Global Quantum Thermometry

Jesus Rubio,1, ∗ Janet Anders,1, 2 and Luis A. Correa1

1Department of Physics and Astronomy, University of Exeter, Stocker Road, Exeter EX4 4QL, United Kingdom.2Institut fur Physik und Astronomie, University of Potsdam, 14476 Potsdam, Germany.

(Dated: 29th September 2021)

A paradigm shift in quantum thermometry is proposed. To date, thermometry has relied on local estimation,which is useful to reduce statistical fluctuations once the temperature is very well known. In order to estimatetemperatures in cases where few measurement data or no substantial prior knowledge are available, we buildinstead a theory of global quantum thermometry. Based on scaling arguments, a mean logarithmic error is shownhere to be the correct figure of merit for thermometry. Its full minimisation provides an operational and optimalrule to post-process measurements into a temperature reading, and it establishes a global precision limit. Weapply these results to the simulated outcomes of measurements on a spin gas, finding that the local approach canlead to biased temperature estimates in cases where the global estimator converges to the true temperature. Theglobal framework thus enables a reliable approach to data analysis in thermometry experiments.

Quantum thermometry aims to improve precision standardsfor temperature sensing in the quantum regime [1–3]. It caninform the design of nanoscale probes [4, 5], the choice ofmeasurement [6–9] and, as we shall see, the post-processingof measured data into an optimal temperature reading. Pre-cision thermometry is rooted in the old problem of interpret-ing temperature fluctuations—in practice, temperature cannotbe accessed directly but rather, estimated from the statisticsof observable properties [10–14]. More generally, estimationtheory allows to devise feasible strategies that can approachthe fundamental precision limits of thermometry. Improvingthermometric accuracy is not only relevant for quantum ther-modynamics [2], but also in any practical application in whichprecise temperature control is necessary.

Strategies for temperature estimation may be classified asfollows. Let T denote a true but unknown temperature, andlet θ ∈ [θ1, θ2] represent a hypothesis about the value of T .Local estimation schemes assume that θ2/θ1 ∼ 1. If no sucha priori hypothesis is required, the scheme is said to be global[15]. One readily sees that, if θ2/θ1 ∼ 1 and T ∈ [θ1, θ2], thenθ ∼ T . Consequently, local strategies allow to reduce statist-ical fluctuations once the temperature is well known [16], butthey cannot address the estimation of unknown temperaturesin full generality. Currently, most literature on quantum ther-mometry focuses on local protocols [1–9].

This is partly due to the widespread use of the Cramer–Raobound (CRB) [17, 18] as the precision benchmark. The stand-ard CRB assumes unbiased estimators, i.e., that the temperat-ure estimates average to T , and it is exactly saturable only fora special class of probability models—the exponential family[16, 19]. To accommodate a wider model set, one can employlocal unbiased estimators [20] which are appropriate when theunknown temperature lies initially on a very narrow interval[16]. More generally, the CRB is approached using asymp-totically large data sets [19, 21], which in turn reduces theestimation error down to a regime in which local strategiesbecome optimal. The applicability of the CRB thus leads toschemes that are useful only in a local sense, excluding caseswhere little is known about the temperature a priori, or whereonly few measurements can be performed. Furthermore, even

if an exact saturation of the CRB were possible, the bound of-ten depends explicitly on the unknown T [3]. In general, localthermometry is thus far too restrictive.

This Letter puts forward a new theory for global quantumthermometry, that is, a theory applicable to estimates based onsmall data sets even if θ lies on a broad range. Here the prob-lem of global temperature estimation is formulated, and fullysolved, within the Bayesian framework [22–24]. We achievethis by assigning a prior probability reflecting the initial stateof information about temperature, and identifying an appro-priate measure of uncertainty (an averaged ‘cost’ or ‘devi-ation’). The latter must also respect the scale invariance of theproblem, and turns out to be a type of mean logarithmic error.Equipped with this measure of uncertainty, we are able to de-rive analytical expressions for the optimal temperature estim-ator and its uncertainty, both of which make no ‘unbiasedness’assumptions. Under certain conditions, local thermometry isrecovered as a special case of this global formalism.

As a means of illustration, we apply the global theory to anon-interacting gas of n spin-1/2 particles. For this example,local thermometry is found unable to predict the true preci-sion scaling when n . 107 (with 5% tolerance). Moreover,the estimator identified as optimal in the local regime can inprinciple yield a much larger uncertainty than its global coun-terpart. To demonstrate the potential usefulness of the globalapproach in the analysis of experimental data, we also simu-late and post-process measurement outcomes for this n-spingas. The global estimator is then found to converge to thetrue temperature after µ ' 102 trials. In contrast, a localanalysis can lead to a biased temperature estimate even forlarge µ. These results show that a paradigm shift towardsBayesian techniques may allow a more robust and signific-antly enhanced optimisation of thermometric protocols, espe-cially in cases where the data are limited [21, 25, 26].

Scale invariance and logarithmic error.— Consider a systemin equilibrium, where the true temperature T is well definedbut unknown. To infer it, one can perform energy measure-ments, which return the value E, given T , according to someconditional probability distribution. This plays the role of a

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likelihood function [19, 22], as it links the measurement pro-cess with the unknown parameter. We denote such functionby p(E|θ), where we recall that θ is our hypothesis for thetrue temperature T . Instead of assuming that θ ∼ T , as localthermometry does, we introduce a prior probability p(θ) as aproxy for our initial state of information about T . It is theninstructive to adopt the limit of complete ignorance, oppositeto (and more general than) local estimation.

