ARTICLE

DOI: 10.1038/s41467-018-04536-7 OPEN - uncertainty relation in

H.J.D. Miller1 & J. Anders1

It is known that temperature estimates of macroscopic systems in equilibrium are most precise when their energy fluctuations are large. However, for nanoscale systems deviations from standard thermodynamics arise due to their interactions with the environment. Here we

1234567890():,; include such interactions and, using quantum estimation theory, derive a generalised ther- modynamic uncertainty relation valid for classical and quantum systems at all coupling strengths. We show that the non-commutativity between the system’s state and its effective energy gives rise to quantum fluctuations that increase the temperature uncertainty. Surprisingly, these additional fluctuations are described by the average Wigner-Yanase- Dyson skew information. We demonstrate that the temperature’s signal-to-noise ratio is constrained by the capacity plus a dissipative term arising from the non-negligible interactions. These findings shed light on the interplay between classical and non-classical fluctuations in quantum thermodynamics and will inform the design of optimal nanoscale thermometers.

1 Department of Physics and Astronomy, University of Exeter, Stocker Road, Exeter EX4 4QL, UK. Correspondence and requests for materials should be addressed to H.J.D.M. (email: [email protected]) or to J.A. (email: [email protected])

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ohr suggested that there should exist a form of com- estimate, and we illustrate our bound with an example of a Bplementarity between temperature and energy in thermo- damped harmonic oscillator. dynamics similar to that of position and momentum in quantum theory1. His reasoning was that in order to assign a Results definite temperature T to a system it must be brought in contact The Wigner-Yanase-Dyson skew information. Our analysis with a thermal reservoir, in which case the energy U of the system throughout the paper will rely on distinguishing between classical fluctuates due to exchanges with the reservoir. On the other hand, and non-classical fluctuations of in quantum to assign a sharp energy to the system it must be isolated from the mechanics, and we first present a framework for quantifying these reservoir, rendering the system’s temperature T uncertain. Based different forms of statistical uncertainty for arbitrary mixed states. on this heuristic argument Bohr conjectured the thermodynamic Let us consider a ^ρ and an A^. Wigner uncertainty relation: and Yanase considered the problem of quantifying the quantum ^ ^ρ 41 1 uncertainty in observable A for the case where is mixed . Δβ  ; ð1Þ However, they observed that the standard measure of uncertainty, ΔU namely the variance Var½^ρ; A^Š := tr½^ρδA^2Š with δA^ = A^ À A^ , − with β = (k T) 1 the inverse temperature. While Eq. (1) has since contains classical contributions due to mixing, and thus fails to B – ^ been derived in various settings2 9, it was Mandelbrot who first fully quantify the non-classical fluctuations in the observable A. based the concept of fluctuating temperature on the theory of This problem can be resolved by finding a quantum measure of ^ ^ statistical inference. Concretely, for a thermal system in canonical uncertainty Q½^ρ; AŠ and classical measure K½^ρ; AŠ such that the equilibrium, Δβ can be interpreted as the standard deviation variance can be partitioned according to associated with estimates of the parameter β. Mandelbrot proved Var½^ρ; A^Š¼Q½^ρ; A^ŠþK½^ρ; A^Š: ð2Þ that Eq. (1) sets the ultimate limit on simultaneous estimates of energy and temperature in classical statistical physics2. fl The notion of uctuating temperature has proved to be fun- 47 fi Following the framework introduced by Luo , these functions damental in the emerging eld of quantum thermometry, where fi advances in nanotechnology now allow temperature sensing at are required to ful l three conditions: (i) both terms should be – ½^ρ; ^Š ½^ρ; ^Š sub-micron scales10 22. Using the tools of quantum