quantifiedequation[1]

$\make@quantifiedequation{\BODY}{#1}$ (1)

# Energy measurements remain thermometrically optimal beyond weak coupling

Jonas Glatthard  Department of Physics and Astronomy, University of Exeter,
Exeter EX4 4QL, United Kingdom<EMAIL_ADDRESS>Karen V. Hovhannisyan
University of Potsdam, Institute of Physics and Astronomy, Karl-Liebknecht-
Str. 24–25, 14476 Potsdam, Germany  Martí Perarnau-Llobet  Département de
Physique Appliquée, Université de Genève, 1211 Genève, Switzerland  Luis A.
Correa  Departamento de Física, Universidad de La Laguna, La Laguna 38203,
Spain  Department of Physics and Astronomy, University of Exeter, Exeter EX4
4QL, United Kingdom  Harry J. D. Miller  Department of Physics and Astronomy,
The University of Manchester, Manchester M13 9PL, United Kingdom
<EMAIL_ADDRESS>
###### Abstract

We develop a general perturbative theory of finite-coupling quantum
thermometry up to second order in probe–sample interaction. By assumption the
probe and sample are in thermal equilibrium, so the probe is described by the
mean-force Gibbs state. We prove that the ultimate thermometric precision can
be achieved—to second order in the coupling—solely by means of local energy
measurements on the probe. Hence, seeking to extract temperature information
from coherences or devising adaptive schemes confers no practical advantage in
this regime. Additionally, we provide a closed-form expression for the quantum
Fisher information, which captures the probe’s sensitivity to temperature
variations. Finally, we benchmark and illustrate the ease of use of our
formulae with two simple examples. Our formalism is completely general and
makes no assumptions about separation of dynamical timescales or the nature of
either the probe or the sample. Therefore, by providing analytical insight
into both the thermal sensitivity and the optimal measurement for achieving
it, our results pave the way for quantum thermometry in setups where finite-
coupling effects cannot be ignored.

##  1 Introduction

Precise measurements of low temperatures are crucial in experiments dealing
with quantum phenomena. Indeed, ultra-low temperatures are routinely achieved
in nanoelectronic systems [1, 2] and cold-atom based platforms [3, 4].
Applications range from the study of fundamental problems in condensed matter
[5, 6, 7] to thermalisation in closed quantum systems [8, 9, 10]; and from the
realisation of thermodynamic cycles [11, 12, 13] to analog quantum simulation
in optical lattices [14, 15], and computation on large-scale programmable
simulators [16, 17].

On the other hand, the application of estimation-theoretic methods to model
thermometric protocols in the ultracold regime has consolidated into the novel
field of quantum thermometry [18, 19]. Its central goals have been to
establish fundamental scaling laws for the signal-to-noise ratio of low-
temperature estimates [20, 21, 22, 23] and to identify design prescriptions
that can make a probe more responsive to temperature variations [24, 25, 26,
27, 28, 29, 30, 23, 31, 32, 33]. A particular focus has been recently placed
on sensing applications with atomic impurities [34, 10, 27, 35, 36], which has
helped to close the gap between theory and applications [35].

Specifically, the quantum Fisher information $\pazocal{F}$ [37] has played a
central role as a figure of merit for quantum thermometry. It governs the
scaling of the best-case signal-to-noise ratio of unbiased temperature
estimates, in the asymptotic limit a of large number of measurements both in
the frequentist approach, through the Crámer–Rao bound [38, 39], and Bayesian
approach, through the van Trees inequality***The Bayesian framework has been
considered within quantum thermometry only recently [40, 41, 42, 43, 44, 45,
46], proving particularly useful when estimating temperatures from scarce data
in real experimental situations [41, 35]. [47, 48]. In this paper, we study
equilibrium probe-based thermometry, where the temperature of the sample is
inferred by measuring a small system—the probe—which is coupled to it so that
both equilibrate [49, 24, 29, 20, 50, 34, 21, 27]. In this setting, we compute
$\pazocal{F}$ and investigate the most informative measurements for probes
that are coupled non-perturbatively to the sample.

In this work, we go beyond the ‘weak’ probe–sample coupling, which is usually
understood as being ‘negligibly small’, such that the equilibrium marginal of
the probe is simply a Gibbs state with respect to the probe’s bare Hamiltonian
at the sample’s temperature. In this case, temperature may be optimally
inferred from energy measurements [24], which offers simplicity and
universality. This is precisely the steady-state prediction of the common
Gorini–Kossakowski–Lindblad–Sudarshan (GKLS) [51, 52, 53] quantum master
equation describing the dynamics of the probe at time $t$. Importantly,
however, the GKLS equation is mathematically rigorous only if the strength of
the probe–sample coupling decays with time and vanishes asymptotically [54].
This means that it cannot reliably describe its own steady state, other than
in the case of asymptotically vanishing dissipation. But perturbative quantum
master equations may encounter even bigger problems in the context of quantum
thermometry. Namely, the assumption that the probe–sample coupling is neglible
tends to break down in the low-temperature regime. There, the temperature $T$
sets a small energy scale that can become comparable to the interaction
Hamiltonian—no matter how small the latter may be next to the bare
Hamiltonians of the sample and the probe [55, 56].

Here, we refer to the ‘finite-coupling’ regime to denote situations in which
the coupling is not negligible. The probe would be then described by the
marginal of the global probe–sample equilibrium state, referred-to as mean-
force Gibbs state—this can be significantly far from ‘Gibbsian’ with respect
to the probe’s bare Hamiltonian [57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67,
68, 69, 70, 71]. While the quantum Fisher information of the probe in this
regime has received considerable attention [72, 29, 73, 20, 50, 21, 34, 27],
not much is known about the optimal thermometric measurement beyond some
specific examples [21, 34, 22]. In this work, we show that energy measurements
remain optimal up to the first two leading orders in the coupling strength.
This is the first general result about the optimal thermometric measurement
beyond the weak-coupling regime and has critical practical importance. Namely,
it establishes that non-diagonal measurements, aimed at extracting temperature
information encoded coherences, do not necessarily confer a real advantage in
the finite-coupling regime, nor does any possible adaptive strategy.
Furthermore, our approach provides general formulas to evaluate thermal
sensitivity explicitly, to second order in the coupling, circumventing the
common pitfalls of the conventional weak coupling assumption.

This paper is structured as follows: In Sec. 2 we introduce the perturbative
formalism of finite-coupling quantum thermometry and present our main results.
We then apply our results to two standard probe–sample models in Sec. 3;
namelly, quantum Brownian motion and the spin–boson model. In Sec. 4 we give
detailed derivations of the formulas presented. Finally, in Sec. 5 we
summarise and draw our conclusions.

##  2 Finite-coupling quantum thermometry

###  2.1 Framework

Let us consider a probe $S$ (for ‘system’) coupled to a sample $B$ (for
‘bath’) with Hamiltonian

$\displaystyle\boldsymbol{H}=\boldsymbol{H}_{\scriptscriptstyle
S}+\boldsymbol{H}_{\scriptscriptstyle B}+\boldsymbol{H}_{\text{int}}.$ (2)

The total system is in a Gibbs state at the inverse temperature $\beta$:

$\displaystyle\tilde{\boldsymbol{\pi}}=\frac{e^{-\beta\boldsymbol{H}}}{\tilde{Z}},$
(3)

with partition function
$\tilde{Z}=\operatorname{tr}{e^{-\beta\boldsymbol{H}}}$, and the goal is to
determine the inverse temperature $\beta$ by measuring only the probe $S$. In
this paper, we work in the units where $\hbar=k=1$, the latter being the
Boltzmann constant. In all what follows, operators are denoted with boldface
symbols. Note that, since only the probe is measured, for our forthcoming
results to hold it is only required that the probe’s state be given by
$\text{tr}_{\scriptscriptstyle B}\left[\tilde{\boldsymbol{\pi}}\right]$—the
actual global state may differ from $\tilde{\boldsymbol{\pi}}$ and be
microcanonical [74, 75, 76, 77, 78, 79], pure [80, 81, 77], or neither [76,
82].

To control the magnitude of the probe–sample interaction, we will henceforth
write the interaction term as
$\boldsymbol{H}_{\text{int}}=\gamma\boldsymbol{V}$, where the dimensionless
parameter $\gamma$ is the coupling strength. The equilibrium state of the
probe, from which the inverse temperature $\beta$ is to be inferred,

$\displaystyle\tilde{\boldsymbol{\pi}}_{\scriptscriptstyle
S}=\frac{\text{tr}_{\scriptscriptstyle
B}\left[e^{-\beta\boldsymbol{H}}\right]}{\tilde{Z}},$ (4)

is the so-called mean-force Gibbs state [64, 69], and it will only coincide
with the bare thermal state of the probe,

$\displaystyle\boldsymbol{\pi}_{\scriptscriptstyle
S}=\frac{e^{-\beta\boldsymbol{H}_{\scriptscriptstyle
S}}}{Z_{\scriptscriptstyle S}},$ (5)

in the limit of vanishing coupling $(\gamma\rightarrow 0)$. Here the partition
function is $Z_{\scriptscriptstyle
S}=\operatorname{tr}{e^{-\beta\boldsymbol{H}_{\scriptscriptstyle S}}}$.