Naively, one would represent complete ignorance as p(θ) ∝1. However, the conditional probability p(E|θ) only dependson θ through the dimensionless ratio E/(kBθ), i.e.,

p(E|θ)dE =f [E/(kBθ)]∫dEf [E/(kBθ)]

dE (1)

for some function f and where kB is Boltzmann’s constant.This implies that we are dealing with a scale estimation prob-lem [22, 23, 27]. Given that T is, at this stage, completelyunknown, so is the scale of the problem. Consistency thusdemands our initial state of information to be invariant un-der transformations E 7→ E′ = γE and θ 7→ θ′ = γθ[11, 22], where γ is a dimensionless constant. In turn, thismeans that our prior must satisfy p(θ)dθ = p(θ′)dθ′, whichleads to the functional equation p(θ) = γ p(γθ) with solutionp(θ) ∝ 1/θ [22, 23]. Indeed, note that the problem could havebeen equivalently formulated in terms of the inverse temperat-ure β = 1/kBθ. Such choice should not alter our prior know-ledge and yet, taking p(θ) ∝ 1 gives p(β) ∝ 1/β2, whilep(θ) ∝ 1/θ correctly leads to p(β) ∝ 1/β [11].

To map a measurement outcome E into a temperature, webuild a point estimator θ(E). Its quality is assessed via somedeviation function D[θ(E), θ] gauging the deviation of θ(E)from θ. Since all the required information is contained in thejoint probability p(E, θ) = p(θ)p(E|θ) ∝ p(E|θ)/θ we writethe average uncertainty of θ(E) as

εD :=

∫dE dθ p(E, θ)D[θ(E), θ]. (2)

The integration over θ accounts for all the available prior in-formation and makes Eq. (2) temperature-independent. Thisis a key feature that, unlike the local approach, leads to well-posed optimisation problems [22].

Our next step is to establish the form of the deviation func-tion D. Let the dimensionless scalar x ∈ (−∞,∞), andlet its prior be p(x) ∝ 1. As discussed, e.g., in [22], thisis a translationally invariant estimation problem, for whicha flat prior represents complete ignorance. The deviation ofx(E) from x is then naturally quantified by the k-distanceD[x(E), x] = |x(E) − x|k. Now we observe that settingx = α log(kBθ/ε0) maps this hypothetical scenario intoour thermometry problem, since p(x)dx = p(θ)dθ impliesp(x) ∝ 1 7→ p(θ) ∝ 1/θ. Here, ε0 is an arbitrary constantwith units of energy included merely for dimension neutral-isation [28], while α is a proportionality factor. Therefore,

D[x(E), x] 7→ D[θ(E), θ] =

∣∣∣∣∣α log

[θ(E)

θ

]∣∣∣∣∣k

, (3)

which is a bona fide scale parameter deviation function: it issymmetric, i.e., D(θ, θ) = D(θ, θ); it respects the invarianceof the problem, that is,D(γθ, γθ) = D(θ, θ); it reaches its ab-solute minimum at θ = θ where it vanishes; and it grows (de-creases) monotonically from (towards) that minimum whenθ > θ (θ < θ). While there may be other functions compat-ible with these conditions, Eq. (3) is certainly a suitable choicefor thermometry. Below we further show that for α = 1 andk = 2, the global framework can be reduced to local thermo-metry assuming one does have prior (local) information. Forthat reason, we will fix α = 1 and k = 2 in the following, andafter Eq. (3) is inserted in Eq. (2), we arrive at

εmle =

∫dE dθ p(E, θ) log2

[θ(E)

θ

]. (4)

We call εmle mean logarithmic error.

Optimal global strategy.— Our goal is to find the temperatureestimator that is optimal with respect to Eq. (4). To do this, wemust minimise εmle over all possible estimators. Since εmle isa functional of θ(E), i.e., εmle = ε[θ(E)], this is achieved bysolving the variational problem

δε[θ(E)] = δ

∫dE L[θ(E), E] = 0, (5)

where L[θ(E), E] :=∫dθp(E, θ) log2 [θ(E)/θ] plays the

role of a Lagrangian. We find that the optimal estimator ϑ(E)minimising Eq. (4) is given by

kBϑ(E)

ε0= exp

[∫dθ p(θ|E) log

(kBθ

ε0

)], (6)

where p(θ|E) = p(E, θ)/p(E) is the posterior probability,given by Bayes theorem, and p(E) =

∫dθ p(E, θ) [29]. In-

serting the optimal estimator ϑ(E) as θ(E) in Eq. (4) furthergives the optimal logarithmic error εopt. The latter can be in-terpreted intuitively when split as

εopt = εp −K, (7)

where εp :=∫dθp(θ) log2 (ϑp/θ) is the uncertainty prior to

any measurement, with optimal (prior) estimate kBϑp/ε0 =exp[∫dθp(θ) log (kBθ/ε0)

], and

K :=

∫dE p(E) log2

[ϑ(E)

ϑp

](8)

can be thought of as the maximal information provided bythe measurement E, on average. The calculations leading toEqs. (6) and (7) are given in the Supplemental Material.

Eqs. (6) and (7) constitute our main result. The former givesan optimal estimator, ϑ(E), requiring no prior assumptionsand directly applicable on a given dataset. Eq. (7) indicates thecorresponding uncertainty, εopt. Since εopt is a true minimum,Eq. (7) also serves as a generalised precision ‘bound’ for tem-perature estimation [30]. Unlike the CRB [3], this holds for

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any estimator, biased or unbiased. In addition, these results,currently written for a single measurement with outcome E,can be trivially adapted to account for any number of repe-titions, i.e., i.i.d. (E1, E2, . . . ) [25]. Global thermometry isthus able to build temperature estimates drawn from abitrarydatasets, including cases with scarce data.