metrology23, non-negative, Q A 0 and K A 0, so that they can be interpreted as forms of statistical uncertainty, (ii) if the state ^ρ is the relation Eq. (1) can also be derived for weakly coupled ½^ρ; ^Š = ½^ρ; ^Š ½^ρ; ^Š¼ quantum systems11,12,14, where the equilibrium state is best pure, then Q A Var A while K A 0 as all uncertainty should be associated to quantum fluctuations alone, (iii) Q½^ρ; A^Š described by the canonical ensemble. Within the grand-canonical ^ρ ensemble the impact of the indistinguishability of quantum par- must be convex with respect to , so that it decreases under classical mixing. Correspondingly, K½^ρ; A^Š must be concave with ticles on the estimation of temperature and the ^ρ has also been explored24. Relation Eq. (1) informs us that when respect to . The following function, known as the WYD skew informa- designing an accurate quantum thermometer one should search 41 for systems with Hamiltonians that produce a large energy tion was shown to be a valid measure of quantum uncertainty: 14 1 À variance . Q ½^ρ; A^Š :¼À tr½½A^; ^ρaŠ½A^; ^ρ1 aŠŠ; a 2ð0; 1Þ; ð3Þ Recently there has been an emerging interest into the effects of a 2 strong coupling on temperature estimation13,15,25. At the nanoscale the strength of interactions between the system and the with the complementary classical uncertainty given by À reservoir may become non-negligible, and the local equilibrium K ½^ρ; A^Š :¼ tr½^ρa δA^ ^ρ1 aδA^Š; a 2ð0; 1Þ: ð4Þ state of the system will not be of Gibbs form26,27. In this regime a thermodynamics needs to be adapted as the equilibrium prop- While conditions (i) and (ii) are easily verified, the convexity/ erties of the system must now depend on the interaction ½^ρ; ^Š ½^ρ; ^Š 28–40 fl concavity of Qa A and Ka A respectively can be proven energy . We will see that the U and its uc- using Lieb’s concavity theorem. tuations ΔU are determined by a modified internal energy ^à The presence of the parameter a demonstrates that there is no operator, denoted by ES, that differs from the bare Hamiltonian unique way of separating the quantum and classical contributions 35,39 of the system . This modification brings into question the to the variance. We here follow the suggestion made in refs. 43,44 validity of Eq. (1) for general classical and quantum systems, and and average over the interval a∈ (0, 1) to define two new the aim of this paper is to investigate the impact of strong cou- quantities: Z pling on the thermodynamic uncertainty relation. 1 Taking into account quantum properties of the effective ½^ρ; ^Š :¼ ½^ρ; ^Š; ð Þ Q A daQa A 5 internal energy operator and its temperature dependence, we here 0 derive the general thermodynamic valid at Z all coupling strengths. Formally this result follows from a general 1 ½^ρ; ^Š :¼ ½^ρ; ^Š: ð Þ upper bound on the quantum Fisher information (QFI) for K A daKa A 6 exponential states. We prove that quantum fluctuations arising 0 from coherences between energy states of the system lead to fl increased uctuations in the underlying temperature. Most It is not only the Q ½^ρ; A^Š and K ½^ρ; A^Š that separate the interestingly, the non-classical modifications to Eq. (1) are a a ^ fi quantum and classical fluctuations of a quantum observable A in quanti ed by the average Wigner-Yanase-Dyson (WYD) skew ^ 41–44 a state ^ρ according to Eq. (2), but also the averaged Q½^ρ; AŠ and information , which is a quantity closely linked to measures ^ of , asymmetry and quantum speed limits45,46. We then K½^ρ; AŠ. This follows from the linearity of the integrals in Eqs. (5) demonstrate that the skew information is also linked to the heat and (6) which also preserve the conditions (i)–(iii). Throughout capacity of the system through a modified fluctuation- the remainder of the paper we will consider Q½^ρ; A^Š and K½^ρ; A^Š as relation (FDR). This result is used to find an upper bound on the the relevant measures of quantum and classical uncertainty, achievable signal-to-noise ratio of an unbiased temperature respectively. While this may appear to be an arbitrary choice, we

