Energy-temperature uncertainty relation in quantum thermodynamics

Energy-temperature uncertainty relation in quantum thermodynamics ARTICLE DOI: 10.1038/s41467-018-04536-7 OPEN Energy-temperature uncertainty relation in quantum thermodynamics 1 1 H.J.D. Miller & J. Anders 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 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 operator 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 heat 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. 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: hm419@exeter.ac.uk) or to J.A. (email: janet@qipc.org) NATURE COMMUNICATIONS (2018) 9:2203 DOI: 10.1038/s41467-018-04536-7 www.nature.com/naturecommunications 1 | | | 1234567890():,; ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/s41467-018-04536-7 ohr suggested that there should exist a form of com- estimate, and we illustrate our bound with an example of a plementarity between temperature and energy in thermo- damped harmonic oscillator. Bdynamics similar to that of position and momentum in quantum theory . 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 observables 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 quantum state ^ρ and an observable A. Wigner uncertainty relation: and Yanase considered the problem of quantifying the quantum uncertainty in observable A for the case where ^ρ is mixed . ð1Þ Δβ  ; However, they observed that the standard measure of uncertainty, ΔU ^ ^ ^ ^ ^ namely the variance Var½^ρ; A := tr½^ρδA  with δA = A  A , −1 contains classical contributions due to mixing, and thus fails to with β = (k T) the inverse temperature. While Eq. (1) has since 2–9 ^ fully quantify the non-classical fluctuations in the observable A. been derived in various settings , it was Mandelbrot who first This problem can be resolved by finding a quantum measure of based the concept of fluctuating temperature on the theory of ^ ^ statistical inference. Concretely, for a thermal system in canonical uncertainty Q½^ρ; A and classical measure K½^ρ; A 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; A¼ Q½^ρ; Aþ K½^ρ; A: ð2Þ that Eq. (1) sets the ultimate limit on simultaneous estimates of energy and temperature in classical statistical physics . The notion of fluctuating temperature has proved to be fun- Following the framework introduced by Luo , these functions damental in the emerging field of quantum thermometry, where are required to fulfil three conditions: (i) both terms should be advances in nanotechnology now allow temperature sensing at ^ ^ non-negative, Q½^ρ; A 0 and K½^ρ; A 0, so that they can be 10–22 23 sub-micron scales . Using the tools of quantum metrology , interpreted as forms of statistical uncertainty, (ii) if the state ^ρ is the relation Eq. (1) can also be derived for weakly coupled ^ ^ ^ pure, then Q½^ρ; A = Var½^ρ; A while K½^ρ; A¼ 0 as all uncertainty 11,12,14 quantum systems , where the equilibrium state is best should be associated to quantum fluctuations alone, (iii) Q½^ρ; A described by the canonical ensemble. Within the grand-canonical must be convex with respect to ^ρ, so that it decreases under ensemble the impact of the indistinguishability of quantum par- classical mixing. Correspondingly, K½^ρ; A must be concave with ticles on the estimation of temperature and the chemical potential respect to ρ. has also been explored . Relation Eq. (1) informs us that when The following function, known as the WYD skew informa- designing an accurate quantum thermometer one should search tion was shown to be a valid measure of quantum uncertainty: for systems with Hamiltonians that produce a large energy variance . a 1a ^ ^ ^ Q ½^ρ; A :¼ tr½½A; ^ρ ½A; ^ρ ; a 2ð0; 1Þ; ð3Þ Recently there has been an emerging interest into the effects of 13,15,25 strong coupling on temperature estimation . At the with the complementary classical uncertainty given by nanoscale the strength of interactions between the system and the a 1a reservoir may become non-negligible, and the local equilibrium ^ ^ ^ K ½^ρ; A :¼ tr½^ρ δA^ρ δA; a 2ð0; 1Þ: ð4Þ 26,27 state of the system will not be of Gibbs form . In this regime 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 ^ ^ ^ ^ concavity of Q ½ρ; A and K ½ρ; A respectively can be proven a a 28–40 energy . We will see that the internal energy U and its fluc- using Lieb’s concavity theorem. tuations ΔU are determined by a modified internal energy The presence of the parameter a demonstrates that there is no operator, denoted by E , that differs from the bare Hamiltonian S unique way of separating the quantum and classical contributions 35,39 43,44 of the system . This modification brings into question the to the variance. We here follow the suggestion made in refs. 