Tight bound on finite-resolution quantum thermometry at low temperatures

Precise thermometry is of wide importance in science and technology in general and in quantum systems in 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 optimal precision scaling with temperature, as the temperature approaches zero. The bound can be saturated by monitoring the non-equilibrium dynamics of a single-qubit probe. We support this finding by accurate numerical simulations of a spin-boson model. Our results are relevant both fundamentally, as they illuminate the ultimate limits to quantum thermometry, and practically, in guiding the development of sensitive thermometric techniques applicable at ultracold temperatures.


I. INTRODUCTION
Sensitive measurements of temperature are essential throughout natural science and modern technology. Increasingly detailed studies of biological, chemical, and physical processes, the miniaturisation of electronics, and emerging quantum technology drive a need for new thermometry techniques applicable at the nanoscale and in regimes where quantum effects become important. Many new approaches are being developed [1][2][3][4][5][6][7][8], however the fundamental limits to precision thermometry are not yet fully understood. Here, we determine a tight bound on the best possible precision with which temperature can be estimated in cold quantum systems, which accounts for limitations due to imperfect measurements.
The classical picture of thermometry is that of a thermometer which is brought into thermal contact with a sample. Observing the state of the thermometer after some time conveys information about the sample temperature. A similar picture can be applied in the quantum regime, where an individual quantum probe, e.g. a two-level system, may interact with a sample system in a thermal state, and subsequently be measured to estimate the temperature. If the probe reaches thermal equilibrium with the sample, or a non-equilibrium steady state, optimal designs of the probe and of the probe-system interaction can be determined [9][10][11][12][13]. Outside of the steady state regime, it was found that access to the transient probe dynamics may outperform the steady-state protocols [14][15][16], that dynamical control acts as a resource [17,18], and that thermometry can in some cases be mapped to a phase estimation problem [19,20]. These findings have spurred further investigations into non-equilibrium thermometry [21][22][23].
Any thermometric technique will be subject to constraints due to finite measurement resolution. In the probe-sample picture, the size of the probe will limit the amount of information which can be extracted about the sample. More generally, any measurement on the sample, implemented using a finitesized apparatus, comes with a lower bound on the attainable * matrj@fysik.dtu.dk resolution of e.g. the system energy spectrum [24][25][26]. Similar restrictions apply in situations where measurements can be made on only part of a large sample [27][28][29], and clearly such finite-resolution constraints must play an important role in formulating fundamental bounds on the attainable thermometric sensitivity.
Here, we derive a bound on the temperature scaling of the smallest error in any possible temperature measurement with finite resolution, as the temperature approaches zero. We furthermore demonstrate that this scaling can be attained using a single-qubit probe, showing that the bound is tight. To derive our bound, we build upon the framework for finite-resolution quantum thermometry introduced by Potts, Brask, and Brunner in [25]. Our results also answer an interesting question left open by this and other previous work, namely whether or not there exist circumstances under which temperature can be estimated precisely in the low-temperature limit -that is, without a diverging absolute error [25,28,30]. We find that the answer is positive. This paper is organized as follows. In Sec. II we introduce a general temperature estimation procedure, and discuss the fundamental precision bounds imposed by the third law of thermodynamics. In Sec. III we propose a finiteresolution criterion, and show how this criterion leads to a tight bound on the attainable precision. In Sec. IV we generalize the framework to include noisy measurements, and finally in Sec. V we investigate a single-qubit thermometer coupled to a bosonic bath, showing that our bound can be saturated in a physical scenario. Our analytical results are supported by numerical simulations of the temperature estimation procedure.