2 NATURE COMMUNICATIONS | (2018) 9:2203 | DOI: 10.1038/s41467-018-04536-7 | www.nature.com/naturecommunications NATURE COMMUNICATIONS | DOI: 10.1038/s41467-018-04536-7 ARTICLE will subsequently prove that the average skew information is Generalised thermodynamic uncertainty relation. We will now intimately connected to thermodynamics. use the results of the previous section to derive an uncertainty relation between energy and temperature for a quantum system Bound on quantum Fisher information for exponential states. strongly interacting with a reservoir. To achieve this we will first We now prove that the average skew information is linked to the discuss the appropriate energy operator for such a system, and quality of a parameter estimate for a quantum exponential state. then proceed to generalise Eq. (1). À^ ^ρ = Aθ = A quantum system S that interacts with a reservoir R is A quantum^ exponential state is of the form θ e Zθ where ÀAθ ^ Zθ ¼ tr½e Š and Aθ is a hermitian operator that is here assumed described by a Hamiltonian to depend analytically on a smooth parameter θ. For any state of ^ ^ ^ ^ ^ ^ ^ HS ∪ R :¼ HS IR þ IS HR þ VS ∪ R; ð9Þ full rank, an operator Aθ can be found such that the state can be expressed in this form, i.e. all full rank states are exponential ^ ^ states. where HS and HR are the bare Hamiltonians of S and R ^ We first recall the standard setup for estimating the parameter respectively, while VS ∪ R is an interaction term of arbitrary 48 ^ θR . First one performs a POVM measurement MðξÞ, where strength. We will consider situations where the environment is ^ fi dξ M^ ðξÞ¼I and ξ denotes the outcomes of the measurement large compared to the system, i.e. the operator norms ful l ^  ^ ; ^ which may be continuous or discrete. The probability of obtaining HR HS VS ∪ R . We make no further assumptions ðξjθÞ¼ ½ ^ ðξÞ^ρ Š ^ a particular outcome is p tr M θ . The measurement about the relative size of the coupling VS ∪ R between the ξ ξ ξ ^ is repeated n times with outcomes { 1, 2,.. n}, and one system and the environment, and the system’s bare energy HS . ~θ ¼ ~θðξ ; ξ ;::ξ Þ constructs a function 1 2 n that estimates the true The global equilibrium state at temperature T for the total ~ Àβ ^ hθi ^ HS ∪ R value of the parameter.R We denote the average estimate by , Hamiltonian S ∪ R is of Gibbs form πShi∪ RðTÞ = e =ZS ∪ R hið::Þ = ξ ¼ ξ ðξ jθÞ ¼ ðξ jθÞð::Þ ^ where d d p p , and assume the −1 ÀβHS ∪ R 1 n 1 n where β = (k T) and ZS ∪ R = trS ∪ R e is the partition h~θi¼θ B estimate is unbiased, i.e. . In this case the mean-squared S ∪ R error in the estimate is equivalent to the variance, which is function for . The Boltzmann constant kB will be set to Δθ2 ¼h~θ2iÀθ2 unity throughout. denoted by . Due to the presence of the interaction term the reduced state of The celebrated quantum Cramér-Rao inequality sets a lower S, denoted π^SðTÞ¼trR½π^S ∪ RðTފ, is generally not thermal with bound on Δθ, optimised over all possible POVMs and estimator ^ 23,48–50 respect to HS, unless the coupling is sufficiently weak, i.e. functions : ^ ^ HS  VS ∪ R . Therefore the partition function determined ^ 1 by HS can no longer be used to calculate the internal energy of Δθ  pffiffiffiffiffiffiffiffiffiffiffiffi ; ð7Þ 35 S nFðθÞ the system . To resolve this issue one can rewrite the state of ÀβH^ à ðTÞ Ã as an effective Gibbs state π^SðTÞ := e S =ZS, where 0 hi1 ^ ÀβHS ∪ R where F(θ) is the QFI. The bound becomes tight in the asymptotic trR e → ∞23 ^ à ð Þ :¼À1 @ ÂÃA; ð Þ limit n . If the exponential state belongs to the so-called HS T ln ^ 10 β ÀβHR ^ ^ ^ trR e ‘exponential family’, which is true if Aθ ¼ θX þ Y for commuting operators X^; Y^, then the bound is also tight in the single-shot 50 28,30–37,39,40 limit (n =ÂÃ1) . The QFI with respect to θ is defined by is the Hamiltonian of mean force . The operator ^2 ^ ^ à FðθÞ :¼ tr ^ρθLθ , where Lθ is the symmetric logarithmic HSðTÞ acts as a temperature-dependent effective Hamiltonian S derivativeÈÉ which uniquely satisfies the operator equation describing the equilibriumhi properties of through the effective 1 ^ 49 à Àβ ^ à ð Þ ∂ ^ρ = ; ^ρ Z ¼ HS T θ θ 2 Lθ θ . Here {..,..} denotes the anti-commutator. partition function S trS e . The free energy associated We now state a general upper bound on F(θ) valid for any à 30,52 with ZS also appears in the open system fluctuation relations . exponential state: ^ S ÀAθ The internal energy of can be computed from this partition Theorem 1: For an exponential state ^ρθ ¼ e =Zθ the QFI with à function via USðTÞ :¼À∂β lnZ . It is straightforward to show respect to the parameter θ is bounded by S that USðTÞ is just the difference between the total energy, ~ ^ US ∪ R ¼À∂β lnZS ∪ R, and the energy of the reservoir, UR ¼ FðθÞK½^ρθ; BθŠ: ð8Þ hi ^ ÀβHR À∂β lnZR with ZR ¼ trR e , in the absence of any coupling ^ ^ S ð Þ = ð ÞÀ ~ ð Þ ð Þ Here K½^ρ ; BθŠ is defined in Eq. (6), and Bθ is the hermitian to , i.e. US T US ∪ R T UR T . In other words, US T is θ ^ ^ S observable Bθ :¼ ∂θAθ. The bound becomes tight in the limits the energy change induced from immersing the subsystem into S ∪ R29,36 where ^ρθ is maximally mixed. the composite state . 35 This theorem demonstrates that the strictly classical fluctua- Seifert has remarked that USðTÞ can be expressed as an ^ ^à tions in Bθ constrain the achievable precision in estimates of θ. expectation value, USðTÞ = ESðTÞ , of the following observable: ‘ ’ Âà The proof of the theorem is outlined in the Methods section. ^à ^ à E ðTÞ :¼ ∂β βH ðTÞ : ð Þ We note that for states σ^θ that fulfil the von-Neumann S S 11 ^ equation ∂θσ^θ ¼Ài½Aθ; σ^θŠ a connection between skew informa- ½σ^ ; ^ Š tion Q1=2 θ Aθ and parameter estimation has previously been 51 ^à made by Luo . While the particular dependence on θ implied by One can interpret ESðTÞ as the effective energy operator this equation is relevant for unitary parameter estimation23, this describing the system, and we will refer to its eigenstates as “the system energy states”. The introduction of this operator allows dependence will not be relevant for temperature estimation since qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiÂà thermal states do not generally fulfil this von-Neumann equation. ^à one to consider fluctuations in the energy ΔUS ¼ Var π^S; E . In contrast, we will see in the next section that Theorem 1 has S ^à ð Þ implications for the achievable precision in determining It is important to note that ES T depends explicitly on the ^ temperature. coupling VS ∪ R and the temperature T.