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: pling on the thermodynamic uncertainty relation. Z Taking into account quantum properties of the effective ^ ^ Q½^ρ; A :¼ daQ ½^ρ; A; ð5Þ internal energy operator and its temperature dependence, we here derive the general thermodynamic uncertainty principle valid at all coupling strengths. Formally this result follows from a general ^ ^ K½^ρ; A :¼ daK ½^ρ; A: ð6Þ upper bound on the quantum Fisher information (QFI) for exponential states. We prove that quantum fluctuations arising from coherences between energy states of the system lead to increased fluctuations in the underlying temperature. Most ^ ^ It is not only the Q ½^ρ; A and K ½^ρ; A that separate the a a interestingly, the non-classical modifications to Eq. (1) are quantum and classical fluctuations of a quantum observable A in quantified by the average Wigner-Yanase-Dyson (WYD) skew 41–44 a state ^ρ according to Eq. (2), but also the averaged Q½^ρ; A and information , which is a quantity closely linked to measures 45,46 ^ K½^ρ; A. This follows from the linearity of the integrals in Eqs. (5) of coherence, asymmetry and quantum speed limits . We then and (6) which also preserve the conditions (i)–(iii). Throughout demonstrate that the skew information is also linked to the heat ^ ^ capacity of the system through a modified fluctuation-dissipation the remainder of the paper we will consider Q½^ρ; A and K½^ρ; A 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 strongly interacting with a reservoir. To achieve this we will first Bound on quantum Fisher information for exponential states. discuss the appropriate energy operator for such a system, and We now prove that the average skew information is linked to the then proceed to generalise Eq. (1). quality of a parameter estimate for a quantum exponential state. θ A quantum system S that interacts with a reservoir R is A quantum exponential state is of the form ^ρ = e =Z where θ θ θ ^ described by a Hamiltonian Z ¼ tr½e  and A is a hermitian operator that is here assumed θ θ to depend analytically on a smooth parameter θ. For any state of ^ ^ ^ ^ ^ ^ ð9Þ H :¼ H  I þ I  H þ V ; ^ S ∪ R S R S R S ∪ R full rank, an operator A can be found such that the state can be expressed in this form, i.e. all full rank states are exponential ^ ^ where H and H are the bare Hamiltonians of S and R states. S R We first recall the standard setup for estimating the parameter respectively, while V is an interaction term of arbitrary S ∪ R θ . First one performs a POVM measurement MðξÞ, where strength. We will consider situations where the environment is ^ ^ large compared to the system, i.e. the operator norms fulfil dξ MðξÞ¼ I and ξ denotes the outcomes of the measurement ^ ^ ^ H  H ; V . We make no further assumptions which may be continuous or discrete. The probability of obtaining R S S ∪ R ^ ^ a particular outcome is pðξjθÞ¼ tr½MðξÞ^ρ . The measurement about the relative size of the coupling V between the S ∪ R is repeated n times with outcomes {ξ ,ξ ,..ξ }, and one ^ 1 2 n system and the environment, and the system’s bare energy H . ~ ~ constructs a function θ ¼ θðξ ; ξ ;::ξ Þ that estimates the true The global equilibrium state at temperature T for the total 1 2 n ~ βH S ∪ R value of the parameter. We denote the average estimate by hθi, Hamiltonian S ∪ R is of Gibbs form π ^ ðTÞ = e =Z R S ∪ R S ∪ R hi wherehi ð::Þ = dξ ¼ dξ pðξ jθÞ¼ pðξ jθÞð::Þ, and assume the −1 βH 1 n 1 n S ∪ R where β = (k T) and Z = tr e is the partition S ∪ R S ∪ R estimate is unbiased, i.e. hθi¼ θ. In this case the mean-squared function for S ∪ R. The Boltzmann constant k will be set to error in the estimate is equivalent to the variance, which is unity throughout. 2 2 2 denoted by Δθ ¼hθ i θ . Due to the presence of the interaction term the reduced state of The celebrated quantum Cramér-Rao inequality sets a lower ^ ^ S, denoted π ðTÞ¼ tr ½π ðTÞ, is generally not thermal with S R S ∪ R bound on Δθ, optimised over all possible POVMs and estimator respect to H , unless the coupling is sufficiently weak, i.e. 23,48–50 functions : ^ ^ H  V . Therefore the partition function determined S S ∪ R 1 by H can no longer be used to calculate the internal energy of Δθ  pffiffiffiffiffiffiffiffiffiffiffiffi ; ð7Þ the system . To resolve this issue one can rewrite the state of S nFðθÞ βH ðTÞ as an effective Gibbs state π ^ ðTÞ := e =Z , where hi 0 1 βH S ∪ R where F(θ) is the QFI. The bound becomes tight in the asymptotic tr e @ A ð10Þ limit n → ∞ . If the exponential state belongs to the so-called H ðTÞ :¼ ln  ; βH β R tr e ^ ^ ^ R ‘exponential family’, which is true if A ¼ θX þ Y for commuting ^ ^ operators X; Y, then the bound is also tight in the single-shot 28,30–37,39,40 limit (n = 1) . The QFI with respect to θ is defined by is the Hamiltonian of mean force . The operator ^ ^ ^ H ðTÞ acts as a temperature-dependent effective Hamiltonian FðθÞ :¼ tr ^ρ L , where L is the symmetric logarithmic θ S θ θ describing the equilibrium properties of S through the effective derivative which uniquely satisfies the operator equation hi 1 49 ^ ^  βH ðTÞ ∂ ^ρ = L ; ^ρ . Here {..,..