furthermore, in this regime
$\dot{\gamma}_{w}\ll\dot{s}_{w}$ as discussed around Eq. (LABEL:eq:low_T)
regime, and the contribution of $\dot{\gamma}_{w}$ to the QFI can also be
neglected. The qubit master equation then becomes

$\begin{split}\frac{\differential}{\differential
t}\rho&=-\mathrm{i}\big{[}\frac{1}{2}w\,\sigma_{z}+H_{LS},\rho\big{]}\\\
&+\gamma_{w}\left(\sigma_{-}\rho\,\sigma_{+}-\frac{1}{2}\big{\\{}\mathinner{|{1}\rangle}\\!\\!\mathinner{\langle{1}|},\rho\big{\\}}\right).\end{split}$
(48)

The Lamb shift contribution to the Hamiltonian reads
$H_{LS}=\begin{pmatrix}\Delta_{T}+\frac{1}{2}\Delta&\\\
&-\Delta_{T}-\frac{1}{2}\Delta\end{pmatrix}$, where we shifted the energy
scale to $\tr H_{LS}=0$. The solution of the master equation is
straightforward, and can be found in Appendix 8.2. From Eq. (48) it is clear
that we are dealing here with the problem of frequency estimation in presence
of spontaneous decay noise ($T_{1}$-relaxation).

We consider three different sensing strategies, described in appendix 8.2 in
detail. The simplest strategy (i) consists in preparing the qubit in the state
$\ket{\Psi_{0}}=\sqrt{1-a}\ket{0}+\sqrt{a}\ket{1}$, letting it evolve for some
finite time $t$, after which the qubit is measured and the procedure is
repeated. For the second strategy (ii) we introduce an auxiliary qubit, which
is not in contact with the sample. The probe and auxiliary qubits are
repeatedly prepared in the state
$\ket{\Psi_{0}}=\sqrt{1-a}\ket{00}+\sqrt{a}\ket{11}$, evolve freely, and are
measured after some time $t$. The entanglement between the two qubits allows
the final measurement to know if a jump $\sigma_{-}$ occurred during the
evolution (such a jump projects the system to the state $\ket{01}$ changing
the parity). The third strategy (iii) requires some continuous control on top
of entanglement with the auxiliary qubit. Here, the qubits are again prepared
in the state of the form $\ket{\Psi_{0}}=\sqrt{1-a}\ket{00}+\sqrt{a}\ket{11}$,
but the parity of the pair is monitored continuously, with repeating the
measurement described by the POVM
$\\{\Pi_{\checkmark}=\mathinner{|{00}\rangle}\\!\\!\mathinner{\langle{00}|}+\mathinner{|{11}\rangle}\\!\\!\mathinner{\langle{11}|},\Pi_{\crossproduct}=\mathds{1}-\Pi_{\checkmark}\\}$
as often as possible. If the error outcome ”$\crossproduct$” occurs the qubit
pair is reinitialised. Otherwise, if no errors are observed for some time $t$
the system is measured and reinitialised. One notes that the coherent part of
the dynamics $-\mathrm{i}[\sigma_{z}\otimes\mathds{1},\rho]$ does note change
the parity of the qubit pair and is unperturbed by the measurements
$\\{\Pi_{\checkmark},\Pi_{\crossproduct}\\}$. The strategy (i) describes the
usual Ramsey interferometer. The strategies (ii) and (iii) build on (i) by
implementing the error-detection schemes considered in [37]. After long enough
running time $\tau\gg t$ all three strategies lead to the QFI of the form

$\mathcal{F}_{\mathcal{E}_{\tau}}=r\,\tau\frac{\dot{s}_{w}^{2}}{\gamma_{w}},$
(49)

with different constants $r$. Optimizing the parameters $a$ and $t$ we find
the values of $r$ given in Table 1. These values are interesting to compare
with the upper-bound of Eq. (42)
$\mathcal{F}_{\mathcal{E}_{\tau}}=16\,\tau\frac{\dot{s}_{w}^{2}}{\gamma_{w}}$.
In the low temperature regime, the $\mathcal{F}_{\mathcal{E}_{\tau}}$ hence
also scales as $T^{2\alpha}$, as seen from Eq. (42), which can be compared to
similar polynomial scalings at low $T$ obtained in the literature via strong
coupling [41, 42, 43]. These results illustrate the potential of quantum
coherent control and entanglement for low-temperature thermometry.

strategy | description | $r$ | $a$
---|---|---|---
(i) | Ramsey interferometer | 1.5 | 0.5
(ii) | +entanglement with ancilla | 2.47 | 0.68
(ii) | \+ continuous control |  $\approx$ 4.16 | 0.83
Table 1: The performance of the thee low-temperature Lamb shift thermometers
with a qubit probe, described in the text. In all cases the QFI is given by
$\mathcal{F}_{\mathcal{E}_{\tau}}=r\,\tau\frac{\dot{s}_{w}^{2}}{\gamma_{w}}$.
Initially the system is prepared in the state
$\ket{\Psi_{0}}=\sqrt{1-a}\ket{0}+\sqrt{a}\ket{1}$ for (i), and
$\ket{\Psi_{0}}=\sqrt{1-a}\ket{00}+\sqrt{a}\ket{11}$ for (ii) and (iii), which
involve an auxiliary qubit.

###  4.2 Collective thermalisation

We now consider a $N$-qubit probe, which is coupled collectively to a sample.
The interaction Hamiltonian takes the form

$\displaystyle H_{I}=J_{x}\otimes B$ (50)

with $J_{x}=\sum_{j}\sigma_{x}^{(j)}$, and $B$ is an unspecified operator
acting on the bath. This type of collective coupling leads to the well known
phenomena of sub/superradiance [44], and has received renewed interest due to
the possibility of realising it in several physical platforms [77]. It is also
worth pointing out that equilibrium thermometry with this collective coupling
has been recently studied in [45]. We now study it in the context of finite-
time non-equilibrium thermometry.

First of all, we note that the fundamental upper bound Eq. (28), obtained
under Born-Markov and secular approximations, yields

$\displaystyle\mathcal{F}_{\mathcal{E}_{\tau|T}}\leq\tau\left(\max_{\omega}\frac{\dot{\gamma}_{-\omega}^{2}}{\gamma_{-\omega}}\right)N^{2}.$
(51)

which holds independently of the specific local Hamiltonian of the probe and
sample (note that the $\gamma_{w}$’s depend on the specific sample into
consideration). Hence, the coupling (50) in principle enables a quadratic
scaling with $N$, as it may be expected from the phenomenon of superradiance.
However, saturating (51) in principle requires engineering the $N$-qubit probe
Hamiltonian of Eq. (29) discussed in Sec. 3.4, i.e the probe needs to behave
as an effective two level system (with a $2^{N-1}$ degeneracy of the upper-
level). This corresponds to an $N$-body interaction between the qubits, and is
a priori very hard to achieve experimentally. In what follows we consider
structurally simpler probes.