Estimates for $\beta$, drawn from a measurement record of length $N$ via an
unbiased estimator, carry an uncertainty $\delta\beta$ such that the signal-
to-noise ratio (SNR) is upper-bounded through the Cramér–Rao inequality [38,
39, 83, 84, 37] as

$\frac{\beta}{\delta\beta}\leq\sqrt{N\beta^{2}\pazocal{F}_{\scriptscriptstyle
S}(\beta)}.$ (6)

Here, $\pazocal{F}_{\scriptscriptstyle S}(\beta)$ is the quantum Fisher
information (QFI), which is given by [83, 84, 37]

$\displaystyle\pazocal{F}_{\scriptscriptstyle
S}(\beta)=\text{tr}_{\scriptscriptstyle
S}\left[\boldsymbol{L}_{\scriptscriptstyle
S}^{2}\tilde{\boldsymbol{\pi}}_{\scriptscriptstyle S}\right],$ (7)

where the symmetric logarithmic derivative $\boldsymbol{L}_{\scriptscriptstyle
S}$ (SLD) is implicitly defined by

$\displaystyle\partial_{\beta}\,\tilde{\boldsymbol{\pi}}_{\scriptscriptstyle
S}=\frac{1}{2}\\{\boldsymbol{L}_{\scriptscriptstyle
S},\tilde{\boldsymbol{\pi}}_{\scriptscriptstyle S}\\},$ (8)

with $\\{\cdot,\cdot\\}$ denoting the anticommutator. Equation (8) is a
Lyapunov equation with respect to $\boldsymbol{L}_{\scriptscriptstyle S}$ and,
since $\tilde{\boldsymbol{\pi}}_{\scriptscriptstyle S}$ is a bounded positive
operator, it has a unique solution, given by [85]

$\displaystyle\boldsymbol{L}_{\scriptscriptstyle
S}=2\int^{\infty}_{0}d\lambda\
e^{-\lambda\tilde{\boldsymbol{\pi}}_{\scriptscriptstyle
S}}(\partial_{\beta}\,\tilde{\boldsymbol{\pi}}_{\scriptscriptstyle
S})e^{-\lambda\tilde{\boldsymbol{\pi}}_{\scriptscriptstyle S}}.$ (9)

The optimal thermometric measurement—namely, the one that minimizes
$\delta\beta$—is the projection onto the eigenbasis of
$\boldsymbol{L}_{\scriptscriptstyle S}$ [83, 84, 37]. This saturates the
Cramér–Rao bound (6) in the $N\to\infty$ limit [37, 48]. For any other
measurement, the best-case scaling of the SNR is given by the ‘classical’
Cramér–Rao bound [38, 39]. For example, when one projects onto the eigenbasis
of $\boldsymbol{H}_{\scriptscriptstyle S}$, then

$\frac{\beta}{\delta\beta}\leq\sqrt{N\beta^{2}\pazocal{I}_{\boldsymbol{H}_{S}}(\beta)},$
(10)

where
$\pazocal{I}_{\boldsymbol{H}_{S}}(\beta)\leq\pazocal{F}_{\scriptscriptstyle
S}(\beta)$ is the (classical) Fisher information associated with this specific
measurement. Here, the bound also becomes tight in the $N\to\infty$ limit
[86]. By definition,

$\displaystyle\pazocal{I}_{\boldsymbol{H}_{S}}(\beta)=\sum\nolimits_{n}\tilde{p}_{n}\,(\partial_{\beta}\ln\tilde{p}_{n})^{2},$
(11)

where
$\tilde{p}_{n}\mathrel{\mathop{\ordinarycolon}}=\bra{\epsilon_{n}}\tilde{\boldsymbol{\pi}}_{\scriptscriptstyle
S}\ket{\epsilon_{n}}$ are the populations of the probe’s equilibrium state in
the eigenbasis $\\{\ket{\epsilon_{n}}\\}_{n}$ of
$\boldsymbol{H}_{\scriptscriptstyle S}$. Equivalently,
$\pazocal{I}_{\boldsymbol{H}_{S}}(\beta)$ can be written as

$\displaystyle\pazocal{I}_{\boldsymbol{H}_{S}}(\beta)=\text{tr}_{\scriptscriptstyle
S}\left[\mathscr{D}_{\boldsymbol{H}_{\scriptscriptstyle
S}}(\boldsymbol{L}_{\scriptscriptstyle
S})^{2}\mathscr{D}_{\boldsymbol{H}_{\scriptscriptstyle
S}}(\tilde{\boldsymbol{\pi}}_{\scriptscriptstyle S})\right],$ (12)

where $\mathscr{D}_{\boldsymbol{H}_{\scriptscriptstyle S}}$ is the dephasing
operation in the eigenbasis of $\boldsymbol{H}_{\scriptscriptstyle S}$:

$\displaystyle\mathscr{D}_{\boldsymbol{H}_{\scriptscriptstyle
S}}(\boldsymbol{Q})\mathrel{\mathop{\ordinarycolon}}=\sum_{n}\ket{\epsilon_{n}}\bra{\epsilon_{n}}\boldsymbol{Q}\ket{\epsilon_{n}}\bra{\epsilon_{n}}.$
(13)

The bare energy measurement defined by $\\{\ket{\epsilon_{n}}\\}_{n}$ is known
to be optimal in the limit $\gamma\rightarrow 0$, where
$\tilde{\boldsymbol{\pi}}_{\scriptscriptstyle
S}\to\boldsymbol{\pi}_{\scriptscriptstyle S}$ and, therefore,
$\boldsymbol{L}_{\scriptscriptstyle
S}\to\left\langle\boldsymbol{H}_{\scriptscriptstyle
S}\right\rangle_{\scriptscriptstyle S}-\boldsymbol{H}_{\scriptscriptstyle S}$
[24]; here and throughout, $\left\langle\cdot\right\rangle_{\scriptscriptstyle
S}\mathrel{\mathop{\ordinarycolon}}=\text{tr}_{\scriptscriptstyle
S}\left[\boldsymbol{\pi}_{\scriptscriptstyle S}\,\cdot\right]$. Furthermore,
the bare energy measurement is optimal at any $\gamma$ as long as
$[\boldsymbol{H}_{\text{int}},\boldsymbol{H}_{\scriptscriptstyle S}]=0$,
because then $[\tilde{\boldsymbol{\pi}}_{\scriptscriptstyle
S},\boldsymbol{H}_{\scriptscriptstyle S}]=0$, and therefore
$[\boldsymbol{L}_{\scriptscriptstyle S},\boldsymbol{H}_{\scriptscriptstyle
S}]=0$, which means that $\\{\ket{\epsilon_{n}}\\}_{n}$ is an eigenbasis for
$\boldsymbol{L}_{\scriptscriptstyle S}$. In general, however,
$\tilde{\boldsymbol{\pi}}_{\scriptscriptstyle S}$, and hence
$\boldsymbol{L}_{\scriptscriptstyle S}$, will not commute with
$\boldsymbol{H}_{\scriptscriptstyle S}$, so the optimal thermometric basis
will differ from $\\{\ket{\epsilon_{n}}\\}_{n}$. Nonetheless, as we prove
below, the bare energy measurement remains as informative as the truly optimal
measurement up to the first two leading orders in the perturbative expansion
in the probe–sample coupling strength $\gamma$.

###  2.2 Results

####  2.2.1 Optimality of local energy measurements at finite coupling

Suppose $\tilde{\boldsymbol{\pi}}_{\scriptscriptstyle S}$ is twice
differentiable in $\gamma$. It can then be Taylor-expanded as

$\displaystyle\tilde{\boldsymbol{\pi}}_{\scriptscriptstyle
S}=\boldsymbol{\pi}_{\scriptscriptstyle
S}+\gamma\,\boldsymbol{p}_{1}+\gamma^{2}\boldsymbol{p}_{2}+O(\gamma^{3}),$
(14)

where $\boldsymbol{p}_{1}$ and $\boldsymbol{p}_{2}$ are some operators and $O$
is the ‘big O’ as per the standard asymptotic notation. In the same manner,
the SLD $\boldsymbol{L}_{\scriptscriptstyle S}$ of
$\tilde{\boldsymbol{\pi}}_{\scriptscriptstyle S}$ [Eq. (9)] will decompose as

$\displaystyle\boldsymbol{L}_{\scriptscriptstyle
S}=-\Delta\boldsymbol{H}_{\scriptscriptstyle
S}+\gamma\,\boldsymbol{l}_{1}+\gamma^{2}\boldsymbol{l}_{2}+O(\gamma^{3}),$
(15)

where $-\Delta\boldsymbol{H}_{\scriptscriptstyle
S}\mathrel{\mathop{\ordinarycolon}}=\left\langle\boldsymbol{H}_{\scriptscriptstyle
S}\right\rangle_{\scriptscriptstyle S}-\boldsymbol{H}_{\scriptscriptstyle S}$
is the SLD of $\boldsymbol{\pi}_{\scriptscriptstyle
S}=\tilde{\boldsymbol{\pi}}_{\scriptscriptstyle S}\big{|}_{\gamma=0}$ and
$\boldsymbol{l}_{1}$, $\boldsymbol{l}_{2}$ are some operators related to
$\boldsymbol{\pi}_{\scriptscriptstyle S}$, $\boldsymbol{p}_{1}$, and
$\boldsymbol{p}_{2}$ through Eq. (9). Plugging these into Eq. (7), we obtain

$\pazocal{F}_{\scriptscriptstyle
S}(\beta)=\,\pazocal{F}^{(0)}_{\scriptscriptstyle
S}(\beta)+\gamma\left[\operatorname{tr}(\boldsymbol{p}_{1}\Delta\boldsymbol{H}_{\scriptscriptstyle
S}^{2})-2\operatorname{tr}(\boldsymbol{l}_{1}\boldsymbol{\pi}_{\scriptscriptstyle
S}\Delta\boldsymbol{H}_{\scriptscriptstyle S})\right]\\\
+\gamma^{2}\left[\operatorname{tr}(\boldsymbol{l}_{1}^{2}\boldsymbol{\pi}_{\scriptscriptstyle
S})+\operatorname{tr}(\boldsymbol{p}_{2}\Delta\boldsymbol{H}_{\scriptscriptstyle
S}^{2})-2\operatorname{tr}(\boldsymbol{l}_{2}\boldsymbol{\pi}_{\scriptscriptstyle
S}\Delta\boldsymbol{H}_{\scriptscriptstyle
S})-\operatorname{tr}(\\{\boldsymbol{p}_{1},\boldsymbol{l}_{1}\\}\Delta\boldsymbol{H}_{\scriptscriptstyle
S})\right]+O(\gamma^{3}),$ (16)

where $\pazocal{F}^{(0)}_{\scriptscriptstyle
S}(\beta)\mathrel{\mathop{\ordinarycolon}}=\operatorname{tr}(\boldsymbol{\pi}_{\scriptscriptstyle
S}\Delta\boldsymbol{H}_{\scriptscriptstyle S}^{2})=\partial_{\beta}^{2}\ln
Z_{\scriptscriptstyle S}$ is the zeroth-order QFI, i.e., that of
$\boldsymbol{\pi}_{\scriptscriptstyle S}$. Now, observing that