II. TEMPERATURE ESTIMATION
We consider a quantum system described by the canonical thermal state ρ β = exp [−βH] /Z β , with H the Hamiltonian operator of the system, and Z β ≡ tr {exp [−βH]} the canonical partition function. The thermal state is parameterized by an inverse temperature β = 1/k B T where k B is the Boltzmann constant. For convenience we adopt units in which k B = 1, such that temperature has the units of energy. arXiv:2001.04096v1 [quant-ph] 13 Jan 2020 Figure 1. Finite measurement resolution is interpreted as an inability to sharply distinguish between consecutive system energy eigenstates. For a macroscopic system with an effectively continuous energy spectrum this constitutes a non-trivial constraint on the attainable thermometric precision.
The task we consider is how to estimate the temperature T of the system. We remark that throughout we consider thermal states where the temperature does not itself fluctuate. However, since temperature is not directly measurable (it is not a quantum mechanical observable), there are fluctuations in any temperature estimate based on indirect measurements.
A general temperature estimation procedure consists of first performing a measurement on the system. The most general N -outcome measurement is represented by a positiveoperator valued measure (POVM) with N elements Π m . Such POVMs capture any possible measurement in quantum mechanics, including scenarios in which information is obtained through a probe interacting with the system, as well as those exploiting quantum coherence [5,14,15]. Each POVM element Π m corresponds to a measurement outcome m which is observed with probability and the resulting probability distribution encodes the system temperature as a statistical parameter. The second step in estimating the temperature is to construct an estimator T est . A general prescription for doing this does not exist [31]. However, it can be shown that for any conceivable estimator the variance is lower bounded through the Cramer-Rao inequality δT 2 est ≥ 1/νF T [32], where ν is the number of independent measurement rounds and is the Fisher information. We note that the Cramer-Rao inequality is asymptotically tight for Bayesian or maximum likelihood estimators [31]. Throughout, motivated by the Cramer-Rao inequality, we adopt the Fisher information as the quantifier of precision. Identifying measurement strategies for which the temperature estimate can achieve minimal variance corresponds to maximizing the Fisher information over all possible measurements (POVMs). This results in a measurement-independent quantity, the quantum Fisher information F Q T [33] . Within the canonical ensemble, it can be shown that a projective measurement of the system energy is optimal [25,34]. The quantum Fisher information is then related to the variance of the system energy where O = tr {Oρ β }. This expression provides a fundamental upper bound on the attainable value of the Fisher information for any measurement at any temperature. As a consequence of the third law of thermodynamics, or more explicitly the assumption that the heat capacity vanishes at zero temperature, the variance of the system energy must vanish at least quadratically in temperature as absolute zero is approached [25]. Hence it follows that T 2 F Q T must vanish in the low-temperature limit, and that the relative error δT 2 est /T 2 must diverge by virtue of the Cramer-Rao inequality. This relation constitutes the ultimate bound on the optimal lowtemperature scaling behaviour of the Fisher information, applicable for any system and for any measurement strategy.

A. Accounting for measurement limitations
In many settings of interest, it is not realistic to implement a projective measurement of the system energy. For instance, whenever the gaps in the energy spectrum are below the energy resolution of the available measurement [28], which happens, e.g., when the system is large enough to appear continuous while the measurement apparatus has a finite size, or whenever only a finite part of the full system can be interacted with within a finite time (see Fig. 1).
Under such conditions of constrained experimental access, it turns out useful to introduce the POVM energies [25] where E m;β may be interpreted as the best guess of the system energy before the measurement, given that outcome m was observed [25]. In the case of projective energy measurements on the system, the POVM energies coincide with the system energy eigenvalues. In general however, the POVM energies are temperature dependent. For convenience we may identify a specific POVM energy E 0;β , defined as the smallest POVM energy in the lowtemperature limit. We can then introduce the POVM energy gaps ∆ m;β ≡ E m;β − E 0;β , which by definition are nonnegative at low temperatures. In terms of these gaps, the Fisher information for a general measurement is given by Similarly to the quantum Fisher information, the above expression takes the form of an energy variance. However for general measurements the energy spectrum of the system is replaced by the spectrum of POVM energies, and the Boltzmann probabilities associated with projective energy measurements are replaced by the POVM probabilities. These changes incorporate restrictions due to limitations of the measurement on top of those imposed by the system itself.
In investigating the scaling behaviour we are implicitly assuming that the Fisher information is a continuous function of temperature, which implies that the POVM energy gaps ∆ m;β must also be continuous functions. Following Ref. [25], we are going to study the scaling behaviour of the Fisher information when the POVM energy gaps have a well-defined power-series expansion in temperature around absolute zero By virtue of Weierstrass' approximation theorem, any continuous function can be approximated arbitrarily well by such a power series [35]. Note that this formulation does not exclude the case of projective energy measurements as this would be described by a series with only the constant term. For more general measurements, however, the expansion might contain non-zero higher-order coefficients. Following Potts et al. [25] we can make use of the relation between the POVM energies and the associated probabilities (Eq. (4)) to write ∆ m;β = −∂ β ln p m;β /p 0;β . Given the power-series expansion of the POVM energy gaps, we can integrate this equation and express the ratio of the probabilities for outcomes m and 0 as where g m is a temperature-independent integration constant. We stress that as a consequence of how we defined E 0;β , the probability p 0;β is the largest probability at zero temperature and must be non-vanishing in this limit. We thus obtain an expression for the probability of obtaining outcome m given fully in terms of the expansion coefficients of the corresponding POVM energy gap (note that the explicit dependence on p 0;β could be avoided by using the fact that the full distribution must be normalised).