####  4.2.1 Non-interacting qubits probe

We start by considering $N$ independent qubits, i.e $H=wJ_{z}$ with
$J_{z}=\frac{1}{2}\sum_{j}\sigma^{(j)}_{z}$ and a tunable frequency $w$. The
eigenstates of $H$ can be conveniently expressed in the collective spin basis.
Here, it will suffice to focus on the set of symmetric states (Dicke
states)777associated to the maximal value
$\frac{N}{2}\left(\frac{N}{2}+1\right)$ of the total spin $\bm{J}^{2}$.

$\displaystyle\ket{\psi_{n}}=\frac{J_{+}}{\sqrt{\Gamma_{n}}}\ket{\psi_{n-1}},\hskip
22.76219pt\Gamma_{n}\equiv n(N+1-n)$ $\displaystyle
H\ket{\psi_{n}}=w\,(n-\frac{N}{2})\ket{\psi_{n}}$ (52)

with $n=0,...,N$, and where $\ket{\psi_{0}}=\ket{0}^{\otimes N}$ is the ground
state and $J^{+}=\sum_{j}\sigma^{(j)}_{+}$ (where
$\sigma_{+}=\outerproduct{1}{0}$). Constraining to the states (4.2.1) is
justified because (i) the dynamics do not mix the subspace spanned by
$\\{\ket{\psi_{n}}\\}$ with the rest of the Hilbert space, and (ii) the states
(4.2.1) have the fastest decay rate, and hence are the most useful for finite-
time thermometry. In this subspace, there are two jump operators in the master
equation reading
$A_{w}=\sum_{n=0}^{N-1}\outerproduct{\psi_{n}}{\psi_{n}}J_{x}\outerproduct{\psi_{n+1}}{\psi_{n+1}}=\sum_{n=0}^{N-1}\sqrt{\Gamma_{n+1}}\outerproduct{\psi_{n}}{\psi_{n+1}}=J_{-}$
for the transition $+w$, and $A_{-w}=A_{w}^{\dagger}=J_{+}$ for the transition
$-w$ (here $J_{\pm}$ operators are restricted to subspace of symmetric
states). Plugging these expressions in (17), we obtain

$\displaystyle\frac{\mathcal{F}_{\mathcal{E}_{\tau|T}}}{\tau}$
$\displaystyle\leq\left\|\sum_{n=0}^{N}\outerproduct{\psi_{n}}{\psi_{n}}\left[\Gamma_{n}\frac{\dot{\gamma}_{w}^{2}}{\gamma_{w}}+\Gamma_{n+1}\frac{\dot{\gamma}_{-w}^{2}}{\gamma_{-w}}\right]\right\|$
$\displaystyle=\frac{N}{2}\left(\frac{N}{2}+1\right)\left(\frac{\dot{\gamma}_{w}^{2}}{\gamma_{w}}+\frac{\dot{\gamma}_{-w}^{2}}{\gamma_{-w}}\right),$
(53)

where the operator norm is saturated for $n=N/2$ for which
$\Gamma_{\frac{N}{2}}=\Gamma_{\frac{N}{2}+1}=\frac{N}{2}\left(\frac{N}{2}+1\right)$,
for simplicity we assume an even $N$ here and in the rest of the section.
Compared to the general upper bound (51), we also observe a quadratic scaling
with $N$ but with a worse prefactor. In particular, since
$0\leq\dot{\gamma}_{w}^{2}/\gamma_{w}\leq\dot{\gamma}_{-w}^{2}/\gamma_{-w}$,
we notice that for large $N$ one loses a factor laying between $1/2$ and $1/4$
as compared to the general upper bound (51). For example, for a bosonic
reservoir with rates

$\begin{split}&\gamma_{w}=gw^{\alpha}N(w)\\\ &\gamma_{-w}=gw^{\alpha}(1\pm
N(w))\end{split}$ (54)

and $N(w)=(e^{\beta w}-1)^{-1}$, it is satisfied that
$\dot{\gamma}_{-w}^{2}/\gamma_{-w}=e^{\beta
w}\dot{\gamma}_{w}^{2}/\gamma_{w}$, so that the ratio between (4.2.1) and (51)
becomes $(1+e^{-\beta w})/4$ at leading order in $N$. It is remarkable that
such a good performance can be obtained with arguably the simplest probe
consisting of non-interacting qubits with $H=wJ_{z}$.

Figure 4: Classical Fisher information for the fast measure-and-prepare
protocol where the state $\ket{\psi_{N/2}}$ evolves for a finite amount of
time $\delta t$ before performing the measurement. The superradiant advantage
$\mathcal{I}\propto N^{2}$ requires $\delta t\propto 1/N^{2}$.

####  4.2.2 The effect of measurement frequency

The bound (4.2.1) can be saturated by the fast measure-and-prepare strategy,
where the probe is continuously reinitialised it in the state
$\ket{\psi_{N/2}}$. In practice, the frequency of the measurements can be
limited and it is relevant to consider suboptimal strategies where the state
evolves for a finite time $\delta t$ before being measured. For a given
$\delta t$ and total sensing time $\tau$, we then consider the fast measure-
and-prepare strategy with a finite measurement frequency, described below.

The probe is initialised in the state $\ket{\psi_{N/2}}$, and is left to
interact with the sample for a time $\delta t$ according to the master
equation

$\displaystyle\frac{\differential}{\differential t}\rho=$
$\displaystyle-i[H,\rho]+\gamma_{w}\left(J_{-}\rho
J_{+}-\frac{1}{2}\\{J_{+}J_{-},\rho\\}\right)$
$\displaystyle+\gamma_{-w}\left(J_{+}\rho
J_{-}-\frac{1}{2}\\{J_{-}J_{+},\rho\\}\right).$ (55)

Focusing on the superradiant subspace, it is convenient to express this as a
stochastic evolution of the probabilities
$p_{n}\equiv\bra{\psi_{n}}\rho\ket{\psi_{n}}$ to occupy each state, given by

$\displaystyle\frac{\differential}{\differential t}p_{m}=$ $\displaystyle-
p_{m}\left(\gamma_{w}\Gamma_{m}+\gamma_{-w}\Gamma_{m+1}\right)$
$\displaystyle+\gamma_{w}p_{m+1}\Gamma_{m+1}+\gamma_{-w}p_{m-1}\Gamma_{m}.$
(56)

After a time $\delta t$, the resulting state $p_{j}=p_{j}(\delta t)$ is
measured, providing the classical Fisher information $\mathcal{I}_{\delta
t}=\sum_{j}\dot{p}_{j}^{2}/p_{j}$.

This process is repeated for $k$ times until $k\delta t=\tau$, yielding a
total Fisher information of $\mathcal{I}_{\rm tot}=k\mathcal{I}_{\delta
t}=\tau\frac{\mathcal{I}_{\delta t}}{\delta t}$. In Fig. 4, we plot
$\mathcal{I}_{\rm tot}/\tau={\mathcal{I}_{\delta t}}/{\delta t}$ as a function
of $\delta t$ for different $N$ and for a bosonic sample with Ohmic spectral
density, $\alpha=1$ in Eq. (54). While Fig. 4 clearly shows the quadratic
advantage for $g\,\delta t\rightarrow 0$, it is also observed that this
requires $g\,\delta t<1/N^{2}$. When this condition is not met, the
superradiant advantage is rapidly lost, and in fact non-superradiant states
($N=1$, prepared in the ground state) become better for $g\delta t\geq 0.1$.