$\displaystyle\operatorname{tr}(\boldsymbol{p}_{1}\Delta\boldsymbol{H}_{\scriptscriptstyle
S}^{2})$
$\displaystyle=\operatorname{tr}(\mathscr{D}_{\boldsymbol{H}_{\scriptscriptstyle
S}}[\boldsymbol{p}_{1}]\Delta\boldsymbol{H}_{\scriptscriptstyle S}^{2}),$
$\displaystyle\operatorname{tr}(\boldsymbol{l}_{1}\boldsymbol{\pi}_{\scriptscriptstyle
S}\Delta\boldsymbol{H}_{\scriptscriptstyle S})$
$\displaystyle=\operatorname{tr}(\mathscr{D}_{\boldsymbol{H}_{\scriptscriptstyle
S}}[\boldsymbol{l}_{1}]\boldsymbol{\pi}_{\scriptscriptstyle
S}\Delta\boldsymbol{H}_{\scriptscriptstyle S}),$

we immediately conclude from Eq. (12) that, up to the first order in $\gamma$,
$\pazocal{F}_{\scriptscriptstyle S}(\beta)$ and
$\pazocal{I}_{\boldsymbol{H}_{S}}(\beta)$ coincide; that is,

$\pazocal{I}_{\boldsymbol{H}_{S}}(\beta)=\pazocal{F}_{\scriptscriptstyle
S}(\beta)+O(\gamma^{2}).$ (17)

However, in general, $\mathscr{D}_{\boldsymbol{H}_{\scriptscriptstyle
S}}[\\{\boldsymbol{p}_{1},\boldsymbol{l}_{1}\\}]\neq\\{\mathscr{D}_{\boldsymbol{H}_{\scriptscriptstyle
S}}[\boldsymbol{p}_{1}],\mathscr{D}_{\boldsymbol{H}_{\scriptscriptstyle
S}}[\boldsymbol{l}_{1}]\\}$, meaning that $\pazocal{F}_{\scriptscriptstyle
S}(\beta)$ and $\pazocal{I}_{\boldsymbol{H}_{S}}(\beta)$ do differ in the
second order. Therefore, we have shown that the bare energy measurement is
thermometrically optimal up to the first two leading orders (i.e., zeroth and
first order), provided that the $O(\gamma)$ term in Eq. (16) is non-zero. In
Appendix A we give explicit examples.

Let us now consider the case where the deviation between the mean-force Gibbs
state $\tilde{\boldsymbol{\pi}}_{\scriptscriptstyle S}$ and the local thermal
state $\boldsymbol{\pi}_{\scriptscriptstyle S}$ appears only at
$O(\gamma^{2})$. This is relevant from the point of view of applications,
since it is the situation most commonly studied in open quantum systems.
Since, in this case $\boldsymbol{p}_{1}=0$, we have that, due to Eq. (9), also
$\boldsymbol{l}_{1}=0$. As a result, the $O(\gamma)$ term in Eq. (16)
vanishes, so that $\pazocal{F}_{\scriptscriptstyle S}(\beta)$ becomes

$\displaystyle\pazocal{F}_{\scriptscriptstyle
S}(\beta)=\pazocal{F}^{(0)}_{\scriptscriptstyle
S}(\beta)+\gamma^{2}\left[\operatorname{tr}(\boldsymbol{p}_{2}\Delta\boldsymbol{H}_{\scriptscriptstyle
S}^{2})-2\operatorname{tr}(\boldsymbol{l}_{2}\boldsymbol{\pi}_{\scriptscriptstyle
S}\Delta\boldsymbol{H}_{\scriptscriptstyle S})\right]+O(\gamma^{3}).$ (18)

As above, we have that

$\displaystyle\operatorname{tr}(\boldsymbol{p}_{2}\Delta\boldsymbol{H}_{\scriptscriptstyle
S}^{2})$
$\displaystyle=\operatorname{tr}(\mathscr{D}_{\boldsymbol{H}_{\scriptscriptstyle
S}}[\boldsymbol{p}_{2}]\Delta\boldsymbol{H}_{\scriptscriptstyle S}^{2}),$
$\displaystyle\operatorname{tr}(\boldsymbol{l}_{2}\boldsymbol{\pi}_{\scriptscriptstyle
S}\Delta\boldsymbol{H}_{\scriptscriptstyle S})$
$\displaystyle=\operatorname{tr}(\mathscr{D}_{\boldsymbol{H}_{\scriptscriptstyle
S}}[\boldsymbol{l}_{2}]\boldsymbol{\pi}_{\scriptscriptstyle
S}\Delta\boldsymbol{H}_{\scriptscriptstyle S}),$

which tells us that, according to Eq. (12), $\pazocal{F}_{\scriptscriptstyle
S}(\beta)$ and $\pazocal{I}_{\boldsymbol{H}_{S}}(\beta)$ coincide up to the
second leading order in $\gamma$. Moreover, as we will show below, if one
additionally imposes the generic assumption (2.2.2), the cubic term in Eq.
(18) also vanishes (see Sec. 4), leaving us with

$\displaystyle\pazocal{I}_{\boldsymbol{H}_{S}}(\beta)=\pazocal{F}_{\scriptscriptstyle
S}(\beta)+O(\gamma^{4}).$ (19)

Equations (17) and (19) constitute our first main result. Namely, the bare
energy measurement is thermometrically optimal up to the first two nonzero
leading orders in $\gamma$. Importantly, Eq. (19) applies to typical open
quantum system scenarios, which means that, in such settings, measurement
optimisation is only necessary at much stronger couplings†††Note, however,
that measurements not commuting with the probe’s bare Hamiltonian can be
advantageous in nonequilibrium thermometry even in the ultraweak coupling
limit [87, 88]..

####  2.2.2 Quantum Fisher information in finite-coupling thermometry

Having proven that energy measurements remain optimal at finite coupling, we
now turn to deriving a closed-form expression for
$\pazocal{F}_{\scriptscriptstyle S}(\beta)$ up to the second order in $\gamma$
for the case of $\boldsymbol{p}_{1}=0$. Here, we outline the derivation of our
formulae. For a detailed derivation one may skip directly to Sec. 4.

We begin by imposing the generic assumption of a factorised probe–sample
interaction; namely, Assumption I: H_int = γ S⊗B, where $\boldsymbol{S}$ and
$\boldsymbol{B}$ are coupling operators belonging to probe and sample,
respectively. Importantly, these are chosen such that
$\boldsymbol{S}\otimes\boldsymbol{B}$ is of the same dimension as
$\boldsymbol{H}_{\scriptscriptstyle S}$ and
$\boldsymbol{H}_{\scriptscriptstyle B}$, to ensure that $\gamma$ is
dimensionless.

As we will show below, under Assumption I, the condition
$\boldsymbol{p}_{1}=0$ is equivalent to Assumption II: ⟨B ⟩_B = 0, where
$\left\langle\boldsymbol{B}\right\rangle_{\scriptscriptstyle
B}\mathrel{\mathop{\ordinarycolon}}=\text{tr}_{\scriptscriptstyle
B}\left[\boldsymbol{\pi}_{\scriptscriptstyle B}\,\boldsymbol{B}\right]$, with
$\boldsymbol{\pi}_{\scriptscriptstyle
B}=e^{-\beta\boldsymbol{H}_{\scriptscriptstyle B}}/Z_{B}$ being the Gibbs
state of the sample with respect to its bare Hamiltonian
$\boldsymbol{H}_{\scriptscriptstyle B}$, and with the same $\beta$ as in Eq.
(3). $Z_{B}=\text{tr}\,e^{-\beta\boldsymbol{H}_{\scriptscriptstyle B}}$ is the
corresponding partition function‡‡‡Note that, whenever $\gamma\neq 0$, the
actual reduced state of $B$, $\text{tr}_{\scriptscriptstyle
S}\left[\tilde{\boldsymbol{\pi}}\right]$, will not coincide with
$\boldsymbol{\pi}_{\scriptscriptstyle B}$. Both
$\boldsymbol{\pi}_{\scriptscriptstyle B}$ and
$\boldsymbol{\pi}_{\scriptscriptstyle S}$ are simply fictitious states that
appear in our perturbative expansions.. Importantly, this assumption is also
almost universally adopted in the theory of open quantum systems. There,
Assumptions I and II are often disregarded as technical. However, they are
crucial from the perspective of thermometry§§§In open quantum systems theory
[53], it is often argued that even when Assumption II is not met, it can be
easily enforced without loss of generality, by re-defining
$\boldsymbol{H}=\boldsymbol{H}_{\scriptscriptstyle
S}^{\prime}(\beta)+\boldsymbol{H}_{\scriptscriptstyle
B}+\boldsymbol{H}_{\text{int}}^{\prime}$, where
$\boldsymbol{H}_{\scriptscriptstyle
S}^{\prime}(\beta)=\boldsymbol{H}_{\scriptscriptstyle
S}+\text{tr}_{\scriptscriptstyle B}\left[\boldsymbol{\pi}_{\scriptscriptstyle
B}\,\boldsymbol{H}_{\text{int}}\right]$ and
$\boldsymbol{H}_{\text{int}}^{\prime}=\boldsymbol{H}_{\text{int}}-\text{tr}_{\scriptscriptstyle
B}\left[\boldsymbol{\pi}_{\scriptscriptstyle
B}\,\boldsymbol{H}_{\text{int}}\right]$. Note, however, that there would be an
explicit $\beta$-dependence in the modified probe Hamiltonian, which does
crucially change the problem from a thermometric viewpoint..

Our main tool is the Taylor expansion for the operator exponential function
[89]. Let $\boldsymbol{Q}$ and $\boldsymbol{R}$ be a pair of dimensionless
linear operators. Then, the function $\mathbb{R}\ni\gamma\mapsto
e^{\boldsymbol{Q}+\gamma\boldsymbol{R}}$ can be cast as [90]

$\displaystyle
e^{\boldsymbol{Q}+\gamma\boldsymbol{R}}=\sum^{\infty}_{n=0}\boldsymbol{Q}_{n}(1)\gamma^{n},$
(20a) where $\boldsymbol{Q}_{0}(s)=e^{s\boldsymbol{Q}}$ and
$\displaystyle\boldsymbol{Q}_{n}(s)\mathrel{\mathop{\ordinarycolon}}=\int^{s}_{0}dt_{1}\
\int^{t_{1}}_{0}dt_{2}\ ......\int^{t_{n-1}}_{0}dt_{n}\
e^{(s-t_{1})\boldsymbol{Q}}\,\boldsymbol{R}\,e^{(t_{1}-t_{2})\boldsymbol{Q}}\boldsymbol{R}\cdots\boldsymbol{R}\,e^{t_{n}\,\boldsymbol{Q}}.$
(20b)

Applying Eq. (20) to $\tilde{\boldsymbol{\pi}}_{\scriptscriptstyle S}$, we
find, after some algebra, that

$\displaystyle\boldsymbol{p}_{1}=-\left\langle\boldsymbol{B}\right\rangle_{\scriptscriptstyle
B}\boldsymbol{\pi}_{\scriptscriptstyle
S}\int_{0}^{\beta}d\beta_{1}e^{\beta_{1}\boldsymbol{H}_{\scriptscriptstyle
S}}[\boldsymbol{S}-\left\langle\boldsymbol{S}\right\rangle_{\scriptscriptstyle
S}]e^{-\beta_{1}\boldsymbol{H}_{\scriptscriptstyle S}},$ (21)

from where it evident that, unless
$\boldsymbol{S}\propto\boldsymbol{\mathbbm{1}}_{\scriptscriptstyle S}$,
$\boldsymbol{p}_{1}=0$ iff
$\left\langle\boldsymbol{B}\right\rangle_{\scriptscriptstyle B}=0$.