} denotes the anti-commutator. S partition function Z ¼ tr e . The free energy associated θ θ 2 θ θ S S We now state a general upper bound on F(θ) valid for any 30,52 with Z also appears in the open system fluctuation relations . exponential state: The internal energy of S can be computed from this partition Theorem 1: For an exponential state ρ ¼ e =Z the QFI with θ θ function via U ðTÞ :¼∂ lnZ . It is straightforward to show S β S respect to the parameter θ is bounded by that U ðTÞ is just the difference between the total energy, U ¼∂ lnZ , and the energy of the reservoir, U ¼ ^ S ∪ R β S ∪ R R FðθÞ K½^ρ ; B : ð8Þ hi βH ∂ lnZ with Z ¼ tr e , in the absence of any coupling β R R R ^ ^ to S, i.e. U ðTÞ = U ðTÞ U ðTÞ. In other words, U ðTÞ is Here K½^ρ ; B  is defined in Eq. (6), and B is the hermitian S S ∪ R R S θ θ θ ^ ^ the energy change induced from immersing the subsystem S into observable B :¼ ∂ A . The bound becomes tight in the limits θ θ θ 29,36 the composite state S ∪ R . where ^ρ is maximally mixed. Seifert has remarked that U ðTÞ can be expressed as an This theorem demonstrates that the strictly classical fluctua- ^ ^ tions in B constrain the achievable precision in estimates of θ. expectation value, U ðTÞ = E ðTÞ , of the following observable: θ S The proof of the theorem is outlined in the ‘Methods’ section. ^ ^ E ðTÞ :¼ ∂ βH ðTÞ : ð11Þ S β S We note that for states σ ^ that fulfil the von-Neumann ^ ^ equation ∂ σ ¼i½A ; σ  a connection between skew informa- θ θ θ θ tion Q ½σ ^ ; A  and parameter estimation has previously been 1=2 θ θ One can interpret E ðTÞ as the effective energy operator made by Luo . While the particular dependence on θ implied by describing the system, and we will refer to its eigenstates as “the this equation is relevant for unitary parameter estimation , this system energy states”. The introduction of this operator allows dependence will not be relevant for temperature estimation since qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi thermal states do not generally fulfil this von-Neumann equation. one to consider fluctuations in the energy ΔU ¼ Var π ^ ; E . S S In contrast, we will see in the next section that Theorem 1 has It is important to note that E ðTÞ depends explicitly on the implications for the achievable precision in determining temperature. coupling V and the temperature T. S ∪ R NATURE COMMUNICATIONS (2018) 9:2203 DOI: 10.1038/s41467-018-04536-7 www.nature.com/naturecommunications 3 | | | ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/s41467-018-04536-7 Our first observation is that, in general, E ðTÞ differs from In standard thermodynamics where the system is described by a Gibbs state the FDR states that the heat capacity is proportional both the bare system Hamiltonian H and the mean force 2 2 ^ to the fluctuations in energy, i.e. C ðTÞ¼ ΔU =T . However, Hamiltonian H ðTÞ. Indeed, this effective energy operator for the S S example studies of open quantum systems of the form Eq. (9) system contains the bare energy part as well as an energetic have shown that the heat capacity can become negative at low contribution from the coupling, E ðTÞ = S 31,32,53,54 temperatures , thus implying it cannot be proportional to ^ ^ ^ ^ H þ ∂ β H ðTÞ H . Moreover, E ðTÞ does not even S β S S S a positive variance in general. ^ ^ commute with H and H ðTÞ. This non-commutativity implies S Our second result indeed shows that there are two additional that the state π ^ ðTÞ exists in a superposition of energy states, contributions to the FDR due to strong-coupling (see ‘Methods’ ^ ^ ^ aside from the trivial situation in which H þ H ; V ¼ 0. section): S R S ∪ R As expected, in the limit of weak coupling E ðTÞ reduces to the S ^ ΔU Q π ^ ; E S S ^ ð14Þ C ðTÞ¼  þ ∂ E ; bare Hamiltonian H . S T S 2 2 T T We are now ready to state the generalised thermodynamic uncertainty relations for strongly coupled quantum systems. 2 2 implying that C ðTÞ can be less than ΔU =T and even negative. 13 S Following the approach taken by De Pasquale et al. , we consider We see that the first correction is due to the quantum fluctuations the QFI F ðβÞ associated with the inverse temperature β. in energy given by the average WYD information Q π ^ ; E , According to the quantum Cramér-Rao bound this functional which only vanishes in the classical limit where quantifies the minimum extent to which the inverse temperature E ðTÞ; π ^ ðTÞ ¼ 0. The second correction is a dissipation term fluctuates from the perspective of S, and we denote these S S stemming from the temperature dependence of the internal fluctuations by Δβ . Given that the state of S takes the form energy operator Eq. (11). Notably this term can still be present in βH ðTÞ π ^ ðTÞ :¼ e =Z we can immediately apply Theorem 1 by the classical limit where the energy operator may depend on ^ ^ identifying B ¼ E ðTÞ with θ = β, leading to θ S temperature if the coupling is non-negligible. As expected both F ðβÞ K π ; E . Applying Eq. (7) for the single-shot case S S S terms can be dropped in the limit of vanishing coupling and the ^ ^ (n = 1) and using the fact that K π ^ ; E = ΔU  Q π ^ ; E ,we standard FDR is recovered. S S S S S obtain the following thermodynamic uncertainty relation: Bound on signal-to-noise ratio for temperature estimates. Let 1 1 Δβ  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  : us denote the uncertainty in the temperature from a given ð12Þ ΔU ^ S ΔU  Q π ^ ; E unbiased estimation scheme by ΔT , with measurements per- S S S S 11,12,14,55 formed on S alone. It is known that in the weak- coupling limit, the optimal signal-to-noise ratio for estimating T from a single measurement is bounded by C ðTÞ: This is the main result of the paper and represents the strong- S coupling generalisation of Eq. (1). It can be seen that