Figure 5: Classical Fisher information of the measure-and-prepare scheme in
the limit $\delta t\rightarrow 0$ for the generalised Hamiltonian $H(b)=\omega
J_{z}/2+bJ_{z}^{2}$. In (a), we compare a scheme where the probe is prepared
in the optimal state with one where it is prepared in the groun state. In this
figure, we vary $b$ while keeping $\omega=1$, $T=1$ and $N=50$, and compare
the results to those obtained. In (b) we instead vary $N$ and different curves
correspond to different values of $b$’s.

Summarising, although the simplest probe allows for a quadratic scaling of the
Fisher information with $N$, this quantum advantage faces two main challenges.
(i) The probe needs to be repeatedly prepared in the state $\ket{\psi_{N/2}}$,
which is highly non-trivial in practice (see e.g. [78]). (ii) As shown in Fig.
4, the time $\delta t$ after which each measurement and state preparation must
be repeated decreases as $\delta t\propto 1/N^{2}$, which becomes increasingly
difficult with increasing probe size $N$.

####  4.2.3 Interacting qubits probe

We will now show that a potential route to overcome both challenges is offered
by engineering slightly more complicated probes. In particular, we now
consider $N$ qubit probes subject to two-body interactions
$\sigma_{z}^{(i)}\sigma_{z}^{(j)}$, the probe Hamiltonian becoming

$H=w\,J_{z}+b\,J_{z}^{2},$ (57)

for tunable parameters $w$ and $b$. This is arguably the simplest probe
Hamiltonian after the non-interacting case (b=0) considered above, and such
two-body interaction are for example implemented in spin squeezing experiments
(one axis twisting [71]). The additional term $bJ_{z}^{2}$ keeps the
eigenvectors $\ket{\psi_{n}}$ unmodified, but changes the spectrum of the
Hamiltonian $H\ket{\psi_{n}}=e_{n}\ket{\psi_{n}}$, which is now given by

$\displaystyle
e_{n}=w\left(n-\frac{N}{2}\right)+b\left(n-\frac{N}{2}\right)^{2}.$ (58)

By deriving the energy $e_{n}$ with respect to $n$ to find it’s local minimum,
one concludes that the ground state $\ket{\psi_{n^{*}}}$ corresponds to

$n^{*}=\text{argmin}_{n}e_{n}\in\left\\{\left\lfloor\frac{N}{2}-\frac{w}{2b}\right\rfloor,\left\lfloor\frac{N}{2}-\frac{w}{2b}+1\right\rfloor\right\\},$
(59)

depending on the parameter values. It is simple to see that $n^{*}$ approaches
$N/2$ as $b$ is increased.

In particular, for $b>w$ the state $\ket{\psi_{N/2}}$ with $e_{N/2}=0$ becomes
the ground state, and the energy gaps to the next levels read

$\begin{split}\Delta&=e_{N/2}-e_{N/2-1}=-(b-w)\\\
\Delta^{\prime}&=e_{N/2}-e_{N/2+1}=-(b+w).\\\ \end{split}$ (60)

Our autonomous strategy of Sec. 3.5, then gives

$\begin{split}\mathcal{I}_{\text{tot}}&=\tau\left(\Gamma_{N/2}\frac{\dot{\gamma}_{\Delta}^{2}}{\gamma_{\Delta}}+\Gamma_{N/2+1}\frac{\dot{\gamma}_{\Delta^{\prime}}^{2}}{\gamma_{\Delta^{\prime}}}\right)\\\
&=\tau\,\frac{N}{2}\left(\frac{N}{2}+1\right)\left(\frac{\dot{\gamma}_{\Delta}^{2}}{\gamma_{\Delta}}+\frac{\dot{\gamma}_{\Delta^{\prime}}^{2}}{\gamma_{\Delta^{\prime}}}\right).\end{split}$
(61)

By choosing $w=0$ and maximizing over $b$ we get the optimal Fisher
information of

$\mathcal{I}_{\text{tot}}=\frac{1}{2}\tau\left(N^{2}+2N\right)\max_{b}\frac{\dot{\gamma}_{-b}^{2}}{\gamma_{-b}},$
(62)

which is only a factor $1/2$ lower as compared to the general upper-bound of
Eq. (51). Thus, engineering a quadratic Hamiltonian allows one to attain half
of the optimal QFI with a simple autonomous strategy: the probe is measured
and reinitialized to the ground state via its coupling to the cold bath (Fig.
2).

In practice, it might be difficult to reach a regime where $b>w$, while
getting some superradiance effect does not require going to the state
$\ket{\psi_{N/2}}$. For this reason we also numerically study the regime of
lower $b$. In Fig. 5 we plot $\mathcal{I}_{\rm tot}/\tau$ given a bosonic
sample and setting $w=1$ and $N=50$. Interestingly, low values of $b\approx
0.05$ provide the best performance, which is close to that obtained with
$\ket{\psi_{N/2}}$ for $b=0$ (here $w$ is not optimized but kept fixed).
Furthermore, in Fig. 5 (b), we show that this conclusion remains true as $N$
is increased.

In conclusion, the quadratic scaling $\mathcal{I}_{\rm tot}\propto N^{2}$ can
in principle be realised through the autonomous scheme described in Fig. 2,
where the probe is also (collectively) coupled to a $T=0$ bath. Furthermore,
from the numerical example we picked one sees that this can be done even when
the quadratic interaction term in the Hamiltonian of Eq. (57) is lower than
the energy splitting of individual qubit by at least an order of magnitude
$b/w<5\%$. In principle, depending on the sample temperature $T$ and the size
of the probe $N$ the required ratio $b/w$ might be much lower, but we leave
this question for future work.

##  5 Outlook and conclusions

In this paper, we considered the task of estimating the temperature of a
sample by letting it interact with a probe for a finite duration. We derived
fundamental bounds on the measurement precision for a large class of processes
where the probe’s evolution is well described by a Markovian master equation.
This was achieved by considering the most general quantum control on the
probe: arbitrary control on its state and Hamiltonian, and the possibility to
realise arbitrary (adaptive, entangling) measurements. In contrast, we assumed
the sample-probe interaction to be fixed, setting a bottleneck for how fast
the temperature of the sample can be imprinted on the state of the probe. In
this sense, our results can be understood as a “causation” speed limit
relating a particular sample-probe interaction with the speed at which a
property of the sample causes a detectable change in the probe (quantified by
the Quantum Fisher Information of the probe with respect to a particular
parameter of the sample). While deriving these results, we realised that
beyond thermometry these limits are valid in a much broader context where a
physical property of a sample is estimated through its interaction with the
probe, whose evolution can be well described by a Markovian evolution. Our
results hence provide a general framework for placing fundamental limits in
finite-time sensing in open (quantum) systems.