B. Low-temperature scaling behaviour
The above model of limited measurements allows us to obtain, by substituting Eqs. (6) and (7) into Eq. (5), an expression for the Fisher information given fully in terms of the POVM energy gaps. Based on this, we can analyse the possible scaling behaviour of the Fisher information, as the system approaches zero temperature. First of all, we note that Eq. (5) can be rewritten as Notice that all terms on the right-hand side are positive, and because of this the scaling behaviour of the Fisher information 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 terms in Eq. (8). We focus on terms that result in sub-exponential scalings, referring the reader to Ref. [25] for a discussion of the remaining terms. For convenience, we define the ground-state set of measurement outcomes, as those for which the probability of obtaining that outcome remains finite at zero temperature (note that the outcome m = 0 is in the ground-state set by definition). From Eq. (7), we see that formally this set can be defined as Ω = {m | ∆ m,0 = ∆ m,1 = 0}. Now consider those terms in the Fisher information above where both outcomes belong to the ground-state set, and assume that the lowest order for which the expansion coefficient of any element in the ground-state set is non-zero is j ≥ 2. To leading order in temperature, the contribution from these terms takes the form These terms in the sum thus vanish at least quadratically, giving at best a constant contribution to the Fisher information.
Notice that if the ground-state set contains only a single outcome (m = 0), then the contribution is identically zero.
Next we consider the terms in the Fisher information where one of the outcomes belong to the ground-state set but the other one does not. To this end, we define the set of outcomes Ω = {m | ∆ m,0 = 0 and ∆ m,1 = 0}, for which the associated probability vanish sub-exponentially as the temperature goes to zero. The set of outcomesΩ has an associated POVM energy coinciding with that of the ground-state set at zero-temperature, but exhibits a linear degeneracy splitting at finite temperature. To leading order in temperature, the contribution from the corresponding terms is which vanishes at a rate determined by the the first-order expansion coefficients ∆ m,1 . It is straightforward to show that all other contributions vanish exponentially in the lowtemperature limit. The (sub-exponential) low-temperature behaviour of the right-hand side of the Fisher information (8), is generally given by the sum of Eq. (9) and Eq. (10). Which of these two dominate depends on the smallest first-order expansion coefficient. If the setΩ is not empty, and at least one element in the set has a value ∆ m,1 < 2(j − 1), where j denotes the lowest order with non-vanishing expansion coefficient within the ground-state set, then the low-temperature behaviour of the Fisher information is captured by In principle the first-order coefficient can take any positive value without violating the scaling bound imposed by the third law of thermodynamics. Notice that a divergent lowtemperature behaviour of the Fisher information can in principle be realised, if ∆ m,1 can take a value smaller than 2.