Hence, the application of Eq. (20) to Eq. (4) up to the second order results
in

$\displaystyle\tilde{\boldsymbol{\pi}}_{\scriptscriptstyle
S}=\boldsymbol{\pi}_{\scriptscriptstyle
S}\big{(}\boldsymbol{\mathbbm{1}}_{\scriptscriptstyle
S}+\gamma^{2}\boldsymbol{X}_{\scriptscriptstyle S}\big{)}+O(\gamma^{4}),$ (22)

where the matrix elements $(\boldsymbol{X}_{\scriptscriptstyle
S})_{nm}=\bra{\epsilon_{n}}\boldsymbol{X}_{\scriptscriptstyle
S}\ket{\epsilon_{m}}$ of the traceless operator
$\boldsymbol{X}_{\scriptscriptstyle S}$ in the eigenbasis of
$\boldsymbol{H}_{\scriptscriptstyle
S}=\sum\nolimits_{n}\epsilon_{n}\ket{\epsilon_{n}}\bra{\epsilon_{n}}$ are

$\displaystyle\begin{split}(\boldsymbol{X}_{\scriptscriptstyle
S})_{nn}&=\int^{\beta}_{0}du\ (\beta-u)\Phi_{\scriptscriptstyle
B}(-iu)\big{(}\sum\nolimits_{k}\phi_{nk}(-iu)-\Phi_{\scriptscriptstyle
S}(-iu)\big{)},\end{split}$ (23a)
$\displaystyle\begin{split}(\boldsymbol{X}_{\scriptscriptstyle S})_{n\neq
m}&=\frac{1}{\Delta_{mn}}\sum\nolimits_{k}(\boldsymbol{S})_{nk}(\boldsymbol{S})_{km}\,\int_{0}^{\beta}du\,\Phi_{\scriptscriptstyle
B}(-iu)\,\left(e^{u\Delta_{km}}\,e^{\beta\Delta_{mn}}-e^{u\Delta_{kn}}\right).\end{split}$
(23b)

We have defined

$\displaystyle\phi_{nk}(x)=|(\boldsymbol{S})_{nk}|^{2}e^{-i\,x\,\Delta_{nk}}.$
(24)

with $\Delta_{nk}=\epsilon_{k}-\epsilon_{n}$ and
$(\boldsymbol{S})_{nk}=\bra{\epsilon_{n}}\boldsymbol{S}\ket{\epsilon_{k}}$. We
have also introduced here the probe and sample auto-correlation functions,
defined as

$\displaystyle\Phi_{\scriptscriptstyle S}(x)$
$\displaystyle\coloneqq\left\langle e^{ix\boldsymbol{H}_{\scriptscriptstyle
S}}\boldsymbol{S}\,e^{-ix\boldsymbol{H}_{\scriptscriptstyle
S}}\boldsymbol{S}\right\rangle_{\scriptscriptstyle
S}=\text{tr}_{\scriptscriptstyle S}\left[\boldsymbol{\pi}_{\scriptscriptstyle
S}\,e^{ix\boldsymbol{H}_{\scriptscriptstyle
S}}\boldsymbol{S}\,e^{-ix\boldsymbol{H}_{\scriptscriptstyle
S}}\boldsymbol{S}\right]$ (25a) $\displaystyle\Phi_{\scriptscriptstyle B}(x)$
$\displaystyle\coloneqq\left\langle e^{ix\boldsymbol{H}_{\scriptscriptstyle
B}}\boldsymbol{B}\,e^{-ix\boldsymbol{H}_{\scriptscriptstyle
B}}\boldsymbol{B}\right\rangle_{\scriptscriptstyle
B}=\text{tr}_{\scriptscriptstyle B}\left[\boldsymbol{\pi}_{\scriptscriptstyle
B}\,e^{ix\boldsymbol{H}_{\scriptscriptstyle
B}}\boldsymbol{B}\,e^{-ix\boldsymbol{H}_{\scriptscriptstyle
B}}\boldsymbol{B}\right].$ (25b)

From these formulas, we see that the condition for the validity of Eq. (22),
$\boldsymbol{X}_{\scriptscriptstyle
S}\ll\boldsymbol{\mathbbm{1}}_{\scriptscriptstyle S}$, is satisfied if
$\gamma^{2}\langle\boldsymbol{S}^{2}\rangle_{\scriptscriptstyle
S}\langle\boldsymbol{B}^{2}\rangle_{\scriptscriptstyle B}\ll T^{2}$. This
shows that, the smaller the temperature $T$, the weaker the coupling needs to
be in order for the perturbative calculations to make sense. An analogue of
Eq. (23) was derived in Ref. [61] and for the special case of linear coupling
to a bosonic bath in Ref. [65].

Thus equipped with Eqs. (22) and (23), we can turn to Eq. (9) and write the
SLD as

$\displaystyle\boldsymbol{L}_{\scriptscriptstyle
S}(\beta)=-\Delta\boldsymbol{H}_{\scriptscriptstyle
S}+\gamma^{2}\sum\nolimits_{nm}\alpha_{nm}(\beta)\ket{\epsilon_{n}}\bra{\epsilon_{m}}+O(\gamma^{4})$
(26a) with $\Delta\boldsymbol{H}_{\scriptscriptstyle
S}\coloneqq\boldsymbol{H}_{\scriptscriptstyle
S}-\left\langle\boldsymbol{H}_{\scriptscriptstyle
S}\right\rangle_{\scriptscriptstyle S}$ and coefficients
$\displaystyle\alpha_{nm}(\beta)=\frac{2\,p_{n}}{p_{n}+p_{m}}\,\partial_{\beta}(\boldsymbol{X}_{\scriptscriptstyle
S})_{nm}+\frac{p_{n}\Delta_{nm}}{p_{n}+p_{m}}\,(\boldsymbol{X}_{\scriptscriptstyle
S})_{nm}.$ (26b)

Here,
$\Delta_{n}\coloneqq\epsilon_{n}-\left\langle\boldsymbol{H}_{\scriptscriptstyle
S}\right\rangle_{\scriptscriptstyle S}$. Also note that the populations
appearing above are

$p_{n}=\bra{\epsilon_{n}}\boldsymbol{\pi}_{\scriptscriptstyle
S}\ket{\epsilon_{n}}=e^{-\beta\epsilon_{n}}/Z_{\scriptscriptstyle S}$ (27)

and should not be confused with the populations of the mean-force Gibbs state

$\tilde{p}_{n}=\bra{\epsilon_{n}}\tilde{\boldsymbol{\pi}}_{\scriptscriptstyle
S}\ket{\epsilon_{n}}=p_{n}+\gamma^{2}p_{n}(\boldsymbol{X}_{\scriptscriptstyle
S})_{nn}+O(\gamma^{4}).$ (28)

We now have the two ingredients needed to compute the QFI from Eq. (7). After
some algebra, this can be compactly written as

$\displaystyle\pazocal{F}_{\scriptscriptstyle
S}(\beta)=\partial^{2}_{\beta}\ln Z_{\scriptscriptstyle
S}+\gamma^{2}\int^{\beta}_{0}du\ \left[\tilde{\Phi}_{\scriptscriptstyle
B}(-iu)\,\partial^{2}_{\beta}\Phi_{\scriptscriptstyle
S}(-iu)+2\,\partial_{\beta}\tilde{\Phi}_{\scriptscriptstyle
B}(-iu)\,\partial_{\beta}\Phi_{\scriptscriptstyle
S}(-iu)\right]+O(\gamma^{4}),$ (29)

with the modified auto-correlation function $\tilde{\Phi}_{\scriptscriptstyle
B}(x)\coloneqq(\beta-ix)\Phi_{\scriptscriptstyle B}(x)$. To second order in
$\gamma$ the QFI can thus be expressed solely in terms of the probe and sample
correlation functions. We emphasise that the validity of Eqs. (26) and (29) is
not, in any way, underpinned by the Born–Markov approximation.

Equations (26) and (29) are our second main result. They can be applied
directly to any open quantum system under finite coupling, provided that
Assumptions I and II are met. And since the systems satisfying these
assumptions span an exceptionally wide range, our closed-form expressions for
$\boldsymbol{L}_{\scriptscriptstyle S}$ and $\pazocal{F}_{\scriptscriptstyle
S}(\beta)$ open up new avenues in quantum thermometry. Note that, when
Assumptions I and II are violated, the QFI will in general pick up a first-
order term, as can be anticipated from Eq. (21). We provide illustrative
examples of that in Appendix A.

Finally, it is often useful to talk in terms of $T$ rather than inverse
temperature $\beta$. Namely, from Eq. (6) we see that the best-case signal-to-
noise ratio (SNR) for estimates of $T$ can be cast as

$\displaystyle\frac{1}{N}\left(\frac{T}{\delta
T}\right)^{2}\leq\beta^{2}\,\pazocal{F}_{\scriptscriptstyle S}(\beta)\coloneqq
C_{\scriptscriptstyle S}(T)+\gamma^{2}\xi(T),$ (30)

with $C_{\scriptscriptstyle S}(T)=\beta^{2}\partial^{2}_{\beta}\ln
Z_{\scriptscriptstyle
S}=\partial_{T}\left\langle\boldsymbol{H}_{\scriptscriptstyle
S}\right\rangle_{\scriptscriptstyle S}$ being the heat capacity of the probe
when in local thermal equilibrium [24], and where $\xi(T)$ follows from the
second-order term in Eq. (29).