We also discussed the attainability of these limits. Whenever the Lamb shift
term in the master equation can be neglected, we showed that they can be
saturated by a fast measure-and-prepare strategy, and developed an autonomous
implementation of this scheme (see Fig. 2). In principle, the presence of a
temperature-dependent Lamb shift can enhance the sensitivity, which however
requires moving away from the fast measurement limit to a frequency estimation
regime, that we illustrated with some examples. This becomes particularly
relevant at low temperatures, where we argued that the Lamb shift can play a
dominant role for finite-time sensing (see Fig. 3).

The results of this paper have been obtained by assuming local control on the
probe and effectively describing the thermal sample through a Markovian master
equation. This framework is of relevance for both theoretical works in non-
equilibrium thermometry [3] as well as for thermometry experiments through
quantum probes [79, 31, 40]. Moving beyond the Markovian approximation, a
challenging future endeavour is to derive similar results in the strong
coupling regime, where the probe-sample evolution is generally described as a
unitary process. Strong coupling naturally enables a faster rate of
information transfer between sample and probe, and hence appears as a
promising resource for non-equilibrium thermometry, whose potential remains
mostly unexplored (see Refs. [41, 42, 5, 43] for works exploiting strong
coupling for equilibrium thermometry). In particular, in order to generalise
the results presented here, ideally one wishes to derive fundamental bounds on
the probe’s QFI after a finite time for a given interaction and sample
Hamiltonian. In general, this appears as an impossible task when assuming full
quantum control on the probe as done here. However, one may hope that some
insights can be obtained in constrained scenarios such as in the the linear-
response regime [80, 81].

When considering microscopic models for the probe-sample interaction, in this
paper we focused on interactions of the form $A\otimes B$ (see Eq. (8)). It
hence remains an open and relevant question to investigate possibilities and
fundamental limits arising when considering more general probe-sample
interactions of the form $\sum_{j}A_{j}\otimes B_{j}$. In this sense, an
exciting question that we leave for future work is whether ballistic or
Heisenberg scaling is possible in thermometry, i.e.,
$\mathcal{F}\propto\tau^{2}$. The fact that the temperature can also be
coherently estimated through the Lamb shift as a frequency (see protocols in
Sec. 4.1) suggests that Heisenberg scaling could be possible in engineered or
exotic reservoirs.

Finally, in this article we took the Quantum/Classical Fisher Information as
figures of merit for thermometry. In the future, it will be interesting to
consider a Bayesian framework for thermometry, and to better understand the
implication of our bounds in this context.

_Acknowledgments -_ We thank Mohammad Mehboudi for inspiring discussions and
feedback. This research was funded by the Swiss National Science Foundation,
via NCCR SwissMap and Ambizione grant PZ00P2-186067.

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##  6 Diffusive scaling of the QFI

Consider the master equation $\frac{\differential\rho}{\differential
t}\rho=\mathcal{L}(\rho)$ of Eq. (6) with the Hamiltonian $H+H_{LS}$ and the
dissipate term given by

$\begin{split}\mathcal{D}(\rho)&=\sum_{\omega}\gamma_{\omega}\left(A_{\omega}\,\rho\,A^{\dagger}_{\omega}-\frac{1}{2}\\{A^{\dagger}_{\omega}A_{\omega},\rho\\}\right)\end{split}$
(63)

A Kraus representation of the infinitesimal channel $e^{\differential
t\mathcal{L}}$ is given in Eq. (12)

$\bm{K}=\begin{pmatrix}\mathds{1}-\differential
t\frac{1}{2}\sum_{\omega}\gamma_{\omega}A_{\omega}^{\dagger}A_{\omega}-\mathrm{i}\,\differential
t\,(H+H_{LS})\\\ \vdots\\\ \sqrt{\differential t\gamma_{\omega}}A_{\omega}\\\
\vdots\end{pmatrix}.$ (64)

Assuming that the jump operators are independent of the parameter, the
derivatives of the Kraus operators with respect to to the parameter read

$\dot{\bm{K}}=\begin{pmatrix}-\differential
t\left(\frac{1}{2}\sum_{\omega}\dot{\gamma}_{\omega}A_{\omega}^{\dagger}A_{\omega}+\mathrm{i}\dot{H}_{LS}\right)\\\
\vdots\\\ \frac{\sqrt{\differential
t}\dot{\gamma}_{\omega}}{2\sqrt{\gamma_{\omega}}}\,A_{\omega}\\\
\vdots\end{pmatrix}.$ (65)

Let us now compute $\bm{K}^{\dagger}(\dot{\bm{K}}+\mathrm{i}h\bm{K})$ up to
the order $O(\differential t)$ to determine if ballistic scaling of the QFI is
possible

$\begin{split}{\bm{K}}^{\dagger}(\dot{\bm{K}}+\mathrm{i}\,h\bm{K})&={\bm{K}}^{\dagger}\dot{\bm{K}}+\mathrm{i}{\bm{K}}^{\dagger}h\bm{K}\\\
\\\ &=\begin{pmatrix}\mathds{1}-\differential
t\frac{1}{2}\sum_{\omega}\gamma_{\omega}A_{\omega}^{\dagger}A_{\omega}+\mathrm{i}\,\differential
t\,(H+H_{LS})\\\ \vdots\\\ \sqrt{\differential
t\gamma_{\omega}}A_{\omega}^{\dagger}\\\
\vdots\end{pmatrix}^{T}\begin{pmatrix}\differential
t\left(\frac{1}{2}\sum_{\omega}\dot{\gamma}_{\omega}A_{\omega}^{\dagger}A_{\omega}-\mathrm{i}\dot{H}_{LS}\right)\\\
\vdots\\\ \frac{\sqrt{\differential
t}\dot{\gamma}_{\omega}}{2\sqrt{\gamma_{\omega}}}\,A_{\omega}\\\ \\\
\vdots\end{pmatrix}+\mathrm{i}{\bm{K}}^{\dagger}h\bm{K}\\\ &=-\differential
t\left(\frac{1}{2}\sum_{\omega}\dot{\gamma}_{\omega}A_{\omega}^{\dagger}A_{\omega}+\mathrm{i}\dot{H}_{LS}\right)+\differential
t\frac{1}{2}\sum_{\omega}\dot{\gamma}_{\omega}A_{\omega}^{\dagger}A_{\omega}+\mathrm{i}{\bm{K}}^{\dagger}h\bm{K}\\\
&=\mathrm{i}\left(-\differential
t\dot{H}_{LS}+{\bm{K}}^{\dagger}h\bm{K}\right).\end{split}$ (66)

The last equation falls in the general context considered in [38], from which
we know that $\|-\differential
t\dot{H}_{LS}+{\bm{K}}^{\dagger}h\bm{K}\|=O(\differential t^{3/2})$ can be
fulfilled if and only if