Recovery of local thermometry.— While, for the sake of gen-erality, T was initially assumed to be completely unknown,one may insert a more localised prior p(θ) from Eq. (4) on-wards [31, 32]. In that case, the hypothesis θ will effectivelylie in a narrow range and the estimator θ(E) will be close to θ.One then has log2 [θ(E)/θ] ' [θ(E)/θ − 1]2, so that Eq. (4)can be approximated as

εmle '∫dθ p(θ)

∆θ2

θ2, (9)

where ∆θ2 :=∫dE p(E|θ)[θ(E) − θ]2. Eq. (9) is the aver-

aged noise-to-signal ratio ∆θ2/θ2 weighted by the prior [33].Since ∆θ2 is the ‘frequentist’ mean square error [16], it maybe lower bounded as [19, 22]

∆θ2 >1

F (θ)

[1 +

∂b(θ)

∂θ

]2+ b(θ)2, (10)

where we have introduced the Fisher information

F (θ) =

∫dE

p(E|θ)

[∂p(E|θ)∂θ

]2, (11)

and the bias b(θ) :=∫dE p(E|θ)[θ(E) − θ]. Provided that

b(θ) ' 0 as is assumed in the local approach [16], we see thatEqs. (9), (10), and p(θ)/θ2 > 0 lead to the Cramer–Rao-likebound

εmle &∫dθ

p(θ)

θ2F (θ):= εcr. (12)

Equality would hold only inasmuch as the assumptions under-pinning Eq. (12) are fulfilled. Namely, b(θ) ' 0 and closenessof θ(E) and θ, which makes εcr a local quantifier even whenθ is integrated. Hence, we have derived a local form of ther-mometry as a limit of the global framework.

Note that the quantifier ε =∫dθ p(θ)F (θ)−1 has been pro-

posed as an attempt to supersede the local paradigm [34, 35].Notwithstanding its merits—Ref. [35] reports results beyondstandard local thermometry—such ε is not scale invariant.This might be ignored if the prior is narrowly concentratedaround some fixed θ′, since then εcr ' [(θ′)2F (θ′)]−1 andε ' F (θ′)−1. That is, one is left in both cases with the Fisherinformation. But this limiting assumption is unnecessary inour truly global and intrinsically scale invariant approach.

Example: Non-interacting spin gas.— We now turn to illus-trate how to find the best thermometric precision one couldpossibly get. Let us consider a gas of n non-interacting spin-1/2 particles with energy gap ~ω in thermodynamic equilib-rium. This could correspond, e.g., to a dilute cloud of impur-ities fully equilibrated with a co-trapped ultra-cold majority

101 102 103 104 105

10-3

10-2

10-1

100

50 100 150 200 250 300 350 400 450 5003

3.5

4

4.5

Figure 1. (colour online) a) Log-log plot of the global optimum εoptin Eq. (7) (solid green) and local Cramer–Rao-like bound in Eq. (12)(dot-dashed blue) for a gas of n non-interacting spin-1/2 particles inthermal equilibrium. As can be seen, the global optimum is lowerthan the local bound unless n → ∞, meaning that the local boundmisses information when n is ‘too small’. Eq. (14) gives an asymp-totic expansion matching the global optimum when n & 102 (dashedpurple). b) Data analysis in global thermometry. We simulated theoutcomes r = (r1, . . . , rµ) of µ energy measurements on the n-spingas, with n = 150 and true temperature (solid red) kBT/(~ω) = 4.We then post-processed r using the global estimator in Eq. (6) (dot-ted blue) and a local estimator initialised at kBθ0/(~ω) = 3 [16, 19](dot-dashed black). The global estimate converges to the true tem-perature after µ ' 102 shots. In contrast, the local theory leads to abiased estimate even for µ ' 500. The global error is calculated viaEq. (4) but with average over the posterior p(θ|r) (see SupplementalMaterial). The standard CRB gives the local error [16, 19].

gas, whose temperature needs to be measured precisely [36–39]. The probability of measuring the total energy E to ber~ω, with r = 0, 1, . . . , n, is [12, 40]

p(r|θ) =

(n

r

)exp[−r~ω/(kBθ)]Z[~ω/(kBθ)]

, (13)

where Z[~ω/(kBθ)] = {exp[−~ω/(kBθ)] + 1}n is the par-tition function. Additionally, suppose that p(θ) ∝ 1/θ isdefined, for instance, on the finite support kBθ/(~ω) ∈[0.1, 10], so that normalisation gives p(θ) = 1/[2θ log(10)].

The optimum εopt can be readily evaluated after insertingthe prior p(θ) and the likelihood (13) into Eq. (7) [41]. Theresult, for n ranging from 10 to 105, is shown in Fig. 1 a).The local limit of this error [cf. Fig. 1 a)] is shown to takethe form εcr ' 51.7/n in the Supplemental Material. Com-paring both, we observe convergence as n → ∞, confirmingthe emergence of local thermometry within the global theory.However, for finite n we see that εcr > εopt. That is, whilethe global estimator extracts all available information for anyn, a local approach leads to information loss when n is small.

4

To gain analytical insight into the scaling of εopt with n,first we note that, in this specific example, εcr − εopt ' bnq .A numerical fit [42] renders the asymptotic expansion

εopt '51.7

n− 143

n5/4, (14)

which, as shown in Fig. 1 a), matches the true optimum εoptwhen n & 102. This in turn allows to assess how large n needsto be for the (local) ∼ 1/n scaling to hold approximately.Consider the equation |εcr[n(τ)]− εopt[n(τ)]| = τ εopt[n(τ)][32], where n(τ) is the minimum number of spins whichdrives the relative error between the optimal and asymptoticcurves below a tolerance τ . Using Eq. (14) gives the con-dition n(τ) ' 58.5 (1 + 1/τ)4. As expected, n(τ) → ∞when τ → 0. But, even when tolerating a 5% deviation,spin numbers of n ∼ 107 are required for local thermo-metry to give a correct scaling. Admittedly, the value ofn(τ) is protocol-dependent [21, 32], but even a simple ex-ample suffices to illustrate the perils of the local framework—information loss (Fig. 1 a)) and failure to produce a valid low-n scaling (Eq. (14)).

