Quantum Precision Thermometry with Weak Measurement

Quantum precision thermometry with weak measurement

Arun Kumar Pati∗ and Chiranjib Mukhopadhyay† Quantum Information and Computation Group, Harish-Chandra Research Institute, HBNI, Allahabad 211019, India

Sagnik Chakraborty‡ and Sibasish Ghosh§ Optics & Quantum Information Group, The Institute of Mathematical Sciences, HBNI, C. I. T. Campus, Taramani, Chennai - 600 113, India

As the minituarization of electronic devices, which are sensitive to temperature, grows apace, sensing of temperature with ever smaller probes is more important than ever. Genuinely quantum mechanical schemes of thermometry are thus expected to be crucial to future technological progress. We propose a new method to measure the temperature of a bath using the weak measurement scheme with a finite dimensional probe. The precision offered by the present scheme not only shows similar qualitative features as the usual Quantum Fisher Information based thermometric protocols, but also allows for flexibility over setting the optimal thermometric window through judicious choice of post selection measurements.

I. INTRODUCTION eral, complex numbers. In the last few years, weak values have been used in the context of measuring quantities which Quantum mechanics lends different status to observable may or may not have quantum mechanical observables as- quantities and parameters of a system, the latter of which are sociated with them, for example, the geometric phase [18], not expressible through Hermitian operators. Among phys- non Hermitian operators [19], density matrix corresponding ical parameters, temperature of a system is perhaps one of to a quantum state [20–24], or the entanglement content of a the most ubiquitously useful. Temperature drives chemical quantum state [25, 26]. The weak value amplification tech- reaction rates [1], control the generation of thermomelectric nique has found recent physical application in observations of current in thermocouples [2], or heat flow between different the spin Hall effect [27], photon trajectories [28], or the time baths [3]. Thus, thermometry, or the science of measuring delay between ultra fast processes [29] as well. Thus, it is temperature, is of paramount importance. This is especially reasonable to ask whether a weak measurement based scheme relevant for modern technological applications and devices, is a viable approach towards thermometry. In this paper, we where the thermal baths may themselves be relatively tiny. provide such a scheme. We mention here that there has been Nanoscale probes, which follow laws of quantum physics, are another theoretical as well as experimental paper by Egan and thus required to measure the temperature of such baths with- Stone [30][31] in the recent past, which introduces the con- out disturbing them too much. Quantum thermometry thus cept of weak thermometry. Experiments demonstrating weak aims at improving the technique of temperature sensing using value enhanced metrological tasks have also ben performed in small probes [4–10]. The analysis of quantum thermometry analogus contexts [32, 33]. so far has focused on seeking to find and saturate the quantum In the present work, we outline how to measure temper- Cramer Rao bound for various partially and fully thermalized ature using weak values with finite dimensional probes and configurations of systems [11, 12]. In addition to ensure non- specifically concentrate on qubit probes. We show the pres- invasiveness, these studies show possible improvement in the ence of an optimal temperature window for the precision of- precision due to quantum effects for non-equilbrium settings. fered by this scheme, which is a feature repeated in the usual Even in the steady state scenario, quantum enhancement in paradigm of quantum thermometry [11, 34–36]. One crucial the precision using a quantum switch has been recently re- advantage offered by the present scheme over previously con- ported [13]. For observables, another technique to experimen- sidered schemes is the ability to shift the optimal precision tally determine them quantitatively has recently come to the window while keeping the probe parameters fixed, and sim- fore, which relies on a so called weak measurement scheme ply changing the post selected state. In addition, we argue arXiv:1901.07415v2 [quant-ph] 16 Jun 2020 [14–17]. Weak measurement is a technique of ascertaining that, the proposal based on the weak value is suitable for mea- information about an observable through a weak interaction surement of a very hot body. If the temperature of a system between the system and the measurement apparatus gener- is very high, then prolonged contact with a measuring appa- ated by the observable. It is then followed by a strong post- ratus may damage the apparatus itself. On the other hand, in selection measurement on the system. One of the most well our weak measurement based scheme, the measuring appara- known facets of weak measurement scheme is the concept of tus is brought into contact with a thermalized qubit of the bath the so called weak values of an observable, which are, in gen- for a very short duration of time, thus potentially saving any damage to the apparatus. [37] The organization of the paper is as follows. SectionII ∗ [email protected] briefly reviews the weak measurement scheme. The main † [email protected] theme of our paper, namely the protocol for measuring tem- ‡ [email protected] perature through weak values, is described in Section III. This § [email protected] is followed by the detailed analysis of precision offered by the 2 present scheme for a qubit probe in SectionIV. SectionV con- tains a complementary QFI based analysis of precision for the weak thermometric protocol. We finally conclude with a few observations and outline some possible future developments. T