##  3 Examples

We now benchmark our Eq. (29) against the exactly solvable Caldeira–Leggett
model for quantum Brownian motion and then, apply it to the spin–boson
problem. In both cases, we model the sample as an infinite collection of
uncoupled harmonic oscillators spanning a quasi-continuum of frequencies—the
most common bath model in open quantum systems. That is,

$\boldsymbol{H}_{\scriptscriptstyle
B}=\sum\nolimits_{k}\omega_{k}\,\boldsymbol{b}^{\dagger}_{k}\boldsymbol{b}_{k},$
(31)

where $\boldsymbol{b}_{k}^{\dagger}$ ($\boldsymbol{b}_{k}$) is the creation
(annihilation) operator at frequency $\omega_{k}$. As it is also customary, we
take the coupling operator $\boldsymbol{B}$ to be

$\boldsymbol{B}=\sum\nolimits_{k}\left(g_{k}\,\boldsymbol{b}_{k}^{\dagger}+g_{k}^{*}\,\boldsymbol{b}_{k}\right),$
(32)

Here, the coupling constants $g_{k}$ encode the overall dissipation strength
(note that $\gamma$ still enters $\boldsymbol{H}_{\text{int}}$). We may
collect them into the spectral density function

$J(\omega)=\pi\,\sum\nolimits_{k}|g_{k}|^{2}\,\delta(\omega-\omega_{k}),$ (33)

where $\delta(\cdot)$ stands here for the Dirac delta. Since we assume that
the sample has a quasi-continuum spectrum, we may give some smooth functional
form to $J(\omega)$.

Using this notation, the bath correlation function evaluates to

$\displaystyle\Phi_{\scriptscriptstyle
B}(-iu)=\frac{1}{\pi}\int^{\infty}_{0}d\omega\
J(\omega)\,\frac{\cosh{\left(\beta\,\omega/2-u\,\omega\right)}}{\sinh{\left(\beta\,\omega/2\right)}}.$
(34)

Furthermore, using Eq. (29) we see that the second-order correction $\xi(T)$
to the SNR from Eq. (30) is given by

$\displaystyle\xi(T)=\int^{\infty}_{0}d\omega\
J(\omega)\,f_{\scriptscriptstyle S}(T,\omega),$ (35a) where $\displaystyle
f_{\scriptscriptstyle S}(T,\omega)=$
$\displaystyle\frac{\beta^{2}}{\pi\sinh{\frac{\beta\,\omega}{2}}}\int^{\beta}_{0}du\
(\beta-u)\,\cosh{\left(\beta\,\omega/2-u\,\omega\right)}$
$\displaystyle\times\Big{[}\partial^{2}_{\beta}\Phi_{\scriptscriptstyle
S}(-iu)+\partial_{\beta}\Phi_{\scriptscriptstyle
S}(-iu)\Big{(}\frac{2}{\beta-u}-\omega\coth\frac{\beta\,\omega}{2}-\omega\tanh\frac{(2u-\beta)\,\omega}{2}\Big{)}\Big{]}.$
(35b)

Eqs. (35) above hold for any spectral density $J(\omega)$, provided that the
bath and the probe–bath coupling are of the standard type defined by Eqs. (31)
and (32). Hence, Eq. (35) holds even where most master equations break down,
e.g., when the bath correlation function happens to be long lived. As we
illustrate below, Eqs. (35) are indeed very practical to quickly evaluate the
finite-coupling corrections to thermal sensitivity, and provide analytic
intuition about their temperature dependence. This is our third result.

Specifically, in what follows we work with a spectral density with variable
Ohmicity $s$ and exponential cutoff; namely,

$\displaystyle J(\omega)=\Omega^{a-s}\,\omega^{s}\,e^{-\omega/\Omega},$ (36)

where $a$ controls the dimension of $J(\omega)$ and therefore depends on the
choice of $\boldsymbol{S}$: if [$\boldsymbol{S}$]$=$[$\omega$]μ, then
$a=1-2\mu$, where [$\cdot$] gives the dimension of the argument.

###  3.1 Quantum Brownian motion

In order to put the accuracy of our second-order formula to the test, we take
a single harmonic oscillator as the probe. That is,

$\displaystyle\boldsymbol{H}_{\scriptscriptstyle
S}=\frac{1}{2}\omega_{0}^{2}\boldsymbol{x}^{2}+\frac{1}{2}\boldsymbol{p}^{2},$
(37)

while the probe operator entering in the interaction Hamiltonian¶¶¶Note that
we are not adding a counter-term to compensate for the distortion of the
harmonic potential due to the coupling to the sample. While it may be
important to do so whenever such renormalisation is not expected to be
physical—especially when the coupling is strong—this effect is irrelevant for
us. in Eq. (2.2.2) is $\boldsymbol{S}=\boldsymbol{x}$. For this choice of
$\boldsymbol{S}$, dimensional analysis imposes the value $a=2$ for the
exponent in Eq. (36). In this simple case, exact expressions can be found for
the steady state of the Brownian oscillator and its thermal sensitivity [29,
34], which allows us to benchmark Eq. (29). Explicit expressions can be found
in Ref. [29].

The only ingredient missing from Eqs. (35) is the probe correlation function,
for which a straighforward calculation yelds

$\displaystyle\Phi_{\scriptscriptstyle
S}(-iu)=\frac{1}{2\omega_{0}}e^{-u\,\omega_{0}}+\frac{1}{\omega_{0}}\frac{\cosh{(u\,\omega_{0})}}{e^{\beta\,\omega_{0}}-1}.$
(38)

In Fig. 1 we compare the signal-to-noise ratio according to Eq. (30) against
the exact result and the thermal sensitivity of a probe in local thermal
equilibrium, which is given by the heat capacity alone. We see that our
second-order formula accurately captures the sensitivity for small coupling.
As already advanced, at very low temperatures, $\beta$ itself sets an energy-
scale compared to which the coupling is no longer perturbative. This results
in our second-order approximation becoming less accurate. On the other hand,
the sensitivity saturates at high temperature, and is correctly approximated
by the assumption of local thermal equilibrium on the probe, as expected.
Hence, our formula provides substantial quantitative advantage in the range of
moderately low temperatures.

Figure 1: Benchmarking Eq. (29) for quantum Brownian motion. The plot shows
the signal-to-noise ratio from our second-order formula (solid blue), the true
SNR ratio from the exact mean-force Gibbs state (dashed black), and from the
infinitesimal coupling limit, i.e. the local Gibbs state (dotted red). The
second-order formula tracks the exact sensitivity well as long as the
probe–sample coupling is the only small parameter. It starts to break down
when the temperature itself becomes a small parameter as $T\rightarrow 0$. We
chose $\omega_{0}=1,\gamma=0.1,s=1$ and $\Omega=100$ ($\hbar=k=1$).

To get further insight into the correction we calculate the high-temperature
expansion of Eq. (35b), which gives

$\displaystyle f_{\scriptscriptstyle
S}(T,\omega)=-\frac{\beta^{2}}{6m\pi\omega}+O(\beta^{4}),$ (39)

doing the integral in Equation (35a) for our spectral density (36) gives

$\displaystyle\xi(T)=-\frac{\Omega^{2}\,\Gamma\left(s\right)}{3m\pi}\,\beta^{2}+O(\beta^{4}),$
(40)

with $\Gamma\left(\cdot\right)$ the Euler gamma function. This is exactly the
lowest-order contribution from the frequency renormalisation due to the finite
interaction with the sample, i.e. the classical mean-force correction [91].

###  3.2 Spin–boson model

We now apply our formula to a case study for which no exact solution is
available—the spin–boson model. In order to study the effects of coherence in
the steady state, we focus on the ‘$\theta$-angled’ spin–boson model. That is,

$\displaystyle\boldsymbol{H}_{\scriptscriptstyle S}$
$\displaystyle=\frac{\epsilon}{2}\boldsymbol{\sigma}_{z},$ (41a)
$\displaystyle\boldsymbol{S}$
$\displaystyle=\cos{\theta}\,\boldsymbol{\sigma}_{z}-\sin{\theta}\,\boldsymbol{\sigma}_{x},$
(41b)

where $\boldsymbol{\sigma}_{\alpha}$ are Pauli operators. For this choice of
$\boldsymbol{S}$, $a=1$. We thus have the probe correlation function

$\displaystyle\Phi_{\scriptscriptstyle
S}(-iu)=\cos^{2}{\theta}+\cosh{\left(\epsilon\,u-\beta\,\epsilon/2\right)}\,\operatorname{sech}{\left(\beta\,\epsilon/2\right)}\,\sin^{2}{\theta}.$
(42)

In Fig. 2 we compare the the SNR ratio as per Eq. (30) for two values of
$\theta$ and, once again, also against the sensitivity of a locally thermal
probe. We also plot the coherence appearing in our approximation of the mean-
force Gibbs state [Eq. (22)], and the coherence showing up in the symmetric
logarithmic derivative, to second order in $\gamma$ [Eq. (26)]. In this case,
however, there is no exact SNR to compare with. We see that the two different
couplings yield slightly different sensitivity, departing from the SNR of the
locally thermal probe. We can also see that, at large enough temperature, the
sensitivity drops to zero, as the spin populations saturate.

Rather counter-intuitively, we can see in the right panel that the
$\theta=\pi/4$ coupling generates temperature-dependent coherences in the
second-order approximation to $\tilde{\boldsymbol{\pi}}_{\scriptscriptstyle
S}$, and in the second-order corrections to
$\boldsymbol{L}_{\scriptscriptstyle S}$. Yet, we have proven that (diagonal)
measurements in the local energy basis $\boldsymbol{H}_{\scriptscriptstyle
S}$—which destroy such temperature information—do still saturate the ultimate
precision bound, at least to second order in $\gamma$ [cf. Eq. (19)]. In other
words, temperature-dependent coherences may only improve precision at higher
orders in finite-coupling quantum thermometry.

Figure 2: Applying Eq. (29) to the spin–boson model. The left panel shows the
SNR from our second-order formula for two types of probe–sample coupling;
namely, $\theta=3\pi/2$, i.e. $\boldsymbol{S}=\boldsymbol{\sigma}_{z}$ (solid
blue) and for $\theta=\pi/4$, i.e. the maximally coherent case (dashed
orange). We compare these to the sensitivity of a locally thermal probe
(dotted red). The top-right panel shows the second-order coherences of the
respective mean-force Gibbs states. The same color coding applies. The bottom-
right plot shows the second-order coherences of the respective symmetric
logarithmic derivatives in the basis of $\boldsymbol{H}_{\scriptscriptstyle
S}$. We emphasize that diagonal energy measurements saturate the second-order
precision bound despite $\tilde{\boldsymbol{\pi}}_{\scriptscriptstyle S}$ and
$\boldsymbol{L}_{\scriptscriptstyle S}$ acquiring temperature-dependent off-
diagonal elements at second order in $\gamma$. The other parameters are the
same as in Fig. 1.