Data analysis in global thermometry.— The global frame-work does not only enable a more comprehensive picture offundamental limits, but, perhaps more importantly, is also areliable tool for experimental data analysis. Consider the fol-lowing protocol: a gas of n spin-1/2 particles is prepared;its energy r~ω is measured; both steps are repeated µ times,generating the outcomes (r1, r2, . . . , rµ) := r. The globalestimate ϑ(r) in Eq. (6) is calculated using the posteriorp(θ|r) ∝ p(r|θ)/θ, with kBθ/(~ω) ∈ [0.1, 10], p(r|θ) =∏µi=1 p(ri|θ) and p(ri|θ) given by Eq. (13). To assess its un-

certainty, the average over p(r, θ) in Eq. (4) is instead takenover p(θ|r), since, in experiments, the outcomes r are known[22, 24]. The resulting error εmle(r) is obviously outcome-dependent, while temperature-independent [43]. Recallingthat the logarithmic error is a noise-to-signal ratio, we intro-duce the Bayesian ‘error bar’ ∆θ(r) := θ(r)

√εmle(r). This

analysis may be compared with the local CRB-based estim-ator θ(r)L = θ0 + [µF (θ0)]−1 ∂ log[p(r|θ0)]/∂θ, with error∆θL = 1/

√µF (θ0) [16, 19]. Here, θ0 is an initial ‘hint’ at

the true temperature T , a prerequisite in local thermometry.We simulated the outcomes r arising from the aforemen-

tioned protocol, for a gas with n = 150 spins and true tem-perature kBT/(~ω) = 4. The local estimate θL ± ∆θL, ini-tialised with the ‘hint’ kBθ0/(~ω) = 3, is shown in Fig. 1 b)(dot-dashed black) to be biased even when µ � 1. This con-trasts with the convergence of the global estimate ϑ ± ∆ϑ(dotted blue) to the true temperature (solid red). One couldargue for a ‘two-step’ method where a part of the data is usedto provide a better θ0 prior to applying local thermometry. Yet,one cannot anticipate how many trials a good seed θ0 requires,nor when such θ0 is sufficiently close to the true temperature.The global framework is instead general, reliable and works atonce. Further evidence is provided in the Supplemental Ma-terial, where a comparison between global thermometry and a

histogram-fitting procedure demonstrates the potential gain ofglobal techniques for experiments with limited data.

Conclusions.— We have demonstrated that local precisionbenchmarks are insufficient whenever few data or no substan-tial prior knowledge are available. On the contrary, a globalapproach is applicable to any temperature-estimation pro-tocol regardless of the measurement record length, and cannaturally account for any degree of prior information. Forinstance, it would be interesting to exploit Eq. (6) to post-process data measured in the nanokelvin regime, which is ex-perimentally accessible with ultracold Bose and Fermi gases[36, 37, 44–48] and relevant for quantum simulation [49]. Inaddition, since Eq. (4) is deduced at the level of probabilitydistributions (i.e., with no explicit consideration of the Bornrule p(E|θ) = Tr[Π(E)ρ(θ)] [50]), Eq. (6) can be appliedalso to classical systems. Finally, note that the key assumptionbehind Eq. (4) is that the parameter is a scale, which makesit applicable beyond temperature estimation (e.g., to estimatebiochemical rates in single-molecule experiments [51–54]).

From a fundamental perspective, combining Eq. (6)—therule to calculate optimal estimates—with the theoretical op-timum (7) provides a powerful tool to address open problemsin thermometry. These include pushing precision limits fur-ther by optimising over the energy spectrum of the probe sys-tem [4, 35], or over measured quantities beyond energy (seeSupplemental Material). Moreover, the global formalism maybe extended to accommodate for the non-equilibrium states[8, 55, 56] resulting from limited access to interacting therm-alised probes, and gives the theoretical support needed to de-rive more fundamental energy–temperature uncertainty rela-tions [57]. Global quantum thermometry has thus potentialto become the new standard for thermometry in the quantumregime.

Acknowledgements.— We thank A. Bayat, W.-K. Mok, K.Bharti, M. Perarnau-Llobet, A. Luis and S. Subramanian forhelpful discussions. J.R. and J.A. acknowledge support fromEPSRC (Grant No. EP/T002875/1), and J.A. acknowledgessupport from EPSRC (Grant No. EP/R045577/1) and theRoyal Society.

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[40] G. A. P. Wyllie, Elementary Statistical Mechanics (Hutchinson,1970).

[41] Numerical algorithms on GitHub https://github.com/jesus-rubiojimenez/QuThermometry-global (2020).

[42] Specifically, a Bayesian fit using log[εcr(n) − εopt(n)] =q log(n) + log(b) + e(n), with Gaussian errors e(n).

[43] In the Supplemental material we prove that Eq. (6) is optimalalso with respect to εmle(r).

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[45] A. Leanhardt, T. Pasquini, M. Saba, A. Schirotzek, Y. Shin,D. Kielpinski, D. Pritchard, and W. Ketterle, Cooling Bose-Einstein condensates below 500 picokelvin, Science 301, 1513(2003).

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Rev. Lett. 103, 170404 (2009).[47] R. Onofrio, Cooling and thermometry of atomic Fermi gases,

Physics-Uspekhi 59, 1129 (2016).[48] C. Carcy, G. Herce, A. Tenart, T. Roscilde, and D. Clement,

Certifying the Adiabatic Preparation of Ultracold Lattice Bo-sons in the Vicinity of the Mott Transition, Phys. Rev. Lett. 126,045301 (2021).

[49] I. Bloch, J. Dalibard, and S. Nascimbene, Quantum simulationswith ultracold quantum gases, Nature Physics 8, 267 (2012).

