Tight bound on finite-resolution quantum thermometry at low temperatures
Mathias R. Jørgensen, Patrick P. Potts, Matteo G. A. Paris, Jonatan B. Brask
TTight Bound on Finite-Resolution Quantum Thermometry at Low Temperatures
Mathias R. Jørgensen , ∗ Patrick P. Potts , Matteo G. A. Paris , and Jonatan B. Brask Department of Physics, Technical University of Denmark, 2800 Kongens Lyngby, Denmark Physics Department and NanoLund, Lund University, Box 118, 22100 Lund, Sweden Quantum Technology Lab, Dipartimento di Fisica ” Aldo Pontremoli ” ,Università degli Studi di Milano, I-20133 Milano, Italy (Dated: October 2, 2020)Precise thermometry is of wide importance in science and technology in general and in quantum systemsin particular. Here, we investigate fundamental precision limits for thermometry on cold quantum systems,taking into account constraints due to finite measurement resolution. We derive a tight bound on the optimalprecision scaling with temperature, as the temperature approaches zero. The bound demonstrates that underfinite resolution, the variance in any temperature estimate must decrease slower than linearly. This scalingcan be saturated by monitoring the non-equilibrium dynamics of a single-qubit probe. We support this findingby numerical simulations of a spin-boson model. In particular, this shows that thermometry with a vanishingabsolute error at low temperature is possible with finite resolution, answering an interesting question left openby previous work. Our results are relevant both fundamentally, as they illuminate the ultimate limits to quantumthermometry, and practically, in guiding the development of sensitive thermometric techniques applicable atultracold temperatures. I. INTRODUCTION
Sensitive measurements of temperature are essentialthroughout natural science and modern technology. Increas-ingly detailed studies of biological, chemical, and physi-cal processes, the miniaturisation of electronics, and emerg-ing quantum technology drive a need for new thermometrytechniques applicable at the nanoscale and in regimes wherequantum effects become important. Many new approachesare being developed [1–12], however the fundamental limitsto precision thermometry are not yet fully understood. Here,we determine a tight bound on the best possible precisionwith which temperature can be estimated in cold quantumsystems, which accounts for limitations due to imperfect mea-surements.The classical picture of thermometry is that of a thermome-ter which is brought into thermal contact with a sample. Ob-serving the state of the thermometer after some time conveysinformation about the sample temperature. A similar picturecan be applied in the quantum regime, where an individualquantum probe, e.g. a two-level system, may interact with asample system in a thermal state, and subsequently be mea-sured to estimate the temperature. If the probe reaches ther-mal equilibrium with the sample, or a non-equilibrium steadystate, optimal designs of the probe and of the probe-systeminteraction can be determined [13–18]. Outside of the steadystate regime, it was found that access to the transient probedynamics may outperform the steady-state protocols [19–21],that dynamical control acts as a resource [22–24], and thatthermometry can in some cases be mapped to a phase esti-mation problem [25, 26]. These findings have spurred furtherinvestigations into non-equilibrium thermometry [27–29].Any thermometric technique will be subject to constraintsdue to finite measurement resolution. In the probe-samplepicture, the size of the probe will limit the amount of informa-tion which can be extracted about the sample. More generally, ∗ [email protected] any measurement on the sample, implemented using a finite-sized apparatus, comes with a lower bound on the attainableresolution of e.g. the system energy spectrum [30–32]. Sim-ilar restrictions apply in situations where measurements canbe made on only part of a large sample [33–35], and clearlysuch finite-resolution constraints must play an important rolein formulating fundamental bounds on the attainable thermo-metric sensitivity.Here, we derive a bound on the precision scaling with tem-perature, as the temperature approaches zero, for thermome-ters with finite energy resolution. Our bound applies to anythermometric technique based on measurements which donot resolve the individual energy levels of the sample en-ergy spectrum. We furthermore demonstrate that this scal-ing can be attained using a single-qubit probe, showing thatthe bound is tight. To derive our bound, we build upon theframework for finite-resolution quantum thermometry intro-duced by Potts, Brask, and Brunner in [31].Our results also demonstrate that thermometry with a van-ishing absolute error at low temperature is possible with finiteresolution, answering an interesting question left open by pre-vious work [31, 34, 36]. For systems with a heat capacitythat vanishes at low temperatures, a property often includedin the third law of thermodynamics, the relative error mustdiverge, regardless of the available resolution [31]. The ab-solute error may either also diverge, stay constant, or vanish,with the latter thus being the best behaviour one can hopefor. However, for gapped systems, even the absolute errorin any unbiased temperature estimate must diverge when thetemperature becomes comparable to the gap [37]. A constantor vanishing absolute error, on the other hand, has been seenin gapless systems, when employing a measurement with acontinuous outcome implying an infinite resolution [34]. Ourresults show that a vanishing absolute error may be obtainedwith a finite-resolution measurement having as little as twooutcomes.This paper is organized as follows. In Sec. II we introducea general temperature estimation procedure, following [31],and discuss the fundamental precision bounds imposed by the a r X i v : . [ qu a n t - ph ] O c t third law of thermodynamics. In Sec. III we propose a finite-resolution criterion, and show how this criterion leads to atight bound on the attainable precision. In Sec. IV we general-ize the framework to include noisy measurements, and finallyin Sec. V we investigate a single-qubit thermometer coupledto a bosonic bath, showing that our bound can be saturatedin a physical scenario. Our analytical results are supportedby numerical simulations of the temperature estimation pro-cedure. II. TEMPERATURE ESTIMATION
We consider a quantum system described by the canoni-cal thermal state ρ β = exp [ − βH ] / Z β , with H the Hamil-tonian operator of the system, and Z β ≡ tr { exp [ − βH ] } the canonical partition function. The thermal state is param-eterized by an inverse temperature β = 1 /k B T where k B isthe Boltzmann constant. For convenience we adopt units inwhich k B = 1 , such that temperature has the units of energy.The task we consider is how to estimate the temperature T ofthe system. We remark that throughout we consider thermalstates where the temperature does not itself fluctuate. How-ever, since temperature is not directly measurable (it is nota quantum mechanical observable), there are fluctuations inany temperature estimate based on indirect measurements. A. Quantifying the estimation precision
A general temperature estimation procedure consists offirst performing a measurement on the system. The most gen-eral N -outcome measurement is represented by a positive-operator valued measure ( POVM ) with N elements Π m . Such POVM s capture any possible measurement in quantum me-chanics, including scenarios in which information is obtainedthrough a probe interacting with the system, as well as thoseexploiting quantum coherence [7, 19, 20]. Each
POVM ele-ment Π m corresponds to a measurement outcome m which isobserved with probability p m ; β = tr { Π m ρ β } , (1)and the resulting probability distribution encodes the systemtemperature as a statistical parameter. The second step in es-timating the temperature is to construct an estimator T est . Ageneral prescription for doing this does not exist [38]. How-ever, it can be shown that for any unbiased estimator the vari-ance is lower bounded through the Cramer-Rao inequality δT est ≥ /ν F T [39], where ν is the number of independentmeasurement rounds and F T ≡ N (cid:88) m =1 p m ; β [ ∂ T ln p m ; β ] , (2)is the Fisher information. We note that the Cramer-Rao in-equality is asymptotically tight for Bayesian or maximumlikelihood estimators [38]. Throughout, motivated by theCramer-Rao inequality, we adopt the Fisher information asthe quantifier of precision. Figure 1. Finite measurement resolution is interpreted as an inabil-ity to sharply distinguish between consecutive system energy eigen-states and results in a non-trivial constraint on the attainable ther-mometric precision. For a macroscopic system with an effectivelycontinuous energy spectrum, any measurement is subject to finiteresolution and thus limited by the bound in Eq. (26).