Apparatus Qubit II. WEAK MEASUREMENT

In theory of weak measurements, a quantum system S is chosen to be in an initial state |ψiiS , called the pre-selected state, and it is made to interact with a measurement appara- FIG. 1. (color online) Schematic of the thermometric scheme. We tus M, prepared in a state |φiM. The interaction is weak in ˆ ˆ assume the bath being a collection of thermalized qubits, and couple strength, and generated by a Hamiltonian Hint = gA ⊗ Px, the measuring apparatus to one of the qubits, which is called the ˆ ˆ where A is a system observable, Px is the momentum operator probe qubit for a very short time. of M and g is a small positive number signifying the strength of the interaction. This is followed by a strong measurement on the system, called the post-selection measurement. The outcome of post-selection measurement that we focus on is degenerate energy eigenvalues E1, E2,..., Ed and the corre- along the state |ψ i. Clearly, the post-selection is a selective f sponding energy eigenstates |ψ1i , |ψ2i ,..., |ψdi. Assume that measurement, as we ignore the other outcomes of the mea- S is in contact with a heat bath of temperature T and S has surement. −βHˆ −βHˆ reached the thermal equilibrium state ρT ≡ e / (Tr[e ]) = The time-evolved state of S + M before post-selection is Pd − Pd − e βEn |ψ i hψ | / e βEn . Here β = 1/(k T), with given by n=1 n n n=1 B h i kB being taken as unity. Prepare the measuring aparatus M in −igAˆ⊗Pˆ x |Ψi = e |ψii ⊗ |φiM a state |φi, having position wave function φ(x). Note that here h i M is considered to be a continuous-variable system, in gen- ≈ ⊗ − ˆ ⊗ ˆ | i ⊗ | i IS IM igA Px ψi φ M , (1) eral. We would like to perform measurement of an observable Aˆ on the system S . up to first order in g. Note that the state |Ψ˜ iSM ≡ (IS ⊗ IM − ˆ ˆ igA ⊗ Px) |ψii ⊗ |φiM is, in general, unnormalized. The re- Consider now evolution of S + M under the action of an duced state of |Ψ˜ i on M is SM interaction Hamiltonian Hînt = gAˆ ⊗ Pˆ x for a small amount of time τ. The interaction strength g is also considered to |φ˜ f i ≈ hψ f |ψii − ig hψ f | Aˆ |ψi Pˆ x |φi , (2) M i M be small – in the regime of weak interaction between S and which on normalizing looks like M. Here Pˆ x is the momentum observable of M canonically ˆ conjugate to the position observable Xˆ. We assume here that |φ i ≈ 1 − igA Pˆ |φi ≈ e−igAw Px |φi , (3) f M w x M M during the time interval [0, τ], S and M are under the action only Hˆ where Aw ≡ hψ f | Aˆ |ψii / hψ f |ψii is called as the weak value of of the Hamiltonian int. This may be fulfilled in different ways: (i) We may decouple the system S at time t = 0 from Aˆ. Note that Aw can also be complex and take values beyond the eigenvalue range of the observable. the heat bath (after S achieves the thermal equilibrium state ρT ) and thereby switch on the interaction Hamiltonian Hînt for Since the weak value Aw is, in general a complex quantity, one must experimentally observe the real as well as imagi- the time duration [0, τ]. (ii) On the other hand, we may think nary parts of the weak value. On measuring certain proper- of assuming here that the the strength g of interaction is much ˆ ties of |φ i , the real and imaginary parts of A can be deter- higher than that of the system Hamiltonian H, so that due to f M w ˆ mined. These properties include shift in position and momen- action of Hint for a small time span τ, it is enough to consider the change in states of S under the action of Hînt only. The tum values compared to that of |φiM, variance of momentum wave-function, rate of change of position wave-function and time span τ should be small enough so that in spite of taking the strength of interaction [38]. Laguerre-Gaussian modes of the strength g of the interaction Hamiltonian Hint being greater an optical beam have also been used to measure the real and than the free Hamiltonian, gτ 1. At the end of the action of imaginary parts of a weak value [39, 40]. Recently it has also the interaction Hamiltonian, the joint state of S + M becomes: been proposed and experimentally demonstrated that the weak value can be inferred from interference visibility and phase −igτAˆ⊗Pˆ x igτAˆ⊗Pˆ x shifts [41]. ρSM(τ) = e (ρT ⊗ |φihφ|)e