[50] Here Π denotes a probability-operator measurement (POM),while ρ is a state operator.

[51] M. D. Baaske, M. R. Foreman, and F. Vollmer, Single-moleculenucleic acid interactions monitored on a label-free microcavitybiosensor platform, Nature nanotechnology 9, 933 (2014).

[52] S. Subramanian, S. Frustaci, and F. Vollmer, Microsecondsingle-molecule enzymology using plasmonically enhanced op-tical resonators, in Frontiers in Biological Detection: FromNanosensors to Systems XII, Vol. 11258, edited by A. Danielli,B. L. Miller, and S. M. Weiss, International Society for Opticsand Photonics (SPIE, 2020) pp. 23 – 30.

[53] S. Subramanian, H. Jones, S. Frustaci, S. Winter, M. van derKamp, V. Arcus, C. Pudney, and F. Vollmer, Sensing enzymeactivation heat capacity at the single-molecule level using gold-nanorod-based optical whispering gallery modes, ACS AppliedNano Materials 4, 4576–4583 (2021).

[54] N. Eerqing, S. Subramanian, J. Rubio, T. Lutz, H.-Y. Wu, J. An-ders, C. Soeller, and F. Vollmer, Comparing Transient Oligo-nucleotide Hybridization Kinetics Using DNA-PAINT and Op-toplasmonic Single-Molecule Sensing on Gold Nanorods, ACS

Photonics 0, null (0), arXiv:2103.07520 (2021).[55] A. De Pasquale, D. Rossini, R. Fazio, and V. Giovannetti, Local

quantum thermal susceptibility, Nature communications 7, 1(2016).

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[62] Note that here we have already adapted the generic notation inthe main text to the specific spin system under consideration.

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SUPPLEMENTAL MATERIAL

Optimisation of the mean logarithmic error

In this work we employ the mean logarithmic error

εmle =

∫dE dθ p(E, θ)log2

[θ(E)

θ

]:= ε[θ(E)] (S1)

as the figure of merit. To optimise it with respect to all possible estimators, first we solve the variational problem

δε[θ(E)] = 0, (S2)

which is mathematically equivalent to require that [61]

d

dαε[θ(E) + αη(E)]

∣∣∣∣α=0

= 0, for all η(E). (S3)

By inserting Eq. (S1) in the left hand side of Eq. (S3), and using the fact that p(E, θ) = p(E)p(θ|E), we find that

d

dαε[θ(E) + αη(E)]

∣∣∣∣α=0

= 2

∫dE dθ p(E, θ) log

[θ(E) + αη(E)

θ

]η(E)

θ(E) + α η(E)

∣∣∣∣α=0

= 2

∫dE

p(E)

θ(E)

∫dθ p(θ|E) log

[θ(E)

θ

]η(E), (S4)

which vanishes for all variations η(r) when ∫dθ p(θ|E) log

[θ(E)

θ

]= 0. (S5)

7

This condition is an equation for θ(E), and its solution can be found straightforwardly if we first rewrite the logarithmic factoras

log

[θ(E)

θ

]= log

[kB θ(E)

ε0

]− log

(kBθ

ε0

), (S6)

arriving at

kB θ(E)

ε0= exp

[∫dθ p(θ|E) log

(kBθ

ε0

)]:=

kBϑ(E)

ε0. (S7)

In other words, we have proven that ϑ(E) makes the mean logarithmic error extremal.Next we wish to verify that θ(E) = ϑ(E) is a minimum, which we may do via the functional version of the second derivative

test. Upon calculating the second variation of the error, evaluated at θ(E) = ϑ(E), we find that

d2

dα2ε[ϑ(E) + αη(E)]

∣∣∣∣α=0

= 2

∫dE dθ p(E, θ)

{1− log

[ϑ(E) + αη(E)

θ

]}[η(E)

ϑ(E) + αη(E)

]2 ∣∣∣∣α=0

= 2

∫dE p(E)

[η(E)

ϑ(E)

]2 ∫dθp(θ|E)

{1− log

[kBϑ(E)

ε0

]+ log

[kBθ

ε0

]}

= 2

∫dE p(E)

[η(E)

ϑ(E)

]2{1− log

[kBϑ(E)

ε0

]+

∫dθ p(θ|E) log

(kBθ

ε0

)}

= 2

∫dE p(E)

[η(E)

ϑ(E)

]2> 0 (S8)

for non-trivial variations (i.e., η(E) 6= 0); consequently, we conclude that ϑ(E) is the optimal estimator that minimises theBayesian uncertainty in Eq. (S1).

The minimum mean logarithmic error can now be found by introducing the optimal estimator ϑ(E) in Eq. (S1). This operationleads to

ε[ϑ(E)] =

∫dE dθ p(E, θ)log2

[ϑ(E)

θ

]

=

∫dE p(E)

∫dθ p(θ|E)

{log

[kBϑ(E)

ε0

]− log

(kBθ

ε0

)}2

=

∫dθ p(θ) log2

(kBθ

ε0

)+

∫dE p(E) log2

[kBϑ(E)

ε0

]− 2

∫dE p(E) log

[kBϑ(E)

ε0

]∫dθ p(θ|E) log

(kBθ

ε0

)

=

∫dθ p(θ) log2

(kBθ

ε0

)−∫dE p(E) log2

[kBϑ(E)

ε0

]:= εopt. (S9)

Interpretation of the logarithmic optimum

It is possible to associate a clear meaning to the optimum in Eq. (S9). To achieve this, let us first address a related but differentquestion: given the prior probability p(θ), and no experimental data, what is the best prior estimate θp for the true temperatureT ? Since we are assuming that no measurement has been yet performed, the logarithmic uncertainty in Eq.(S1) needs to besubstituted by