It is interesting to note that, in this case, the correction $\xi(T)$ to the
optimal SNR has a definite sign—it is always negative. Indeed, the factor in
brackets in Eq. (35b) is negative for all $\omega$, while the spectral density
is obviously positive for all $\omega$. Hence, the weaker the probe–sample
coupling the better for this type of probe. For completeness, we calculate the
high–$T$ limit of Eq. (35b). We get

$\displaystyle f_{\scriptscriptstyle
S}(T,\omega)=-\frac{2\sin^{2}\left(\theta\right)\epsilon^{2}}{3\pi\omega}\,\beta^{3}+O(\beta^{5})$
(43)

which, after integration as per Eq. (35a), yields

$\displaystyle\xi(T)=-\frac{4\sin^{2}\left(\theta\right)\epsilon^{2}\Omega\Gamma\left(s\right)}{3\pi}\,\beta^{3}+O(\beta^{5}).$
(44)

This highlights what can already be seen in Fig. 2—that here, the vanishing-
coupling limit gives a far better approximation to the thermal sensitivity at
moderate-to-large temperature, when compared with the quantum Brownian motion
example.

##  4 Derivation

###  4.1 Mean-force Gibbs state

In this section we elaborate on the derivation of the results presented in
Sec. 2. The first step in our search for an explicit expression for
$\pazocal{F}_{\scriptscriptstyle S}(\beta)$ will be to obtain a second-order
approximation of the reduced probe state
$\tilde{\boldsymbol{\pi}}_{\scriptscriptstyle S}$, following [61]. Applying
Eq. (20) to $e^{-\beta\boldsymbol{H}}$ leads to:

$\displaystyle e^{-\beta\boldsymbol{H}}=$ $\displaystyle
e^{-\beta(\boldsymbol{H}_{\scriptscriptstyle
S}+\boldsymbol{H}_{\scriptscriptstyle
B})}\bigg{(}\boldsymbol{\mathbbm{1}}-\gamma\int^{\beta}_{0}d\beta_{1}\
e^{\beta_{1}(\boldsymbol{H}_{\scriptscriptstyle
S}+\boldsymbol{H}_{\scriptscriptstyle
B})}\boldsymbol{V}e^{-\beta_{1}(\boldsymbol{H}_{\scriptscriptstyle
S}+\boldsymbol{H}_{\scriptscriptstyle B})}$
$\displaystyle+\gamma^{2}\int^{\beta}_{0}d\beta_{1}\
\int^{\beta_{1}}_{0}d\beta_{2}\
e^{\beta_{1}(\boldsymbol{H}_{\scriptscriptstyle
S}+\boldsymbol{H}_{\scriptscriptstyle
B})}\boldsymbol{V}e^{-(\beta_{1}-\beta_{2})(\boldsymbol{H}_{\scriptscriptstyle
S}+\boldsymbol{H}_{\scriptscriptstyle
B})}\boldsymbol{V}e^{-\beta_{2}(\boldsymbol{H}_{\scriptscriptstyle
S}+\boldsymbol{H}_{\scriptscriptstyle B})}\bigg{)}+O(\gamma^{3}).$ (45)

Taking the trace yields an approximation for the total partition function

$\tilde{Z}=Z_{\scriptscriptstyle S}Z_{\scriptscriptstyle
B}-\gamma\,\beta\,\text{tr}_{\scriptscriptstyle
S}\left[e^{-\beta\boldsymbol{H}_{\scriptscriptstyle
S}}\boldsymbol{S}\right]\text{tr}_{\scriptscriptstyle
B}\left[e^{-\beta\boldsymbol{H}_{\scriptscriptstyle
B}}\boldsymbol{B}\right]\\\ +\gamma^{2}\int^{\beta}_{0}d\beta_{1}\
\int^{\beta_{1}}_{0}d\beta_{2}\ \text{tr}_{\scriptscriptstyle
S}\left[e^{-\beta\boldsymbol{H}_{\scriptscriptstyle
S}}e^{(\beta_{1}-\beta_{2})\boldsymbol{H}_{\scriptscriptstyle
S}}\boldsymbol{S}\,e^{-(\beta_{1}-\beta_{2})\boldsymbol{H}_{\scriptscriptstyle
S}}\boldsymbol{S}\right]\\\ \times\text{tr}_{\scriptscriptstyle
B}\left[e^{-\beta\boldsymbol{H}_{\scriptscriptstyle
B}}e^{(\beta_{1}-\beta_{2})\boldsymbol{H}_{\scriptscriptstyle
B}}\boldsymbol{B}\,e^{-(\beta_{1}-\beta_{2})\boldsymbol{H}_{\scriptscriptstyle
B}}\boldsymbol{B}\right]+O(\gamma^{3}),$ (46)

where Assumption I [Eq. (2.2.2)] has been applied. Using
$(1+x)^{-1}=1-x+O(x^{2})$, Assumption II [Eq. (2.2.2)], and the definition of
probe and sample auto-correlation functions from Eq. (25), we find

$\displaystyle\frac{1}{\tilde{Z}}=\frac{1}{Z_{\scriptscriptstyle
S}Z_{\scriptscriptstyle B}}\bigg{(}1-\gamma^{2}\int^{\beta}_{0}d\beta_{1}\
\int^{\beta_{1}}_{0}d\beta_{2}\ \Phi_{\scriptscriptstyle
S}(-i(\beta_{1}-\beta_{2}))\Phi_{\scriptscriptstyle
B}(-i(\beta_{1}-\beta_{2}))\bigg{)}+O(\gamma^{4}),$ (47)

Multiplying Eqs. (4.1) and (47), collecting terms, and taking the partial
trace over the sample, we find that the first order correction vanishes, so
that

$\displaystyle\tilde{\boldsymbol{\pi}}_{\scriptscriptstyle
S}=\frac{1}{\tilde{Z}}\text{tr}_{\scriptscriptstyle
B}\left[e^{-\beta\boldsymbol{H}}\right]=\boldsymbol{\pi}_{\scriptscriptstyle
S}\big{(}\boldsymbol{\mathbbm{1}}_{\scriptscriptstyle
S}+\gamma^{2}\boldsymbol{X}_{\scriptscriptstyle S}\big{)}+O(\gamma^{4}),$

where we have introduced the operator

$\boldsymbol{X}_{\scriptscriptstyle S}=\int^{\beta}_{0}d\beta_{1}\
\int^{\beta_{1}}_{0}d\beta_{2}\ \Phi_{\scriptscriptstyle
B}(-i(\beta_{1}-\beta_{2}))\\\
\times\big{[}e^{\beta_{1}\boldsymbol{H}_{\scriptscriptstyle
S}}\boldsymbol{S}e^{-(\beta_{1}-\beta_{2}))\boldsymbol{H}_{\scriptscriptstyle
S}}\boldsymbol{S}e^{-\beta_{2}\boldsymbol{H}_{\scriptscriptstyle
S}}-\Phi_{\scriptscriptstyle
S}(-i(\beta_{1}-\beta_{2}))\boldsymbol{\mathbbm{1}}_{\scriptscriptstyle
S}\big{]},$ (48)

Note that $\text{tr}_{\scriptscriptstyle
S}\left[\boldsymbol{X}_{\scriptscriptstyle
S}\boldsymbol{\pi}_{\scriptscriptstyle S}\right]=0$ as required by
normalisation.

Writing the diagonal elements of the operator
$(\boldsymbol{X}_{\scriptscriptstyle
S})_{nn}\coloneqq\bra{\epsilon_{n}}\boldsymbol{X}_{\scriptscriptstyle
S}\ket{\epsilon_{n}}$ in the eigenbasis of $\boldsymbol{H}_{\scriptscriptstyle
S}=\sum\nolimits_{n}\epsilon_{n}\ket{\epsilon_{n}}\bra{\epsilon_{n}}$ gives

$(\boldsymbol{X}_{\scriptscriptstyle S})_{nn}=\int^{\beta}_{0}d\beta_{1}\
\int^{\beta_{1}}_{0}d\beta_{2}\ \Phi_{\scriptscriptstyle
B}(-i(\beta_{1}-\beta_{2}))\\\
\times\big{[}\sum\nolimits_{k}\phi_{nk}(-i(\beta_{1}-\beta_{2}))-\Phi_{\scriptscriptstyle
S}(-i(\beta_{1}-\beta_{2}))\big{]},$ (49)

which can be turned into Eq. (23) by introducing the new variables
$u=\beta_{1}-\beta_{2}$ and $v=\beta_{1}+\beta_{2}$ and performing the
integral over $v$; namely

$\displaystyle(\boldsymbol{X}_{\scriptscriptstyle S})_{nn}$
$\displaystyle=\frac{1}{2}\,\int_{0}^{\beta}du\,\int_{u}^{2\beta-u}dv\,\Phi_{\scriptscriptstyle
B}(-iu)\big{(}\sum\nolimits_{k}\phi_{nk}(-iu)-\Phi_{\scriptscriptstyle
S}(-iu)\big{)}$ $\displaystyle=\int^{\beta}_{0}du\
(\beta-u)\Phi_{\scriptscriptstyle
B}(-iu)\big{(}\sum\nolimits_{k}\phi_{nk}(-iu)-\Phi_{\scriptscriptstyle
S}(-iu)\big{)},$

In turn, the off-diagonal elements of $\boldsymbol{X}_{\scriptscriptstyle S}$
are

$\displaystyle(\boldsymbol{X}_{\scriptscriptstyle
S})_{nm}=\sum\nolimits_{k}(\boldsymbol{S})_{nk}(\boldsymbol{S})_{km}\int^{\beta}_{0}d\beta_{1}\
e^{\beta_{1}\Delta_{kn}}\int^{\beta_{1}}_{0}d\beta_{2}\
e^{\beta_{2}\Delta_{mk}}\Phi_{\scriptscriptstyle B}(-i(\beta_{1}-\beta_{2}))$
(50)

Again, we may introduce $u=\beta_{1}-\beta_{2}$ and $v=\beta_{1}+\beta_{2}$
and integrate over $v$, which gives