Identifying measurement strategies for which the temper-ature estimate can achieve minimal variance corresponds tomaximizing the Fisher information over all possible measure-ments (
POVM s). This results in a measurement-independentquantity, the quantum Fisher information F QT [40] . Withinthe canonical ensemble, it can be shown that a projective mea-surement of the system energy is optimal [31, 37]. The quan-tum Fisher information is then related to the variance of thesystem energy T F QT = (cid:10) H (cid:11) − (cid:104) H (cid:105) , (3)where (cid:104) O (cid:105) = tr { Oρ β } . This expression provides a fun-damental upper bound on the attainable value of the Fisherinformation for any measurement at any temperature. As aconsequence of the third law of thermodynamics, or more ex-plicitly the assumption that the heat capacity vanishes at zerotemperature, the variance of the system energy must vanishat least quadratically in temperature as absolute zero is ap-proached [31]. Hence it follows that T F QT must vanish inthe low-temperature limit, and that the relative error δT est /T must diverge by virtue of the Cramer-Rao inequality. Thisrelation constitutes the ultimate bound on the optimal low-temperature scaling behaviour of the Fisher information, ap-plicable for any system and for any measurement strategy. B. Accounting for measurement limitations
In many settings of interest, it is not realistic to imple-ment a projective measurement of the system energy. Forinstance, whenever the gaps in the energy spectrum are be-low the energy resolution of the available measurement [34],which happens, e.g., when the system is large enough to ap-pear continuous while the measurement apparatus has a finitesize, or whenever only a finite part of the full system can beinteracted with within a finite time (see Fig. 1). Under suchconditions of constrained experimental access, it is useful tointroduce the
POVM energies [31] E m ; β ≡ p m ; β tr { Π m Hρ β } , (4)where E m ; β may be interpreted as the best guess of the sys-tem energy before the measurement, given that outcome m was observed [31]. In the case of projective energy measure-ments on the system, the POVM energies coincide with thesystem energy eigenvalues. In general however, the
POVM energies are temperature dependent.For convenience we may identify a specific
POVM en-ergy E β , defined as the smallest POVM energy in the low-temperature limit. We can then introduce the
POVM energygaps ∆ m ; β ≡ E m ; β − E β , which by definition are non-negative at low temperatures. In terms of these gaps, theFisher information for a general measurement is given by F T = (cid:80) m p m ; β ∆ m ; β − ( (cid:80) m p m ; β ∆ m ; β ) T . (5)Similarly to the quantum Fisher information, the above ex-pression takes the form of an energy variance. However forgeneral measurements the energy spectrum of the system isreplaced by the spectrum of POVM energies, and the Boltz-mann probabilities associated with projective energy mea-surements are replaced by the
POVM probabilities. Thesechanges incorporate restrictions due to limitations of the mea-surement on top of those imposed by the system itself.In investigating the scaling behaviour we are implicitly as-suming that the Fisher information is a continuous function oftemperature, which implies that the
POVM energy gaps ∆ m ; β must also be continuous functions. Following Ref. [31], weare going to study the scaling behaviour of the Fisher in-formation when the POVM energy gaps have a well-definedpower-series expansion in temperature around absolute zero ∆ m ; β = ∆ m, + ∞ (cid:88) k =1 ∆ m,k β − k . (6)By virtue of Weierstrass’ approximation theorem, any contin-uous function can be approximated arbitrarily well by such apower series [41]. Note that this formulation does not excludethe case of projective energy measurements as this would bedescribed by a series with only the constant term. For moregeneral measurements, however, the expansion might containnon-zero higher-order coefficients.Following Potts et al. [31] we can make use of the rela-tion between the POVM energies and the associated probabil-ities (Eq. (4)) to write ∆ m ; β = − ∂ β ln p m ; β /p β . Given thepower-series expansion of the POVM energy gaps, we can in-tegrate this equation and express the ratio of the probabilitiesfor outcomes m and 0 as p m ; β p β = g m e − β ∆ m, β − ∆ m, ∞ (cid:89) k =1 e ∆ m,k +1 β − k /k , (7)where g m is a temperature-independent integration constant.We stress that as a consequence of how we defined E β , the probability p β is the largest probability at zero temperatureand must be non-vanishing in this limit. We thus obtain anexpression for the probability of obtaining outcome m givenfully in terms of the expansion coefficients of the correspond-ing POVM energy gap (note that the explicit dependence on p β could be avoided by using the fact that the full distribu-tion must be normalised). C. Low-temperature scaling behaviour
The above model of limited measurements allows us to ob-tain, by substituting Eqs. (6) and (7) into Eq. (5), an expres-sion for the Fisher information given fully in terms of the
POVM energy gaps. Based on this, we can analyse the possi-ble scaling behaviour of the Fisher information, as the systemapproaches zero temperature. First of all, we note that Eq. (5)can be rewritten as F T = 12 T (cid:88) m,n p m ; β p n ; β ( β ∆ m ; β − β ∆ n ; β ) . (8)Notice that all terms on the right-hand side are positive, andbecause of this the scaling behaviour of the Fisher informa-tion is determined by the term in the sum (or the set of terms)which vanishes least rapidly as the temperature goes to zero.We now consider the scaling that arises from different termsin Eq. (8). We focus on terms that result in sub-exponentialscalings, referring the reader to Ref. [31] for a discussion ofthe remaining terms.For convenience, we define the ground-state set of mea-surement outcomes, as those for which the probability of ob-taining that outcome remains finite at zero temperature (notethat the outcome m = 0 is in the ground-state set by defi-nition). From Eq. (7), we see that formally this set can bedefined as Ω = { m | ∆ m, = ∆ m, = 0 } . Now considerthose terms in the Fisher information above where both out-comes belong to the ground-state set. To leading order intemperature, the contribution from these terms takes the form T (cid:88) m,n ∈ Ω p m ; β p n ; β (∆ m,j − ∆ n,j ) T j − , (9)where j labels the lowest order for which the expansion co-efficient of any element in the ground-state set is non-zero( j ≥ ). These terms in the sum thus vanish at least quadrat-ically, giving at best a constant contribution to the Fisher in-formation. Notice that if the ground-state set contains only asingle outcome ( m = 0 ), then the contribution is identicallyzero.Next we consider the terms in the Fisher