∫dθp(θ)log2(θp/θ), so that θp is now a number rather than a function. Then, solving the simple optimisation

problem

d

dθp

∫dθ p(θ) log2

(θpθ

)= 0,

d2

dθ2p

∫dθ p(θ) log2

(θpθ

)> 0 (S10)

shows that the optimal prior estimate is

kB θpε0

= exp

[∫dθ p(θ) log

(kBθ

ε0

)]:=

kBϑpε0

, (S11)

8

with optimal prior uncertainty

εp =

∫dθ p(θ) log2

(ϑpθ

)=

∫dθ p(θ) log2

(kBθ

ε0

)−[∫

dθ p(θ) log

(kBθ

ε0

)]2. (S12)

Note that we have used

log

(ϑpθ

)= log

(kBϑpε0

)− log

(kBθ

ε0

). (S13)

Returning to our original problem, we can now manipulate the optimum in Eq. (S9) as

εopt =

∫dθ p(θ) log2

(kBθ

ε0

)− 2

∫dθ p(θ) log

(kBθ

ε0

)log

(kBϑpε0

)+

∫dθ p(θ) log2

(kBϑpε0

)

−∫dE p(E) log2

[kBϑ(E)

ε0

]+ 2

∫dθ p(θ) log

(kBθ

ε0

)log

(kBϑpε0

)−∫dθ p(θ) log2

(kBϑpε0

)

=

∫dθ p(θ) log2

(ϑpθ

)−∫dE p(E) log2

[kBϑ(E)

ε0

]

+ 2

∫dE p(E) log

[kBϑ(E)

ε0

]log

(kBϑpε0

)−∫dE p(E) log2

(kBϑpε0

)

=

∫dθ p(θ) log2

(ϑpθ

)−∫dE p(E) log2

[ϑ(E)

ϑp

]:= εp −K, (S14)

where we have employed Eq. (S12), the fact that∫dθ p(θ) =

∫dE p(E) = 1 and∫

dθ p(θ) log

(kBθ

ε0

)=

∫dE p(E)

∫dθp(θ|E)log

(kBθ

ε0

)=

∫dE p(E) log

[kBϑ(E)

ε0

], (S15)

which stems from Eq. (S7). According to the last line in Eq. (S14), the quantity K is a logarithmic distance between the optimalestimator ϑ(E) and the optimal prior estimate ϑp. As such, the more different from ϑp the optimal estimator ϑ(E) is, the largertheir logarithmic distance becomes, which reduces the optimal error εopt because K is subtracted from the prior uncertainty εp.Hence, we may interpret K as a measure of the maximal information that the measurement outcome E can provide on averagewith respect to the information that was already available prior to performing the experiment.

Bayesian analogue of the noise-to-signal ratio

In the main text we argue that the notion of locality exploited in local estimation theory [16] may be implemented in theBayesian framework by imposing that θ(E) is generally close to θ. In that case, and using that p(E, θ) = p(θ)p(E|θ), the meanlogarithmic error in Eq. (S1) can be Taylor-expanded around θ(E) = θ and approximated as

εmle '∫dE dθ p(E, θ)

[θ(E)

θ− 1

]2=

∫dθ

p(θ)

θ2

∫dE p(E|θ)

[θ(E)− θ

]2:=

∫dθ p(θ)

∆θ2

θ2, (S16)

which is a Bayesian analogue of the noise-to-signal ratio ∆θ2/θ2. This result differs from that by Phillies [10] and Prosper [11],who instead constructed a Bayesian noise-to-signal ratio by using the moments of the posterior probability p(θ|E). However,this ignores the necessity of acknowledging the nature of the unknown parameter not only via the prior, but also when choosingthe deviation function, while our method takes both of these into account.

Asymptotics for a gas of non-interacting spins-1/2 particles

Let

p(r|θ) =

(n

r

)exp[−r~ω/(kBθ)]Z[~ω/(kBθ)]

(S17)

9

be the probability, conditioned on the hypothesis θ, of measuring the total energy of a gas of n non-interacting spins-1/2 particlesto be r~ω [12, 40], with r = 0, 1, . . . , n and partition function

Z

(~ωkBθ

)=

n∑r=0

(n

r

)exp

(−r~ωkBθ

)=

n∑r=0

(n

r

)[exp

(− ~ωkBθ

)]r1(n−r) =

[exp

(− ~ωkBθ

)+ 1

]n. (S18)

According to our results in the main text, the calculation of the Bayesian optimum in Eq. (S9) may be approximated as

εopt ' εcr =

∫dθ

p(θ)

θ2F (θ)(S19)

when n� 1. In our case, the Fisher information becomes

F (θ) =∑r

1

p(r|θ)

[∂p(r|θ)∂θ

]2=

n∑r=0

(n

r

)exp

(r~ωkBθ

)Z

(~ωkBθ

)(∂

∂θ

{exp [−r~ω/(kBθ)]Z[~ω/(kBθ)]

})2

=

n∑r=0

(n

r

)exp [−r~ω/(kBθ)]Z[~ω/(kBθ)]

{r~ωkBθ2

− Z ′[~ω/(kBθ)]Z[~ω/(kBθ)]

}2

=

{Z ′[~ω/(kBθ)]Z[~ω/(kBθ)]

}2

− 2~ωkBθ2

Z ′[~ω/(kBθ)]Z[~ω/(kBθ)]

n∑r=0

(n

r

)exp [−r~ω/(kBθ)]Z[~ω/(kBθ)]

r +

(~ωkBθ2

)2 n∑r=0

(n

r

)exp [−r~ω/(kBθ)]Z[~ω/(kBθ)]

r2

=exp[~ω/(kBθ)]

n

{Z ′[~ω/(kBθ)]Z[~ω/(kBθ)]