$\dot{H}_{LS}\in\mathrm{span}\left\\{\mathds{1},\sqrt{\gamma_{\omega}}(A_{\omega}+A_{\omega}^{\dagger}),\sqrt{\gamma_{\omega}}\mathrm{i}(A_{\omega}-A_{\omega}^{\dagger}),\sqrt{\gamma_{\omega}\gamma_{\omega^{\prime}}}(A_{\omega}^{\dagger}A_{\omega^{\prime}}+A_{\omega^{\prime}}^{\dagger}A_{\omega}),\sqrt{\gamma_{\omega}\gamma_{\omega^{\prime}}}\mathrm{i}(A_{\omega}^{\dagger}A_{\omega^{\prime}}-A_{\omega^{\prime}}^{\dagger}A_{\omega})\right\\}.$
(67)

One can see this explicitly by writing the Gauge matrix in the form
$h=\begin{pmatrix}h_{00}&\bm{h}^{\dagger}\\\ \bm{h}&\bar{h}\end{pmatrix}$ with
a complex vector $\bm{h}=(\dots h_{\omega}\dots)$ and an Hermitian matrix
$\bar{h}$, so that in the leading order

${\bm{K}}^{\dagger}h\bm{K}=h_{00}+\sum_{\omega}\sqrt{\differential
t\gamma_{\omega}}(h_{\omega}A_{\omega}+h_{\omega}^{*}A_{\omega}^{\dagger})+\sum_{\omega\omega^{\prime}}\differential
t\sqrt{\gamma_{\omega}\gamma_{\omega^{\prime}}}(\bar{h}_{\omega\omega^{\prime}}A_{\omega}^{\dagger}A_{\omega^{\prime}}+\bar{h}_{\omega\omega^{\prime}}^{*}A_{\omega^{\prime}}^{\dagger}A_{\omega}).$
(68)

Let us now add the assumption the the Lamb shift Hamiltonian is block-diagonal
$H_{LS}=\sum_{\varepsilon}\Pi_{\varepsilon}H_{LS}\Pi_{\varepsilon}$, which
brings additional structure relating the jump operators $A_{\omega}$ and
$H_{LS}$. Recall that the jump operators are off-diagonal

$A_{\omega}=\sum_{(\varepsilon,\varepsilon^{\prime})\in
P_{\omega}}\Pi_{\varepsilon}A_{\omega}\Pi_{\varepsilon}^{\prime},$ (69)

while $H_{LS}$ (and thus $\dot{H}_{LS}$) is block-diagonal

$\dot{H}_{LS}=\sum_{\varepsilon}\Pi_{\varepsilon}\dot{H}_{LS}\Pi_{\varepsilon}.$
(70)

It follows that $\dot{H}_{LS}$ is orthogonal to all the jump operators (with
respect to the Hilbert-Schmidt product). Furthermore, a product of two jump
operators for $\omega\neq\omega^{\prime}$

$A_{\omega}^{\dagger}A_{\omega^{\prime}}=\sum_{(\varepsilon,\varepsilon^{\prime})\in
P_{\omega}}\sum_{(\epsilon,\epsilon^{\prime})\in
P_{\omega^{\prime}}}\Pi_{\varepsilon^{\prime}}A_{\omega}\Pi_{\varepsilon}\,\Pi_{\epsilon}A_{\omega^{\prime}}\Pi_{\epsilon^{\prime}}=\sum_{(\varepsilon,\varepsilon^{\prime})\in
P_{\omega}}\sum_{(\epsilon,\epsilon^{\prime})\in
P_{\omega^{\prime}}}\Pi_{\varepsilon^{\prime}}A_{\omega}\Pi_{\varepsilon}\,A_{\omega^{\prime}}\Pi_{\epsilon^{\prime}}\delta_{\varepsilon\epsilon}$
(71)

is also off diagonal
$\Pi_{\epsilon}A_{\omega}^{\dagger}A_{\omega^{\prime}}\Pi_{\varepsilon}=0$,
because each pair $(\varepsilon,\varepsilon^{\prime})$ can only appear for one
$\omega$. Thus $A_{\omega}^{\dagger}A_{\omega^{\prime}}$ for
$\omega\neq\omega^{\prime}$ are also orthogonal to $\dot{H}_{LS}$. Hence the
only possibility for $\dot{H}_{LS}$ to fall in the span of the noise operators
in Eq. (67) is that

$\dot{H}_{LS}\in\mathrm{span}\left\\{\mathds{1},\gamma_{\omega}A_{\omega}^{\dagger}A_{\omega}\right\\}$
(72)

since $A_{\omega}^{\dagger}A_{\omega}$ are the only block-diagonal
contributions.

In the context of our microscopic model where the Lamb shift term the form

$\dot{H}_{LS}=\sum_{\omega}\dot{s}_{\omega}A_{\omega}^{\dagger}A_{\omega}.$
(73)

Clearly, a sufficient for the condition (72) it is sufficient that all the
rates $\gamma_{\omega}$ are nonzero whenever $\dot{s}_{\omega}$ is nonzero.

##  7 Upper-bound on the QFI

We now assume
$\dot{H}_{LS}\in\mathrm{span}\left\\{\mathds{1},\gamma_{\omega}A_{\omega}^{\dagger}A_{\omega}\right\\}$
and derive an upper-bound on the QFI. Given the assumption we can write

$\dot{H}_{LS}=h_{\mathds{1}}\mathds{1}+\sum_{\omega}h_{\omega}A_{\omega}^{\dagger}A_{\omega},$
(74)

where $h_{\omega}=0$ for $\gamma_{\omega}=0$ such that the ration
$\frac{h_{\omega}}{\gamma_{\omega}}$ is well defined. Then
$\|\bm{K}^{\dagger}(\dot{\bm{K}}+\mathrm{i}h\bm{K})\|=O(\differential
t^{3/2})$ is fullfilled by the choice $h=\begin{pmatrix}h_{0}\differential
t&\\\ &\mathrm{diag}[h_{\omega}/\gamma_{\omega}]\end{pmatrix}$. To set an
upper-bound on QFI we have to compute
$\|(\dot{{\bm{K}}}^{\dagger}-\mathrm{i}\bm{K}h)(\dot{\bm{K}}+\mathrm{i}\,h\bm{K})\|$.
To do so, we write to the leading orders

$\begin{split}(\dot{\bm{K}}+\mathrm{i}\,h\bm{K})&=\begin{pmatrix}-\differential
t\left(\sum_{\omega}\frac{1}{2}\dot{\gamma_{\omega}}A^{\dagger}_{\omega}A_{\omega}+\mathrm{i}\dot{H}_{LS}\right)\\\
\vdots\\\ \sqrt{\differential
t}\frac{\dot{\gamma}_{\omega}}{2\sqrt{\gamma_{\omega}}}A_{\omega}\\\
\vdots\end{pmatrix}+\begin{pmatrix}\mathrm{i}\,h_{0}\differential
t\,\mathds{1}\\\ \vdots\\\
\mathrm{i}\frac{h_{\omega}}{\gamma_{\omega}}\sqrt{\differential
t\gamma_{\omega}}A_{\omega}\\\ \vdots\end{pmatrix}\\\
&=\begin{pmatrix}-\differential
t\left(\sum_{\omega}(\frac{1}{2}\dot{\gamma_{\omega}}+\mathrm{i}h_{\omega}\gamma_{\omega})A^{\dagger}_{\omega}A_{\omega}\right)\\\
\vdots\\\ \sqrt{\differential
t}\frac{\dot{\gamma}_{\omega}+2\mathrm{i}h_{\omega}}{2\sqrt{\gamma_{\omega}}}A_{\omega}\\\
\vdots\end{pmatrix}\end{split}.$ (75)