≈ (IS ⊗ IM − igτAˆ ⊗ Pˆ x)(ρT ⊗ |φi hφ|)

III. ASSESSING TEMPERATURE THROUGH WEAK × (IS ⊗ IM + igτAˆ ⊗ Pˆ x) VALUES ≈ ρT ⊗ |φi hφ| − igτ[Aˆ ⊗ Pˆ x, ρT ⊗ |φi hφ|]. (4)

(16). One of the advantages of the present scheme is that the where 0 ≤ δ 1, and |χi is a random pure qubit state with measuring apparatus is brought into contact with the heat bath corresponding Bloch sphere parameters θ, and φ. Physically for a very short time, hence reducing the chance of damage to this indicates imperfect thermalization, which is experimen- the apparatus. We add here that the rationale behind choosing tally relevant, especially in situations where the thermalization the operator A in this form is the following, if A is chosen as timescale is not extremely fast compared to the time available δ being along the z-axis of the Bloch sphere, there is no infor- for sensing. The corresponding weak value is denoted by Aw. mation to be obtained about the temperature from the weak Now, an experimentalist may use the formula (15) to find the value. In case of thermalization, the azimuthal symmetry of apparent inverse temperature β˜ of the bath as the state in the Bloch sphere picture means it is equally feasible choosing any operator A along the x − y plane of the Bloch ˜ 2 (δ) sphere, thus we may assume the above form of A without loss β = arctanh(−iAw ) (18) of generality. Once A is fixed, the azimuthal symmetry of the E2 − E1 problem is lost, and one has to choose the post selected state It is of obvious practical interest to us that the inferred value carefully. Above, we assumed an example of the post selected of temperature does not change wildly if the thermalization is state, however a more general analysis of the precision of our imperfect. In order to quantify this, we invoke the idea of scheme depending upon different post-selections have been quantifying a relative error, which is the difference between performed in the following sections. the temperature furnished from imperfect thermalization, and the genuine temperature of the bath, divided by the bath temperature, and scaled by the imperfection δ. The squared rel- IV. PRECISION IN MEASUREMENT OF TEMPERATURE ative error introduced through imperfect thermalization may thus be taken to be |β˜ − β|2/(δ2β2). It is this quantity we shall It is natural to wonder about the optimal temperature win- concentrate upon. To remove the effect of |χi, which may con- dow where the present scheme works best, that is, most pre- ceivably capture some information about the state in which the cisely. The usual procedure for determining the precision of probe was initialized, we shall finally average over the pure quantum thermometers is through finding the corresponding states |χi. We will write E2 − E1 as the gap ∆ from here on. (δ) quantum Cramer Rao bound. In this section, we adopt a dif- Performing a perturbation expansion for Aw around δ = 0, ferent approach. We restrict to qubit systems for simplicity. and retaining terms upto first order in δ, the expression for β˜ However, the present analysis can be extended to higher di- is given by mensions in a similar fashion. ! 2 β∆ β˜ = arctanh c1 tanh − ic2 , (19) A. Precision analysis for imperfect thermalization ∆ 2

2 |hψ f |χi| hψ f |A|χihχ|ψ f i where c1 = 1 − δ , and c2 = δ . Note that, Let us assume that the initial pre-measurement state is not hψ f |ρT |ψ f i hψ f |ρT |ψ f i exactly a thermal state, but very close to it, and written in the c1 and c2 are functions of the Bloch sphere angles {θ, φ} of the following manner state χ. Thus, averaging over them, the resulting root mean squared relative error Nβ for the particular weak measurement strategy adopted in the previous section, is written as a func- (δ) ρT = (1 − δ)ρT + δ|χ(θ, φ)ihχ(θ, φ)|, (17) tion of inverse temperature as

" # 2 1 |β˜ − β|2 1 Z π Z 2π 2 β∆ N2 = sin θdθdφ = arctanh c (θ, φ) tanh − ic (θ, φ) − 1 sin θdθdφ. (20) β 2 2 2 1 2 4π β δ 4πδ θ=0 φ=0 β∆ 2 "