}2

=1

θ2

(~ω

2kBθ

)2n

cosh2[~ω/(2kBθ)], (S20)

where we have used the identities

n∑r=0

(n

r

)exp

(−r~ωkBθ

)r =

kBθ2

~ωZ ′(

~ωkBθ

),

n∑r=0

(n

r

)exp

(−r~ωkBθ

)r2 =

(kBθ

2

~ω

)2 {Z ′[~ω/(kBθ)]}2

Z[~ω/(kBθ)]

{1 +

exp [~ω/(kBθ)]n

}, (S21)

with

Z ′(

~ωkBθ

)=

n~ωkBθ2

Z[~ω/(kBθ)]exp[~ω/(kBθ)] + 1

, (S22)

which can be found by differentiating

n∑r=0

(n

r

)exp

(−r~ωkBθ

)(S23)

with respect to θ once and twice, respectively. Then, inserting Eq. (S20) into Eq. (S19), for kBθ/(~ω) ∈ [0.1, 10], leads to

εcr =2

n log(10)

(kB~ω

)2 ∫ 10~ω/kB

0.1~ω/kBdθ θ cosh2

(~ω

2kBθ

)=

2

n log(10)

∫ 10

0.1

dy y cosh2

(1

2y

)' 51.7

n, (S24)

where kBθ/(~ω) := y and the value for the last integral has been found numerically. In summary, we have shown that,asymptotically, εopt ' εcr = 51.7/n, as stated in the main discussion.

10

Data analysis in global thermometry

Protocol

Consider the measurement protocol:

1. a gas of n non-interacting spin-1/2 particles is prepared with statistics according to p(r|θ) in Eq. (S17);

2. its total energy r~ω is measured; and

3. both steps are repeated µ times, generating the outcomes (r1, . . . , rµ) := r.

The likelihood function representing the information associated with this protocol is

p(r|θ) =

µ∏i=1

p(ri|θ) =

[µ∏i=1

(n

ri

)]exp

(−µr~ωkBθ

)[exp

(− ~ωkBθ

)+ 1

]−µn, (S25)

where r :=∑µi=1 ri/µ.

Simulation of outcomes

For the purpose of illustrating how an analysis of experimental data would proceed using the global framework in this work,here we simulate the measurement outcomes (r1, . . . , rµ). First we construct the discrete cumulative distribution

r∑m=0

p(m|T ) := fn,T (r), (S26)

where p(m|T ) is given by Eq. (S17), T is the true temperature, and 0 6 r 6 n. By construction, 0 6 fn,T (r) 6 1. Next, weimpose fn,T (r) = u, where u is a uniformly distributed random number between 0 and 1. Inverting this relationship we get

r = f−1n,T (u), (S27)

where f−1n,T is defined such that f−1n,T (fn,T (x)) = x and fn,T (f−1n,T (y)) = y. Finally, by generating a string of µ random numbers(u1, . . . , uµ) between 0 and 1, we arrive at

(r1, . . . , rµ) = (f−1n,T (u1), . . . , f−1n,T (uµ)). (S28)

The data in the main text was simulated with a numerical implementation of this procedure. We chose kBT/(~ω) = 4, n = 150,and µ = 500, and the random numbers were produced using the rand function in MATLAB.

Global thermometry in experiments

Consider the logarithmic deviation function D[θ(r), θ] = log2[θ(r)/θ]. To the theorist who is searching for fundamentalprecision limits, both the correct hypothesis θ and the measurement outcomes r that a given scheme may produce are unknown.Hence, theoretical studies such as that in the first part of our work require that we average log2[θ(r)/θ] over both temperaturesand outcomes weighted by the joint probability p(r, θ), which leads to the uncertainty quantifier [62]

εmle =∑r

∫dθ p(r, θ) log2

[θ(r)

θ

]. (S29)

The situation an experimentalist faces is, however, different. Since, in experiments, the measurement outcomes r are known(here, because we simulated them), the uncertainty quantifier for this new situation is instead

εmle(r) :=

∫dθ p(θ|r) log2

[θ(r)

θ

], (S30)

11

where p(θ|r) is the posterior probability given, in this case, by

p(θ|r) ∝ p(θ)p(r|θ) ∝ 1

θexp

(−µr~ωkBθ

)[exp

(− ~ωkBθ

)+ 1

]−µn, (S31)

with kBθ/(~ω) ∈ [0.1, 10]. As expected, the temperature is still integrated—it is unknown—, but the final error depends nowon r. Note that the deviation function is still D[θ(r), θ] = log2[θ(r)/θ].

A natural question is whether the estimator in Eq. (S7) is optimal also with respect to the uncertainty quantifier in Eq. (S30).The next section answers in the affirmative—even when theorists and experimentalists need to evaluate their uncertainties dif-ferently because their initial information differs, both use the same deviation function and will find the same optimal estimator.For an extended discussion, see, e.g., sections 13.8 and 13.8 of Ref. [22], and section 3.2 of Ref. [24].

Optimal strategy

The first derivative of Eq. (S30) reads

dεmle(r)

dθ(r)= 2

∫dθ

p(θ|r)

θ(r)

{log

[kB θ(r)

ε0

]− log

(kBθ

ε0

)}. (S32)

Imposing dεmle(r)/dθ(r) = 0 leads to

kB θ(r)

ε0= exp

[∫dθ p(θ|r) log

(kBθ

ε0

)]:=

kBϑ(r)

ε0, (S33)

which is the same estimator in Eq. (S7) but written in the notation of the spin gas. By inserting Eq. (S33) in the expressionfor the second derivative d2εmle(r)/dθ(r)2, one further finds that the latter is positive. Therefore, the estimator in Eq. (S33)(or, equivalently, Eq. (S7)) leads to a minimal error and is thus optimal with respect to both the theory-motivated uncertainty inEq. (S29), and the experiment-based uncertainty in Eq. (S30).