Which makes it easy to compute

$(\dot{{\bm{K}}}^{\dagger}-\mathrm{i}\bm{K}h)(\dot{\bm{K}}+\mathrm{i}\,h\bm{K})=\differential
t\sum_{\omega}\left|\frac{\dot{\gamma}_{\omega}+2\mathrm{i}h_{\omega}}{2\sqrt{\gamma_{\omega}}}\right|^{2}A_{\omega}^{\dagger}A_{\omega}=\frac{\differential
t}{4}\sum_{\omega}\frac{\dot{\gamma}_{\omega}^{2}+4h^{2}_{\omega}}{\gamma_{\omega}}A_{\omega}^{\dagger}A_{\omega}.$
(76)

Taking the norm gives

$\left\|(\dot{{\bm{K}}}^{\dagger}-\mathrm{i}\bm{K}h)(\dot{\bm{K}}+\mathrm{i}\,h\bm{K})\right\|=\frac{\differential
t}{4}\left\|\sum_{\omega}\frac{\dot{\gamma}_{\omega}^{2}+4h_{\omega}^{2}}{\gamma_{\omega}}A_{\omega}^{\dagger}A_{\omega}\right\|.$
(77)

Using Eq. (11), we obtain a bound on the QFI

$\mathcal{F}_{\mathcal{E}_{\tau|T}}\leq\tau\left\|\sum_{\omega}\frac{\dot{\gamma}_{\omega}^{2}+4h^{2}_{\omega}}{\gamma_{\omega}}A_{\omega}^{\dagger}A_{\omega}\right\|$
(78)

For $H_{LS}=\sum_{\omega}\dot{s}_{\omega}A_{\omega}^{\dagger}A_{\omega}$
suggests a particularly simple choice of the coefficients
$h_{\omega}=\dot{s}_{\omega}$, leaging to

$\mathcal{F}_{\mathcal{E}_{\tau|T}}\leq\tau\left\|\sum_{\omega}\frac{\dot{\gamma}_{\omega}^{2}+4\dot{s}_{\omega}}{\gamma_{\omega}}A_{\omega}^{\dagger}A_{\omega}\right\|$
(79)

Finally, in the in absence of temperature dependent Lamb shift
$\dot{H}_{LS}=0$, or for measurement schemes that are insensitive to it, we
get the bound

$\mathcal{F}_{\mathcal{E}_{\tau|T}}\leq\tau\left\|\sum_{\omega}\frac{\dot{\gamma}_{\omega}^{2}}{\gamma_{\omega}}A_{\omega}^{\dagger}A_{\omega}\right\|.$
(80)

##  8 App: Qubit probe example

###  8.1 Derivation of the optimal bound

To find the value of the parameter $x$ that gives the tightest bound in Eq.
(41), we write it as

$\mathcal{F}_{\mathcal{E}_{\tau}|T}\leq\tau\,\min_{x}\text{max}\big{\\{}f_{-}(x)=\frac{\dot{\gamma}_{w}^{2}+4(\dot{s}_{w}+x)^{2}}{\gamma_{-w}},\,f_{+}(x)=\frac{\dot{\gamma}_{w}^{2}+4(\dot{s}_{w}-x)^{2}}{\gamma_{w}}\Big{\\}}.$
(81)

The two function $f_{\pm}(x)$ are ”upwards” parabolas centered at
$x=\pm\dot{s}_{w}$, with the minimal values
$f_{-}(-\dot{s}_{w})=\frac{\dot{\gamma}_{w}^{2}}{\gamma_{-w}}\geq
f_{+}(\dot{s}_{w})=\frac{\dot{\gamma}_{w}^{2}}{\gamma_{w}}$. To solve the
minmax we distinguish two different regimes.

On the one hand (i), if the minimal value of $f_{-}(x)$, attained at
$x=-\dot{s}_{w}$, is larger than $f_{+}(-\dot{s}_{w})$ the best upper bound is
clearly given by $f_{-}(-\dot{s}_{w})$.

On the other (ii) if $f_{-}(-\dot{s}_{w})<f_{+}(-\dot{s}_{w})$, the best bound
is attained when the two functions are equal $f_{-}(x)=f_{+}(x)$ for
$x\in[-\dot{s}_{w},\dot{s}_{w}]$. Taking $\dot{s}_{w}\geq 0$ without loss of
generality (otherwise one can flip its sign), such an $x$ is straightforward
to find

$x_{*}(w)=-\frac{4\dot{s}_{w}(\gamma_{w}+\gamma_{-w})-\sqrt{64\dot{s}_{w}^{2}\gamma_{w}\gamma_{-w}-\dot{\gamma}_{w}^{2}(\gamma_{w}-\gamma_{-w})^{2}}}{\gamma_{w}-\gamma_{-w}}.$
(82)

And implies the optimal upper-bound
$f_{+}(x_{*}(w))=f_{-}(x_{*}(w))=\frac{16\dot{s}_{w}}{\gamma_{w}-\gamma_{-w}}(-x_{*}(w)).$
In summary, combining (i) and (ii) we find the form of the tightest upper-
bound

$\mathcal{F}_{\mathcal{E}_{\tau}|T}\leq\tau\begin{cases}\frac{16\dot{s}_{w}(-x_{*}(w))}{\gamma_{w}-\gamma_{-w}}&\frac{\dot{\gamma}_{w}^{2}}{\gamma_{-w}}<\frac{\dot{\gamma}_{w}^{2}+16\dot{s}^{2}}{\gamma_{w}}\\\
\frac{\dot{\gamma}_{w}^{2}}{\gamma_{-w}}&\frac{\dot{\gamma}_{w}^{2}}{\gamma_{-w}}\geq\frac{\dot{\gamma}_{w}^{2}+16\dot{s}^{2}}{\gamma_{w}}\end{cases}$
(83)

with the expression of $x_{*}(w)$ given in Eq. (82).

