#  Fundamental limits in Bayesian thermometry and attainability via adaptive
strategies

Mohammad Mehboudi<EMAIL_ADDRESS>Département de Physique
Appliquée, Université de Genève, 1211 Geneva, Switzerland  Mathias R.
Jørgensen<EMAIL_ADDRESS>Department of Physics, Technical University of
Denmark, 2800 Kongens Lyngby, Denmark  Stella Seah  Département de Physique
Appliquée, Université de Genève, 1211 Geneva, Switzerland  Jonatan B. Brask
Department of Physics, Technical University of Denmark, 2800 Kongens Lyngby,
Denmark  Jan Kołodyński  Centre for Quantum Optical Technologies, Centre of
New Technologies, University of Warsaw, 02-097 Warsaw, Poland  Martí Perarnau-
Llobet<EMAIL_ADDRESS>Département de Physique Appliquée,
Université de Genève, 1211 Geneva, Switzerland

###### Abstract

We investigate the limits of thermometry using quantum probes at thermal
equilibrium within the Bayesian approach. We consider the possibility of
engineering interactions between the probes in order to enhance their
sensitivity, as well as feedback during the measurement process, i.e.,
adaptive protocols. On the one hand, we obtain an ultimate bound on
thermometry precision in the Bayesian setting, valid for arbitrary
interactions and measurement schemes, which lower bounds the error with a
quadratic (Heisenberg-like) scaling with the number of probes. We develop a
simple adaptive strategy that can saturate this limit. On the other hand, we
derive a no-go theorem for non-adaptive protocols that does not allow for
better than linear (shot-noise-like) scaling even if one has unlimited control
over the probes, namely access to arbitrary many-body interactions.

Introduction.—Preparing quantum systems at low temperatures is an essential
task for development of quantum technologies Celi  _et al._ (2016); Bloch  _et
al._ (2008, 2012). Measuring temperature precisely is necessary to validate
cooling and ensure the performance of quantum protocols, and has been
demonstrated in cutting-edge experiments Leanhardt  _et al._ (2003); Bouton
_et al._ (2020); Scigliuzzo  _et al._ (2020); Olf  _et al._ (2015); Gati  _et
al._ (2006); Ronzani  _et al._ (2018); Spiegelhalder  _et al._ (2009); Tan
_et al._ (2017); Adam  _et al._ (2021). However, measuring the temperature of
quantum systems is often challenging. On the one hand, due to the scarcity of
thermal fluctuations at such low temperatures, the relative statistical error
on thermometry, i.e. the signal-to-noise ratio, can be exceptionally notable.
On the other hand, the fragility of quantum systems, requires additional
forward planning in order to minimise disturbance while maximising the
information obtained. The theory of quantum thermometry is built to address
this pivotal task Mehboudi  _et al._ (2019a); De Pasquale and Stace (2018).

When quantum systems are used as thermometers or probes, they offer resources
such as entanglement, coherence, and many-body interactions, that can improve
precision Jevtic  _et al._ (2015); Correa  _et al._ (2015); Paris (2015); De
Pasquale  _et al._ (2016); Razavian  _et al._ (2019); Correa  _et al._ (2017);
Hovhannisyan and Correa (2018); Mehboudi  _et al._ (2015); Mirkhalaf  _et al._
(2021); Hofer  _et al._ (2017); Seah  _et al._ (2019); Mehboudi  _et al._
(2019b); Paz-Silva  _et al._ (2017); Campbell  _et al._ (2017); Mitchison  _et
al._ (2020); Latune  _et al._ (2020); Potts  _et al._ (2019); Jørgensen  _et
al._ (2020); Planella  _et al._ (2020). To date, such enhancements have been
developed in the context of local thermometry, aiming at designing a
thermometer that detects the smallest temperature variations around a known
temperature Mehboudi  _et al._ (2019a); De Pasquale and Stace (2018). In many
practical situations, however, one might not know the temperature accurately
beforehand. Rather, one has only limited prior knowledge about the temperature
of the sample. Under such circumstances, Bayesian estimation is a more
suitable approach to thermometry, and has been the subject of a few recent
studies Rubio  _et al._ (2020); Alves and Landi (2021).

The goal of this work is to set the ultimate bounds of Bayesian equilibrium
thermometry, and to develop adaptive strategies to saturate them. It is
insightful to first recall analogous results in the local approach to
equilibrium thermometry Mehboudi  _et al._ (2019a); De Pasquale and Stace
(2018). In that case, for any unbiased estimator ${\tilde{\theta}}$ of the
temperature $\theta_{0}$, the mean square error is inversely proportional to
the heat capacity of the probe: $\Delta{\tilde{\theta}}\propto 1/C$ Jahnke
_et al._ (2011); Paris (2015); Correa  _et al._ (2015); Mehboudi  _et al._
(2015). For $n$-body probes, $C$ can scale super-extensively with $n$ in the
vicinity of a critical point, with the ultimate bound $C\approx n^{2}/4$
Correa  _et al._ (2015); Płodzień  _et al._ (2018)—a quadratic scaling with
the number of resources reminiscent of the Heisenberg scaling in quantum
metrology Giovannetti  _et al._ (2004). Here, we show that similar bounds hold
in the Bayesian approach, but that adaptive strategies are needed to saturate
them in contrast to the local case. In fact, we prove that any non-adaptive
strategy necessarily leads to $\Delta{\tilde{\theta}}\propto 1/n$ for
sufficiently large $n$—i.e., a shot-noise-like scaling Giovannetti  _et al._
(2004)—a no-go result that holds even when arbitrary control over the $n$-body
probe Hamiltonian is allowed. This shows that adaptive measurement strategies
are a crucial ingredient for optimal quantum thermometry whenever the
temperature value is a priori not perfectly known.

Figure 1: Schematic representation of the adaptive scenario. A total of $N$
probes are used in groups of $n$ to estimate the temperature of the sample,
$\theta_{0}$. Initially, our prior temperature distribution is given by
$p(\theta)$, according to which we choose the Hamiltonian of the first $n$
probes to be $H_{n}^{(1)}$ that minimises the expected mean square logarithmic
error. The probes interact and thermalise with the sample followed by an
energy measurement, yielding an outcome, say $x_{1}$. Our knowledge about the
temperature will be reflected in the posterior distribution $p(\theta|x_{1})$.
This will be used as the prior for the second round—in order to find the
optimal Hamiltonian $H_{n}^{(2)}$. This process is repeated $m=N/n$ times. In
contrast, in the non-adaptive scenario the Hamiltonian is fixed
$H_{n}^{(k)}=H_{n}~{}\forall k$.