III. SCALING BOUND FOR LARGE SYSTEMS
In this section, we propose a finite-resolution criterion characterizing realistic measurements. We then go on to show how this criterion leads to a lower bound on the first-order coefficient ∆ m,1 . Furthermore we present an example of a measurement saturating the finite-resolution bound, showing that the bound is tight.
Recall that we are interested in a physical regime in which the system has an effectively continuous energy spectrum. Operationally this means that any measurement on the system can only resolve energy differences much larger than the energy gaps of the system energy spectrum. In this regime, it is convenient to work with the system density of states , where the sum is over distinct system energy eigenvalues and d k is the corresponding degeneracy. Throughout, we adopt the convention that the smallest system energy eigenvalue is set to zero ( 0 = 0). Now, we introduce a filtered density of states D m for each measurement outcome m, as the system density of states filtered through the corresponding POVM element where 1 k is the projection operator onto the eigenspace with energy k . Notice that the sum of all the filtered densities of states adds up to the total density of states. Furthermore, we introduce the continuous filter function f m ( ), for- the straight-line segments connecting these values. In addition we note that the density of states can be expressed as the rate of change of the number of states with energy below σ( ) = k d k θ( − k ), where θ denotes the Heaviside step function. Given these, the filtered density of states decompose into the product where the filter function fully characterizes the implemented measurement. Importantly we notice that the function σ( ) is non-decreasing for all energies. If we compute the Laplace transform in β of the filtered density of states, the result takes the form of a Stieltjes integral over a measure given by σ( ) The last equality can be obtained directly from equation (1), and relates the Laplace transformed filtered density of states to the product of the probability and the canonical partition function. Notice that the measure σ( ) is a discontinuous function of energy. For macroscopic systems the measure can often be approximated by an effective continuous measure, when σ( ) and f m ( ) vary on widely separated energy scales. To see this, we first define the averaged measure with respect to an energy window ω by which for non-zero ω is a continuous function of energy except at = 0, and which tends to a differentiable function of energy as ω is increased. The inclusion of the step function at zero energy is important if we are to capture the zero temperature limit correctly, since it ensures that the ground-state of the averaged model coincides with that of the exact model. For the purposes of low-temperature thermometry only the low-energy behaviour is of importance, and to leading order in energy we adopt an effective measure given by where d 0;ω is an effective ground-state degeneracy and α ω ,γ ω are positive, real-valued constants. The coefficient γ ω characterize the low-energy growth in the total number of states with energy less than . If we compute the Laplace transform with respect to this averaged measure (which now takes the form of a standard Riemann integral) we obtain to leading order in energŷ The averaged measure tends to overestimate the number of low-energy states as ω is increased, however this effect becomes negligible in the limit where the inverse temperature β is small compared to 1/ω. Now if we assume that f m ( ) does not vary significantly across an energy range ω, thenD m (β) is well approximated by the averaged function D m;ω (β). More quantitatively we can state this condition in the form of an inequality which bounds the rate of change of the filter function with energy. For macroscopic systems we can take the limit ω → 0, and in this case we are going to adopt the following finiteresolution criterion (FRC): FRC: In the limit of a macroscopic system, the filter function f m ( ) tends to a continuous, rightdifferentiable function of the system energy.
This is nothing more than a restatement of equation (18) for vanishing ω, which restricts the rate of change of the filter function to a finite value. We note that at = 0, the filter function may be discontinuous and Eq. (18) tends to the right derivative for ω → 0.

A. Finite-resolution bound
Having characterized what we mean by a finite-resolution measurement, we ask what the consequences of our finiteresolution criterion are for the behaviour of the POVM energy gaps in the macroscopic limit. By making use of equation (7), we obtain the relation (we now drop the dependence on the energy window ω and write simply d 0 ,α and γ) where for convenience we have defined the transfer function Now this is a relation at the level of the Laplace-transformed, filtered densities of states. We can invert equation (19) into a relationship directly between the filtered densities of states by taking the inverse Laplace transform of both sides. By applying the Laplace convolution theorem [37,38], we derive the relation We now focus on the specific case of m ∈Ω. For these outcomes, the inverse Laplace transform can be computed straightforwardly, and to leading order in energy we obtain where Γ(∆ m,1 ) denotes the Gamma function [37]. As we saw in the preceding section, the outcomes withinΩ are exactly the ones with potential to provide optimal low-temperature scaling of the Fisher information.
Recall, that the reference outcome m = 0, was chosen such that the associated probability approaches a constant value at zero temperature. This implies that the overlap of the POVM element Π 0 with the system ground state is non-zero, and therefore f 0 (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 case we find from equations (17) and (21) that to leading order in energy Based on this expression we can infer constraints on the linear coefficient. First, the requirement that f m (0) = 0 gives the weakest constraint ∆ m,1 > γ. This simply expresses the fact that the Fisher information is upper bounded by the the quantum Fisher information, which scales as T γ−2 for a density of states scaling as γ−1 . Further, the finite-resolution criterion restricts the rate of change to be bounded, d d f m ( ) < ∞. This implies a tightened scaling bound which directly implies that the Fisher information may not diverge any more rapidly than 1/T (achieved in the limit γ → 0). Note that a diverging Fisher information in the low-temperature limit can only be realized through a σ( ) that grows sub-linearly with energy, i.e., γ < 1. As an example of a system exhibiting such a sub-linear growth we mention systems of massive non-interacting particles at zero chemical potential [25], for such systems γ = 1/2 for one-dimensional geometries.