the bound on the uncertainty in the inverse temperature is increased ð15Þ C ðTÞ: ΔT whenever quantum energy fluctuations are present. These additional fluctuations are quantified by the non-negative Q π ^ ; E , increasing which implies a larger lower bound on S S This bound is tight for a single measurement of T and implies Δβ . One recovers the usual uncertainty relation when Q π ; E S S S that precise measurements of the temperature require a large heat can be neglected, which is the case when the interaction capacity. The result follows straightforwardly from the quantum commutes with the bare Hamiltonians of S and R or when the ^ Cramér-Rao inequality and the standard FDR. interaction is sufficiently weak. We note that Q π ^ ; E vanishes Using our modified FDR Eq. (14), we here give the strong- for classical systems and Eq. (12) reduces to the original coupling generalisation of the bound Eq. (15). Considering uncertainty relation Eq. (1), but with energy fluctuations ^ ^ estimates of T rather than the inverse temperature β, a simple quantified by E instead of the bare Hamiltonian H . S S change of variables reveals that the QFI with respect to T is If one repeats the experiment n times, then the uncertainty in pffiffiffi related to that of β, F ðβÞ¼ T F ðTÞ. From Theorem 1 we again the estimate can be improved by a factor of 1= n . We remark S S ^ ^ have T F ðTÞ K π ; E , and combining this with Eqs. (14) that in the weak coupling limit, where H ðTÞ’ H , the state of S S S S S S and (7) we obtain: belongs to the exponential family, and hence the bound on Δβ becomes tight for a single measurement in agreement with Eq. 2 ^ ^ ð16Þ (1). However, when V is non-negligible the Hamiltonian of  C ðTÞ ∂ E : S T S ∪ R S ΔT mean force cannot generally be expressed in the linear form ^ ^ ^ βH ðTÞ = βX þ Y . This means in general it is necessary to take S S 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 volume of the system, the coupling limits. In these regimes the POVM saturating Eq. (16)is heat capacity is defined as the temperature derivative of the given by the maximum-likelihood estimator measured in the 31,32 internal energy U ðTÞ , i.e. basis of the relevant symmetric logarithmic derivative .We stress that Eq. (16) is valid in the classical limit, in which case it is ∂U ð13Þ C ðTÞ :¼ : always tight. We remark that the RHS of Eq. (16) can alternatively ∂T be expressed in terms of the skew information, in which case 2 2  2 ðT=ΔT Þ  ΔU =T  Q π ^ ; E =T . S S S S 4 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 Application to damped harmonic oscillator. While the bound operator, n ^ ¼ ^a ^a , with annihilation operator ^a ¼ T T T T Eq. (16) is tight in the high temperature limit, for general open qffiffiffiffiffi quantum systems the accuracy of the bound is not known. We i ^ ^ x þ p with A = M ω . T T T 2h A show that the bound is very good for the example of a damped The internal energy operator is now obtained by straightfor- harmonic oscillator linearly coupled to N harmonic oscillators in ward differentiation, see Eq. (11), and given by 34,56,57 the reservoir . Experimentally, such a model describes the 58 2 behaviour of nano-mechanical resonators and BEC impu- ^ ^ a þ a ^ 2 2 T T 59 Mω ^x ð19Þ rities . Here the system Hamiltonian is H ¼ þ , while ^ ^ E ðTÞ¼ α H ðTÞ g ; 2M 2 S T S T 2 2 2 P 2 ^p M ω ^x N j j j j the reservoir Hamiltonian is H ¼ þ and the R j¼1 2M 2 ′ ′ ω A interaction term is given by T T where α ¼ 1  T and g ¼ hω T . Using this operator we T T T ω A ! T T X λ obtain analytic expressions for C ðTÞ, F ðTÞ, Q π ^ ; E and S S S S V ¼ λ ^x  ^x þ ^x : ð17Þ S ∪ R j j 2 ^ ∂ E in Supplementary Note 3. 2M ω j j S j¼1 Figure 1 shows the square root of the average skew information Q π ^ ; E in units of ħω as a function of temperature for different S S coupling strengths. As expected we see that the quantum To allow a fully analytical solution, the reservoir frequencies fluctuations in energy vanish in the high temperature limit, while are chosen equidistant, ω = jΔ and the continuum limit is taken fluctuations grow with increased coupling strengths due to so that Δ → 0 (and N → ∞). The coupling constants are chosen as rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi increased non-commutativity between E ðTÞ and the state π ^ of 2γM Mω Δ S S j j 34 D the Drude-Ullersma spectrum , λ ¼ , where γ 2 2 ^ j ^ π ω þω the oscillator. Interestingly we see that Q π ; E decays D j S S exponentially to zero in the low temperature limit, implying that is the damping coefficient controlling the interaction strength and the state of the oscillator commutes with the internal energy ω is a large cutoff frequency. operator in this regime. Whether this is a general feature or As shown by Grabert et al. , the resulting Hamiltonian of specific to the example here remains an open question. mean force for the oscillator can be parameterised by a Figure 2 shows the optimal signal-to-noise ratio for estimating temperature-dependent mass and frequency, T determined by the Cramér-Rao bound Eq. (7), 2 2 2 p M ω ^x 1 2 T T ^ ðT=ΔT Þ ¼ T F ðTÞ, as a function of temperature T and H ðTÞ¼ þ ¼ hω n ^ þ ; ð18Þ S opt S S T T 2M 2 2 coupling strength γ. The bound we derived