Preliminaries and setup.— We consider estimation of temperature of a sample,
denoted as $\theta_{0}$, given some prior distribution $p(\theta)$ reflecting
our initial knowledge on $\theta_{0}$. We assume we have at our disposal $N$
copies of a $d$-dimensional system that we use as probes. The temperature is
inferred by letting the probes reach thermal equilibrium at $\theta_{0}$, and
then measuring them—corresponding to the framework of equilibrium thermometry
Mehboudi  _et al._ (2019a); De Pasquale and Stace (2018). In order to
establish fundamental bounds, we assume full control on the Hamiltonian of the
probes, and in particular the ability to make them interact. Therefore,
alternatively one can think of a $d^{N}$-dimensional probe, which constitutes
our resource.

The thermometry process is divided into $m$ rounds, each involving $n=N/m$
probes. Every round consists of: (I) preparation of the $n$-body probe, (II)
interaction with the sample and thermalisation, (III) measurement/data
acquisition, and (IV) data analysis (see Fig. 1). In the first round, we start
by engineering the Hamiltonian $H_{n}^{(1)}$ of the $n$-body probe into any
desired configuration based on the prior distribution $p(\theta)$. That is, we
arrange the energy distribution of the $n$-body probe to become most sensitive
to the relevant temperature range. Next, in step (II), this $n$-body system is
put in contact with the sample, reaching a Gibbs state
$\omega_{\theta_{0}}(H_{n}^{(1)})=\exp[-H_{n}^{(1)}/{\theta}_{0}]/Z$, with
$Z={\rm Tr}(\exp[-H_{n}^{(1)}/{\theta}_{0}])$ the partition function. Then, in
step (III), a measurement is performed that yields an outcome $x_{1}$. We
focus on energy measurements since they are optimal as the Gibbs state is
diagonal in the energy basis. In the data analysis (step (IV)), the
measurement outcome is used to obtain the posterior distribution through
Bayes’ rule:

$p(\theta|x_{1})=\frac{p(x_{1}|\theta)p(\theta)}{p(x_{1})},$ (1)

where $p(x|\theta)$ is the likelihood function (which depends on the
temperature and the Hamiltonian), $p(\theta)$ is the prior probability
distribution on $\theta$, and $p(x)=\int\\!{\rm
d}\theta\,p(\theta)\,p(x|\theta)$ is the outcome probability. The next round
proceeds in an analogous way, but replacing the prior $p(\theta)$ by
$p(\theta|x_{1})$ and $H_{n}^{(1)}$ by $H_{n}^{(2)}$. Likewise, in round
$k>1$, $p(\theta)$ is replaced by $p(\theta|{\bf x}_{k-1})$ with ${\bf
x}_{k-1}\equiv\\{x_{k-1},...,x_{2},x_{1}\\}$ and $H_{n}^{(1)}$ is replaced by
$H_{n}^{(k)}$. Such a strategy is adaptive since $H_{n}^{(k)}$ depends on
${\bf x}_{k-1}$. In contrast, a non-adaptive strategy satisfies
$H_{n}^{(k)}=H_{n}$ $\forall k$, where $H_{n}$ is chosen according to the
initial prior $p(\theta)$ only. At the end of the thermometry process (round
$m$), the final estimate ${\tilde{\theta}}({\bf x}_{m})$ of $\theta_{0}$ is
computed.

In order to gauge the quality of the estimator, we need to introduce some
figure of merit, or an error quantifier that describes how far
${\tilde{\theta}}$ is from $\theta_{0}$, on average. A natural measure which
is suitable for equilibrium probes is the expected mean square logarithmic
error (EMSLE) (see Rubio  _et al._ (2020) for justification and the
accompanying paper Jørgensen  _et al._ (2021) for a deeper analysis and
generalisation)

$\displaystyle\text{\footnotesize{EMSLE}}\coloneqq\int{\rm d}\theta\hskip
1.42262ptp(\theta)\int{\rm d}{\bf x}_{m}\hskip 1.42262ptp({\bf
x}_{m}|\theta)\ln^{2}\left[\frac{{\tilde{\theta}}({\bf
x}_{m})}{\theta}\right],$ (2)

with $d{\bf x}_{m}\coloneqq dx_{m}...dx_{1}$. Moreover,

$\displaystyle{\tilde{\theta}}({\bf x}_{m})=\exp\left[\int{\rm
d}\theta\frac{p(\theta)p({\bf x}_{m}|\theta)}{p({\bf
x}_{m})}\ln\theta\right],$ (3)

is the optimal temperature estimator, i.e., it minimises EMSLE Rubio  _et al._
(2020).

We wish to find lower bounds for EMSLE, as well as optimal strategies to
saturate them, for both adaptive and non-adaptive measurements. More
precisely, our aim is to minimise EMSLE as a function of the number $N$ of
probes, with $N=mn$. We will pay particular attention to the relevant case
where $m\gg 1$ is large (asymptotic regime) but $n$ is limited due to e.g.
experimental limitations on the amount of probes that can be collectively
processed. In this case, we will focus on the scaling of EMSLE with $n$ for a
fixed but large $m$.

Main results.—Our main results are (i) an ultimate precision limit for
Bayesian thermometry that holds for both adaptive and non-adaptive strategies,
which in principle allows for a quadratic (Heisenberg-like) scaling with $n$,
(ii) a no-go theorem that forbids super-extensive scaling in any non-adaptive
scenario, and (iii) an adaptive strategy that reaches the ultimate limit.
These results are derived in what follows (some technical details are given in
the Appendix).

Given the prior distribution $p(\theta)$, one may utilise the Van Trees
inequality Van Trees (2004); Trees and Bell (2007) to construct a lower bound
on the estimation error after $m$ rounds of the protocol

$\displaystyle\text{\footnotesize{EMSLE}}^{-1}\leqslant Q[p(\theta)]$
$\displaystyle+\sum_{k=1}^{m}\int d\textbf{x}_{k-1}\hskip
1.42262ptp(\textbf{x}_{k-1})\int d\theta\hskip 1.42262pt\hskip
1.42262ptp(\theta|{\bf x}_{k-1})\hskip 1.42262ptC(\theta;H_{n}^{(k)}),$ (4)

where $p(\theta|{\textbf{x}_{0}})=p(\theta)$,
$p({\textbf{x}_{0}})=p({\textbf{x}_{0}}|\theta)=1$, and $\int
d{\textbf{x}_{0}}=1$ are introduced to compress our notation. Here,
$Q[p(\theta)]$ quantifies the prior information given by