Data analysis in local thermometry

In the absence of an asymptotically large number of measurement outcomes (i.e., µ is finite and possibly small), local estim-ation leads to estimates of the form [16, 19]

θL(r)±∆θL =

{θ0 +

1

µF (θ0)

∂ log[p(r|θ)]∂θ

∣∣∣∣∣θ=θ0

}± 1√

µF (θ0), (S34)

where θ0 is an initial ‘hint’ at the true temperature.From equation Eq. (S25) we see that

∂ log[p(r|θ)]∂θ

=µ~ωkBθ2

{r − n

exp[~ω/(kBθ)] + 1

}. (S35)

By inserting this and the Fisher information (S20) in Eq. (S34), we find that the local estimate associated with the n-spin gas is

θL(r) = θ0

{1 +

kBθ0~ω

[4r

ncosh2

(~ω

2kBθ0

)− exp

(− ~ωkBθ0

)− 1

]}(S36)

with uncertainty

∆θL =1√µn

2kBθ20

~ωcosh

(~ω

2kBθ0

). (S37)

The local estimator examined in the main text is based on the choice kBθ0/(~ω) = 3, which is close to the true temperaturekBT/(~ω) = 4. All other simulation parameters were chosen as in indicated in the previous section.

12

50 100 150 200 2500

5

10

15

-10 -5 0 5 10

0

0.05

0.1

0.15

-10 -5 0 5 10

0

0.05

0.1

0.15

-10 -5 0 5 10

0

0.05

0.1

0.15

Figure 2. (colour online) a) Global estimate (shaded blue) from position measurements x on a quantum harmonic oscillator in the canonicalensemble. This is to be compared with the fit of position histograms to a Gaussian profile A exp[−x2/(2σ2)], from where three temperatureestimates (black circles), for µ = 50, 100 and 200, have ben generated. Specifically, these are found by inverting the relationship betweenvariance σ and temperature θ in Eq. (S39). As can be observed, the global estimates oscillate around the true temperature (solid red) evenwhen the number of trials µ is low. In contrast, the asymptotic estimates are biased towards larger temperatures unless µ & 200. Plots (b-d)show the aforementioned histograms (red circles) together with the associated Gaussian fits (solid black). The uncertainty areas (shaded black)have been calculated by propagating the standard errors of the estimates for A and σ.

Thermometry beyond energy measurements

Scale-invariant global thermometry is not restricted to protocols that measure energy. For example, consider a harmonicoscillator with mass m, angular frequency ω and coordinate q. Measuring the position, the probability that the dimensionlessposition coordinate x := q

√mω/~ lies between x and x+ dx is given, in the canonical ensemble, by [63]

p(x|θ)dx =exp{−x2/[2σ(θ)2]}√

2πσ(θ)2dx, (S38)

where

σ(θ) =

√1

2coth

(~ω

2kBθ

). (S39)

We see that, even when energy is no longer the measured quantity, Eq. (S38) only depends on the ratio of energies ~ω/2 andkBθ. As such, the arguments leading to the prior p(θ) ∝ 1/θ and the deviation function D[θ(x), θ] = log2[θ(x)/θ] apply.

To verify that global thermometry renders the correct temperature in this scenario, we simulated a string of outcomes x =(x1, . . . , xµ) drawn from the thermal distribution in Eq. (S38), with true temperature kBT/(~ω) = 6. Given the posteriorprobability p(θ|x) ∝ p(θ) Πµ

i=1p(xi|θ), with kBθ/(~ω) ∈ [0.1, 10], the calculation of the global estimate ϑ(x) ± ∆ϑ(x) =

ϑ(x)[1 ±√εmle(x)] proceeds as in the main text. The result, represented as a blue shaded area in Fig. 2 a), shows how the

global estimate oscillates around the correct temperature (red solid line) and converges from µ ∼ 100 trials.

13

Global estimation vs histogram fitting

To provide further motivation for the application of global techniques in experiments, let us compare the new global approachwith the standard practice of fitting probability distributions, which is an asymptotic estimation method. Let fT (x)dx be therelative frequency of the dimensionless coordinate lying between x and x + dx when the true temperature is T . If µ � 1, itis often reasonable to assume that fT (x)dx ∼ p(x|θ = T )dx. In turn, this justifies fitting a histogram constructed with theoutcomes x to the Gaussian profile in Eq. (S38). Such a procedure yields an estimate σ ± ∆σ for the variance in Eq. (S39);we can then estimate the temperature by inverting the functional relationship between σ and θ, finding kB θF (x)/(~ω) =1/[2 arcoth(2σ2)]. The associated uncertainty ∆θF (x) can further be calculated by propagating the standard error ∆σ.

Considering the same dataset x employed in the Bayesian analysis (Fig. 2 a), blue), we constructed the histograms shown inFigs. 2 b) - d) for the first µ = 200, 100 and 50 data points, respectively. The asymptotic estimates are kB(θF ±∆θF )/(~ω) =7±1, 8±3 and 10±6, and these are shown in Fig. 2 a) as black circles with error bars. While these asymptotic estimates correctlyconverge to the true temperature as the number of data points increases, they are noticeably biased towards large temperatureswhen the number of trials µ is small. In contrast, not only are the global counterparts (blue) in Fig. 2 a) closer to the correcttemperature even when the number of trials is as low as µ = 50, but they also include values both above and under the truetemperature, as one would expect from a reliable limited-data estimate.

This example, together with the local analysis for a gas of spin-1/2 particles considered in previous sections, constitutes asolid demonstration that global thermometry can have an appreciably better performance than standard techniques. Moreover,we note that global techniques bypass the necessity of constructing histograms, thus holding great potential for applicationswhere the number of trials is small. Importantly, spatially resolved position measurements of impurity atoms thermalising witha majority condensate have been experimentally demonstrated by using an optical lattice to ‘freeze’ the impurities in place withmicrometre resolution prior to fluorescence imaging [64].