in Eq. (16) given by the heat capacity and an additional dissipation term is also where M and ω are given through the expectation values of p T T plotted and shows very good agreement with the optimum 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 0.08 / = 0.5 √Q / = 0.2 0.06 0.6 / = 0.1 0.04 0.4 ΔT 0.02 0.2 0.00 0 5 10 15 20 10 T Fig. 1 Skew information for the damped oscillator. Plot of quantum Fig. 2 Bound on temperature signal-to-noise ratio. The coloured plot shows rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi hi the optimal signal-to-noise ratio ðT=ΔT Þ of an unbiased temperature ^ S energetic fluctuations Q π ; E =hω for the damped oscillator as a opt S S estimate for the damped oscillator, as a function of temperature T and hi function of T/ħω for different coupling strengths γ. Here Q π ^ ; E is the coupling strength γ. This optimal measurement is determined by the quantum Fisher information, which places an asymptotically achievable average Wigner-Yanase-Dyson skew information for the effective energy lower bound on the temperature fluctuations ΔT through the Cramér-Rao operator E . These fluctuations are present when the state of the oscillator S 2 inequality . The mesh plot shows the upper bound on ðT=ΔT Þ derived S opt π ^ ðTÞ is not diagonal in the basis of E due to the non-negligible interaction S S 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 coupling γ leads to an increase in the skew information. The quantum capacity of the system at arbitrary coupling strengths. It can be seen that fluctuations are most pronounced at low temperatures where the thermal the upper bound becomes tight in both the high temperature and weak energies become comparable to the oscillator spacing, T ’ h ω.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 38,68,69 T → 0, albeit with different rates of decay. Outside of these limits where correlations play a prominent role . The results establish a connection between abstract measures of quantum the difference between the bound and ðT=ΔT Þ has a opt information theory, such as the QFI and skew information, and a maximum, and at the temperature and coupling for which this material’s effective thermodynamic properties. This provides a maximum occurs the bound is roughly 30% greater than 2 starting point for future investigations into nanoscale thermo- ðT=ΔT Þ . S 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 Proof of Theorem 1. Here we provide a sketch for the proof of the bound Eq. (8) influence fluctuations 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. Let us first denote the spectral decomposition of the state ^ρ ¼ θ ^ e =Z by ^ρ ¼ p ψ ψ , where the eigenstates satisfy A ψ ¼ λ ψ . the well-known complementarity relation Eq. (1) between energy θ θ n n n n θ n n n From this decomposition the QFI can be expressed as follows : and temperature to all interaction strengths. This derivation is based on a bound on the QFI for exponential states which we ψ ∂ ^ρ ψ n θ θ m FðθÞ¼ 2 : ð20Þ prove in Theorem 1. As Theorem 1 is valid for any state of full- p þ p n m n;m 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 exponential operator : spread in energy, unbiased estimates of the underlying tempera- hi ture are limited to a greater extent due to coherences between ^ ^ ^ A ð1aÞ A a A θ θ ^ θ ð21Þ ∂ e :¼ dae ∂ ½A e : θ θ θ energy states. These coherences only arise for quantum systems beyond the weak coupling assumption. We found that these Combining Eqs. (20) and (21) eventually yields additional temperature fluctuations are quantified by the average f ðp ; p Þ WYD skew information, thereby establishing a link between 2 n m 2 FðθÞ¼ ΔB þ ðp þ p Þ B ; ð22Þ θ n m nm p þ p quantum and classical forms of statistical uncertainty in nanos- n m n<m cale thermodynamics. With coherence now understood to be an ^   ^ ^ where B ¼ ψ ∂ A ψ , ΔB ¼ Var ^ρ ; B and nm n θ θ m θ θ θ important resource in the performance of small-scale heat 60,61 2ðp  p Þ engines , our findings suggest that the skew information could n m fpðÞ ; p :¼ : ð23Þ n m lnðÞ p =p be used to unveil further non-classical aspects of quantum ther- n m modynamics. This complements previous results that connect Similarly, expanding in the basis ψ leads to an expression for the average 51 n skew information to both unitary phase estimation and quan- skew information with respect to the operator B : tum speed limits . X Q ^ρ ; B ¼ðÞ ðp þ p Þ f ðp ; p Þ B : θ n m n m θ nm ð24Þ Our second result Eq. (14) is a generalisation of the well- n<m known FDR to systems beyond the weak coupling regime. This further establishes a connection between the skew information Using the inequality and the system’s heat capacity C ðTÞ. Proving that the heat f ðp ; p Þ n m 1; ð25Þ capacity, with its strong coupling corrections, vanishes in the p þ p n m zero-temperature limit in accordance with the third law of ther- ^ ^ and comparing Eqs. (22) and (24) yields FðθÞ ΔB  Q ^ρ ; B . The theorem modynamics remains an open question. The appearance of the θ θ θ ^ ^ ^ then follows from the fact that ΔB ¼ K ^ρ ; B þ Q ^ρ ; B according to Eq. (2). skew information in Eq. (14) suggests that quantum coherences θ θ θ θ θ may play a role in ensuring its validity. Recent