B. Tightness of bound
We now illustrate that the proposed finite-resolution bound is tight. Consider a binary measurement which resolves the system ground state exponentially well in the sense that it has POVM elements where κ > 0. Note that the overlap of Π 0 with the system energy eigenstates decays exponentially away from zero. This feature makes is straightforward to write down the filtered density of states. Focusing on m = 1 we find where nothing has been assumed about the form of the system density of states. We thus see that the corresponding filter function takes the form f 1 ( ) = κ +O( 2 ) to leading order in energy. If we adopt the density of states introduced in the preceding subsection, that is D( ) = d 0 δ( ) + αγ γ−1 + O( γ ), then upon comparison with equation (23) we find ∆ 1,1 = 1 + γ. Hence the binary exponential resolution measurement saturates the finite-resolution bound.
For good measure we now show how the same conclusion can be derived directly from the probabilities. The probability of obtaining outcome m = 0 can be written in terms of the system partition function as Substituting the probabilities p 0;β and p 1;β = 1 − p 0;β into the general form of the Fisher information (Eq. 2), one finds that The partition function is given by the Laplace transform of the density of states, hence we find Z β = d 0 exp (αγΓ(γ)β −γ /d 0 ) (in App. C we show how this form of the partition function describes a system of non-interacting bosonic modes). From this form of the partition function we can derive the low-temperature behaviour of the average energy If we substitute these into the above Fisher information, we find that to leading order in temperature (assuming that κ/β 1) which takes the form of Eq. (11) with ∆ 1,1 = 1 + γ and g 1 = ακγ 2 Γ(γ). Since γ can in principle take any positive value, the exponential-resolution measurement saturates the finiteresolution bound and attains a Fisher information scaling as 1/T in the limit γ → 0.

IV. GENERALIZATION TO NOISY MEASUREMENTS
In this section we extend the thermometry framework above to include noisy measurements. As the framework is general, one might ask if noise effects are not already accounted for. The answer is that in principle noise effects are described. However, for some noisy measurements, the POVM energy gap does not have a Taylor expansion. An impractically large number of terms must then be included in the expansion (6), to accurately capture the low-temperature behaviour, as the series becomes approximate.
To model noisy measurements, we consider the case where the observed outcomes m correspond to coarse graining over a fine-grained POVM with elements Π mµ . The probability of observing m is then Physically this could correspond to a measurement implemented using a sensor, where only a subset of the sensor degrees of freedom (or a subspace of the full sensor Hilbert space) is experimentally accessible. If we were to compute the Fisher information directly using the fine-grained distribution p mµ , we recover the noiseless results, and obtain an upper bound on the Fisher information computed from the coarse-grained distribution. This fact follows directly from the relation between the relative entropy of two probability distributions differing by an infinitesimal temperature δT and the Fisher information Since the relative entropy is monotonically decreasing under coarse-graining [39], we conclude that noise always reduces the Fisher information. The question we now address is, how it impacts the attainable scaling with temperature. Following the approach developed above, we introduce the fine-grained POVM energies which may be interpreted as the best guess of the system energy before the measurement, given the outcome (m, µ) [25]. For convenience we identify the smallest POVM energy in the low-temperature limit with the outcome E 00;β , and then define the fine-grained POVM energy gap ∆ mµ;β ≡ E mµ;β − E 00;β , which by definition is non-negative at low temperatures. Modelling the fine-grained POVM energy gaps by a power-series expansion around zero temperature as in Eq. (6), we are led to a probability distribution identical to (7), but with m replaced by the compound index mµ. Now, since the Fisher information is not defined with respect to the fine-grained probabilities, but rather with respect to the coarse-grained probabilities, it turns out that the relevant energies are the coarse-grained POVM energy gaps defined by In terms of these, the Fisher information can be written in the same 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 Notice that all terms in the sum are positive. Hence, the scaling behaviour of the Fisher information is determined by the term (or set of terms) which vanishes least rapidly as the temperature approaches zero. From Eq. (34), we can anticipate that fine-grained energy gaps that have a Taylor expansion may result in coarsegrained gaps that do not. This may result in qualitatively different behaviour of the fine-and coarse-grained Fisher information. In particular, noise may render the scaling of the Fisher information worse. In appendix B we discuss in general terms how noise impacts the attainable Fisher information scaling, here we illustrate the effect of noise with an example.