$Q[p(\theta)]\coloneqq\int d\theta\
p(\theta)\left[1+\theta\partial_{\theta}\log p(\theta)\right]^{2},$ (5)

while the second term takes into account the information acquired through all
measurements. The heat capacity of the probe at round $k$ of the measurement
is denoted $C(\theta;H_{n}^{(k)})$, with the Hamiltonian $H_{n}^{(k)}$
designed according to the prior and the information acquired so far. Recall
that, by definition,
$C(\theta;H_{n})\coloneqq\partial_{\theta}E(\theta;H_{n})$ where
$E(\theta;H)={\rm Tr}[H\omega_{\theta}(H)]$ is the energy of the probe at
thermal equilibrium. In order to bound Eq. ( Fundamental limits in Bayesian
thermometry and attainability via adaptive strategies ), we first define the
maximum of the integrand over $\\{H_{n}^{(k)}\\}_{k=1}^{m}$ for a given
specific trajectory ${\bf x}_{m}$:

$\displaystyle\Gamma({\bf x}_{m})\coloneqq$
$\displaystyle\max_{{\\{H_{n}^{(k)}\\}}_{k}}\,\sum_{k=1}^{m}\int d\theta\hskip
1.42262ptp(\theta|{\bf x}_{k-1})C(\theta;H_{n}^{(k)})$
$\displaystyle\leq\sum_{k=1}^{m}\int d\theta\hskip 1.42262ptp(\theta|{\bf
x}_{k-1})C_{D}=mC_{D}$ (6)

where $C_{D}\coloneqq\max_{H_{n}}C(\theta;H_{n})$, i.e., the maximum heat
capacity of an $n$-body probe, and in the last line we used that $C_{D}$ is
independent of $\theta$ (see Correa  _et al._ (2015) and the Appendix for an
explicit expression for $C_{D}$). Furthermore, it is bounded by
$C_{D}\leq\frac{n^{2}}{4}\log^{2}d$, which is quickly saturated as $n$ is
increased. Putting everything together, we obtain from ( Fundamental limits in
Bayesian thermometry and attainability via adaptive strategies ) the following
chain of inequalities:

$\displaystyle\text{\footnotesize{EMSLE}}^{-1}$ $\displaystyle\leqslant
Q[p(\theta)]+mC_{D}$ $\displaystyle\leqslant
Q[p(\theta)]+m\frac{n^{2}}{4}\log^{2}d.$ (7)

This gives an ultimate bound on Bayesian thermometry [Result (i)], which both
adaptive and non-adaptive strategies should respect. This bound implies that
any Bayesian thermometry protocol is ultimately limited by a quadratic
Heisenberg-like scaling.

The ultimate bound ( Fundamental limits in Bayesian thermometry and
attainability via adaptive strategies ) becomes tight and can be saturated by
adaptive strategies in the regime $m\gg 1$ (see results below). However, let
us now prove that any non-adaptive strategy fails to saturate it, and in fact
$\text{\footnotesize{EMSLE}}^{-1}$ can increase at most linearly with $n$. To
this end, we take few steps back and start with the second term in (
Fundamental limits in Bayesian thermometry and attainability via adaptive
strategies ). Using that the Hamiltonian remains constant throughout the
protocol, $H_{n}^{(k)}=H_{n}$ $\forall k$, this reads

$\displaystyle{\bar{\Gamma}}$
$\displaystyle\coloneqq\sum_{k=1}^{m}\iint\\!d\theta
d\textbf{x}_{k-1}\,p(\theta)\,p(\textbf{x}_{k-1}|\theta)\,C(\theta;H_{n})$
$\displaystyle=m\int\\!d\theta\,p(\theta)\,C(\theta;H_{n}).$ (8)

Integrating by parts—recall that
$C(\theta;H_{n})=\partial_{\theta}E(\theta;H_{n})$—and maximising over $H_{n}$
gives

${\bar{\Gamma}}\leqslant m\max_{H_{n}}\int d\theta\
[-\partial_{\theta}p(\theta)]E(\theta;H_{n}),$ (9)

where we assumed that $p(\theta)E(\theta;H_{n})$ is smooth and vanishes at the
boundaries. By defining ${\cal R}$ as the temperature domain where
$\partial_{\theta}p(\theta)\leqslant 0$ we have

$\displaystyle{\bar{\Gamma}}$ $\displaystyle\leqslant m\max_{H_{n}}\int_{\cal
R}d\theta\ [-\partial_{\theta}p(\theta)]E(\theta;H_{n})$
$\displaystyle\leqslant m\int_{\cal R}d\theta\
[-\partial_{\theta}p(\theta)]\max_{H_{n}}E(\theta;H_{n})$ (10)

To make further progress, in the Appendix we derive an upper bound on the
energy of an $n$-body system at thermal equilibrium (with total dimension
$d^{n}$):

$\displaystyle\max_{H_{n}}E(\theta;H_{n})$ $\displaystyle\leq\theta\hskip
1.42262pt{\cal W}_{n}\leq\theta\hskip 1.42262ptn\log d$ (11)

where ${\cal W}_{n}\coloneqq W[(d^{n}-1)/e]$ with $W$ being the Lambert W
function, and the second equality is saturated as $n\gg 1$. Plugging these
results back into ( Fundamental limits in Bayesian thermometry and
attainability via adaptive strategies ) we obtain a no-go theorem for non-
adaptive strategies [Result (ii)]

$\displaystyle\text{\footnotesize{EMSLE}}^{-1}\overset{\mathrm{non-
adaptive}}{\leqslant}Q[p(\theta)]+f[p(\theta)]mn\log\\!d,$ (12)

where $f[p(\theta)]=\int_{\cal R}d\theta\ [-\partial_{\theta}p(\theta)]\theta$
is a functional of the prior. Crucially, the bound (12) implies that, even
with arbitrary control over the $n$-body Hamiltonian, one cannot go above a
linear scaling in $n$ with non-adaptive strategies (compare with the general
bound ( Fundamental limits in Bayesian thermometry and attainability via
adaptive strategies )).