resource-theoretic 62,63 Derivation of the fluctuation-dissipation relation. Here we briefly outline the derivations of the third law could provide a possible avenue proof of the modified FDR Eq. (14). First note that according to the definition Eq. for exploring the impact of coherences. (5), the classical uncertainty in energy is given by By applying the FDR to temperature estimation we derive our third result, Eq. (16), an upper bound on the optimal signal-to-  1a  a ^ ^ ^ K π ^ ; E ¼ datr π ^ δE π ^ δE ; ð26Þ S S S S S S noise ratio expressed in terms of the system’s heat capacity. Notably the bound implies that when designing a probe to ^ ^ ^ where δE :¼ E ðTÞ E ðTÞ . By using the integral expression Eq. (21), we S S S measure T, its bare Hamiltonian and interaction with the sample show in Supplementary Note 2 that this equation for K π ; E is equivalent to S S should be chosen so as to both maximise C ðTÞ whilst mini- ^ ^ K π ^ ; E ¼ T tr E ∂ π ^ : ð27Þ ^ S S S T S mising the additional dissipation term ∂ E . It is an interesting T S open question to consider the form of Hamiltonians that achieve Using the product rule for the differential of an operator, along with the equation this optimisation in the strong coupling scenario. Furthermore, ^ ^ K π ^ ; E = ΔU  Q π ^ ; E given by Eq. (5), completes the derivation. S S S S S one expects that improvements to low-temperature thermometry resulting from strong interactions, such as those observed in , Data availability. The data that support the findings of this study are available from the corresponding author upon reasonable request. will be connected to the properties of the effective internal energy operator. 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Energy-temperature uncertainty relation in quantum thermodynamics

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ARTICLE DOI: 10.1038/s41467-018-04536-7 OPEN Energy-temperature uncertainty relation in quantum thermodynamics 1 1 H.J.D. Miller & J. Anders 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 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 operator 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 heat 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. 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: hm419@exeter.ac.uk) or to J.A. (email: janet@qipc.org) NATURE COMMUNICATIONS (2018) 9:2203 DOI: 10.1038/s41467-018-04536-7 www.nature.com/naturecommunications 1 | | | 1234567890():,; ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/s41467-018-04536-7 ohr suggested that there should exist a form of com- estimate, and we illustrate our bound with an example of a plementarity between temperature and energy in thermo- damped harmonic oscillator. Bdynamics similar to that of position and momentum in quantum theory . 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 observables 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 quantum state ^ρ and an observable A. Wigner uncertainty relation: and Yanase considered the problem of quantifying the quantum uncertainty in observable A for the case where ^ρ is mixed . ð1Þ Δβ  ; However, they observed that the standard measure of uncertainty, ΔU ^ ^ ^ ^ ^ namely the variance Var½^ρ; A := tr½^ρδA  with δA = A  A , −1 contains classical contributions due to mixing, and thus fails to with β = (k T) the inverse temperature. While Eq. (1) has since 2–9 ^ fully quantify the non-classical fluctuations in the observable A. been derived in various settings , it was Mandelbrot who first This problem can be resolved by finding a quantum measure of based the concept of fluctuating temperature on the theory of ^ ^ statistical inference. Concretely, for a thermal system in canonical uncertainty Q½^ρ; A and classical measure K½^ρ; A 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; A¼ Q½^ρ; Aþ K½^ρ; A: ð2Þ that Eq. (1) sets the ultimate limit on simultaneous estimates of energy and temperature in classical statistical physics . The notion of fluctuating temperature has proved to be fun- Following the framework introduced by Luo , these functions damental in the emerging field of quantum thermometry, where are required to fulfil three conditions: (i) both terms should be advances in nanotechnology now allow temperature sensing at ^ ^ non-negative, Q½^ρ; A 0 and K½^ρ; A 0, so that they can be 10–22 23 sub-micron scales . Using the tools of quantum metrology , interpreted as forms of statistical uncertainty, (ii) if the state ^ρ is the relation Eq. (1) can also be derived for weakly coupled ^ ^ ^ pure, then Q½^ρ; A = Var½^ρ; A while K½^ρ; A¼ 0 as all uncertainty 11,12,14 quantum systems , where the equilibrium state is best should be associated to quantum fluctuations alone, (iii) Q½^ρ; A described by the canonical ensemble. Within the grand-canonical must be convex with respect to ^ρ, so that it decreases under ensemble the impact of the indistinguishability of quantum par- classical mixing. Correspondingly, K½^ρ; A must be concave with ticles on the estimation of temperature and the chemical potential respect to ρ. has also been explored . Relation Eq. (1) informs us that when The following function, known as the WYD skew informa- designing an accurate quantum thermometer one