information whereone of the outcomes belong to the ground-state set but theother one does not. To this end, we define the set of outcomes ˜Ω = { m | ∆ m, = 0 and ∆ m, (cid:54) = 0 } , for which the asso-ciated probability vanish sub-exponentially as the tempera-ture goes to zero. The set of outcomes ˜Ω has an associated POVM energy coinciding with that of the ground-state set atzero-temperature, but exhibits a linear degeneracy splitting atfinite temperature. To leading order in temperature, the con-tribution from the corresponding terms is T (cid:88) m ∈ ˜Ω g m ∆ m, T ∆ m, , (10)which vanishes at a rate determined by the the first-orderexpansion coefficients ∆ m, . It is straightforward to showthat all other contributions vanish exponentially in the low-temperature limit.The (sub-exponential) low-temperature behaviour of theright-hand side of the Fisher information (8), is generallygiven by the sum of Eq. (9) and Eq. (10). Which of thesetwo dominate depends on the smallest first-order expansioncoefficient. If the set ˜Ω is not empty, and at least one elementin the set has a value ∆ m, < j − , where j denotes thelowest order with non-vanishing expansion coefficient withinthe ground-state set, then the low-temperature behaviour ofthe Fisher information is captured by F T = (cid:88) m ∈ ˜Ω g m ∆ m, T ∆ m, − . (11)In principle the first-order coefficient can take any positivevalue without violating the scaling bound imposed by thethird law of thermodynamics (ensuring divergence of the rel-ative error). Notice that even a divergent low-temperature be-haviour of the Fisher information can in principle be realised,if ∆ m, can take a value smaller than 2. III. SCALING BOUND FOR LARGE SYSTEMS
In this section, we propose a finite-resolution criterioncharacterizing realistic measurements. We aim to capture anysituation in which the available measurements cannot resolvethe individual gaps in the system energy spectrum, whichtherefore appears continuous. Below, we make this statementprecise. We then go on to show how this criterion leads to alower bound on the first-order coefficient ∆ m, , constrainingthe low-temperature scaling of the error in any temperatureestimation scheme. Furthermore we present an example of ameasurement saturating the finite-resolution bound, showingthat the bound is tight. A. Finite-resolution criterion
In the regime where the system has an effectively contin-uous energy spectrum (as the measurement only resolves en-ergy differences much larger than the gaps in the spectrum,it is convenient to work with the system density of states D ( (cid:15) ) ≡ (cid:80) k d k δ ( (cid:15) − (cid:15) k ) , where the sum is over distinct sys-tem energy eigenvalues and d k is the corresponding degener-acy. Throughout, we adopt the convention that the smallestsystem energy eigenvalue is set to zero ( (cid:15) = 0 ).Now, we introduce a filtered density of states D m for eachmeasurement outcome m , as the system density of states fil-tered through the corresponding POVM element D m ( (cid:15) ) ≡ (cid:88) k d k δ ( (cid:15) − (cid:15) k ) tr (cid:20) Π m (cid:15) k d k (cid:21) , (12) where (cid:15) k is the projection operator onto the eigenspace withenergy (cid:15) k . Notice that the sum of all the filtered densi-ties of states adds up to the total density of states. Further-more, we introduce the continuous filter function f m ( (cid:15) ) , for-mally defined by the values f m ( (cid:15) k ) = tr [Π m (cid:15) k /d k ] andthe straight-line segments connecting these values. In addi-tion we note that the density of states can be expressed as therate of change of the number of states with energy below (cid:15)σ ( (cid:15) ) = (cid:80) k d k θ ( (cid:15) − (cid:15) k ) , where θ denotes the Heaviside stepfunction. Given these, the filtered density of states decom-pose into the product D m ( (cid:15) ) = f m ( (cid:15) ) dσ ( (cid:15) ) d(cid:15) , (13)where the filter function fully characterizes the implementedmeasurement. Importantly we notice that the function σ ( (cid:15) ) isnon-decreasing for all energies. If we compute the Laplacetransform in β of the filtered density of states, the result takesthe form of a Stieltjes integral over a measure given by σ ( (cid:15) ) [42] ˆ D m ( β ) ≡ (cid:90) ∞ dσ ( (cid:15) ) f m ( (cid:15) ) e − β(cid:15) = Z β p m ; β . (14)The last equality can be obtained directly from equation (1),and relates the Laplace transformed filtered density of statesto the product of the probability and the canonical partitionfunction. Notice that the measure σ ( (cid:15) ) is a discontinuousfunction of energy.For macroscopic systems the measure can often be approx-imated by an effective continuous measure, when σ ( (cid:15) ) and f m ( (cid:15) ) vary on widely separated energy scales. To see this,we first define the averaged measure with respect to an en-ergy window ω by σ ω ( (cid:15) ) = θ ( (cid:15) ) 1 ω (cid:90) (cid:15) + ω/ (cid:15) − ω/ dsσ ( s ) , (15)which for non-zero ω is a continuous function of energy ex-cept at (cid:15) = 0 , and which tends to a differentiable function ofenergy as ω is increased. The inclusion of the step functionat zero energy is important if we are to capture the zero tem-perature limit correctly, since it ensures that the ground-stateof the averaged model coincides with that of the exact model.For the purposes of low-temperature thermometry only thelow-energy behaviour is of importance, and to leading orderin energy we adopt an effective measure given by dσ ω ( (cid:15) ) = (cid:2) d ω δ ( (cid:15) ) + α ω γ ω (cid:15) γ ω − + O ( (cid:15) γ ω ) (cid:3) d(cid:15), (16)where d ω is an effective ground-state degeneracy and α ω , γ ω are positive, real-valued constants. The coefficient γ ω > characterizes the low-energy growth in the total number ofstates with energy less than (cid:15) .If we compute the Laplace transform with respect to thisaveraged measure (which now takes the form of a standardRiemann integral) we obtain to leading order in energy ˆ D m ; ω ( β ) = d ω f m (0)+ α ω γ ω (cid:90) ∞ d(cid:15)(cid:15) γ ω − f m ( (cid:15) ) e − β(cid:15) . (17)The averaged measure tends to overestimate the number oflow-energy states as ω is increased, however this effect be-comes negligible in the limit ω (cid:28) T . Now if we assume that f m ( (cid:15) ) does not vary significantly across an energy range ω ,then ˆ D m ( β ) is well approximated by the averaged function ˆ D m ; ω ( β ) . More quantitatively we can state this condition inthe form of an inequality | f m ( (cid:15) + ω ) − f m ( (cid:15) ) | ω (cid:28) ω , (18)which bounds the rate of change of the filter function withenergy. For macroscopic systems we can take the limit ω → , and in this case we are going to adopt the following finite-resolution criterion (FRC) : FRC:
In the limit of a macroscopic system, thefilter function f m ( (cid:15) ) tends to a continuous, right-differentiable function of the system energy. This is nothing more than a restatement of equation (18) forvanishing ω , which restricts the rate of change of the filterfunction to a finite value. We note that at (cid:15) = 0 , the filterfunction may be discontinuous and Eq. (18) tends to the rightderivative for ω → . B. Finite-resolution bound
Having characterized what we mean by a finite-resolutionmeasurement, we ask what the consequences of our finite-resolution criterion are for the behaviour of the
POVM energygaps in the macroscopic limit. By making use of equation (7),we obtain the relation (we now drop the dependence on theenergy window ω and write simply d , α and γ ) ˆ D m ( β ) = ˆ G m ( β ) ˆ D ( β ) , (19)where for convenience we have defined the transfer function ˆ G m ( β ) ≡ g m e − β ∆ m, β − ∆ m, ∞ (cid:89) k =1 e ∆ m,k +1 β − k /k . (20)Now this is a relation at the level of the Laplace-transformed,filtered densities of states. We can obtain a relationship di-rectly between the filtered densities of states by taking theinverse Laplace transform of both sides of Eq. (19). By ap-plying the Laplace convolution theorem [43, 44], we derivethe relation D m ( (cid:15) ) = (cid:90) (cid:15) ds G m ( (cid:15) − s ) D ( s ) . (21)We now focus on the specific case of m ∈ ˜Ω . For theseoutcomes, the inverse Laplace transform can be computedstraightforwardly, and to leading order in energy we obtain G m ( (cid:15) ) = g m Γ(∆ m, ) (cid:15) ∆ m, − + O ( (cid:15) ∆ m, ) , (22) where Γ(∆ m, ) denotes the Gamma function [43]. As we sawin the preceding section, the outcomes within ˜Ω are exactlythe ones with potential to provide optimal low-temperaturescaling of the Fisher information.Recall, that the reference outcome m = 0 , was chosen suchthat the associated probability approaches a constant value atzero temperature. This implies that the overlap of the POVM element Π with the system ground state is non-zero, andtherefore f (0) is non-zero. On the other hand for outcomes m ∈ ˜Ω the probability vanishes in the low-temperature limit,implying a vanishing overlap f m (0) = 0 . Hence in this casewe find from equations (17) and (21) that to leading order inenergy f m ( (cid:15) ) = g m d f (0) αγ Γ(∆ m, ) (cid:15) ∆ m, − γ + O ( (cid:15) ∆ m, +1 − γ ) . (23)Based on this expression we can infer constraints on the linearcoefficient. First, the requirement that f m (0) = 0 gives theweakest constraint ∆ m, > γ . This simply expresses the factthat the Fisher information is upper bounded by the the quan-tum Fisher information, which scales as T γ − for a density ofstates scaling as (cid:15) γ − . Further, the finite-resolution criterionrestricts the rate of change to be bounded, dd(cid:15) f m ( (cid:15) ) < ∞ .This implies a tightened scaling bound ∆ m, ≥ γ, for m ∈ ˜Ω . (24)Since γ > by definition, this implies that the Fisher infor-mation must grow slower than /T , i.e., lim T → T F T = 0 . (25)Further note that a diverging Fisher information in the low-temperature limit can only be realized through a σ ( (cid:15) ) thatgrows sub-linearly with energy, i.e., γ < . As an example ofa system exhibiting such a sub-linear growth we mention sys-tems of massive non-interacting particles at zero chemical po-tential [31]. For such systems γ = 1 / for one-dimensionalgeometries.By virtue of the Cramer-Rao bound, Eq. (25) implies thatthe absolute error (squared) must vanish more slowly than T lim T → δT T = ∞ . (26)The equivalent Eqs. (25) and (26) constitute the main resultof our paper. They imply that for an effectively continuousspectrum, the low-temperature scaling of the precision is notonly bounded by the third law, which demands a divergingrelative error, but by a tighter bound. Interestingly, our boundstill allows for a vanishing absolute error, a scenario that canbe physically realized as illustrated below. C. Proving tightness of bound
We now illustrate that the proposed finite-resolution boundis tight. Consider a binary measurement which resolves thesystem ground state exponentially well in the sense that it has
POVM elements Π = e − κH , Π = − e − κH , (27)where κ > . Note that the overlap of Π with the system en-ergy eigenstates decays exponentially away from zero. Thisfeature makes is straightforward to write down the filtereddensity of states. Focusing on m = 1 we find D ( (cid:15) ) = (cid:2) − e − κ(cid:15) (cid:3) D ( (cid:15) ) , (28)where nothing has been assumed about the form of the sys-tem density of states. We thus see that the corresponding filterfunction takes the form f ( (cid:15) ) = κ(cid:15) + O ( (cid:15) ) to leading order inenergy. If we adopt the density of states introduced in the pre-ceding subsection, that is D ( (cid:15) ) = d δ ( (cid:15) ) + αγ(cid:15) γ − + O ( (cid:15) γ ) ,then upon comparison with equation (23) we find ∆ , =1 + γ . Hence the binary exponential resolution measurementsaturates the finite-resolution bound.For good measure we now show how the same conclusioncan be derived directly from the probabilities. The probabilityof obtaining outcome m = 0 can be written in terms of thesystem partition function as p β = Z κ + β Z − β . (29)Substituting the probabilities p β and p β = 1 − p β intothe general form of the Fisher information (Eq. 2), one findsthat T F T = Z κ + β Z β − Z κ + β (cid:16) (cid:104) H (cid:105) β − (cid:104) H (cid:105) κ + β (cid:17) . (30)The partition function is given by the Laplace trans-form of the density of states, hence we find Z β = d exp ( αγ Γ( γ ) β − γ /d ) (in App. A we show how this formof the partition function describes a system of non-interactingbosonic modes). From this form of the partition function wecan derive the low-temperature behaviour of the average en-ergy (cid:104) H (cid:105) β = αγ Γ( γ ) d β − (1+ γ ) . (31)If we substitute these into the above Fisher information,we find that to leading order in temperature (assuming that κ/β (cid:28) ) F T = ακγ Γ( γ )(1 + γ ) T γ − + O ( T γ ) , (32)which takes the form of Eq. (11) with ∆ , = 1 + γ and g = ακγ Γ( γ ) . Since γ can in principle take any posi-tive value, the exponential-resolution measurement saturatesthe finite-resolution bound and asymptotically attains a Fisherinformation scaling as /T in the limit γ → . IV. GENERALIZATION TO NOISY MEASUREMENTS
In this section we extend the thermometry frameworkabove to include noisy measurements. As the framework is general, one might ask if noise effects are not already ac-counted for. The answer is that in principle noise effectsare described. However, for some noisy measurements, the
POVM energy gap does not have a Taylor expansion. Whileone may still approximate the energy gap by a polynomial,a physically appealing extension of the formalism allows forcircumventing this approximation. We find that our boundgiven in Eq. (26) also holds for noisy measurements.