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^Ã Our first observation is that, in general, ESðTÞ differs from In standard thermodynamics where the system is described by ^ both the bare system Hamiltonian HS and the mean force a Gibbs state the FDR states that the is proportional ^ Ã fl ð Þ¼Δ 2= 2 Hamiltonian HSðTÞ. Indeed, this effective energy operator for the to the uctuations in energy, i.e. CS T US T . However, system contains the bare energy part as well as an energetic example studies of open quantum systems of the form Eq. (9) ^Ã ð Þ = have shown that the heat capacity can become negative at low contributionÂÃÀÁ from the coupling, ES T 31,32,53,54 ^ þ ∂ β ^ Ã ð ÞÀ ^ ^Ã ð Þ , thus implying it cannot be proportional to HS β HS T HS . Moreover, ES T does not even a positive variance in general. ^ ^ Ã ð Þ commute with HS and HS T . This non-commutativity implies Our second result indeed shows that there are two additional π^ ð Þ that the state S T exists in a superpositionÂÃ of energy states, contributions to the FDR due to strong-coupling (see ‘Methods’ ^ ^ ^ aside from the trivial situation in which HS þ HR; VS ∪ R ¼ 0. section): ^Ã ð Þ ÂÃ As expected, in the limit of weak coupling ES T reduces to the Δ 2 π^ ; ^Ã ^ ð Þ¼ US À Q S ES þ ∂ ^Ã ; ð Þ bare Hamiltonian HS. CS T ES 14 T2 T2 T We are now ready to state the generalised thermodynamic uncertainty relations for strongly coupled quantum systems. ð Þ Δ 2= 2 13 implying that CS T can be less than US T and even negative. Following the approach taken by De Pasquale et al. , we consider fi fl ðβÞ β We see that the rst correction is due to the quantum uctuationsÂÃ the QFI FS associated with the inverse temperature . π^ ; ^Ã According to the quantum Cramér-Rao bound this functional in energy given by the average WYD information Q S ES , quantifies the minimum extent to which the inverse temperature whichÂÃ only vanishes in the classical limit where ^Ã ð Þ; π^ ð Þ ¼ fluctuates from the perspective of S, and we denote these ES T S T 0. The second correction is a dissipation term fluctuations by ΔβS. Given that the state of S takes the form stemming from the temperature dependence of the internal ÀβH^ Ã ðTÞ Ã energy operator Eq. (11). Notably this term can still be present in π^SðTÞ :¼ e S =ZS we can immediately apply Theorem 1 by ^ ¼ ^Ã ð Þ θ = β the classical limit where the energy operator may depend on identifyingÂÃBθ ES T with , leading to temperature if the coupling is non-negligible. As expected both ðβÞ π^ ; ^Ã FS K S ES . Applying Eq.ÂÃ (7) for the single-shotÂÃ case terms can be dropped in the limit of vanishing coupling and the ^Ã 2 ^Ã (n = 1) and using the fact that K π^S; ES = ΔUS À Q π^S; ES ,we standard FDR is recovered. obtain the following thermodynamic uncertainty relation: Bound on signal-to-noise ratio for temperature estimates. Let Δβ  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1  1 : S ÂÃΔ ð Þ us denote the uncertainty in the temperature from a given 2 ^ ^Ã US 12 ΔUS À Q πS; ES unbiased estimation scheme by ΔTS, with measurements per- formed on S alone. It is known11,12,14,55 that in the weak- coupling limit, the optimal signal-to-noise ratio for estimating T ð Þ This is the main result of the paper and represents the strong- from a single measurement is bounded by CS T :  coupling generalisation of Eq. (1). It can be seen that the bound 2 T on the uncertainty in the inverse temperature is increased  ð Þ: ð15Þ Δ CS T whenever quantum energy fluctuations are present. These TS additionalÂÃfluctuations are quantified by the non-negative ^Ã Q π^S; ES , increasing which implies a larger lower boundÂÃ on ^Ã ΔβS. One recovers the usual uncertainty relation when Q π^S; ES This bound is tight for a single measurement of T and implies can be neglected, which is the case when the interaction that precise measurements of the temperature require a large heat commutes with the bare Hamiltonians of S andÂÃR or when the capacity. The result follows straightforwardly from the quantum ^Ã interaction is sufficiently weak. We note that Q π^S; E vanishes Cramér-Rao inequality and the standard FDR. S fi for classical systems and Eq. (12) reduces to the original Using our modi ed FDR Eq. (14), we here give the strong- uncertainty relation Eq. (1), but with energy fluctuations coupling generalisation of the bound Eq. (15). Considering ^Ã ^ β quantified by ES instead of the bare Hamiltonian HS. estimates of T rather than the inverse temperature , a simple If one repeats the experiment n times, then thepffiffiffi uncertainty in change of variables reveals that the QFI with respect to T is β ðβÞ¼ 4 ð Þ the estimate can be improved by a factor of 1= n48. We remark related to that of , ÂÃFS T FS T . From Theorem 1 we again ^ Ã ^ 4 ð Þ π^ ; ^Ã that in the weak coupling limit, where HSðTÞ’HS, the state of S have T FS T K S ES , and combining this with Eqs. (14) belongs to the exponential family, and hence the bound on ΔβS and (7) we obtain:  becomes tight for a single measurement in agreement with Eq. 2 T ^  ð ÞÀ ∂ ^Ã : ð16Þ (1). However, when VS ∪ R is non-negligible the Hamiltonian of Δ CS T T ES mean force cannot generally be expressed in the linear form TS ^ Ã ^ ^ βHSðTÞ = βXS þ YS. This means in general it is necessary to take the asymptotic limit in order to saturate Eq. (12). This is our third result and demonstrates that the optimal signal-to-noise ratio for estimating the temperature of S is Fluctuation-Dissipation relation beyond weak-coupling.We bounded by both the heat capacity and the added dissipation now detail the impact of strong interactions on the heat capacity term, which can be both positive or negative. This bound is of the quantum system and the implications for the precision of independently tight in both the high temperature and weak- temperature measurements. For a fixed of the system, the fi coupling limits. In these regimes the POVM saturating Eq. (16)is heat capacity is de ned as the temperature derivative of the given by the maximum-likelihood estimator measured in the ð Þ31,32 internal energy US T , i.e. basis of the relevant symmetric logarithmic derivative48.We ∂ US ð Þ stress that Eq. (16) is valid in the classical limit, in which case it is CSðTÞ :¼ : 13 ∂T always tight. We remark that the RHS of Eq. (16) can alternatively be expressed in terms ofÂÃ the skew information, in which case 2 2 2 ^Ã 2 ðT=ΔTSÞ  ΔUS=T À Q π^S; ES =T .