###  8.2 Solving the master equation

In this section we solve the master equation of the qubit probe of Sec.4.1 in
the low temperature regime $T\ll w$. We will assume a pure initial state
$\ket{\Psi_{0}}$. The master equation takes the form

$\frac{\differential}{\differential
t}\rho=-\mathrm{i}\big{[}\frac{1}{2}\tilde{w}\,\sigma_{z},\rho\big{]}+\gamma_{w}\left(\underbrace{\sigma_{-}\rho\,\sigma_{+}-\frac{1}{2}\big{\\{}\mathinner{|{1}\rangle}\\!\\!\mathinner{\langle{1}|},\rho\big{\\}}}_{=\mathcal{D}(\rho)}\right).$
(84)

with $\tilde{w}=w+2\Delta_{T}+\Delta$. One notes that in this equation the
unitary dynamics commute with the dissipation
$[\sigma_{z},\mathcal{D}(\rho)]=\mathcal{D}\left([\sigma_{z},\rho]\right)$.
Hence, we can decompose the evolution as ”pure” dissipation
$\frac{\differential}{\differential t}\rho=\gamma_{w}\mathcal{D}(\rho)$ acting
on the state for time $t$ followed by unitary evolution
$\frac{\differential}{\differential
t}\rho=-\frac{\mathrm{i}}{2}\tilde{w}[\sigma_{z},\rho]$ of the same duration.
The unitary dynamics results in the unitary operator

$U_{t}=\exp(-\frac{\mathrm{i}}{2}t\,\tilde{w}\,\sigma_{z}).$ (85)

The stochastic dynamics projects the system on the ground state
$\mathinner{|{0}\rangle}\\!\\!\mathinner{\langle{0}|}\propto\sigma_{-}\rho(t)\sigma_{+}$
with the rate
$\tr\mathinner{|{1}\rangle}\\!\\!\mathinner{\langle{1}|}\rho(t)$. At each time
it gives result to two possible branches, either the jump $\sigma_{-}$ has
already occurred and the system remains in
$\rho_{\text{j}}(t)=p_{\text{j}}(t)\mathinner{|{0}\rangle}\\!\\!\mathinner{\langle{0}|}$,
or it has not yet occurred $\rho_{\text{nj}}(t)$. In the no-jump branch the
dynamics is given by $\frac{\differential}{\differential
t}\rho_{\text{nj}}(t)=-\frac{\gamma_{w}}{2}\\{\mathinner{|{1}\rangle}\\!\\!\mathinner{\langle{1}|},\rho_{\text{nj}}(t)\\}$,
and is straightforward to solve

$\rho_{\text{nj}}(t)=\mathinner{|{\Psi_{\text{nj}}(t)}\rangle}\\!\\!\mathinner{\langle{\Psi_{\text{nj}}(t)}|}\qquad\ket{\Psi_{\text{nj}}(t)}=e^{-\frac{t}{2}\gamma_{\omega}\mathinner{|{1}\rangle}\\!\\!\mathinner{\langle{1}|}}\ket{\Psi_{0}}$
(86)

The probability that a jump has occurred before $t$ is thus

$p_{\mathrm{j}}(t)=1-\tr\rho_{\text{nj}}(t)=1-\tr
e^{-\gamma_{w}t\mathinner{|{1}\rangle}\\!\\!\mathinner{\langle{1}|}}\rho_{0}.$
(87)

Plugging in the coherent evolution given by $U_{t}$, the state at time $t$ is

$\rho_{t}=U_{t}\rho_{\mathcal{D}}(t)U_{t}^{\dagger}\qquad\rho_{\mathcal{D}}(t)=p_{\text{j}}(t)\mathinner{|{0}\rangle}\\!\\!\mathinner{\langle{0}|}+\rho_{\text{nj}}(t)$
(88)

We compute the QFI of the state as $\mathcal{F}_{\rho_{t}}=\tr\dot{\rho}_{t}L$
where $L$ is the symmetric logarithmic derivative
$\frac{1}{2}\\{\rho_{t},L\\}=\dot{\rho}_{t}$. In the frame rotating with
$U_{t}$ one has

$\rho_{t}=\rho_{\mathcal{D}}(t)\qquad\dot{\rho}_{t}=-\frac{\mathrm{i}}{2}t\dot{\tilde{w}}[\sigma_{z},\rho_{\mathcal{D}}(t)],$
(89)

where $\dot{\tilde{w}}=2\dot{\Delta}_{T}=2\dot{s}_{w}$.

###  8.3 Ramsey scheme

Let us now consider a measurement scheme where the probe is prepared is
subject to free evolution in Eq. (84) for time $t$, after which it is mearused
and reinitialised. Without loss of generality we assume that the probe is
prepare in a state in th x-z plane of the Bloch sphere
$\ket{\Psi_{0}}=\sqrt{1-a}\ket{0}+\sqrt{a}\ket{1}$, such that

$\rho_{0}=\begin{pmatrix}1-a&\sqrt{a-a^{2}}\\\
\sqrt{a-a^{2}}&a\end{pmatrix}=(\sqrt{1-a}\ket{0}+\sqrt{a}\ket{1})(\sqrt{1-a}\bra{0}+\sqrt{a}\bra{1})$
(90)

with $a\in[0,1]$. From Eqs. (88-89) we get

$\begin{split}\rho_{t}=\begin{pmatrix}1-e^{-t\gamma_{w}}a&e^{-\frac{\gamma_{w}}{2}t}\sqrt{a-a^{2}}\\\
e^{-\frac{\gamma_{w}}{2}t}\sqrt{a-a^{2}}&e^{-{\gamma_{w}}t}a\end{pmatrix}\qquad\dot{\rho}_{t}=2\dot{s}_{w}te^{-\frac{\gamma_{w}}{2}t}\sqrt{a-a^{2}}\sigma_{y}.\end{split}$
(91)

With these expressions one verifies that the SLD is given by
$L=\dot{s}_{w}te^{-\frac{\gamma_{w}}{2}t}\sqrt{a-a^{2}}\sigma_{x}$ and the QFI
$\mathcal{F}_{\rho_{t}}=\tr\dot{\rho}_{t}L=16{\dot{s}}_{w}^{2}t^{2}e^{-\gamma_{w}t}(a-a^{2}).$
The QFI is maximized to

$\mathcal{F}_{\rho_{t}}=4{\dot{s}}_{w}^{2}t^{2}e^{-\gamma_{w}t}$ (92)

for $\rho_{0}=\mathinner{|{+}\rangle}\\!\\!\mathinner{\langle{+}|}$, i.e.
$a=\frac{1}{2}$. It remains to determint the optimal duration between the
preparation and the measurement $t_{\mathrm{opt}}$, i.e. which maximizes the
QFI rate

$t_{\mathrm{opt}}=\mathrm{argmax}_{t}\,\frac{\mathcal{F}_{\rho_{t}}}{t}=\frac{1}{\gamma_{w}}.$
(93)

The best strategy is then to measure the state of the probe after
$t_{\mathrm{opt}}$, and reinitialize it in the state $\ket{+}$ after the
measurement. For a long enough running time given the QFI becomes

$\mathcal{F}_{\mathcal{E}_{\tau}}=\tau\,\frac{\mathcal{F}_{\rho_{t_{\mathrm{opt}}}}}{t_{\mathrm{opt}}}=\tau\,\frac{4}{e}\frac{\dot{s}_{w}^{2}}{\gamma_{w}}\approx
1.5\,\tau\frac{\dot{s}_{w}^{2}}{\gamma_{w}}$ (94)

In comparison, the upper-bound of Eq. (42) reads
$\mathcal{F}_{\mathcal{E}_{\tau}}\leq 16\tau\
\frac{\dot{s}_{w}^{2}}{\gamma_{w}}$ in the limit
$\dot{s}_{w}\gg\dot{\gamma}_{w}$, and is larger by an order of magnitude.