A. Noisy temperature measurements
To model noisy measurements, we consider the case wherethe observed outcomes m correspond to coarse graining overa fine-grained POVM with elements Π mµ . The probability ofobserving m is then p m ; β = (cid:88) µ p mµ ; β = (cid:88) µ tr { Π mµ ρ β } . (33)Physically this could correspond to a measurement imple-mented using a sensor, where only a subset of the sensordegrees of freedom (or a subspace of the full sensor Hilbertspace) is experimentally accessible. If we were to computethe Fisher information directly using the fine-grained distri-bution p mµ , we recover the noiseless results, and obtain anupper bound on the Fisher information computed from thecoarse-grained distribution. This fact follows directly fromthe relation between the relative entropy of two probabilitydistributions differing by an infinitesimal temperature δT andthe Fisher information D ( p T || p T + δT ) = F T δT as δT → . (34)Since the relative entropy is monotonically decreasing undercoarse-graining [45], we conclude that noise always reducesthe Fisher information.The question we now address is, how it impacts the attain-able scaling with temperature. Following the approach devel-oped above, we introduce the fine-grained POVM energies E mµ ; β ≡ p mµ ; β tr { Π mµ ρ β } , (35)which may be interpreted as the best guess of the systemenergy before the measurement, given the outcome ( m, µ ) [31]. For convenience we identify the smallest POVM en-ergy in the low-temperature limit with the outcome E β ,and then define the fine-grained POVM energy gap ∆ mµ ; β ≡ E mµ ; β − E β , which by definition is non-negative at lowtemperatures. Modelling the fine-grained POVM energy gapsby a power-series expansion around zero temperature as inEq. (6), we are led to a probability distribution identical to(7), but with m replaced by the compound index mµ .Since the Fisher information is not defined with respectto the fine-grained probabilities, but rather with respect tothe coarse-grained probabilities, the relevant energies are thecoarse-grained POVM energy gaps defined by ∆ ( c ) m ; β ≡ (cid:88) µ p mµ ; β p m ; β ∆ mµ ; β . (36) Figure 2. Illustration of filtered density of state for anoisy binary exponential resolution measurement using D ( (cid:15) ) = L − (cid:2) exp (cid:0) αβ − (cid:1)(cid:3) (dotted red line) with α = 0 . . (a) The whitenoise measurement corresponds to swapping the observed measure-ment outcomes with some probability, such that each coarse-grainedoutcome has contributions both from elements within and elementsnot within the ground-state set. The dashed green lines gives D and D (their sum is shown with the solid green line), and theblue dashed-dotted lines correspond to elements D and D (withtheir sum given by the solid blue line). (b) In App. B we show thatan alternative noise model consists of a mixing of several similarmeasurement outcomes. In the specific case depicted here, the fine-grained outcomes to be summed are almost identical except for pro-jecting onto slightly different energy distributions. In terms of these, the Fisher information can be written in thesame form as the fine-grained Fisher information of Eq. (8),but with the fine-grained probability and the fine-grained
POVM energy gaps replaced by their coarse-grained versions F T = 12 T (cid:88) m,n p m ; β p n ; β (cid:16) β ∆ ( c ) m ; β − β ∆ ( c ) n ; β (cid:17) . (37)Notice that all terms in the sum are positive. Hence, the scal-ing behaviour of the Fisher information is determined by theterm (or set of terms) which vanishes least rapidly as the tem-perature approaches zero.From Eq. (36), we can anticipate that fine-grained en-ergy gaps that have a Taylor expansion may result in coarse-grained gaps that do not. This may result in qualitatively dif-ferent behaviour of the fine- and coarse-grained Fisher infor-mation. In particular, noise may render the scaling of theFisher information worse. In appendix B we discuss in gen-eral terms how noise impacts the attainable Fisher informa-tion scaling. In particular, we show that the noise can neverresult in a better scaling for the Fisher information, implyingthat the bound given in Eq. (26) also holds for noisy measure-ments. Here we illustrate the effect of noise with an example. B. Illustration of noisy measurement
A simple example illustrating noise is obtained by addingwhite noise to the binary, exponential-resolution measure-ment of Sec. III C. That is, we study a binary
POVM defined by Π = η exp ( − κH ) + (1 − η ) / . To understand howthis noise model arises from coarse graining a fine-grainedmeasurement, we consider the fine-grained POVM Π = 1 + η e − κH , Π = 1 − η (cid:0) − e − κH (cid:1) , Π = 1 + η (cid:0) − e − κH (cid:1) , Π = 1 − η e − κH , (38)such that Π = Π + Π and Π = Π + Π . As in thenoiseless case, we suppose that the average energy exhibitsa power-law behaviour (cid:104) H (cid:105) β = αβ − (1+ γ ) at low tempera-tures in the macroscopic limit, with α and γ both positive.The corresponding partition function (at low temperatures) isthen Z β = exp ( αβ − γ /γ ) . For the fine-grained measure-ment outcomes, we find that to leading order in temperature(assuming that κ/β (cid:28) and η < ), the POVM energy gapswith respect to the reference E β , take the form ∆ β = ∆ β = 0 , ∆ β = ∆ β = (1 + γ ) T + O ( T γ ) . (39)We see that the fine-grained measurement outcomes have anassociated set of POVM energy gaps that have a Taylor seriesin the low-temperature limit. Furthermore, they exhibit a lin-ear degeneracy splitting. It then follows from Eq. (11) thatthe Fisher information takes the form F T = ακ (1 + γ ) T γ − + O ( T γ ) , (40)which is equivalent to the noiseless form found above[cf. Eq. (32)]. Notice that when having access to the fine-grained distribution, both the POVM energies and the result-ing Fisher information is independent of the parameter η quantifying the amount of white noise.The picture changes when considering the coarse-grainedenergy gap (Eq. (36)). To leading order in temperature this isgiven by ∆ ( c )1; β = 1 + η − η ακ (1 + γ ) T γ + O ( T γ ) . (41)Notice that in contrast to the fine-grained energy gaps, thiscoarse-grained gap does not have a Taylor expansion. Com-puting the Fisher information over the coarse-grained gapsand probabilities (making use of Eq. (37)) gives F T = 4 η − η ( ακ (1 + γ )) T γ + O ( T γ ) . (42)This example thus illustrates how noise can result in a coarse-grained gap that has no Taylor expansion and how this mayresult in a different (worse) scaling for the Fisher informationat low-temperatures. Qualitatively we can understand the al-tered scaling by studying the coarse-grained filtered densityof states. For the example considered here we have D ( (cid:15) ) = f ( (cid:15) ) D ( (cid:15) ) = 1 + η e − κ(cid:15) D ( (cid:15) ) , D ( (cid:15) ) = f ( (cid:15) ) D ( (cid:15) ) = 1 − η (cid:0) − e − κ(cid:15) (cid:1) D ( (cid:15) ) , (43)and under coarse-graining these are added together. Noticethat whereas the filter function f ( (cid:15) ) goes to zero as (cid:15) → ,this is not true of f ( (cid:15) )+ f ( (cid:15) ) (the same feature is found forthe m = 1 outcomes). Hence in this case the noise removesoutcomes from the set ˜Ω , resulting in the worse scaling (notethat a vanishing filter function at (cid:15) = 0 implies a vanishingprobability at T = 0 and vice versa, cf. Eq. (14)). This effectis illustrated in Fig.2a. In App. B we study an alternativenoise model. In this model each coarse-grained outcome canbe seen as the sum of several similar (in the sense of preparingsimilar energy distributions) fine-grained outcomes. This isillustrated in Fig.2b.The noisy framework put forward here shows that ourfinite-resolution bound, as well as the results of Ref. [31] ap-ply for any POVM that can be written as a coarse grainingover a fine-grained
POVM which has a spectrum with a welldefined Taylor series. As the coarse-grained
POVM itself maynot have a spectrum with a well defined Taylor series, thisextends the applicability of the results of Ref. [31] (as longas we do not want to rely on approximate Taylor series in thespirit of the Weierstrass theorem).