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Application to damped harmonic oscillator. While the bound y operator, n^ ¼ ^a a^ , with annihilation operator ^a ¼ Eq. (16) is tight in the high temperature limit, for general open qffiffiffiffiffiT T T T quantum systems the accuracy of the bound is not known. We AT ^ þ i ^ = ω 2h x A p with AT MT T. show that the bound is very good for the example of a damped The internalT energy operator is now obtained by straightfor- harmonic oscillator linearly coupled to N harmonic oscillators in 34,56,57 ward differentiation, see Eq. (11), and given by the reservoir . Experimentally, such a model describes the  58 2 behaviour of nano-mechanical resonators and BEC impu- ^2 þ ^y ^2 a a 59 ^ p Mω2^x2 Ã Ã T T ð19Þ rities . Here the system Hamiltonian is HS ¼ þ , while ^ ð Þ¼α ^ ð ÞÀ ; 2M 2 ES T T HS T gT P ^2 ω2^2 2 ^ N pj Mj j xj the reservoir Hamiltonian is HR ¼ ¼ þ and the j 1 2Mj 2 ω′ ′ interaction term is given by T AT where α ¼ 1 À ω T and g ¼ hω T . Using this operator we ! T T T T AT ÂÃ XN 2 ^Ã λ obtain analytic expressions for CSðTÞ, FSðTÞ, Q π^S; E and ^ ¼ Àλ ^ ^ þ j ^2 : ð Þ S VS ∪ R jx xj x 17 ^Ã ω2 ∂ ES in Supplementary Note 3. j¼1 2Mj j T ÂÃFigure 1 shows the square root of the average skew information ^Ã Q π^S; ES in units of ħω as a function of temperature for different To allow a fully analytical solution, the reservoir frequencies coupling strengths. As expected we see that the quantum fl are chosen equidistant, ω = jΔ and the continuum limit is taken uctuations in energy vanish in the high temperature limit, while j fl so that Δ → 0 (and N → ∞). The couplingr constantsffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi are chosen as uctuations grow with increased coupling strengths due to ^Ã ð Þ π^ 2γM Mω2Δ ω2 increased non-commutativity between ES T andÂÃ the state S of 34 λ ¼ j j D γ Ã the Drude-Ullersma spectrum , j π ω2 þω2, where π^ ; ^ D j the oscillator. Interestingly we see that Q S ES decays is the damping coefficient controlling the interaction strength and exponentially to zero in the low temperature limit, implying that ω D is a large cutoff frequency. the state of the oscillator commutes with the internal energy As shown by Grabert et al.56, the resulting Hamiltonian of operator in this regime. Whether this is a general feature or mean force for the oscillator can be parameterised by a specific to the example here remains an open question. temperature-dependent mass and frequency, Figure 2 shows the optimal signal-to-noise ratio for estimating  T determined by the Cramér-Rao bound Eq. (7), ^2 2 2 Ã p M ω ^x 1 2 2 ^ ð Þ¼ þ T T ¼ ω ^ þ ; ð Þ ðT=ΔTSÞ ¼ T FSðTÞ, as a function of temperature T and HS T h T nT 18 opt 2MT 2 2 coupling strength γ. The bound we derived in Eq. (16) given by the heat capacity and an additional dissipation term is also ω ^2 where MT and T are given through the expectation values of p plotted and shows very good agreement with the optimum ^2 and x in the global thermal state, see Supplementary Note 3 for estimation scheme quantified by the QFI. The bound clearly detailed expressions. In its diagonal form the mean-force becomes tight in the high-temperature limit (T → ∞) independent Hamiltonian contains a temperature-dependent number of the coupling strength. Conversely the bound is also tight in the weak-coupling limit (γ → 0) independent of the temperature. The