Let us now give some physical intuition behind the no-go result Eq. (12). It
has already been noted in the literature that engineered probes for
thermometry show enhanced sensitivity only in a small temperature range
$\Delta$ Correa  _et al._ (2015); Mehboudi  _et al._ (2019a); Campbell  _et
al._ (2018); Román-Ancheyta  _et al._ (2019); Mok  _et al._ (2020). In fact,
finite-size scaling theory tells us that if $C\propto n^{1+\alpha}$, then
$\Delta\propto n^{-\gamma}$ with $\gamma\geq\alpha$ in order to ensure that
the energy density of an equilibrium state remains finite Huang (2009) (see in
this sense Eq. (11)). This implies that, for any $p(\theta)$ with a finite
width (independent of $n$), the figure of merit $\int{\rm d}\theta\hskip
1.42262ptp(\theta)C(\theta)$ entering into (8) can grow at most linearly with
$n$ for sufficiently large $n$. In other words, optimal $n$-body probes
require priors with a width smaller than $\mathcal{}O(1/n)$ to obtain super-
linear scaling, and conversely a finite width in $p(\theta)$ will eventually
kill any super-linear scaling. The no-go result (12) makes this intuition
rigorous. We also note that (12) is a particular case of a more general family
of bounds where the prior-dependent quantity $f[p(\theta)]$ can be replaced by
other functionals which can become tighter for some $p(\theta)$; this is
discussed in the Appendix.

The above reasoning also explains why adaptive protocols can potentially
saturate ( Fundamental limits in Bayesian thermometry and attainability via
adaptive strategies ). By updating the prior $p(\theta)$ to the posterior
$p(\theta|{\bf x}_{j-1})$ in each step of the process ($j=1,...,m$), the
posterior can stay inside the optimal region for sufficiently large $m$, thus
enabling super-linear thermometry precision. Furthermore, as a result of
sharpening the prior knowledge, we are getting closer to a local thermometry
scenario Jørgensen  _et al._ (2021). This suggests using optimal probes for
local thermometry as an ansatz for the Bayesian thermometry with adaptive
strategies. The optimal thermometer in the local scenario is an effective two-
level system with $d^{n}-1$-fold degeneracy in the excited state, with an
energy gap that depends both on $n$ and the temperature Correa  _et al._
(2015). For large $n$ its heat capacity reads
$C_{D}\approx(n^{2}/4)\log^{2}d$, as already mentioned Płodzień  _et al._
(2018). Inspired by this, at the $k$th round we restrict to the class of
Hamiltonians $H_{n}^{(k)}$ with the aforementioned two-level structure, and
tune the energy gap such that the EMSLE (2) is minimised. As we shall see in
the example below, we are able to achieve a quadratic scaling with $n$ and
saturate ( Fundamental limits in Bayesian thermometry and attainability via
adaptive strategies ) through this strategy [Result (iii)].

---
Figure 2: Left—Contour plot of the prior versus the number of simulation
rounds $m$ (logarithmic scale), and the ratio $\theta/\theta_{0}$ between the
estimated $\tilde{\theta}$ and true temperature $\theta_{0}$. As $m$
increases, the prior sharpens around the true temperature, and
${\tilde{\theta}}/\theta_{0}$ approaches unity. Here, we have set $n=1$,
$\alpha=1$, $\theta_{\min}=1$, and $\theta_{\max}=10$ in arbitrary units.
Right—Loglog plot of the expected mean square logarithmic error (EMSLE)
attained by the adaptive strategy vs. the total number of qubits $N$. Dark
solid lines represent different values of $n$. They show that, for
sufficiently large $N$, the bigger $n$ is the smaller the error can get. The
red-dashed line is the (not necessarily tight) bound on non-adaptive
strategies: only the shaded area can be achieved using non-adaptive protocols.
One can cross the border with adaptive strategies for $n>10$. Figure 3:
(Dashed red) Loglog plot of the normalised expected mean square logarithmic
error (EMSLE) after $m$ rounds of the adaptive scheme—for sufficiently large
$m$—vs $n$. This shows that for large enough $n$ the error vanishes
quadratically with $n$, which can be better seen from the inset. (Blue) The
minimum achievable EMSLE given by the r.h.s of ( Fundamental limits in
Bayesian thermometry and attainability via adaptive strategies ). The perfect
agreement shows the efficiency of the proposed adaptive protocol.

Case study.—The results presented here are valid for a broad class of priors,
but in what follows we stick to a specific choice in order to illustrate their
usage. In any relevant application of thermometry, the temperature is known a
priori to lie within a certain range, i.e.,
$\theta_{\min}\leqslant\theta_{0}\leqslant\theta_{\max}$. We use a family of
probability distributions that are suitable in this case and were proposed in
Li  _et al._ (2018):

$p(\theta)=\frac{1}{k_{\alpha}(\theta_{\max}-\theta_{\min})}\left[e^{\alpha\sin^{2}\left(\pi\frac{\theta-\theta_{\min}}{\theta_{\max}-\theta_{\min}}\right)}-1\right]$
(13)

with

$k_{\alpha}\coloneqq e^{\alpha/2}I_{0}(\alpha/2)-1,$ (14)

where $I_{0}$ is the modified Bessel function of the first kind. In the limit
$\alpha\rightarrow-\infty$ the above prior becomes a constant over the domain,
while in the limit $\alpha\rightarrow 0$ we have
$p(\theta)\propto\sin^{2}(2\theta)$.

The adaptive strategy works as follows. We consider as a resource $N$ qubits,
which are divided in $m$ groups of $n$ qubits. In each group, the $n$-qubit
Hamiltonian is engineered to become a two-level system with degeneracy
$(2^{n}-1)$ and with a tunable gap $\epsilon$ (recall that this structure is
optimal for local thermometry). In the first round, we tune the gap to
$\epsilon^{(1)}$ to minimise the single shot EMSLE, that is we set $m=1$ in
(2)

$\displaystyle\text{\footnotesize{EMSLE}}=\int{\rm d}\theta\hskip
1.42262ptp(\theta)\sum_{x}\hskip
1.42262ptp(x|\theta)\ln^{2}\left[\frac{{\tilde{\theta}}(x)}{\theta}\right],$
(15)

where now $x\in\\{0,1\\}$ corresponds to observing the ground or the excited
state. Then, we measure the energy of the system. Given the outcome $x_{1}$ is
observed, we update the prior to $p(\theta)\rightarrow p(\theta|x_{1})$, and
implement the same procedure to choose $\epsilon^{(2)}$ in the second round
(i.e., we minimise (15) replacing $p(\theta)\rightarrow p(\theta|x_{1})$).
This process is repeated until all probes are used.