A. Illustration of noisy measurement
A simple example illustrating noise is obtained by adding white noise to the binary, exponential-resolution measurement of Sec. III B. That is, we study a binary POVM defined by Π 0 = η exp (−κH) + (1 − η)1/2. To understand how this noise model arises from coarse graining a fine-grained measurement, we consider the fine-grained POVM such that Π 0 = Π 00 + Π 01 and Π 1 = Π 10 + Π 11 . As in the noiseless case, we suppose that the average energy exhibits a power-law behaviour H β = αβ −(1+γ) at low temperatures in the macroscopic limit, with α and γ both positive. The corresponding partition function (at low temperatures) is then Z β = exp (αβ −γ /γ). For the fine-grained measurement outcomes, we find that to leading order in temperature (assuming that κ/β 1 and η < 1), the POVM energy gaps with respect to the reference E 00;β , take the form ∆ 00;β = ∆ 11;β = 0, We see that the fine-grained measurement outcomes have an associated set of POVM energy gaps that have a Taylor series in the low-temperature limit. Furthermore, they exhibit a linear degeneracy splitting. It then follows from Eq. (11) that the Fisher information takes the form which is equivalent to the noiseless form found above [cf. Eq. (30)]. Notice that when having access to the finegrained distribution, both the POVM energies and the resulting Fisher information is independent of the parameter η quantifying the amount of white noise. The picture changes when considering the coarse-grained energy gap (Eq. (34)). To leading order in temperature this is given by Notice that in contrast to the fine-grained energy gaps, this coarse-grained gap does not have a Taylor expansion. Computing the Fisher information over the coarse-grained gaps and probabilities (making use of Eq. (35)) gives This example thus illustrates how noise can result in a coarsegrained gap that has no Taylor expansion and how this results in a different (worse) scaling for the Fisher information at low-temperatures. Qualitatively we can understand why this is the case by studying the coarse-grained filtered density of states. For the example considered here we have D 00 ( ) = f 00 ( )D( ) = 1 + η 2 e −κ D( ), and under coarse-graining these are added together. Notice that whereas the filter function f 01 ( ) goes to zero as → 0, this is not true of f 00 ( )+f 01 ( ) (the same feature is found for the m = 1 outcomes). Hence in this case the noise removes outcomes from the setΩ, resulting in the worse scaling (note that a vanishing filter function at = 0 implies a vanishing probability at T = 0 and vice versa, cf. Eq. (14)). This effect is illustrated in Fig.2a. In App. B we show that an alternative noise model consist of a mixing of similar measurement outcomes in which each coarse-grained outcome can be seen as the sum of several similar fine-grained outcomes, this is illustrated in Fig.2b. In the case considered here, what is meant by similar outcomes is that the fine-grained outcomes are almost identical except for preparing slightly different energy distributions.
The noisy framework put forward here shows that our finite-resolution bound, as well as the results of Ref. [25] apply for any POVM that can be written as a coarse graining over a fine-grained POVM which has a spectrum with a well defined Taylor series. As the coarse-grained POVM itself may not have a spectrum with a well defined Taylor series, this extends the applicability of the results of Ref. [25] (as long as we do not want to rely on approximate Taylor series in the spirit of the Weierstrass theorem).