$\displaystyle(\boldsymbol{X}_{\scriptscriptstyle
S})_{nm}=\frac{1}{\Delta_{mn}}\sum\nolimits_{k}(\boldsymbol{S})_{nk}(\boldsymbol{S})_{km}\,\int_{0}^{\beta}du\,\Phi_{\scriptscriptstyle
B}(-iu)\,\left(e^{u\Delta_{km}}\,e^{\beta\Delta_{mn}}-e^{u\Delta_{kn}}\right)\
\ \ \ (n\neq m).$ (51)

###  4.2 Symmetric logarithmic derivative

We now use the expression for the mean-force Gibbs state to obtain the SLD via
Eq. (9). To do so, we differentiate Eq. (22) and use
$\left\langle\boldsymbol{H}_{\scriptscriptstyle
S}\right\rangle_{\scriptscriptstyle S}=-\partial_{\beta}\ln
Z_{\scriptscriptstyle S}$ to get

$\displaystyle\partial_{\beta}\,\tilde{\boldsymbol{\pi}}_{\scriptscriptstyle
S}=-\Delta\boldsymbol{H}_{\scriptscriptstyle
S}\,\boldsymbol{\pi}_{\scriptscriptstyle
S}+\gamma^{2}\partial_{\beta}(\boldsymbol{\pi}_{\scriptscriptstyle
S}\boldsymbol{X}_{\scriptscriptstyle S})+O(\gamma^{4}).$ (52)

Turning to the operator
$e^{-\lambda\tilde{\boldsymbol{\pi}}_{\scriptscriptstyle S}}$ and combining
Eqs. (20) and (22) leads to

$\displaystyle e^{-\lambda\tilde{\boldsymbol{\pi}}_{\scriptscriptstyle
S}}=e^{-\lambda\boldsymbol{\pi}_{\scriptscriptstyle
S}}\left(\boldsymbol{\mathbbm{1}}_{\scriptscriptstyle
S}-\gamma^{2}\boldsymbol{\pi}_{\scriptscriptstyle
S}\int^{\lambda}_{0}d\lambda^{\prime}\
e^{\lambda^{\prime}\boldsymbol{\pi}_{\scriptscriptstyle
S}}\boldsymbol{X}_{\scriptscriptstyle
S}\,e^{-\lambda^{\prime}\boldsymbol{\pi}_{\scriptscriptstyle
S}}\right)+O(\gamma^{4}).$ (53)

Using this, we can finally compute the SLD. From Eqs. (52) and (53) we get

$e^{-\lambda\tilde{\boldsymbol{\pi}}_{\scriptscriptstyle
S}}(\partial_{\beta}\,\tilde{\boldsymbol{\pi}}_{\scriptscriptstyle
S})e^{-\lambda\tilde{\boldsymbol{\pi}}_{\scriptscriptstyle
S}}=-\Delta\boldsymbol{H}_{\scriptscriptstyle
S}\,\boldsymbol{\pi}_{\scriptscriptstyle
S}e^{-2\lambda\boldsymbol{\pi}_{\scriptscriptstyle S}}\\\
+\gamma^{2}\boldsymbol{\pi}_{\scriptscriptstyle
S}\int^{\lambda}_{0}d\lambda^{\prime}\
e^{-(\lambda-\lambda^{\prime})\boldsymbol{\pi}_{\scriptscriptstyle
S}}\boldsymbol{X}_{\scriptscriptstyle
S}\,e^{-(\lambda+\lambda^{\prime})\boldsymbol{\pi}_{\scriptscriptstyle
S}}\Delta\boldsymbol{H}_{\scriptscriptstyle
S}\,\boldsymbol{\pi}_{\scriptscriptstyle S},\\\
+\gamma^{2}e^{-\lambda\boldsymbol{\pi}_{\scriptscriptstyle
S}}\partial_{\beta}(\boldsymbol{\pi}_{\scriptscriptstyle
S}\boldsymbol{X}_{\scriptscriptstyle
S})\,e^{-\lambda\boldsymbol{\pi}_{\scriptscriptstyle S}}\\\
+\gamma^{2}\Delta\boldsymbol{H}_{\scriptscriptstyle
S}\,\boldsymbol{\pi}_{\scriptscriptstyle
S}^{2}e^{-2\lambda\boldsymbol{\pi}_{\scriptscriptstyle
S}}\int^{\lambda}_{0}d\lambda^{\prime}\
e^{\lambda^{\prime}\boldsymbol{\pi}_{\scriptscriptstyle
S}}\boldsymbol{X}_{\scriptscriptstyle
S}\,e^{-\lambda^{\prime}\boldsymbol{\pi}_{\scriptscriptstyle
S}}+O(\gamma^{4})$ (54)

We can now recover Eq. (26) using Eqs. (9) and working again in the eigenbasis
of $\boldsymbol{H}_{\scriptscriptstyle S}$. Namely,

$\displaystyle\boldsymbol{L}_{\scriptscriptstyle
S}(\beta)=-\Delta\boldsymbol{H}_{\scriptscriptstyle
S}+\gamma^{2}\sum\nolimits_{nm}\alpha_{nm}(\beta)\ket{\epsilon_{n}}\bra{\epsilon_{m}}+O(\gamma^{4}),$

with coefficients

$\displaystyle\alpha_{nm}(\beta)=\frac{2\,p_{n}}{p_{n}+p_{m}}\,\partial_{\beta}(\boldsymbol{X}_{\scriptscriptstyle
S})_{nm}+\frac{p_{n}\Delta_{nm}}{p_{n}+p_{m}}\,(\boldsymbol{X}_{\scriptscriptstyle
S})_{nm}.$

Here, we have used the fact that $\partial_{\beta}\,p_{n}=p_{n}\,(\langle
H_{\scriptscriptstyle S}\rangle_{\scriptscriptstyle
S}-\epsilon_{n})=-p_{n}\,\Delta_{n}$ (not to be confused with the energy gaps
$\Delta_{nk}$). As long as we know the sample correlation function
$\Phi_{\scriptscriptstyle B}(x)$ and the eigenstates of
$\boldsymbol{H}_{\scriptscriptstyle S}$, we will be able to obtain the
elements $(\boldsymbol{X}_{\scriptscriptstyle S})_{nm}$ from Eqs. (23) and
(51), and thus, the SLD as per Eq. (26).

###  4.3 Quantum Fisher information

The QFI can now be directly computed by using $\pazocal{F}_{\scriptscriptstyle
S}(\beta)=\text{tr}_{\scriptscriptstyle
S}\left[\boldsymbol{L}_{\scriptscriptstyle
S}^{2}\,\tilde{\boldsymbol{\pi}}_{\scriptscriptstyle S}\right]$. Collecting
terms up to second order yields

$\displaystyle\pazocal{F}_{\scriptscriptstyle
S}(\beta)=\partial^{2}_{\beta}\ln Z_{\scriptscriptstyle
S}+\gamma^{2}\sum\nolimits_{n}\left[p_{n}\Delta_{n}^{2}(\boldsymbol{X}_{\scriptscriptstyle
S})_{nn}-2\,p_{n}\Delta_{n}\alpha_{nn}\right]+O(\gamma^{4}),$ (55)

where $\partial^{2}_{\beta}\ln Z_{\scriptscriptstyle
S}=\langle(\Delta\boldsymbol{H}_{\scriptscriptstyle S})^{2}\rangle_{S}$ and
$\alpha_{nn}=\partial_{\beta}(\boldsymbol{X}_{\scriptscriptstyle S})_{nn}$, by
Eq. (26b).

We can still obtain the much more elegant expression (29), in terms of probe
and sample auto-correlation functions alone. Noting that

$\displaystyle\Phi_{S}(-iu)=\sum\nolimits_{nk}\phi_{nk}(-iu)$

is easy to see that

$\displaystyle\sum\nolimits_{n}p_{n}\Delta_{n}(\boldsymbol{X}_{\scriptscriptstyle
S})_{nn}$ $\displaystyle=-\partial_{\beta}\Phi_{\scriptscriptstyle S}(-iu)$
(56a)
$\displaystyle\sum\nolimits_{n}p_{n}\Delta_{n}^{2}(\boldsymbol{X}_{\scriptscriptstyle
S})_{nn}$ $\displaystyle=\partial_{\beta}^{2}\Phi_{\scriptscriptstyle
S}(-iu).$ (56b)

Furthermore, since $\sum\nolimits_{n}p_{n}\,\Delta_{n}=0$, we see that
$\partial_{\beta}(\boldsymbol{X}_{\scriptscriptstyle S})_{nn}$ reduces to

$\displaystyle\partial_{\beta}(\boldsymbol{X}_{\scriptscriptstyle
S})_{nn}=-\partial_{\beta}\tilde{\Phi}_{\scriptscriptstyle
B}(-iu)\,\partial_{\beta}\Phi_{\scriptscriptstyle S}(-iu),$ (57)

where, recall that $\tilde{\Phi}_{\scriptscriptstyle
B}(-iu)=(\beta-u)\,\Phi_{\scriptscriptstyle B}(-iu)$. Hence, Eq. (55) can be
recast as desired:

$\displaystyle\pazocal{F}_{\scriptscriptstyle
S}(\beta)=\partial^{2}_{\beta}\ln Z_{\scriptscriptstyle
S}+\gamma^{2}\int^{\beta}_{0}du\ \left[\tilde{\Phi}_{\scriptscriptstyle
B}(-iu)\,\partial^{2}_{\beta}\Phi_{\scriptscriptstyle
S}(-iu)+2\,\partial_{\beta}\tilde{\Phi}_{\scriptscriptstyle
B}(-iu)\,\partial_{\beta}\Phi_{\scriptscriptstyle
S}(-iu)\right]+O(\gamma^{4}).$

###  4.4 Classical Fisher information for energy measurements

Finally, we connect with our general result in Eq. (19). To that end, we use
the definition (11) of the (classical) Fisher information associated to energy
measurements of $\tilde{\boldsymbol{\pi}}_{\scriptscriptstyle S}$ on the basis
of the bare Hamiltonian $\boldsymbol{H}_{\scriptscriptstyle S}$. Namely,

$\displaystyle\pazocal{I}_{\boldsymbol{H}_{S}}(\beta)=\sum\nolimits_{n}\tilde{p}_{n}\,(\partial_{\beta}\ln\tilde{p}_{n})^{2}.$
(58)

Plugging in the mean-force Gibbs state populations from Eq. (28) gives

$\displaystyle\pazocal{I}_{\boldsymbol{H}_{S}}(\beta)$
$\displaystyle=\sum\nolimits_{n}\tilde{p}_{n}(\partial_{\beta}\ln\tilde{p}_{n})^{2}$
$\displaystyle=\partial_{\beta}^{2}\ln{Z_{\scriptscriptstyle
S}}+\gamma^{2}\sum\nolimits_{n}\left[p_{n}\Delta_{n}^{2}(\boldsymbol{X}_{\scriptscriptstyle
S})_{nn}-2\Delta_{n}\partial_{\beta}(\boldsymbol{X}_{\scriptscriptstyle
S})_{nn}\right]+O(\gamma^{4}).$

Here, we have used the identities
$\ln\tilde{p}_{n}=p_{n}+\gamma^{2}(\boldsymbol{X}_{\scriptscriptstyle
S})_{nn}+O(\gamma^{4})$, $\partial_{\beta}\ln p_{n}=-\Delta_{n}$, and the fact
that $\sum\nolimits_{n}p_{n}(\partial_{\beta}\ln
p_{n})^{2}=\partial_{\beta}^{2}\ln Z_{\scriptscriptstyle S}$. As expected, we
thus recover Eq. (55), meaning that local energy measurements on the probe are
as efficient as the truly optimal measurements in finite-coupling quantum
thermometry. This statement holds generally to second order in the coupling
for any thermometric setup. The only underpinning assumption is that probe and
sample have a separable coupling, and that the sample operator averages to
zero when in a local thermal state. This choice for the dissipative
interaction is indeed widespread in open quantum systems.