V. SINGLE-QUBIT PROBEA. Measurement protocol
We now illustrate our results by considering temperatureestimation of a system of non-interacting bosons using a sin-gle qubit as a probe. The system is described by a spectrum ofsingle-particle energies ω k (we take (cid:126) = 1 ). Consider the fol-lowing measurement strategy: (i) first we initialise the probequbit in its ground state | (cid:105) , (ii) then an interaction is turnedon between the probe and the system for a short time t , and(iii) we perform a projective measurement of the qubit energy.Given this protocol, the probability of finding the qubit in theexcited state | (cid:105) is p β = tr (cid:110) (cid:104) | U † t | (cid:105)(cid:104) | U t | (cid:105) ρ β (cid:111) . (44)We take the time-evolution operator U t to be generated by atime-independent Hamiltonian H = (cid:88) k ω k a † k a k + Ω2 σ z + H int , (45)where a † k , a k denotes the bosonic creation and annihilationoperators. The probe qubit is characterised by the three Paulioperators { σ x , σ y , σ z } , and we take the probe energy to beproportional to the σ z operator.Computing the outcome probabilities requires specifyingan interaction Hamiltonian and determining the resulting dy-namics. This task is complicated by the fact that the low-temperature and short-time regime is generally not accessiblevia standard Markovian master equations [15, 46]. However,if the interaction time is sufficiently short we can make ana-lytical progress by approximating the probability up to sec-ond order in t . In this case we find that p β = t tr {(cid:104) | H int | (cid:105)(cid:104) | H int | (cid:105) ρ β } + O ( t ) . (46) We consider a linear interaction Hamiltonian consisting ofan excitation-preserving part and a non-excitation-preservingpart. Introducing the raising and lowering operators σ ± = ( σ x ± iσ y ) for the probe qubit, the interaction Hamiltoniantakes the form H int = (cid:88) k g k (cid:104) σ + a k + σ − a † k (cid:105) + (cid:88) k λ k (cid:104) σ − a k + σ + a † k (cid:105) , (47)where { g k , λ k } are real-valued coupling coefficients. In thelimit of a macroscopic system, these coupling coefficientsare taken to approach continuous functions. Physically thismeans that the interaction cannot selectively probe an individ-ual system mode (ensuring that the finite resolution criterionis satisfied).Given H int , it becomes straightforward to show fromEq. (46) that the excited-state probability at short times takesthe form p β = t (cid:88) k (cid:0) g k + λ k (cid:1) n β ( ω k ) + t (cid:88) k λ k , (48)where n β ( ω k ) denotes the Bose-Einstein distribution. Wesee that the probability consists of two contributions: atemperature-dependent term, in which the probability is di-rectly related to the occupation of the bath modes, anda temperature-independent term. The presence of thetemperature-independent term means that the probability offinding the probe qubit in the excited state is generally non-zero even at arbitrarily low temperatures. As in the examplein Sec. IV B, this prevents a scaling of the form of Eq. (11)and can be captured by our framework for noisy thermometry. B. Excitation-preserving interaction
We now focus on the excitation-preserving case ( λ k = 0 ),and consider an interaction characterised by a continuousspectral density of the form ρ ( ω ) = (cid:88) k g k δ ( ω − ω k ) = 2 αω − sc ω s e − ω/ω c , (49)where α is the dissipation strength, s is the ohmicity and ω c is the cutoff energy [46–49]. The sum in the excited-stateprobability (48) is then replaced by an integral, which can besolved analytically. In the low-temperature limit we find p β = 2 α ( ω c t ) Γ(1 + s ) (cid:18) Tω c (cid:19) s + O ( T s ) . (50)We see that this protocol gives a probability vanishing sub-exponentially as the temperature goes to zero, and comparingwith the general expression Eq. (7), we see that to lowest or-der, the POVM gap scales as ∆ = (1 + s ) T . The case of anexcitation-preserving interaction can thus (for short time atleast) be described within our noiseless thermometry frame-work. Figure 3. Numerically computed Fisher information for (a) the sub-Ohmic ( s = 1 / , and (b) the Ohmic ( s = 1) spin-boson model,with δt = 0 . / Ω , α = 0 . and ω c = 10Ω . The solid black linesdisplay the short-time analytical results at time Ω t = 0 . , show-ing good agreement with the numerical simulations. In case (b) thesimulations exhibit a quadratic temperature scaling at low tempera-tures, while in case (a) a linear scaling is obtained. The solid greyline gives the Fisher information obtained from the steady state ofa secular Born-Markov master equation, which scales exponentiallyat low temperatures [31]. From the value of the linear expansion coefficient, ∆ , =1+ s , it follows that for ohmicity approaching zero, the finite-resolution bound ∆ , ≥ is approached. The correspondingFisher information scales as F T ∝ T s − and thus divergesfor sub-Ohmic baths in the low-temperature limit. This servesas an illustration that the finite-resolution bound is in princi-ple attainable via an excitation-preserving interaction in theshort-time limit, and thus the bound is tight. Realising suchan excitation-preserving interaction may however be chal-lenging. C. Excitation-non-preserving interaction
We now turn to the arguably more realistic excitation non-preserving case. The case λ k = g k corresponds to the well-known spin-boson model [47–50]. Adopting the same spec-tral density as above, the excited-state probability in this casetakes the form p β = 4 α ( ω c t ) Γ(1 + s ) (cid:18) Tω c (cid:19) s + 2 α ( ω c t ) Γ(1 + s ) + O ( T s ) . (51) In contrast to the excitation-preserving case, this probabil-ity does not in general correspond to the noiseless version ofEq. (7) since the POVM energy gap ∆ ∝ T s +2 , does nothave a Taylor expansion for arbitrary s at low temperatures.However, as shown in App. C, this scenario can be describedusing a fine-grained POVM with energy gaps that do have aTaylor expansion. Therefore, the scenario is captured by thenoisy framework.Given the probability (51), a short calculation shows thatthe Fisher information has a low-temperature scaling givenby F T ∝ T s . Again, this is in full agreement with the gen-eral noisy theory developed above. Within the spin-bosonmodel, the Fisher information thus vanishes quadratically foran Ohmic spectral density with s = 1 , and linearly for a sub-Ohmic spectral density with s = 1 / .To corroborate the analytical results based on the short-time approximation, we turn to a numerical simulation of theFisher information for the spin-boson model. To perform thesimulations we made use of the recently developed tensor-network TEMPO algorithm and its extension to multi-timemeasurement scenarios [51, 52]. Details of the simulationsare provided in