Now, neglecting the second and the higher order coeffficients of generality, that the energy gap ∆ = 1. The expression for of δ, we note that the expression for relative RMS error is root mean squared relative error Nβ is given by written as q β β∆ − − 2 2 − 4|v tanh + iv |2 (13 cos 2ν) cosh β 4 cos 2ξ sin ν cosh ( 2 ) (3 + cos 2ν) 2 1 1 2 2 . Nβ = sin θdθdφ (21) 2 4π 2 2 2 β∆ 2 3β cosh(β/2) + cos ξ sinh(β/2) [1 − tanh (β/2)] ∆ β (1 − tanh 2 ) " (22) where v1 = (1 − c1)/δ, and v2 = c2/δ. At this point, let us fix the observable A = −i|ψ1ihψ2| + i|ψ2ihψ1|, as in the previous For specific choices of the post selected state, the anove equa- section, and assume the arbitrary post selected state |ψ f i = tion yields the expression for error, and consequently, the in- iν cos(ξ/2)|ψ1i + e sin(ξ/2)|ψ2i. We also assume, without loss verse quantity signifies the precision of measurement. For ex- 5

0.7

|ψf〉 = √(1-ε) |ψ1〉 + √ε |ψ2〉 1 0.6 |ψf〉 = √2 (|ψ1〉 + |ψ2〉)

0.5

1/Nβ 0.3

0.2

0.1 0 1 2 T 4 5 6

FIG. 2. (color online) Precision of the scheme plotted against the temperature for the speciﬁc choice of post-selected states |ψ i = FIG. 4. (color online) The magnitude of precision at optimal tem- √ √ f 1 − |ψ i + |ψ i (green dotted curve), |ψ i = √1 (|ψ i + |ψ i) perature is plotted against the parameters ξ and ν of the post selected 1 2 f 2 1 2 (red dashed curve). Energy gap between ground and excited states is state |ψ f i. unity in each case. = 0.01 is assumed.

FIG. 3. (color online) The temperature at which the optimal precision is achieved is plotted against the parameters ξ and ν of the post selected state |ψ f i.

1 ample, assuming |ψ f i = √ (|ψ i + |ψ i), as in the previous 2 1 2 section, the relative error reads as s 1 + 4 cosh β + 3 cosh 2β N+ = . (23) β 9β2

Fig.2 shows that the precision, which is defined as the inverse of the relative error, attains a relatively narrow peak at a finite temperature, which determines the best operating window of the scheme. While the qubit thermometric schemes [8, 11, 13, 34] based on the strong measurement are different in conception than the present scheme, it is nonetheless note- worthy that the phenomenon of a narrow peak in the precision corresponding to an optimal temperature window is present in both the cases. We also note that for the QFI based analysis of optimal qubit thermometric probes with unit energy FIG. 5. (color online) The magnitude of precision is plotted against the parameters ξ and ν of the post selected state |ψ i for (above) low ≈ f gap, the peak is situated at T 0.41 [11, 34], which is ob- temperature T = 0.05, and (below) high temperature T = 100. tained through solving the transcendental equation [11, 34] e1/T = (1 + 2T)/(1 − 2T). In comparison, in the present 6 scheme, for a specific post-selected state |ψ f i, the location of the peak for optimal precision is obtained by the vanishing of the first derivative with respect to the inverse temperature β of the expression in (22). In particular, for the specific example |ψi = √1 (|ψ i + |ψ i) discussed in the previous section, the 2 1 2 equation takes the form