##  5 Conclusions

We have developed a theory of finite-coupling quantum thermometry, focusing on
the second-order correction terms. On the one hand, we have obtained closed-
form expressions for the symmetric logarithmic derivative, which fixes the
optimal basis in which to measure the probe; and the quantum Fisher
information, which gauges the responsiveness of the probe to temperature
fluctuations on the sample. Our formulae make minimal assumptions on the
probe–bath coupling and can be readily applied to any setup by simply
computing the probe and sample auto-correlation functions. In particular, we
make no assumptions on the nature of the sample, so that our framework is
applicable even when standard open-system weak-coupling tools—based on the
Born–Markov approximation—break down.

On the other hand, we have proved that projective measurements in the local
energy basis of the probe are always optimal to the first two leading orders
in the probe–sample coupling. That is, even if temperature information may be
encoded in the off-diagonal elements of the probe’s marginal state, the
advantage in extracting and processing such information would only show up in
higher order terms. Hence, measurement optimisation is unnecessary for quantum
thermometry at moderate couplings.

Finally, we have illustrated the use of our approach with two relevant
examples—the exactly solvable Caldeira–Legett model (which served as
benchmark) and the spin–boson model. Specifically, we have derived an explicit
expression for the second-order correction to the signal-to-noise ratio for
temperature estimation, suitable for any open-system model which uses a
canonical linear bath with quasi-continuum spectrum; regardless of the
specific spectral density. Such formula can thus be of independent interest.

Our results open up new directions in finite-coupling thermometry. Namely, our
formulae can easily deal with thermometry on more realistic non-linear sample
models, which remain virtually unexplored to date. They can also facilitate
the treatment of complex probe–sample interactions, such as the ones appearing
when modelling an atomic impurity in a Bose–Einstein condensate [34], without
the need for linearisation. Crucially, having established the optimality of
energy measurements up to second order in the coupling strength highlights the
importance of directing practical efforts towards measuring the probe in its
local basis. Furthermore, guaranteeing that optimal thermometry may be
attained using a fixed temperature-independent basis also eliminates the need
for more complex adaptive schemes.

Interestingly, lifting our Assumption II on the vanishing expectation value of
the interaction in the stationary state of the sample would result in
inhomogeneous terms, which might introduce a thermometric advantage in terms
of additional temperature dependence. Studying how this may be exploited in
practice will be the subject of future study.

## Acknowledgements

JG is supported by a scholarship from CEMPS at the University of Exeter. LAC
acknowledges the Spanish Ramón y Cajal program (Fellowship: RYC2021-325804-I)
funded by MCIN/AEI/10.13039/501100011033 and “NextGenerationEU”/PRTR. KVH
acknowledges support by the University of Potsdam startup funds. MPL
acknowledges support by the the Swiss National Science Foundation through an
Ambizione Grant No. PZ00P2-186067. HM acknowledges support from the Royal
Commission for the Exhibition of 1851. LAC, KVH, MPL and HM acknowledge the US
National Science Foundation (Grant No. NSF PHY1748958) and thank the Kavli
Institute for Theoretical Physics for their warm hospitality during the
program “Thermodynamics of quantum systems: Measurement, engines, and
control”, during which this (long overdue) project started.

##  Appendix A Examples with
$\left\langle\boldsymbol{B}\right\rangle_{\scriptscriptstyle B}\neq 0$

In this short Appendix, we give explicit examples in which Assumption II in
Eq. (2.2.2)—namely
$\left\langle\boldsymbol{B}\right\rangle_{\scriptscriptstyle B}=0$—is not
satisfied. Specifically, we begin by illustrating how, in this case, the QFI
and SLD may pick up terms of $O(\gamma)$. To that end, let us consider the
two-qubit model

$\boldsymbol{H}=\boldsymbol{H}_{\scriptscriptstyle
S}+\boldsymbol{H}_{\scriptscriptstyle
B}+\boldsymbol{H}_{\text{int}}=\omega\,\boldsymbol{\sigma}_{z,1}\otimes\boldsymbol{\mathbbm{1}}_{2}+\omega\,\boldsymbol{\mathbbm{1}}_{1}\otimes\boldsymbol{\sigma}_{z,2}+\gamma\,\boldsymbol{\sigma}_{z,1}\otimes\boldsymbol{\sigma}_{z,2},$

where the sub-indices label the qubits, ‘1’ being the probe and ‘2’ being the
sample. Also, in all the expressions that follow, we shall set $\omega=1$ for
simplicity.

The corresponding mean-force Gibbs state can be expanded as

$\tilde{\boldsymbol{\pi}}_{\scriptscriptstyle
S}=\boldsymbol{\pi}_{\scriptscriptstyle
S}+\gamma\,p_{1}\,\boldsymbol{\sigma}_{z,1}+\gamma^{2}p_{2}\,\boldsymbol{\sigma}_{z,1}+O(\gamma^{3}),$

with $p_{1}=\frac{1}{2}\,\beta\,\operatorname{sech}^{2}{\beta}\,\tanh{\beta}$
and
$p_{2}=\frac{1}{2}\,\beta^{2}\,\operatorname{sech}^{2}{\beta}\,\tanh^{3}{\beta}$.
As we anticipated, in this case the QFI does feature a linear term, namely

$\pazocal{F}_{\scriptscriptstyle
S}(\beta)=\partial_{\beta}^{2}\ln{Z_{S}}+\operatorname{sech}^{4}{\beta}\left[\beta(\cosh{2\beta}-3)-\sinh{2\beta}\right]\,\gamma+O(\gamma^{2}),$

and so does the symmetric logarithmic derivative

$\boldsymbol{L}_{\scriptscriptstyle
S}=-\Delta\boldsymbol{H}_{\scriptscriptstyle
S}+l_{1}\,\boldsymbol{\sigma}_{z,1}+O(\gamma^{2}),$

with
$l_{1}=1+\tanh{\beta}+\operatorname{sech}^{2}{\beta}\,(\beta+2\beta\,\tanh{\beta}-1)$.
We know from Sec. 2.2.1 that the classical Fisher information for measurements
in the local energy basis and the QFI must agree in the zeroth and first order
in $\gamma$. However, as noted in Sec. 2.1, due to the coupling in this toy
model being ’non-dissipative’, i.e.,
$[\boldsymbol{H}_{\text{int}},\boldsymbol{H}]=0$,
$\boldsymbol{L}_{\scriptscriptstyle S}$ is diagonal in the
$\boldsymbol{H}_{\scriptscriptstyle S}$ basis at all orders. Hence, energy
measurements are always optimal in this particular case.

Let us now turn to a more general scenario. Namely,

$\boldsymbol{H}=\boldsymbol{H}_{\scriptscriptstyle
S}+\boldsymbol{H}_{\scriptscriptstyle
B}+\boldsymbol{H}_{\text{int}}=\omega\,\boldsymbol{\sigma}_{z,1}\otimes\boldsymbol{\mathbbm{1}}_{2}+\omega\,\boldsymbol{\mathbbm{1}}_{1}\otimes\boldsymbol{\sigma}_{z,2}+\gamma\,\boldsymbol{\Pi}_{1}\otimes\boldsymbol{\sigma}_{x,2},$

where $\boldsymbol{\Pi}=\ket{1}\bra{1}$. In this case, we can see that

$\displaystyle\tilde{\boldsymbol{\pi}}_{\scriptscriptstyle
S}=\boldsymbol{\pi}_{\scriptscriptstyle
S}+\gamma\,p_{1}\,\boldsymbol{\sigma}_{x,1}+\gamma^{2}\,p_{2}\,\boldsymbol{\sigma}_{z,1}+O(\gamma^{3}),$

with coefficients

$\displaystyle p_{1}$
$\displaystyle=\frac{1}{4}\,\tanh{\beta}\,(\tanh{\beta}-1)$ $\displaystyle
p_{2}$
$\displaystyle=\frac{1}{8}\,\operatorname{sech}{\beta}\,(\tanh{\beta}-1)\,(\beta\,\operatorname{sech}{\beta}-\sinh{\beta}).$

In particular, note how the mean-force Gibbs state does feature
$\beta$-dependent coherences.

The corresponding symmetric logarithmic derivative is

$\boldsymbol{L}_{\scriptscriptstyle
S}=-\Delta\boldsymbol{H}_{\scriptscriptstyle
S}+\gamma\,l_{1}\,\boldsymbol{\sigma}_{x,1}+O(\gamma^{2}),$

with $l_{1}=\frac{1}{2}\,\operatorname{sech}^{2}{\beta}\,(2\tanh{\beta}-1)$,
meaning that the optimal measurement is non-diagonal in the energy basis
already to first order in the coupling strength. And yet, as we proved in full
generality in Sec. 2.2.1, the classical Fisher information of local energy
measurements and the quantum Fisher information must agree to second leading
order. Indeed, direct calculation shows that

$\pazocal{F}_{\scriptscriptstyle
S}(\beta)-\pazocal{I}_{\boldsymbol{H}_{S}}(\beta)=\frac{1}{4}\,\operatorname{sech}^{4}{\beta}\,(1-2\tanh{\beta})^{2}\,\gamma^{2}+O(\gamma^{3}).$

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