###  8.4 Error detection scheme

Next, let us consider the case where the evolution is subject to quantum
control. In particular, we introduce an auxiliary and prepare the initial
state

$\ket{\Psi_{0}}=\sqrt{1-a}\ket{00}+\sqrt{a}\ket{11}.$ (95)

First, consider the case where no control operations are performed on the
state between the preparation and the measurement. The Lindbladian in Eq. (84)
acts trivially on the auxilliary qubit, and the evolution of the probe and
ancilla system is thus almost identical with the case described above. The
only difference is that the jump operator $\sigma_{-}$ (acting on the probe
only) leaves the system in the state

$(\sigma_{-}\otimes\mathds{1})\rho(t)(\sigma_{+}\otimes\mathds{1})\propto\mathinner{|{0}\rangle}\\!\\!\mathinner{\langle{0}|}\otimes\mathinner{|{1}\rangle}\\!\\!\mathinner{\langle{1}|},$
(96)

which has a different _parity_ as compared to the initial state. While in the
no-jump branch the parity of the initial state is preserved, i.e. in the jump
branch the state evolves to

$\ket{\Psi_{\text{nj}}^{\prime}(t)}=\sqrt{1-a}\ket{00}+\sqrt{a}e^{-\frac{\gamma_{w}}{2}t}\ket{11},$
(97)

And the final state is now a direct sum

$\rho_{\mathcal{D}}(t)=\mathinner{|{\Psi_{\text{nj}}^{\prime}(t)}\rangle}\\!\\!\mathinner{\langle{\Psi_{\text{nj}}^{\prime}(t)}|}\oplus
p_{\mathrm{j}}(t)\mathinner{|{01}\rangle}\\!\\!\mathinner{\langle{01}|}$ (98)

rather then a mixture. Futhremore, note that the coherenet evolution
$U_{t}=\exp\left(-\frac{\mathrm{i}}{2}t\,\tilde{w}\sigma_{z}\otimes\mathds{1}\right)$,
preserves the parity. Hence, by measuring the parity
$\\{\Pi_{\checkmark},\Pi_{\crossproduct}\\}$, with

$\begin{split}\Pi_{\checkmark}&=\mathinner{|{00}\rangle}\\!\\!\mathinner{\langle{00}|}+\mathinner{|{11}\rangle}\\!\\!\mathinner{\langle{11}|}\\\
\Pi_{\crossproduct}&=\mathinner{|{01}\rangle}\\!\\!\mathinner{\langle{01}|}+\mathinner{|{10}\rangle}\\!\\!\mathinner{\langle{10}|},\end{split}$
(99)

of the two-qubit state at any time, one can learn if a jump has occurred
without disturbing the coherent part of the evolution. From

$\begin{split}\rho_{t}=\mathinner{|{\Psi_{\text{nj}}^{\prime}(t)}\rangle}\\!\\!\mathinner{\langle{\Psi_{\text{nj}}^{\prime}(t)}|}\oplus
p_{\mathrm{j}}(t)\mathinner{|{01}\rangle}\\!\\!\mathinner{\langle{01}|}\qquad\dot{\rho}_{t}=2\dot{s}_{w}te^{-\frac{\gamma_{w}}{2}t}\sqrt{a-a^{2}}\,\mathrm{i}(\outerproduct{11}{00}-\outerproduct{00}{11})\end{split}$
(100)

we compute

$\mathcal{F}_{\rho_{t}}=16\frac{(1-a)at^{2}{\dot{s}}_{w}^{2}}{(1-a)e^{t\gamma_{w}}+a}.$
(101)

To find the optimal initial state $a$ and sensing time $t$, we again optimize
the QFI rate

$\mathcal{F}_{\mathcal{E}_{\tau}}=\tau\max_{a,t}\frac{\mathcal{F}_{\rho_{t}}}{t}\approx
2.47\,\tau\frac{\dot{s}_{w}^{2}}{\gamma_{w}},$ (102)

and obtained for an initial state with $a\approx 0.68$. We see that in
improves over the previous strategy.

###  8.5 Fast error detection scheme

As the parity measurement commutes with the evolution, a more involved control
strategy is to monitor the occurrence of jumps all the time. In this case when
an error occurs, the state of the system is projected to $\ket{0,1}$ and
carries no information on the temperature (recall that we neglect the
dependence of the jump rate on the temperature $\dot{\gamma}_{w}\approx 0$
which is exponentially suppressed). If a jump is detected one reinitialises
the state of the probe and auxiliary qubits immediately. This allows one to
save some ”sensing” time.

The probability that no jumps occurs between $t=0$, when the state
$\ket{\Psi_{0}}=\sqrt{1-a}\ket{00}+\sqrt{a}\ket{11}$ is prepare, and $t$ reads

$p_{\mathrm{nj}}(t)=\tr\mathinner{|{\Psi_{\mathrm{nj}}(t)}\rangle}\\!\\!\mathinner{\langle{\Psi_{\mathrm{nj}}(t)}|}=1-a+a\,e^{-\gamma_{w}t}.$
(103)

While the probability that the first jump happens during the infinitesimal
time interval $[t,t+\differential t]$ is

$r_{\mathrm{j}}(t)=-\frac{\differential}{\differential
t}p_{\mathrm{nj}}(t)\differential t=a\gamma_{w}e^{-\gamma_{w}t}.$ (104)

Hence, if the strategy is to wait for a time $T$ before measuring the system
conditional on no-errors, the average duration of a single trial is

$\langle{t}\rangle_{T}=p_{\text{nj}}(T)T+\int_{0}^{T}\differential
t\,r_{\mathrm{j}}(t)t=\frac{a\left(1-e^{-\gamma_{w}T}\right)+(1-a)T\gamma_{w}}{\gamma_{w}}.$
(105)

It is the average duration of the between the preparation and the first
reinitializtion. In addition we can compute the average QFI of a single trial.
The only nonzero contribution to QFI, comes from the branch where no error
occurs and we have already computed it in the previous section

$\langle{\mathcal{F}}\rangle_{T}=16\frac{(1-a)aT^{2}{\dot{s}}_{w}^{2}}{(1-a)e^{T\gamma_{w}}-a}.$
(106)

For long sensing times we can estimate the QFI rate by the ratio between the
average QFI and the average duration of a trial

$\mathcal{F}_{\mathcal{E}_{\tau}}\approx\tau\max_{a,T}\frac{\langle{\mathcal{F}}\rangle_{T}}{\langle{t}\rangle_{T}}\approx
4.16\,\tau\frac{\dot{s}_{w}^{2}}{\gamma_{w}},$ (107)

obtained for $a\approx 0.83$.