In our simulations, we apply the adaptive process for a given $\theta_{0}$
sampled from $p(\theta)$, which yields a trajectory as illustrated in the left
panel of Fig. 2. We see that the prior peaks around the true temperature as
$m$ increases, and the estimated temperature gets closer to the true
temperature, i.e., ${\tilde{\theta}}/\theta_{0}\to 1$. The average over a
large amount of trajectories enables us to compute EMSLE in Eq. (2) with high
accuracy (in the numerical simulations, we consider $\mathcal{O}(1000/m)$
trajectories, which is enough to ensure convergence). This is used in the
right panel of Fig. 3, where we plot EMSLE in the adaptive scenario for
various values of $n$, benchmarked against the no-go bound for non-adaptive
scenarios—only the shaded area can be accessed by non-adaptive strategies
given any $n\leq N$. We see that as we increase $n$ the error gets smaller for
large enough $N$. In particular, there exist some threshold $n$ (in this
example $n>10$) for which one can beat the no-go bound via adaptive
strategies. As an example, given $N=10^{3}$ and $\theta_{\rm max}/\theta_{\rm
min}=10$ in Eq. (13) (with $\alpha=1$), adaptive strategies using $n\approx
10$ interacting qubits outperform arbitrary non-adaptive strategies, even ones
involving full control of all the $10^{3}$ qubits.

Finally, we ask whether the adaptive strategy can reach the Heisenberg-like
scaling, $\text{\footnotesize{EMSLE}}^{-1}\propto mn^{2}$. To this aim, we
study the behaviour of the error with the resources $n$ for a sufficiently
large number of repetitions $m$. The results are depicted in Fig. 3, where we
see Eq. ( Fundamental limits in Bayesian thermometry and attainability via
adaptive strategies ) is saturated and therefore the proposed adaptive scheme
reaches the ultimate bound on thermometry.

Conclusions and future directions.—We derived fundamental limitations of the
Bayesian approach to equilibrium thermometry, which shows a Heisenberg-like
quadratic scaling with the number of probes. We showed that non-adaptive
strategies cannot saturate this bound and, in fact, are limited to shot-noise-
like scaling whenever the initial prior is not sharp. We also constructed a
simple adaptive protocol which can saturate the ultimate bound, thus
highlighting the crucial role of adaptivity in quantum thermometry. This role
is importantly different to the setting of Bayesian quantum phase-estimation
protocols Berry and Wiseman (2000), in which the corresponding Heisenberg
limit that applies to most general adaptive protocols Górecki  _et al._ (2020)
can be attained by resorting only to measurements being adaptively varied in
between the phase-encoding channel uses Wiseman  _et al._ (2009). In contrast,
in equilibrium thermometry the form of probe states (Gibbs) and measurement
(energy-basis) is fixed, and it is the thermalising channel (i.e. the probe
Hamiltonian) that must be adaptively adjusted for the quadratic scaling
indicated by the local approach Correa  _et al._ (2015) to become reachable.

While these results were obtained by taking the expected mean square
logarithmic error as the precision quantifier, it is important to note that
any reasonable error measure leads to the same main conclusions Rubio  _et
al._ (2020); Jørgensen  _et al._ (2021). The optimal strategy relies on using
a highly degenerate two-level system as a probe, as in the local regime Correa
_et al._ (2015). Although this is a very idealised Hamiltonian, Hamiltonians
that achieve similar sensitive precision have been proposed in the low-
temperate regime through a one-dimensional fermionic system Płodzień  _et al._
(2018), and in an upcoming work we will discuss a Hamiltonian probe reaching
quadratic scaling with $n$ based on the Ising model.

We leave as interesting future work the establishment of similar Bayesian
bounds in nonequilibrium thermometry Jevtic  _et al._ (2015); Cavina  _et al._
(2018); Hovhannisyan  _et al._ (2021) and critical metrology Frérot and
Roscilde (2018); Rams  _et al._ (2018).

Acknowledgements.—We gratefully thank J. Rubio and L. A. Correa for fruitful
discussions in an early stage of this work. M.M. and M.P-L. acknowledge
financial support from the Swiss National Science Foundation (NCCR SwissMAP
and Ambizione grant PZ00P2-186067). J.K. acknowledges the Foundation for
Polish Science within the “Quantum Optical Technologies” project carried out
within the International Research Agendas programme cofinanced by the European
Union under the European Regional Development Fund. MRJ and JBB acknowledge
support by the Independent Research Fund Denmark.

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##  Appendix A Derivation of the Van Trees inequality

##  A.0.1 Preliminaries

We consider a continuous Euclidean one-dimensional parameter space
$\Lambda\subseteq\mathbb{R}$. For a Euclidean space, a suitable function
measuring the distance between parameter values is the absolute difference,
i.e.

$\mathcal{D}(\tilde{\lambda},\lambda)=|\tilde{\lambda}-\lambda|\ \ \text{for}\
\ \tilde{\lambda},\lambda\in\Lambda.$ (16)

Within the Bayesian approach to parameter estimation we start from a prior
probability density $p(\lambda)$ over the parameter space $\Lambda$. The prior
probability is updated as measurement data is acquired. Given measurement data
$\textbf{x}_{m}=\left\\{x_{1},...,x_{m}\right\\}$, Bayes’ theorem allows us to
express the update at measurement step $k$ as

$\displaystyle
p(\lambda|\textbf{x}_{k})=\frac{p(x_{k}|\lambda,\textbf{x}_{k-1})p(\lambda|\textbf{x}_{k-1})}{p(x_{k}|\textbf{x}_{k-1})},$
(17)

where in order to compress the notation we let
$p(x_{1}|\lambda,\textbf{x}_{0})=p(x_{1}|\lambda)$,
$p(\lambda|\textbf{x}_{0})=p(\lambda)$, and
$p(x_{1}|\textbf{x}_{0})=p(x_{1})$. Here, $p(x_{k}|\lambda,\textbf{x}_{k-1})$
is the likelihood function associated with the implemented measurement. Note
that this might be conditional on past measurement outcomes. We have also
defined the marginal density

$p(x_{k}|\textbf{x}_{k-1})\coloneqq\int d\lambda\
p(x_{k}|\lambda,\textbf{x}_{k-1})p(\lambda|\textbf{x}_{k-1}).$ (18)

For later convenience we introduce the joint probability density
$p(\lambda,\textbf{x}_{k})=p(x_{k}|\lambda,\textbf{x}_{k-1})p(\lambda,\textbf{x}_{k-1})$,
where it is understood that $p(\lambda,\textbf{x}_{0})=p(\lambda)$, and
therefore $p(\textbf{x}_{0})=1$ and $p(\textbf{x}_{0}|\lambda)=1$. Applying
equation (17) iteratively, we can write the posterior distribution resulting
from the full measurement trajectory as

$\displaystyle
p(\lambda|\textbf{x}_{m})=\frac{p(\textbf{x}_{m}|\lambda)p(\lambda)}{p(\textbf{x}_{m})},$
(19)

where we have defined

$\displaystyle
p(\textbf{x}_{m}|\lambda)=\prod_{k=1}^{m}p(x_{k}|\lambda,\textbf{x}_{k-1})$
(20) $\displaystyle
p(\textbf{x}_{m})=\prod_{k=1}^{m}p(x_{k}|\textbf{x}_{k-1}).$ (21)