 0.10 / = 1 0  5 0.08   / = 0.5 10 √ / = 0.2 Q 0.06 0.6  h   / = 0.1 T 2 0.04 0.4 ΔT 2 0.02 0.2

0.00 0 0 5 10 15 20 5 T  10 T h h Fig. 1 Skew information for the damped oscillator. Plot of quantum rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffihi Fig. 2 Bound on temperature signal-to-noise ratio. The coloured plot shows ^Ã the optimal signal-to-noise ratio ðT=ΔT Þ2 of an unbiased temperature energetic fluctuations Q π^S; ES =hω for the damped oscillator as a S opt hi estimate for the damped oscillator, as a function of temperature T and ^Ã function of T/ħω for different coupling strengths γ. Here Q π^S ; ES is the coupling strength γ. This optimal measurement is determined by the average Wigner-Yanase-Dyson skew information for the effective energy quantum Fisher information, which places an asymptotically achievable lower bound on the temperature fluctuations ΔT through the Cramér-Rao operator E^Ã . These fluctuations are present when the state of the oscillator S S inequality13. The mesh plot shows the upper bound on ðT=ΔT Þ2 derived π^ ðTÞ E^Ã S opt S is not diagonal in the basis of S due to the non-negligible interaction here from the generalised thermodynamic uncertainty relation Eq. (16). between the system and reservoir. The plot shows that increasing the This uncertainty relation links the temperature fluctuations to the heat γ coupling leads to an increase in the skew information. The quantum capacity of the system at arbitrary coupling strengths. It can be seen that fl uctuations are most pronounced at low temperatures where the thermal the upper bound becomes tight in both the high temperature and weak T ’ hω become comparable to the oscillator spacing, .As coupling limits expected, the skew information decreases to zero in both the high temperature and weak coupling limits

NATURE COMMUNICATIONS | (2018) 9:2203 | DOI: 10.1038/s41467-018-04536-7 | www.nature.com/naturecommunications 5 ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/s41467-018-04536-7 optimum and the bound both converge exponentially to zero as standard thermodynamics with negligible interactions and those T → 0, albeit with different rates of decay. Outside of these limits where correlations play a prominent role38,68,69. The results ð =Δ Þ2 establish a connection between abstract measures of quantum the difference between the bound and T TS opt has a information theory, such as the QFI and skew information, and a maximum, and at the temperature and coupling for which this ’ maximum occurs the bound is roughly 30% greater than material s effective thermodynamic properties. This provides a 2 starting point for future investigations into nanoscale thermo- ðT=ΔTSÞ . opt dynamics, extending into the regime where the weak coupling assumption is not justified. Discussion Methods In this paper we have shown how non-negligible interactions fl fl Proof of Theorem 1. Here we provide a sketch for the proof of the bound Eq. (8) in uence uctuations in temperature at the nanoscale. Our main on the QFI for exponential states. The full derivation can be found in Supple- result Eq. (12) is a thermodynamic uncertainty relation extending mentary Note 1. LetP us first denote the spectral decomposition of the state ^ρ θ ¼ À^ Aθ = ^ρ ¼ ψ ψ ^ ψ ¼ λ ψ the well-known complementarity relation Eq. (1) between energy e Zθ by θ n pn n n , where the eigenstates satisfy Aθ n n n . From this decomposition the QFI can be expressed as follows50: and temperature to all interaction strengths. This derivation is based on a bound on the QFI for exponential states which we X ψ ∂ ^ρ ψ 2 ðθÞ¼ n θ θ m : ð Þ F 2 þ 20 prove in Theorem 1. As Theorem 1 is valid for any state of full- n;m pn pm rank, the bound will be of interest to other areas of quantum metrology. Our uncertainty relation shows that for a given finite To proceed we utilise the following integral expression for the derivative of an spread in energy, unbiased estimates of the underlying tempera- exponential operator70: hi Z ^ 1 ^ ^ ture are limited to a greater extent due to coherences between ÀAθ Àð1ÀaÞ Aθ ^ Àa Aθ ∂θ e :¼À dae ∂θ½AθŠe : ð21Þ energy states. These coherences only arise for quantum systems 0 beyond the weak coupling assumption. We found that these additional temperature fluctuations are quantified by the average Combining Eqs. (20) and (21) eventually yields X f 2ðp ; p Þ WYD skew information, thereby establishing a link between ðθÞ¼Δ^2 þ n m Àð þ Þ 2 ; ð Þ F Bθ þ pn pm Bnm 22 quantum and classical forms of statistical uncertainty in nanos- n