App. D. Making use of this algorithm has thebenefit that the temperature derivative of the excited state canitself be expressed as a tensor network and computed to thesame level of accuracy as the probability itself.Results for the Ohmic and the sub-Ohmic cases are shownin Fig. 3. Generally we find that the short-time approx-imation provides a good description of the observed scal-ing behaviour at sufficiently short times. Even more inter-esting we note that the scaling behaviour predicted withinthe short-time approximation is valid even at times well be-yond the regime in which the short-time approximation isexpected to hold ( αδt Γ(1 + s ) ω c (cid:28) ). This indicatesthat the predicted precision scaling is experimentally relevant,even without the requirement of being able to probe the non-equilibrium qubit dynamics at very short-times. Notice thatthe low-temperature Fisher information tends to initially in-crease with time as information about the environment state isextracted by the qubit. After some time the low-temperatureFisher information starts to decrease. This can be understoodas the qubit reaching a stationary state, such that a one-timemeasurement performed on the qubit can no longer probe therelaxation dynamics induced by the coupling with the thermalbath (see also [2, 13]).Finally, we note that at sufficiently low temperatures thesimulated Fisher information differs from the Markovian re-sult, even for the rather weak coupling and long times con-sidered here. A similar effect was observed in the context oftemperature estimation via the Kubo-Martin-Schwinger-likerelations obeyed by emission and absorption spectra of mul-tichromophoric systems [53]. There it was pointed out thatfaithfully recovering the temperature from observed spec-tra requires taking into account system-environment correla-tions. This is true even at very low coupling strengths, wherethese correlations are generally weak.0 VI. CONCLUSION
In this paper we have discussed precision scaling for ther-mometry in cold quantum systems. In particular, we have in-vestigated how finite measurement resolution, meaning thatstates that are close in energy cannot be perfectly distin-guished, impacts the precision scaling. We have proposed afinite-resolution criterion characterising such measurements.Based on this, we derived a tightened bound on the scal-ing of the Fisher information. Furthermore, we showed thatthis bound is tight as it can be saturated via both an ex-ponential resolution measurement as well as an excitation-preserving, single-qubit measurement on a sample of non-interacting bosons. We validated the approximations involved in demonstrating tightness for the single-qubit measurementby performing a numerical simulation of the sub-Ohmic spin-boson model. Here, we provided an illustration of a Fisherinformation scaling linearly with temperature. Interestingly,as far as we are aware, this is the best scaling which hasbeen found in any concrete physical model subject to finite-resolution constraints.
ACKNOWLEDGMENTS
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Consider a collection of non-interacting bosonic modeswith Hamiltonian H = (cid:80) k ω k b † k b k . The partition functionof this system takes the form ln Z β = − (cid:88) k ln (cid:2) − e − βω k (cid:3) . (A1)In the continuum limit, the sum over modes can be approx-imated by the integral over a continuous density of modes g ( ω ) ln Z β = − (cid:90) ∞ dωg ( ω ) ln (cid:2) − e − βω (cid:3) . (A2)Expanding the logarithm in powers of e − βω and re-scalingeach term in the resulting series, we can write the above as ln Z β = (cid:90) ∞ dω (cid:20) g ( ω ) + g ( ω/ ... (cid:21) e − βω (A3)In the low-temperature limit, this integral is dominated by thelow-energy part of the density of modes. If we assume thatat low-energies the density of modes takes the form g ( ω ) = αω γ , where α and γ are positive constants, then the integraltakes the form ln Z β = α (cid:32) ∞ (cid:88) n =1 n γ (cid:33) (cid:90) ∞ dωω γ e − βω = αζ ( γ + 1)Γ( γ + 1) β − (1+ γ ) , (A4)where ζ denotes the Riemann zeta function and Γ the gammafunction. Thus Z β = exp (cid:16) αζ ( γ + 1)Γ( γ + 1) β − (1+ γ ) (cid:17) . (A5)This expression has the general form used in the main text ifwe make the identification γ → γ .2 Appendix B: Scaling behaviour for the noisy model
In the low-temperature limit, the dominant fine-grainedprobabilities are those with a vanishing zeroth-order coeffi-cient in the
POVM energy-gap expansion, and only coarse-grained probabilities containing contributions from suchterms are relevant. For convenience we introduce two sets offine-grained outcomes: First Ω m = { µ | ∆ mµ, = ∆ mµ, =0 } , which is the set of fine-grained outcomes giving a non-vanishing contribution to the coarse-grained probability ofobtaining outcome m . Second, ˜Ω m = { µ | ∆ mµ, = 0 and ∆ mµ, (cid:54) = 0 } , which is the set of fine-grained outcomes forwhich the contribution to the coarse-grained probability for m vanishes sub-exponentially. Lastly, to simplify the laterdiscussion, we denote the specific outcome within ˜Ω m whichrealises the smallest value of the first-order coefficient by ˜ µ m .We now note that if there exists some coarse-grained out-come m such that Ω m is empty while ˜Ω m is non-empty, thenthe arguments presented for the noiseless case also apply tothe noisy case, and the same optimal scaling behaviour of theFisher information can be attained. Thus, in this case, thenoise is not detrimental for the scaling. On the other hand,if no such m exists, then we refer to detrimental noise (as-suming that ˜Ω m is non-empty for at least one outcome). Fordetrimental noise we are then left with outcomes for which Ω m is non-empty, while ˜Ω m may or may not be non-empty.We now show that detrimental noise results in a worse scal-ing compared to the noise-free scenario. This implies thatour finite-resolution bound is also applicable to noisy mea-surements.Consider the right-hand side of Eq. (37) for the case ofdetrimental noise. For terms where both ˜Ω m and ˜Ω n areempty, the scaling behaviour is identical with that of the cor-responding noiseless terms (Eq. (9)), except that the noiselesscoefficients of the POVM energy gap must be replaced by thecoarse-grained version ∆ ( c ) m,j ≡ (cid:88) µ ∈ Ω m p mµ ; β p m ; β ∆ mµ,j . (B1)If a coarse-grained second-order POVM energy gap exists(that is ∆ ( c ) m, − ∆ ( c ) n, (cid:54) = 0 for some m and n ), then thesame scaling behaviour of the Fisher information as givenby Eq. (9) is attainable and this scaling is optimal (note thatthe probabilities considered here tend to nonzero constants atzero temperature). If a second-order gap does not exist, thenthe optimal scaling is instead provided by terms for which ˜Ω m is non-empty for some m . A straightforward calculationshows