3 cosh β − 1 β = β , (24) 3 sinh β − 2 tanh 2 which has the solution T ≈ 0.79. The temperatures corresponding to optimal precision for other choices of post- selected states are depicted in Fig3. Interestingly, for every |ψ f i, the corresponding optimal temperature is significantly higher than 0.41. This indicates the possibility that the present scheme may be better than the strong measurement based one based one for the relevant temperature range. It is natural to FIG. 6. (color online) The magnitude of precision at optimal tem- | i wonder about the post-selected state ψ f which corresponds perature is plotted against the temperature T and the polar angle ξ of to maximum precision. From Fig4 as well as Fig5, it may the post selected state |ψ f i. be concluded that for any arbitrary temperature, |ψ f i close to |ψ1i maximizes the precision. Here, let us add a note of cau- 1.1 0.35 tion, from the definition, the weak value is independent of the 1.0 0.30 temperature when |ψ f i is exactly |ψ1i, hence measuring the 0.9 0.25 0.8 weak value furnishes no information about the temperature. 0.20 0.7 Thus, we must take |ψ f i to be extremely close to |ψ1i, but not 0.15 0.6 identically equal. 0.10 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 ξ ξ π π B. Precision analysis for unsharp post-selection FIG. 7. (color online) Left- optimal temperature, and Right- precision at optimal temperature, as a function of the polar angle ξ of the Let us now consider another potential source of error in the post measurement state |ψ f i. present scheme, that is, the post selection may be unsharp. Let us assume the unsharp post-measurement qubit state in the form where ξ is the polar angle of the pure post selection state |ψ f i. 1 For the specific choice |ψ f i = √ (|ψ i + |ψ i) in the last sec- ρ() = (1 − )|ψ ihψ | + 11 (25) 2 1 2 f f f 2 tion, this amounts to the following expression for precision, which is defined as the inverse of the error Thus, the corresponding perturbed weak value of the observable A is given by 4 1/Nβ(|+i) = h i, (29) β 1 − tanh2 β () " !# 2 Tr(ρ f AρT ) Tr(Aρ ) − 1 A ≈ A T O 2 w = () w 1 + + ( ) Aw hψ f |ρT |ψ f i Tr(ρ f ρT ) It can be seen from Fig.6 that the precision, defined as the (26) inverse of the relative error, attains a peak at some tempera- Now, using (15), the corresponding expression for shifted in- ture, which determines the corresponding optimal temperature ˜ 2 () verse temperature is β = ∆ arctanh(−iAw ). From which, the window for thermometry. Fig.7 reveals that the optimal tem- formula for the squared relative error incurred is given by perature for the scheme varies from Topt ≈ 0.54 to Topt ≈ 1.12. Solution of the following transcendental equation determines the location of the optimal temperature T ∗ for any given post- 2 2 |β˜ − β| 4 Tr(AρT ) − 1 selected state |ψ i with the polar angle ξ N2 = = . f β 2 2 2 β ∆2β2[1 − tanh (β∆/2)]2 hψ f |ρT |ψ f i 1 ∗ 1 (27) sinh ∗ − T (1 + cosh ∗ ) cos ξ = T T (30) As in the previous section, if one chooses A = −i|ψ1ihψ2| + ∗ 1 1 T sinh T ∗ − cosh T ∗ + 2 i|ψ2ihψ1|, then, assuming ∆ = 1 without loss of generality, the corresponding expression for relative error reads as An interesting feature we observe in Fig7 is that there is a shift in the optimal temperature window towards the right, and 4 Nβ = , (28) higher temperatures is associated with the reduction in the op- h 2 β i β β 1 − tanh 2 (1 + cos ξ tanh 2 ) timal precision attainable through the present scheme. This is 7 in line with the intuition that, as a distinctly quantum mechan- 5 ical scheme, the weak measurement protocol should work best φ = 0, Δ = 1 in the low temperature regime. 4 θ = π / 4

n θ = π / 2 o i

s θ = 3π / 4 i

V. QUANTUM FISHER INFORMATION BASED ANALYSIS c 3 e

OF PRECISION OF WEAK THERMOMETRY PROTOCOL r P

d 2 Until now, we have looked at the robustness of precision of e l the weak measurement based thermometric protocol. In this a c section, we present the complementary analysis of thermo- S 1 metric precision in this protocol through the usual quantum estimation theoretic methods. Let us ﬁrst recall the relevant bound on ﬂuctuation ∆u of estimation of a single parameter u, 0 which is known as the quantum Cramer-Rao bound (QCRB). 0 0.2 0.4 T 0.6 0.8 1