V. SINGLE-QUBIT PROBE
We now focus on estimating the temperature of a system of non-interacting bosons using a single qubit as a probe. The system is described by a spectrum of single-particle energies ω k (we take = 1). Consider the following measurement strategy: (i) first we initialise the probe qubit in its ground state |0 , (ii) then an interaction is turned on 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 the excited state |1 is We take the time-evolution operator U t to be generated by a time-independent Hamiltonian where a † k , a k denotes the bosonic creation and annihilation operators. The probe qubit is characterised by the three Pauli operators {σ x , σ y , σ z }, and we take the probe energy to be proportional to the σ z operator.
Computing the outcome probabilities now requires specifying an interaction Hamiltonian and determining the resulting dynamics. This task is complicated by the fact that the low-temperature and short-time regime is generally not accessible via standard Markovian master equations [11,40]. However, if the interaction time is sufficiently short we can make analytical progress by approximating the probability up to second order in t. In this case we find that We consider a linear interaction Hamiltonian consisting of an excitation-preserving part and a non-excitation-preserving part. Introducing the raising and lowering operators σ ± = 1 2 (σ x ± iσ y ) for the probe qubit, the interaction Hamiltonian takes the form where {g k , λ k } are real-valued coupling coefficients. In the limit of a macroscopic system, these coupling coefficients are taken to approach continuous functions. Physically this means that the interaction cannot implement a sharp displacement of an individual system mode.
Given H int , it becomes straightforward to show from Eq. (44) that the excited-state probability at short times takes the form where n β (ω k ) denotes the Bose-Einstein distribution. We see that the probability consists of two contributions: a temperature-dependent term, in which the probability is directly related to the occupation of the bath modes, and a temperature-independent term. The presence of the temperature-independent term means that the probability of finding the probe qubit in the excited state is generally nonzero even at arbitrarily low temperatures. As in the example in Sec. IV A, this prevents a scaling of the form of Eq. (11) and can be captured by our framework for noisy thermometry.

A. Excitation-preserving interaction
We now focus on the excitation-preserving case (λ k = 0), and consider an interaction characterised by a continuous spectral density of the form where α is the dissipation strength, s is the ohmicity and ω c is the cutoff energy [40][41][42][43]. The sum in the excited-state probability (46) is then replaced by an integral, which can be solved analytically. In the low-temperature limit we find We see that this protocol gives a probability vanishing subexponentially as the temperature goes to zero, and comparing with the general expression Eq. (7), we see that to lowest order, the POVM gap scales as ∆ 1 = (1 + s) T . The case of an excitation-preserving interaction can thus (for short time at least) be described within our noiseless thermometry framework.
From the value of the linear expansion coefficient, ∆ 1,1 = 1+s, it follows that for ohmicity approaching zero, the finiteresolution bound ∆ 1,1 ≥ 1 is approached. The corresponding Fisher information scales as T s−1 and thus diverges for sub-Ohmic baths in the low-temperature limit. This serves as an illustration that the finite-resolution bound is in principle attainable via an excitation-preserving interaction in the shorttime limit, and thus the bound is tight. Realising such an excitation-preserving interaction may however be challenging.

B. Excitation-non-preserving interaction
We now turn to the arguably more realistic excitation nonpreserving case. The case λ k = g k corresponds to the wellknown spin-boson model [41][42][43][44]. Adopting the same spectral density as above, the excited-state probability in this case takes the form In contrast to the excitation-preserving case, this probability does not in general correspond to the noiseless version of Eq. (7) since the POVM energy gap ∆ 1 ∝ T s+2 , does not have a Taylor expansion for arbitrary s at low temperatures.
However, as shown in App. D, this scenario can be described using a fine-grained POVM with energy gaps that do have a Taylor expansion. Therefore, the scenario is captured by the noisy framework.
Given the probability (49), a short calculation shows that the Fisher information has a low-temperature scaling given by F T ∝ T 2s . Again, this is in full agreement with the general noisy theory developed above. Within the spin-boson model, the Fisher information thus vanishes quadratically for an Ohmic spectral density with s = 1, and linearly for a sub-Ohmic spectral density with s = 1/2.
To corroborate the analytical results based on the shorttime approximation, we turn to a numerical simulation of the Fisher information for the spin-boson model. To perform the simulations we made use of the recently developed tensornetwork TEMPO algorithm and its extension to multi-time measurement scenarios [45,46]. Details of the simulations are provided in App. A. Making use of this algorithm has the benefit that the temperature derivative of the excited state can itself be expressed as a tensor network and computed to the same level of accuracy as the probability itself.
Results for the Ohmic and the sub-Ohmic cases are shown in Fig. 3. Generally we find that the short-time approximation provides a good description of the observed scaling behaviour at sufficiently short times. Even more interesting we note that the scaling behaviour predicted within the short-time approximation is valid even at times well beyond the regime in which the short-time approximation is expected to hold (αδt 2 Γ(1 + s)ω 2 c 1). This indicates that the predicted precision scaling is experimentally relevant, even without the requirement of being able to probe the nonequilibrium qubit dynamics at very short-times. Notice that the low-temperature Fisher information tends to initially increase with time as information about the environment state is extracted by the qubit. After some time the low-temperature Fisher information starts to decrease. This can be understood as the qubit reaching a stationary state, such that a one-time measurement performed on the qubit can no longer probe the relaxation dynamics induced by the coupling with the thermal bath.
Finally, we note that at sufficiently low temperatures the simulated Fisher information differs from the Markovian result, even for the rather weak coupling and long times considered here. A similar effect was observed in the context of temperature estimation via the Kubo-Martin-Schwinger-like relations obeyed by emission and absorption spectra of multichromophoric systems [47]. There it was pointed out that faithfully recovering the temperature from observed spectra requires taking into account system-environment correlations. This is true even at very low coupling strengths, where these correlations are generally weak.