should search tion was shown to be a valid measure of quantum uncertainty: for systems with Hamiltonians that produce a large energy variance . a 1a ^ ^ ^ Q ½^ρ; A :¼ tr½½A; ^ρ ½A; ^ρ ; a 2ð0; 1Þ; ð3Þ Recently there has been an emerging interest into the effects of 13,15,25 strong coupling on temperature estimation . At the with the complementary classical uncertainty given by nanoscale the strength of interactions between the system and the a 1a reservoir may become non-negligible, and the local equilibrium ^ ^ ^ K ½^ρ; A :¼ tr½^ρ δA^ρ δA; a 2ð0; 1Þ: ð4Þ 26,27 state of the system will not be of Gibbs form . In this regime 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 ^ ^ ^ ^ concavity of Q ½ρ; A and K ½ρ; A respectively can be proven a a 28–40 energy . We will see that the internal energy U and its fluc- using Lieb’s concavity theorem. tuations ΔU are determined by a modified internal energy The presence of the parameter a demonstrates that there is no operator, denoted by E , that differs from the bare Hamiltonian S unique way of separating the quantum and classical contributions 35,39 43,44 of the system . This modification brings into question the to the variance. We here follow the suggestion made in refs. 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: pling on the thermodynamic uncertainty relation. Z Taking into account quantum properties of the effective ^ ^ Q½^ρ; A :¼ daQ ½^ρ; A; ð5Þ internal energy operator and its temperature dependence, we here derive the general thermodynamic uncertainty principle valid at all coupling strengths. Formally this result follows from a general ^ ^ K½^ρ; A :¼ daK ½^ρ; A: ð6Þ upper bound on the quantum Fisher information (QFI) for exponential states. We prove that quantum fluctuations arising from coherences between energy states of the system lead to increased fluctuations in the underlying temperature. Most ^ ^ It is not only the Q ½^ρ; A and K ½^ρ; A that separate the a a interestingly, the non-classical modifications to Eq. (1) are quantum and classical fluctuations of a quantum observable A in quantified by the average Wigner-Yanase-Dyson (WYD) skew 41–44 a state ^ρ according to Eq. (2), but also the averaged Q½^ρ; A and information , which is a quantity closely linked to measures 45,46 ^ K½^ρ; A. This follows from the linearity of the integrals in Eqs. (5) of coherence, asymmetry and quantum speed limits . We then and (6) which also preserve the conditions (i)–(iii). Throughout demonstrate that the skew information is also linked to the heat ^ ^ capacity of the system through a modified fluctuation-dissipation the remainder of the paper we will consider Q½^ρ; A and K½^ρ; A 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 strongly interacting with a reservoir. To achieve this we will first Bound on quantum Fisher information for exponential states. discuss the appropriate energy operator for such a system, and We now prove that the average skew information is linked to the then proceed to generalise Eq. (1). quality of a parameter estimate for a quantum exponential state. θ A quantum system S that interacts with a reservoir R is A quantum exponential state is of the form ^ρ = e =Z where θ θ θ ^ described by a Hamiltonian Z ¼ tr½e  and A is a hermitian operator that is here assumed θ θ to depend analytically on a smooth parameter θ. For any state of ^ ^ ^ ^ ^ ^ ð9Þ H :¼ H  I þ I  H þ V ; ^ S ∪ R S R S R S ∪ R full rank, an operator A can be found such that the state can be expressed in this form, i.e. all full rank states are exponential ^ ^ where H and H are the bare Hamiltonians of S and R states. S R We first recall the standard setup for estimating the parameter respectively, while V is an interaction term of arbitrary S ∪ R θ . First one performs a POVM measurement MðξÞ, where strength. We will consider situations where the environment is ^ ^ large compared to the system, i.e. the operator norms fulfil dξ MðξÞ¼ I and ξ denotes the outcomes of the measurement ^ ^ ^ H  H ; V . We make no further assumptions which may be continuous or discrete. The probability of obtaining R S S ∪ R ^ ^ a particular outcome is pðξjθÞ¼ tr½MðξÞ^ρ . The measurement about the relative size of the coupling V between the S ∪ R is repeated n times with outcomes {ξ ,ξ ,..ξ }, and one ^ 1 2 n system and the environment, and the system’s bare energy H . ~ ~ constructs a function θ ¼ θðξ ; ξ ;::ξ Þ that estimates the true The global equilibrium state at temperature T for the total 1 2 n ~ βH S ∪ R value of the parameter. We denote the average estimate by hθi, Hamiltonian S ∪ R is of Gibbs form π ^ ðTÞ = e =Z R S ∪ R S ∪ R hi wherehi ð::Þ = dξ ¼ dξ pðξ jθÞ¼ pðξ jθÞð::Þ, and assume the −1 βH 1 n 1 n S ∪ R where β = (k T) and Z = tr e is the partition S ∪ R S ∪ R estimate is unbiased, i.e. hθi¼ θ. In this case the mean-squared function for S ∪ R. The Boltzmann constant k will be set to error in the estimate is equivalent to the variance, which is unity throughout. 