##  A.0.2 Mean-square distance estimation

An estimation theory is a prescription for specifying a parameter estimate
$\bar{\lambda}(\textbf{x}_{m})$, computed as a function of the data, and for
providing a measure of the confidence in the computed estimate. Here we employ
the framework of mean-square distance (MSD) estimation, in which the
confidence in an estimate is gauged by the posterior MSD:

$\displaystyle\text{\footnotesize{MSD}}(\textbf{x}_{m})$
$\displaystyle\coloneqq\int d\lambda\
p(\lambda|\textbf{x}_{m})\mathcal{D}(\bar{\lambda}(\textbf{x}_{m}),\lambda)^{2},$
(22) $\displaystyle=\int d\lambda\ p(\lambda|\textbf{x}_{m})\
|\bar{\lambda}(\textbf{x}_{m})-\lambda|^{2},$

where the second equality follows as we are considering a Euclidean parameter
space. Given a measure of the confidence in an estimate, it is natural to find
the maximum confidence estimator. It can be shown, by minimizing Eq. (22) with
respect to $\bar{\lambda}(\textbf{x}_{m})$, that the choice of estimator which
minimizes the posterior MSD is the posterior mean

$\bar{\lambda}(\textbf{x}_{m})=\int d\lambda\
p(\lambda|\textbf{x}_{m})\lambda.$ (23)

In what follows, we exclusively consider the posterior mean, i.e. the minimal
mean-square distance (MMSD) estimator. Since the MSD is a stochastic quantity
defined for a single measurement trajectory, it is common to consider the
expected mean-square distance (EMSD):

$\text{\footnotesize{EMSD}}:=\int d\lambda d\textbf{x}_{m}\
p(\lambda,\textbf{x}_{m})\ |\bar{\lambda}(\textbf{x}_{m})-\lambda|^{2},$ (24)

which is obtained by averaging the MSD over the marginal distribution
$p(\textbf{x}_{m})$. As the name suggest, this quantity gives the MSD which
would be obtained on average, if the true parameter value is sampled from the
prior probability.

##  A.0.3 The Van-Trees inequality

We now give a derivation of a Bayesian Cramér-Rao bound on the EMSD, in
particular we consider the Van Trees inequality Van Trees (2004). To derive
the Bayesian bound we first define the quantity

$\displaystyle\mathcal{H}\coloneqq\int d\lambda d\textbf{x}_{m}$
$\displaystyle\sqrt{p(\lambda,\textbf{x}_{m})}\
(\bar{\lambda}(\textbf{x}_{m})-\lambda)$ (25)
$\displaystyle\times\sqrt{p(\lambda,\textbf{x}_{m})}\ \partial_{\lambda}\log
p(\lambda,\textbf{x}_{m}),$

where differentiability of the posterior distribution is implicitly assumed.
The above quantity is clearly defined to motivate an application of the
Cauchy-Schwarz inequality. From a direct evaluation of the above integral we
find

$\displaystyle\mathcal{H}$ $\displaystyle=\int d\lambda d\textbf{x}_{m}\
(\bar{\lambda}(\textbf{x}_{m})-\lambda)\partial_{\lambda}p(\lambda,\textbf{x}_{m})$
(26) $\displaystyle=1+\int
d\textbf{x}_{m}\left\\{\left[\bar{\lambda}(\textbf{x}_{m})-\lambda\right]p(\lambda,\textbf{x}_{m})\right\\}_{\lambda\in\mathcal{B}(\Lambda)},$

where $\mathcal{B}(\Lambda)$ denotes the boundaries of the parameter space
$\Lambda$. In most cases of interest the boundary term vanish. Here we take
the vanishing of the boundary term as a constraint on the class of models
considered, i.e.

$\displaystyle p(\lambda,\textbf{x}_{m})=0\ \ \text{for}\ \
\lambda\in\mathcal{B}(\Lambda)$ (27) $\displaystyle\lambda
p(\lambda,\textbf{x}_{m})=0\ \ \text{for}\ \ \lambda\in\mathcal{B}(\Lambda).$
(28)

Given these boundary conditions it follows that $\mathcal{H}=1$. If we return
to the definition of $\mathcal{H}$, i.e. equation (25), and apply the Cauchy-
Schwarz inequality, then we obtain the Van Trees inequality:

$\displaystyle\text{\footnotesize{EMSD}}^{-1}$ $\displaystyle\leqslant\int
d\lambda d\textbf{x}_{m}\
p(\lambda,\textbf{x}_{m})\left[\partial_{\lambda}\log
p(\lambda,\textbf{x}_{m})\right]^{2}$ (29) $\displaystyle=Q[p(\lambda)]+\int
d\lambda d\textbf{x}_{m}\
p(\lambda,\textbf{x}_{m})\left[\partial_{\lambda}\log
p(\textbf{x}_{m}|\lambda)\right]^{2},$

where the second equality follows directly from the decomposition
$p(\lambda,\textbf{x}_{m})=p(\textbf{x}_{m}|\lambda)p(\lambda)$ of the joint
probability distribution, and we have defined the so-called Bayesian
information of the prior distribution (which quantifies the prior information
about the parameter $\lambda$) as Van Trees (2004):

$Q[p(\lambda)]\coloneqq\int d\lambda\ p(\lambda)\left[\partial_{\lambda}\log
p(\lambda)\right]^{2}.$ (30)

We can put the Van Trees inequality into the form employed in the main text by
decomposing the likelihood function using equation (20), and then rewriting
the expression using Bayes theorem:

$\displaystyle\text{\footnotesize{EMSD}}^{-1}$ $\displaystyle\leqslant
Q[p(\lambda)]+\int d\lambda d\textbf{x}_{m}\
p(\lambda,\textbf{x}_{m})\sum_{k=1}^{m}\left[\partial_{\lambda}\log
p(x_{k}|\lambda,\textbf{x}_{k-1})\right]^{2}$ (31)
$\displaystyle=Q[p(\lambda)]+\sum_{k=1}^{m}\int d\lambda d\textbf{x}_{k-1}\
p(\lambda,\textbf{x}_{k-1})\int
dx_{k}p(x_{k}|\lambda,\textbf{x}_{k-1})\left[\partial_{\lambda}\log
p(x_{k}|\lambda,\textbf{x}_{k-1})\right]^{2}$
$\displaystyle=Q[p(\lambda)]+\sum_{k=1}^{m}\int
d\textbf{x}_{k-1}p(\textbf{x}_{k-1})\int d\lambda\
p(\lambda|\textbf{x}_{k-1})\ h_{k}(\lambda)$

where

$h_{k}(\lambda)\coloneqq\int
dx_{k}p(x_{k}|\lambda,\textbf{x}_{k-1})\left[\partial_{\lambda}\log
p(x_{k}|\lambda,\textbf{x}_{k-1})\right]^{2}.$ (32)

is just the Fisher Information of the distribution
$p(x_{k}|\lambda,\textbf{x}_{k-1})$ evaluated with respect to the parameter
$\lambda$ Van Trees (2004). Note that the Fisher information $h_{k}(\lambda)$
is generally conditioned on the past measurement trajectory
$\textbf{x}_{k-1}$—a fact that we suppress in the notation for simplicity.