that the contribution from such terms takes the form [ g m ˜ µ m ∆ m ˜ µ m , ] (cid:80) µ ∈ Ω m g mµ T m ˜ µm, − , (B2)which should be summed over all outcomes m for whichboth Ω m and ˜Ω m are non-empty. Assuming that the finite-resolution criterion applies ( ∆ m ˜ µ m , ≥ ), this contributionis at best constant. Hence under the conditions of finite res-olution and detrimental noise, a diverging Fisher informationis impossible. As a second example of a noisy measurement we can con-sider the coarse-graining of a fine-grained measurement ofthe form Π = 12 e − κH , Π = 1 − η e − κH , Π = 12 (cid:0) − e − κH (cid:1) , Π = 12 − − η e − κH . (B3)This fine-grained model is illustrated in Fig. 2b. For this mea-surement we find ∆ β = ∆ β = 0 and ∆ β ≈ (1 + γ ) T + (1 + γ ) ακT γ ∆ β ≈ (1 + γ ) ακη T γ . (B4)Hence, as in the previous example, the fine-grained measure-ment gives a Fisher information scaling as T γ − to leadingorder, and the coarse-grained measurement gives a T γ scal-ing, F T = (2 − η ) ( ακ ) η (1 + γ ) T γ + O (cid:0) T γ (cid:1) . (B5)Thus the same scaling behaviour of the Fisher information isobserved for this alternative example of a noisy model. Notethat both models exhibit detrimental noise which results inthe different scalings for the fine- and coarse-grained Fisherinformation. Appendix C: The non-excitation-preserving interaction as anoisy POVM
From Eq. (46), we find that the
POVM elements can be writ-ten as Π = t (cid:104) | H int | (cid:105)(cid:104) | H int | (cid:105) , (C1)and Π = 1 − Π . In the thermal state under consideration,there are no coherences between different bosonic modes andthere is no squeezing. Therefore, many terms in Eq. (C1) donot contribute to the probabilities. Dropping these terms, wecan write a slightly simpler POVM that results in the exactsame probabilities, capturing the full effect of the measure-ment ˜Π = t (cid:88) k (cid:0) g k + λ k (cid:1) a † k a k + t (cid:88) k λ k , (C2)and ˜Π = 1 − ˜Π . This POVM has an energy gap that hasno Taylor expansion, scaling as T s in the low temperaturelimit for g k = λ k and the spectral density given in Eq. (49).We can however write the POVM in Eq. (C2) as a coarsegraining over the fine-grained
POVM (note the similarity toEq. (38)) ˜Π = 1 + η X, ˜Π = 1 − η − X ) , ˜Π = 1 + η − X ) , ˜Π = 1 − η X, (C3)3such that ˜Π = ˜Π + ˜Π and ˜Π = ˜Π + ˜Π . Here weintroduced η = 1 − t (cid:88) k λ k , (C4)and X = t (cid:80) k ( g k + λ k ) a † k a k − t (cid:80) k λ k . (C5)The fine-grained POVM elements are of the same form as the
POVM elements for the excitation-preserving case. Indeed,setting λ k = 0 , only ˜Π and ˜Π remain finite but do notchange their form. We therefore find the same POVM gaps asfor the excitation-preserving case ∆ = ∆ = 0 , ∆ = ∆ = (1 + s ) T. (C6)The Fisher information for the fine-grained POVM thus scalesas T s − . The coarse grained POVM gap is determined byEq. (36) and reads ∆ = p p + p (1 + s ) T, (C7)which scales as T s +2 for the scenario considered in the maintext. Appendix D: Tensor network simulation
Here we provide details of the numerical methods behindthe result shown in Fig. 3. We consider the ground state prob-ability p ( k )0; β = tr (cid:110) ˆ P U kδt (cid:104) ˆ P ⊗ ρ β (cid:105)(cid:111) , (D1)where ˆ P is a projection operator onto the qubit ground state | (cid:105) , and we have decomposed the unitary evolution into k -steps of duration δt . Furthermore we consider the spin-bosonmodel ˆ H = (cid:88) k ω k ˆ a † k ˆ a k + 12 Ωˆ σ z + 12 ˆ σ x (cid:88) k g k (cid:16) a k + a † k (cid:17) . (D2)The spin-boson model can be numerically simulated usingrecently developed tensor network methods [51, 52]. Takingeach unitary step to be of a short duration we can make theapproximation (Trotter-Suzuki decomposition) U δt = W δt/ V δt W δt/ + O ( δt ) , (D3)where W δt = exp (cid:16) − iδt ( ˆ H − Ωˆ σ z / (cid:17) describes the in-fluence of the sample on the probe qubit, and V δt =exp ( − iδt Ωˆ σ z / describes the free evolution of the probequbit. As the interaction term is diagonal in the eigenstatesof the operator ˆ σ x , we can expand the ground state probabil-ity in terms of these eigenstates. This gives rise to a discrete Feynman-Vernon Influence functional, which can be summedanalytically. The ground state probability then takes the form p ( k )0; β = (cid:88) { α } ˆ P α k +1 V α k α k +1 δt ... V α α δt ˆ P α × (cid:104) Π ki =1 Π ij =1 A α i α j β (cid:105) (cid:2) Π kl =0 δ α l +1 ,α l (cid:3) . (D4)where we have introduced a compound index α = ( s, r ) ofspin-x eigenvalues, δ α i ,α j denotes the Kronecker delta func-tion, ˆ P α = (cid:104) s | ˆ P | r (cid:105) , and V are the Liouville operators repre-senting the free dynamics of the ancilla qubit V αα (cid:48) δt = (cid:104) s | V δt | s (cid:48) (cid:105)(cid:104) r (cid:48) | V † δt | r (cid:105) . (D5)The influence tensors, A α i α j β , describe the influence of thesample on the state of the qubit and contain all the tempera-ture dependence of the probability. For linearly coupled mod-els, the individual tensors depend only on the time separation ( i − j ) δt/ . The influence tensors are given by A α i α j β = e − ( s i − r i )( η i − j s j − η ∗ i − j r j ) , (D6)expressed in terms of the memory kernel elements η i − j = (cid:40)(cid:82) t i t i − (cid:82) t j t j − dt (cid:48) dt (cid:48)(cid:48) C ( t (cid:48) − t (cid:48)(cid:48) ) , i (cid:54) = j (cid:82) t i t i − (cid:82) t (cid:48) t i − dt (cid:48) dt (cid:48)(cid:48) C ( t (cid:48) − t (cid:48)(cid:48) ) , i = j , (D7)which are themselves defined in terms of the bath auto-correlation function C ( t ) = 1 π (cid:90) ∞ dωρ ( ω ) cosh [ ω ( β − it )]sinh ( βω/ . (D8)The bath auto-correlation function is given in terms of thespectral density ρ ( ω ) introduced in the main text.The attainable temperature estimation precision dependsnot only on the ground state probability, but also on thederivative of this probability. Computing the derivative of thedistribution with respect to the inverse temperature gives ∂ β p ( k )0 β = k (cid:88) i =1 i (cid:88) j =1 µ ij (cid:88) { α } (cid:2) Π kl =0 δ α l +1 ,α l (cid:3) × ˆ P α k +1 V α k α k +1 δt ... V α α δt ˆ P α × (cid:104) Π ki =1 Π ij =1 A α i α j β (cid:105) α − i α − j (D9)where we have defined α − = s − r . It turns out that thesame tensor network methods used to compute the proba-bility can be used to compute the derivative of the proba-bility. Furthermore we have defined µ ij = − ∂ β η i − j , thesquare of which gives the Fisher information scaling at low-temperatures. At low temperatures, all the temperature de-pendence of the ground-state probability comes from thesecoefficients. We can approximate them by the series µ ij = αδt β γ +2 × (cid:20) Γ( γ + 2) − δt β ( i − j ) Γ( γ + 4)+ δt β ( i − j ) Γ( γ + 6) − ... (cid:21)(cid:21)