Notably the bound implies that when designing a probe to δ^Ã :¼ ^Ã ð ÞÀ ^Ã ð Þ where ES ES T ES T . By using the integral expressionÂÃ Eq. (21), we measure T, its bare Hamiltonian and interaction with the sample π^ ; ^Ã ð Þ show in Supplementary Note 2 that this equation for K S ES is equivalent to should be chosen so as to both maximise CS T whilst mini- ÂÃ ÂÃ Ã π^ ; ^Ã ¼ 2 ^Ã ∂ π^ : ð Þ ∂ ^ K S ES T tr ES T S 27 mising the additional dissipation term T ES . It is an interesting open question to consider the form of Hamiltonians that achieve UsingÂÃ the product ruleÂÃ for the differential of an operator, along with the equation ^Ã 2 ^Ã this optimisation in the strong coupling scenario. Furthermore, K π^S ; ES = ΔUS À Q π^S ; ES given by Eq. (5), completes the derivation. one expects that improvements to low-temperature thermometry 25 resulting from strong interactions, such as those observed in , Data availability. The data that support the findings of this study are available will be connected to the properties of the effective internal energy from the corresponding author upon reasonable request. operator. In particular, it is clear from Eq. (16) that any improved scaling of the QFI at low temperatures must be determined by the Received: 31 January 2018 Accepted: 8 May 2018 ð Þ ∂ ^Ã relative scaling of CS T and T ES , and exploring this further remains a promising direction of research. Advancements in nanotechnology now enable temperature sensing over micro- scopic spatial resolutions64,65, and understanding how to exploit interactions between a probe and its surroundings will be crucial References to the development of these nanoscale thermometers. 1. Bohr, N. Faraday lecture. Chemistry and the quantum theory of atomic The presented approach opens up opportunities for exploring constitution. J. Chem. Soc. 1, 349–384 (1932). the intermediate regime between the limiting cases66,67 of

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Acknowledgements We would like to thank Pieter Kok for insightful discussions and Simon Horsley for a Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in critical reading of the manuscript. H.J.D.M. is supported by EPSRC through a Doctoral published maps and institutional affiliations. Training Grant. J.A. acknowledges support from EPSRC, grant EP/M009165/1, and the Royal Society. This research was supported by the COST network MP1209 “Thermo- dynamics in the quantum regime”. Open Access This article is licensed under a Creative Commons Author contributions Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give J.A. suggested corrections to the thermodynamic properties of strongly coupled systems appropriate credit to the original author(s) and the source, provide a link to the Creative and supervised the project. H.J.D.M. proved the Theorem and the generalised thermo- Commons license, and indicate if changes were made. The images or other third party dynamic uncertainty relation. J.A. performed the example calculation. Both authors material in this article are included in the article’s Creative Commons license, unless discussed the findings, drew the conclusions and wrote the paper. indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory Additional information regulation or exceeds the permitted use, you will need to obtain permission directly from Supplementary Information accompanies this paper at https://doi.org/10.1038/s41467- the copyright holder. To view a copy of this license, visit http://creativecommons.org/ 018-04536-7. licenses/by/4.0/.

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