1 FIG. 8. (color online) Scaled precision F˜ plotted against tempera- ∆u ≥ , (31) p ture for various choices of post-selected state parameter θ. The en- nFu(ρu) ergy gap ∆ is taken to be unity, and the azimuthal angle φ of the where F is the quantum Fisher Information for the state ρu, post-selected state is assumed to be zero. and n is the number of runs. For the single parameter estimation case, it is always possible to saturate the lower bound. We now remember that the ﬁnal state of the pointer after the post- optimal temperature window by shifting the choice of the post selection in state |ψ f i is given in (6), which is a pure state. For selection. As earlier, attempt to shift this optimal temperature a pure state |ψui with the corresponding parameter u, we recall window towards the right, i.e., to higher temperatures by ju- that the QFI is given by [43] diciously choosing the post-selection state, inevitably results in a loss of optimal precision. Thus, the QFI based analysis yields the same qualitative picture as the analysis performed ˙ ˙ ˙ 2 Fu = 4hψu|ψui − 4|hψu|ψui| , (32) earlier. where |ψ˙ui denotes the ﬁrst derivative of the state |ψui with respect to the parameter u. Putting this in (6), we obtain the following expression for QFI for temperature T after a little VI. DISCUSSIONS algebra We have presented a protocol for measuring temperature of !2 a quantum mechanical bath through measuring weak values 2 2 dAw 2 FT = g τ (ξ − ξ ) (33) on a probe which is in thermal equilibrium with the bath. Our dT protocol relies on careful choice of Hamiltonian, which is gen- ∗ ˆ ˆ erating the weak interaction, and the basis of post-selection igAw Px ˆ −igAw Px 2 2gτIm(Aw) Here ξ = hφ|e Pxe |φi = |hφ|xi| e . Thus, measurement. Although our protocol is applicable to probes upto leading order, the square-root of QFI is proportional to of any dimension, we restrict to qubit probes and compared our result with other thermometric schemes which employ

dρT dρT strong POVM measurements on the probe and maximization hψ |A |ψ i hψ | |ψ ihψ |Aρ |ψ i p dA f f f f f T f of quantum Fisher information (QFI). We find that there is a F ∝ F˜ = w = dT − dT T 2 dT hψ f |ρT |ψ f i (hψ f |ρT |ψ f i) narrow operational window of temperature values which can be optimally measured through our protocol, a feature charac- (34) teristic to other usual thermometric schemes. Moreover, in our ˆ Once more, we assume A = σˆ y unless otherwise mentioned. protocol the peak value of the optimal temperature window θ Expressing an arbitrary post-selected state as |ψ f i = cos 2 |0i+ is shifted towards higher temperature compared to the usual θ iφ ˜ sin 2 e |1i, we obtain the expression for scaled precision F as schemes. It may be helpful to investigate, in future works, being the possible reason behind such a phenomenon. Interestingly, we find a trade off between shifting the optimal temperature q window to higher temperatures and the optimal precision. 2e∆/T ∆ sin θ cos2 φ + cos2 θ sin2 φ We expect that the present work will turn out to be use- F˜ = (35) 2 ful in the context of improving techniques for precise and T 2 1 − cos θ + e∆/T (1 + cos θ) efficient temperature estimation of nanoscale thermodynamic Some specific illustrations for different values of θ are demon- systems. Several issues however, are left for future investi- strated in Fig.8. The existence of an optimal temperature win- gation. Firstly, a comprehensive description of the present dow is once more observed, as is the feature of a shift in the scheme for higher dimensional probes, including harmonic 8 oscillator probes, remains to be explored. In particular, we showed in the present work that the location of the optimal " # 2 operating window depends on choice of post-selection, even 2 β∆ = arctanh c1(θ, φ) tanh − ic2(θ, φ) − 1 for a probe with fixed energy spectrum. It is of some interest β∆ 2 to determine whether the location of operating window can " # 2 2 β∆ be made even more tunable by changing the post-selection in = arctanh (1 − δv1) tanh − iδv2) − 1 the higher dimensional cases. Additionally, some other ques- β∆ 2 " !# 2 tions emerge from the idea of weak thermometry. Firstly, in 2 β∆ β∆ the usual strong measurement based approach to thermometry, = arctanh tanh − δ v1 tanh + iv2 − 1 β∆ 2 2 optimizing the precision for a qudit probe leads to the optimal energy spectrum of the probe to be a highly unphysical one, (36) namely, a gapped ground state and all the other energy eigen- Expanding the last expression in terms of δ and retaining only states are energetically degenerate [11]. It is an open problem upto first order yields the following expression for the inte- to figure out the form of the optimal energy spectrum of the grand probe in the arbitrary dimensional case, and check whether the corresponding Hamiltonian is experimentally feasible.. Addi-   2 tionally, finding the information disturbance tradeoff [7] for δ(v tanh β∆ + iv 2 β∆ 1 2 2  thermometers working with weak measurement schemes is =  −  − 1 β∆  2 − 2 β∆  an interesting avenue of future work. Depending upon the 1 tanh 2 2 β∆ 2 choice of the actual physical systems for the probe, the op- 4δ |v1 tanh + iv2| erating windows for the precise measurement of temperature = 2 , (37) 2 2 − 2 β∆ 2 will have different signatures for systems under consideration, ∆ β (1 tanh 2 ) which we would like to explore in future. Finally, the weak which is the integrand in eqn. (21) of the main text. The value based thermometric scheme relies on having an identi- integration on the next step is performed on Wolfram Mathe- cally prepared ensemble, and hence suffers from the drawback matica, and the simplified result for our choice of observable that single shot measurement results can not achieve arbitrary A is given out by the expression in eqn. (22). precision, and further improvement of thermometric precision Derivation of eqn. (23) from eqn. (22) – From eqn. (22), with finite round of measurements based on the weak ther- for our specific choice of post-selection, the RMS error reads mometry scheme should be useful to perform in future. s 12 cosh β − 4 Nβ = 2 2 β 2 β 2 9β cosh 2 (1 − tanh 2 ) s (12 cosh β − 4) cosh2 β = 2 ACKNOWLEDGEMENTS 9β2 s (6 cosh β − 2 + 6 coshβ −2 cosh β = CM acknowledges a doctoral fellowship from the Depart- 9β2 ment of Atomic Energy, Government of India, as well as fund- s ing from INFOSYS. AKP and CM wish to thank The Institute 3(1 + cosh 2β) + 4 cosh β − 2 = of Mathematical Sciences, Chennai, for their kind hospitality 9β2 while hosting them for a visit, during which this work was s completed. We would like to thank the anonymous referees 1 + 4 cosh β + 3 cosh 2β = (38) for their constructive criticism. 9β2