##  Appendix B Application to equilibrium thermometry

##  B.0.1 Preliminaries

We now turn our attention to equilibrium probe thermometry. Let
$\theta\in\Theta$ denote the sample temperature, where $\Theta$ is the space
of temperatures. We consider a measurement consisting of first thermalizing
the $n$-qudit probe system, described by a Hamiltonian operator $H_{n}^{(k)}$,
and then performing a projective energy measurement of the probe. The probe at
measurement step $k$ is found in the thermal Gibbs state

$\displaystyle\omega(\theta;H_{n}^{(k)})=\frac{e^{-H_{n}^{(k)}/k_{B}\theta}}{\mathcal{Z}_{n}^{(k)}}$
(33)

with $\mathcal{Z}_{n}^{(k)}={\rm Tr}(e^{-H_{n}^{(k)}/k_{B}\theta})$, and
$k_{B}$ is Boltzmann’s constant. For convenience, the ground-state energy is
set to zero. The definitions of the average probe energy and the probe heat
capacity are:

$\displaystyle E(\theta;H_{n}^{(k)})\coloneqq{\rm
Tr}(H_{n}^{(k)}\omega(\theta;H_{n}^{(k)})),$ (34) $\displaystyle
C(\theta;H_{n}^{(k)})\coloneqq\frac{{\rm d}E(\theta;H_{n}^{(k)})}{{\rm
d}\theta}.$ (35)

##  B.0.2 Mean-square logarithmic error

The MSD estimation theory developed in the preceding sections is defined with
respect to a Euclidean parameter space $\Lambda$. In the case of equilibrium
probe thermometry, the space of temperatures is not a Euclidean parameter
space. However, in the specific case of equilibrium probe thermometry of a
thermalizing channel, the space of temperatures can be mapped into a Euclidean
space by taking the logarithm Rubio  _et al._ (2020); Jørgensen  _et al._
(2021)

$\lambda(\theta)=\log(\theta).$ (36)

The EMSD then takes the form of an expected mean-square logarithmic error
(EMSLE) studied in the main text.

##  B.0.3 Van Trees inequality in thermometry

For the sake of generality we will stick to an arbitrary parameterization
$\lambda(\theta)$, i.e. a one-to-one map $\lambda:\Theta\rightarrow\Lambda$,
which is assumed to be differentiable. The Fisher information transforms under
a change of parameterization as

$h_{k}(\lambda)=(d\theta/d\lambda)^{2}h_{k}(\theta).$ (37)

Here, the data is obtained via projective energy measurements of the probe
system. In fact, this is the optimal measurement maximising the Fisher
information, which thus constitutes then the so-called quantum Fisher
information being directly related to the heat capacity of the probe, i.e.
Correa  _et al._ (2015):

$\displaystyle C(\theta;H_{n}^{(k)})=\theta^{2}h_{k}(\theta),$ (38)

which is a functional of the probe Hamiltonian. In terms of the probe heat
capacity the posterior averaged Fisher information introduced in the preceding
section takes the form

$\text{\footnotesize{EMSD}}^{-1}\leqslant Q[p(\theta)]+\sum_{k=1}^{m}\int
d\textbf{x}_{k-1}p(\textbf{x}_{k-1})\int d\theta\ p(\theta|\textbf{x}_{k-1})\
\left[\frac{1}{\theta}\frac{d\theta}{d\lambda}\right]^{2}C(\theta;H_{n}^{(k)}).$
(39)

where we have made use of the parameterization invariance of the probability
density, i.e. $d\lambda p(\lambda|\textbf{x})=d\theta p(\theta|\textbf{x})$,
which is a requirement on a well-defined probability density. For convenience,
we define

$J_{\lambda}(\theta)\coloneqq\left[\frac{1}{\theta}\frac{d\theta}{d\lambda}\right],$
(40)

and note that in the specific case of $\lambda(\theta)=\log(\theta)$ it
follows that $J_{\lambda}(\theta)=1$, in which case we recover the form of the
Van Trees inequality given in the main text. Lastly, we note that when working
with the logarithmic parameterization, the Bayesian information of the prior
takes the form

$Q[p(\theta)]=\int d\theta\,p(\theta)\left[1+\partial_{\theta}\log
p(\theta)\right]^{2}$ (41)

We are interested in the optimal probe design, and formally define the
optimization problem

$\displaystyle\Gamma(\textbf{x}_{k-1})\coloneqq\max_{H_{n}^{(k)}}\int d\theta\
p(\theta|\textbf{x}_{k-1})J_{\lambda}(\theta)^{2}C(\theta;H_{n}^{(k)})$ (42)

Note that $\Gamma$ is in general a functional of the past measurement
trajectory, i.e., the optimal probe structure depends on the prior knowledge
of the parameter to be estimated. In the following sections we derive model-
independent upper bounds on $\Gamma$.

##  B.0.4 Model-independent super-extensive upper bound on $\Gamma$

In this section we derive a super-extensive bound on $\Gamma(\textbf{x})$.
Starting with Eq. (42), we note that since the integrand is positive we can
provide an upper bound by moving from a global maximization to a local
maximization, i.e.