VI. CONCLUSION
In this paper we have discussed precision scaling for thermometry in cold quantum systems. In particular, we have investigated how finite measurement resolution, meaning that states that are close in energy cannot be perfectly distinguished, impacts the precision scaling. We have proposed a finite-resolution criterion characterising such measurements. Based on this, we derived a tightened bound on the scaling of the Fisher information. Furthermore, we showed that this bound is tight as it can be saturated via both an exponential resolution measurement as well as an excitationpreserving, single-qubit measurement on a sample of noninteracting bosons. We validated the approximations involved in demonstrating tightness for the single-qubit measurement by performing a numerical simulation of the sub-Ohmic spinboson model. Here, we provided an illustration of a Fisher information scaling linearly with temperature. Interestingly, as far as we are aware, this is the best scaling which has been found in any concrete physical model subject to finiteresolution constraints.

ACKNOWLEDGMENTS
MRJ and JBB were supported by the Independent Research Fund of Denmark. PPP acknowledges funding from the European Union's Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie Grant Agreement No. 796700. noise is not detrimental for the scaling. On the other hand, if no such m exists, then we refer to detrimental noise (assuming thatΩ m is non-empty for at least one outcome). For detrimental noise we are then left with outcomes for which Ω m is non-empty, whileΩ m may or may not be non-empty.
Consider the right-hand side of Eq. (35) for the case of detrimental noise. For terms where bothΩ m andΩ n are empty, the scaling behaviour is identical with that of the corresponding noiseless terms (Eq. (9)), except that the noiseless coefficients of the POVM energy gap must be replaced by the coarse-grained version n,2 = 0 for some m and n), then the same scaling behaviour of the Fisher information as given by Eq. (9) is attainable and this scaling is optimal (note that the probabilities considered here tend to nonzero constants at zero temperature). If a second-order gap does not exist, then the optimal scaling is instead provided by terms for which Ω m is non-empty for some m. A straightforward calculation shows that the contribution from such terms takes the form which should be summed over all outcomes m for which both Ω m andΩ m are non-empty. Assuming that the finiteresolution criterion applies (∆ mμm,1 ≥ 1), this contribution is at best constant. Hence under the conditions of finite resolution and detrimental noise, a diverging Fisher information is impossible. As a second example of a noisy measurement we can consider the coarse-graining of a fine-grained measurement of the form Π 00 = 1 2 e −κH , Π 01 = 1 − η 2 e −κH , This fine-grained model is illustrated in Fig. 2b. For this measurement we find ∆ 00;β = ∆ 01;β = 0 and ∆ 10;β ≈ (1 + γ)T + (1 + γ)ακT 2+γ Hence, as in the previous example, the fine-grained measurement gives a Fisher information scaling as T γ−1 to leading order, and the coarse-grained measurement gives a T 2γ scaling, Thus the same scaling behaviour of the Fisher information is observed for this alternative example of a noisy model. Note that both models exhibit detrimental noise which results in the different scalings for the fine-and coarse-grained Fisher information.