Derivation of eqn. (24) – Optimizing the RMS error re- quires the derivative with respect to temperature ∂βNβ to van- ish. Hence, 9β2 (6 sinh 2β + 4 sinh β) = 18β (1 + 3 cosh 2β + 4 cosh β) ; APPENDIX 1 + 3 cosh 2β + 4 cosh β V β = ; 3 sinh 2β + 2 sinh β In this appendix, we provide further details on calculations 4 cosh2 β (3 cosh β − 1) V β = 2 ; in sec. IV. Unless otherwise mentioned, equation numbers 6 sinh β cosh β + 2 sinh β will correspond to equation numbers in the main text. 3 cosh β − 1 V β = (39) Derivation of eqn. (21) - The integrand of eqn. (20) is 3 sinh β − 2 tanh β given, according to the notation in the main text, as 2 9

[1] S. Arrhenius, Z. Phys. Chemie 4U(1), 96 (1889). [34] T. M. Stace, Phys. Rev. A 82, 011611 (2010). [2] F. J. DiSalvo, Science 285, 703 (1999), [35] D. Xie, C. Xu, and A. M. Wang, Quantum Information Pro- http://science.sciencemag.org/content/285/5428/703.full.pdf. cessing 16, 155 (2017). [3] H. Callen, Thermodynamics and an Introduction to Thermo- [36] M. G. A. Paris, Journal of Physics A: Mathematical and Theo- statistics (Wiley, 1985). retical 49, 03LT02 (2016). [4] P. P. Hofer, J. B. Brask, M. Perarnau-Llobet, and N. Brunner, [37] According to the formulation of the weak measurement proto- Phys. Rev. Lett. 119, 090603 (2017). col, described in section III, it is enough to consider gτ 1, [5] P. A. Ivanov, arXiv e-prints , arXiv:1809.01451 (2018), even if we allow hot probes. Note that, for performing the post- arXiv:1809.01451 [quant-ph]. selection strong measurement, the assumption gτ 1 may [6] A. De Pasquale and T. M. Stace, arXiv e-prints , be dropped. Nevertheless, the issue of the measuring appara- arXiv:1807.05762 (2018), arXiv:1807.05762 [quant-ph]. tus getting destroyed due to the hot probe can be circumvented [7] L. Seveso and M. G. A. Paris, Phys. Rev. A 97, 032129 (2018). by considering a diﬀerent measuring apparatus. [8] M. Mehboudi, A. Sanpera, and L. A. Correa, arXiv e-prints , [38] G. Mitchison, R. Jozsa, and S. Popescu, Phys. Rev. A 76, arXiv:1811.03988 (2018), arXiv:1811.03988 [quant-ph]. 062105 (2007). [9] M. Brunelli, S. Olivares, and M. G. A. Paris, Phys. Rev. A 84, [39] H. Kobayashi, K. Nonaka, and Y. Shikano, Phys. Rev. A 89, 032105 (2011). 