$\Gamma(\textbf{x})\leqslant\int d\theta\
p(\theta|\textbf{x})J^{2}_{\lambda}(\theta)\ \max_{H}\ C(\theta;H).$ (43)

The problem of maximizing the heat capacity, over all possible probe
Hamiltonians at a given temperature, has been solved by Correa et al. Correa
_et al._ (2015). The solution can be formulated as the temperature-independent
tight upper bound

$C(\theta;H)\leqslant\left[\frac{\xi_{D}}{2}\right]^{2}-1,$ (44)

where $\xi_{D}$ is the solution to the transcendental equation

$e^{\xi_{D}}=\left(D-1\right)\frac{\xi_{D}+2}{\xi_{D}-2}.$ (45)

This equation does not have a closed form solution. However, a general feature
of the solution is that $\xi_{D}>\log(D-1)$, and that $\xi_{D}$ approach
$\log(D-1)$ from above as $D$ becomes large. From this it follows that
$\Gamma(\textbf{x})$ satisfies the super-extensive upper bound

$\displaystyle\Gamma(\textbf{x})$
$\displaystyle\leqslant\left(\xi_{D}/2-1\right)\left(\xi_{D}/2+1\right)\int
d\theta\ p(\theta|\textbf{x})J^{2}_{\lambda}(\theta),$ (46)

which grows super-extensively in $\log(D)$. If we average $\Gamma(\textbf{x})$
over the past measurement trajectory we find

$\int
d\textbf{x}p(\textbf{x})\Gamma(\textbf{x})\leqslant\left(\xi_{D}/2-1\right)\left(\xi_{D}/2+1\right)\left\langle
J^{2}_{\lambda}\right\rangle_{\text{prior}}\equiv C_{D}\left\langle
J^{2}_{\lambda}\right\rangle_{\text{prior}},$ (47)

where we have defined

$\left\langle J^{2}_{\lambda}\right\rangle_{\text{prior}}=\int d\theta\
p(\theta)J^{2}_{\lambda}(\theta).$ (48)

This bound is expected to be approximately tight in the limit where the prior
is local with respect to the width of the heat capacity. As we will see in the
next section, designing a probe with a critical heat capacity at a certain
temperature, i.e. one attaining the maximal heat capacity, will result in the
width of the heat capacity decreasing as $1/\log(D)$. We thus see that
saturating the super-extensive bound requires a prior probability distribution
confined to a domain
$\theta\in\left[\theta_{c}-\Delta/2,\theta_{c}+\Delta/2\right]$ where
$\theta_{c}$ is the critical temperature and $\Delta=1/\log(D)$. As $D$
increase this corresponds to an increasing amount of prior information.

##  B.0.5 Tight upper bound on the thermal energy density

In this section we want to derive an upper bound on the thermal energy at a
given temperature for any probe structure, subject to the dimensionality
constraint $\dim{H}=D$ on the considered probes. We will find that the thermal
energy density is upper bounded by the temperature. Define the maximum thermal
energy for any probe structure as

$E_{\max}(\theta)\coloneqq\max_{H}\ E(\theta;H),$ (49)
$E(\theta;H)\coloneqq\text{Tr}\left[H\omega(\theta;H)\right],$ (50)

where $\omega(\theta;H)$ is a thermal state at temperature $\theta$. We denote
the energy eigenvalues of the probe Hamiltonian by
$\left\\{\varepsilon_{l}\right\\}$, and for convenience set the ground-state
energy to zero. If we take the derivative of the thermal energy, and equate to
zero we obtain the condition

$\varepsilon_{l}=\theta+E(\theta;H)\eqqcolon\varepsilon,$ (51)

which implies a $D-1$ degeneracy in the first excited state. Evaluating the
above condition for this probe structure leads to a transcendental equation
for $\varepsilon/\theta$ which can be solved. The result is the temperature-
dependent upper bound

$\displaystyle E(\theta;H)\leqslant\theta\mathcal{W}_{D}$ (52)
$\displaystyle\mathcal{W}_{D}\coloneqq W\left(\frac{D-1}{e}\right),$ (53)

where $W$ denotes the product logarithm, also called the Lambert W function.
In the limit of large $D$ the behaviour of the product logarithm is such that
$\mathcal{W}_{D}$ tends asymptotically to $\log(D)$ from below. We stress that
the above bound on the thermal energy can be saturated by an effective two
level probe with a $D-1$ degenerate excited state, and a temperature-dependent
energy gap.

##  B.0.6 Alternative extensive bound for the non-adaptive scenario

Our alternative bound follows the exact same procedure, except we first recall
that the thermal energy can be expressed as
$\theta^{2}\partial_{\theta}\Psi(\theta;H)$, where the Massieu potential reads
$\Psi(\theta;H)\coloneqq\log\mathcal{Z}(\theta;H)$ with
$\mathcal{Z}(\theta;H)$ being the partition function of the probe. Starting
from Eq. (8) of the main text and by performing twice integration by parts we
get

$\displaystyle{\bar{\Gamma}}$
$\displaystyle=m\int\\!d\theta\,p(\theta)\,C(\theta;H_{n})=-m\int d\theta\
\left[\partial_{\theta}p(\theta))\right]E(\theta;H)$ $\displaystyle=\ m\int
d\theta\
\left[\partial_{\theta}\left(\theta^{2}\partial_{\theta}p(\theta)\right)\right]\Psi(\theta;H),$
(54)

where again we take the vanishing and differentiablity of the boundary terms
in both integrations—that is $p(\theta)E(\theta;H_{n})$ and
$\theta^{2}\partial_{\theta}p(\theta)\Psi(\theta;H)$—as a restriction on the
choice of parameterization. We can derive an upper bound on the optimal
solution by noting that $\Psi(\theta;H)\geqslant 0$—recall that the ground
state energy is set to zero—and by introducing
$\bar{\mathcal{R}}=\left\\{\theta\mid\partial_{\theta}\left(\theta^{2}\partial_{\theta}p(\theta)\right)\geqslant
0\right\\}$. Then

$\displaystyle{\bar{\Gamma}}$ $\displaystyle\leqslant m\max_{H}\
\int_{\bar{\mathcal{R}}}d\theta\
\left[\partial_{\theta}\left(\theta^{2}\partial_{\theta}p(\theta)\right)\right]\Psi(\theta;H).$
(55)

As the integrand is now positive we can maximize the Massieu potential
locally. Since the logarithm is monotonically increasing in its argument, this
corresponds to substituting the largest value of the partition function, i.e.
the Hilbert space dimension. The bound then takes the form

$\displaystyle{\bar{\Gamma}}$ $\displaystyle\leqslant
m\log(D)\int_{\bar{\mathcal{R}}}d\theta\
\left[\partial_{\theta}\left(\theta^{2}\partial_{\theta}p(\theta)\right)\right]$
(56) $\displaystyle\leqslant
m\log(D)\left\\{\theta^{2}\partial_{\theta}p(\theta)\right\\}_{\bar{\mathcal{R}}}$
$\displaystyle\eqqcolon m\log(D)g[p(\theta)],$

where $g[p(\theta)]$ is a functional of the prior distribution but independent
of the probe. This gives two complementary bounds on ${\bar{\Gamma}}$, i.e.
one expressed in terms of $f[p(\theta)]$ as presented in the main text, and
one in terms of $g[p(\theta)]$. Which of these two is tighter depends on the
specific prior.