053816 (2014). [10] M. Brunelli, S. Olivares, M. Paternostro, and M. G. A. Paris, [40] J. Dressel and A. N. Jordan, Phys. Rev. Lett. 109, 230402 Phys. Rev. A 86, 012125 (2012). (2012). [11] L. A. Correa, M. Mehboudi, G. Adesso, and A. Sanpera, Phys. [41] G. Nirala, S. Narayan Sahoo, A. K. Pati, and U. Sinha, arXiv Rev. Lett. 114, 220405 (2015). e-prints , arXiv:1807.09014 (2018), arXiv:1807.09014 [quant- [12] V. Cavina, L. Mancino, A. De Pasquale, I. Gianani, M. Sbros- ph]. cia, R. I. Booth, E. Roccia, R. Raimondi, V. Giovannetti, and [42] L. Vaidman, American Journal of Physics 60, 182 (1992), M. Barbieri, Phys. Rev. A 98, 050101 (2018). https://doi.org/10.1119/1.16940. [13] C. Mukhopadhyay, M. K. Gupta, and A. K. Pati, arXiv e-prints [43] D. Petz and C. Ghinea, in Quantum Probability and Related , arXiv:1812.07508 (2018), arXiv:1812.07508 [quant-ph]. Topics, edited by R. Rebolledo and M. Orszag (2011) pp. 261– [14] Y. Aharonov, D. Z. Albert, and L. Vaidman, Phys. Rev. Lett. 281, arXiv:1008.2417 [quant-ph]. 60, 1351 (1988). [15] A. Feizpour, X. Xing, and A. M. Steinberg, Phys. Rev. Lett. 107, 133603 (2011). [16] J. Harris, R. W. Boyd, and J. S. Lundeen, Phys. Rev. Lett. 118, 070802 (2017). [17] A. N. Jordan, J. Mart´ınez-Rincon,´ and J. C. Howell, Phys. Rev. X 4, 011031 (2014). [18] E. Sjqvist, Physics Letters A 359, 187 (2006). [19] A. K. Pati, U. Singh, and U. Sinha, Phys. Rev. A 92, 052120 (2015). [20] J. S. Lundeen and C. Bamber, Phys. Rev. Lett. 108, 070402 (2012). [21] E. Haapasalo, P. Lahti, and J. Schultz, Phys. Rev. A 84, 052107 (2011). [22] M. Malik, M. Mirhosseini, M. P. Lavery, J. Leach, M. J. Pad- gett, and R. W. Boyd, Nature communications 5, 3115 (2014). [23] G. I. Viza, J. Mart´ınez-Rincon,´ G. B. Alves, A. N. Jordan, and J. C. Howell, Phys. Rev. A 92, 032127 (2015). [24] J. Sinclair, M. Hallaji, A. M. Steinberg, J. Tollaksen, and A. N. Jordan, Phys. Rev. A 96, 052128 (2017). [25] M. Tukiainen, H. Kobayashi, and Y. Shikano, Phys. Rev. A 95, 052301 (2017). [26] S. Goswami, S. Chakraborty, S. Ghosh, and A. S. Majumdar, Phys. Rev. A 99, 012327 (2019). [27] O. Hosten and P. Kwiat, Science 319, 787 (2008). [28] S. Kocsis, B. Braverman, S. Ravets, M. J. Stevens, R. P. Mirin, L. K. Shalm, and A. M. Steinberg, Science 332, 1170 (2011). [29] G. Strubi¨ and C. Bruder, Phys. Rev. Lett. 110, 083605 (2013). [30] P. Egan and J. A. Stone, Opt. Lett. 37, 4991 (2012). [31] We thank an anonymous referee for bringing this work to our attention. [32] P. B. Dixon, D. J. Starling, A. N. Jordan, and J. C. Howell, Phys. Rev. Lett. 102, 173601 (2009). [33] D. J. Starling, P. B. Dixon, A. N. Jordan, and J. C. Howell, Phys. Rev. A 82, 063822 (2010).