2 2 2 denoted by Δθ ¼hθ i θ . Due to the presence of the interaction term the reduced state of The celebrated quantum Cramér-Rao inequality sets a lower ^ ^ S, denoted π ðTÞ¼ tr ½π ðTÞ, is generally not thermal with S R S ∪ R bound on Δθ, optimised over all possible POVMs and estimator respect to H , unless the coupling is sufficiently weak, i.e. 23,48–50 functions : ^ ^ H  V . Therefore the partition function determined S S ∪ R 1 by H can no longer be used to calculate the internal energy of Δθ  pffiffiffiffiffiffiffiffiffiffiffiffi ; ð7Þ the system . To resolve this issue one can rewrite the state of S nFðθÞ βH ðTÞ as an effective Gibbs state π ^ ðTÞ := e =Z , where hi 0 1 βH S ∪ R where F(θ) is the QFI. The bound becomes tight in the asymptotic tr e @ A ð10Þ limit n → ∞ . If the exponential state belongs to the so-called H ðTÞ :¼ ln  ; βH β R tr e ^ ^ ^ R ‘exponential family’, which is true if A ¼ θX þ Y for commuting ^ ^ operators X; Y, then the bound is also tight in the single-shot 28,30–37,39,40 limit (n = 1) . The QFI with respect to θ is defined by is the Hamiltonian of mean force . The operator ^ ^ ^ H ðTÞ acts as a temperature-dependent effective Hamiltonian FðθÞ :¼ tr ^ρ L , where L is the symmetric logarithmic θ S θ θ describing the equilibrium properties of S through the effective derivative which uniquely satisfies the operator equation hi 1 49 ^ ^  βH ðTÞ ∂ ^ρ = L ; ^ρ . Here {..,..} denotes the anti-commutator. S partition function Z ¼ tr e . The free energy associated θ θ 2 θ θ S S We now state a general upper bound on F(θ) valid for any 30,52 with Z also appears in the open system fluctuation relations . exponential state: The internal energy of S can be computed from this partition Theorem 1: For an exponential state ρ ¼ e =Z the QFI with θ θ function via U ðTÞ :¼∂ lnZ . It is straightforward to show S β S respect to the parameter θ is bounded by that U ðTÞ is just the difference between the total energy, U ¼∂ lnZ , and the energy of the reservoir, U ¼ ^ S ∪ R β S ∪ R R FðθÞ K½^ρ ; B : ð8Þ hi βH ∂ lnZ with Z ¼ tr e , in the absence of any coupling β R R R ^ ^ to S, i.e. U ðTÞ = U ðTÞ U ðTÞ. In other words, U ðTÞ is Here K½^ρ ; B  is defined in Eq. (6), and B is the hermitian S S ∪ R R S θ θ θ ^ ^ the energy change induced from immersing the subsystem S into observable B :¼ ∂ A . The bound becomes tight in the limits θ θ θ 29,36 the composite state S ∪ R . where ^ρ is maximally mixed. Seifert has remarked that U ðTÞ can be expressed as an This theorem demonstrates that the strictly classical fluctua- ^ ^ tions in B constrain the achievable precision in estimates of θ. expectation value, U ðTÞ = E ðTÞ , of the following observable: θ S The proof of the theorem is outlined in the ‘Methods’ section. ^ ^ E ðTÞ :¼ ∂ βH ðTÞ : ð11Þ S β S We note that for states σ ^ that fulfil the von-Neumann ^ ^ equation ∂ σ ¼i½A ; σ  a connection between skew informa- θ θ θ θ tion Q ½σ ^ ; A  and parameter estimation has previously been 1=2 θ θ One can interpret E ðTÞ as the effective energy operator made by Luo . While the particular dependence on θ implied by describing the system, and we will refer to its eigenstates as “the this equation is relevant for unitary parameter estimation , this system energy states”. The introduction of this operator allows dependence will not be relevant for temperature estimation since qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi thermal states do not generally fulfil this von-Neumann equation. one to consider fluctuations in the energy ΔU ¼ Var π ^ ; E . S S In contrast, we will see in the next section that Theorem 1 has It is important to note that E ðTÞ depends explicitly on the implications for the achievable precision in determining temperature. coupling V and the temperature T. S ∪ R NATURE COMMUNICATIONS (2018) 9:2203 DOI: 10.1038/s41467-018-04536-7 www.nature.com/naturecommunications 3 | | | ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/s41467-018-04536-7 Our first observation is that, in general, E ðTÞ differs from In standard thermodynamics where the system is described by a Gibbs state the FDR states that the heat capacity is proportional both the bare system Hamiltonian H and the mean force 2 2 ^ to the fluctuations in energy, i.e. C ðTÞ¼ ΔU =T . However, Hamiltonian H ðTÞ. Indeed, this effective energy operator for the S S example studies of open quantum systems of the form Eq. (9) system contains the bare energy part as well as an energetic have shown that the heat capacity can become negative at low contribution from the coupling, E ðTÞ = S 31,32,53,54 temperatures , thus implying it cannot be proportional to ^ ^ ^ ^ H þ ∂ β H ðTÞ H . Moreover, E ðTÞ does not even S β S S S a positive variance in general. ^ ^ commute with H and H ðTÞ. This non-commutativity implies S Our second result indeed shows that there are two additional that the state π ^ ðTÞ exists in a superposition of energy states, contributions to the FDR due to strong-coupling (see ‘Methods’ ^ ^ ^ aside from the trivial situation in which H þ H ; V ¼ 0. section): S R S ∪ R As expected, in the limit of weak coupling E ðTÞ reduces to the S ^ ΔU Q π ^ ; E S S ^ ð14Þ C ðTÞ¼  þ ∂ E ; bare Hamiltonian H . S T S 2 2 T T We are now ready to state the generalised thermodynamic uncertainty relations for strongly coupled quantum systems. 2 2 implying that C ðTÞ can be less than ΔU =T and even negative. 13 S Following the approach taken by De Pasquale et al. , we consider We see that the first correction is due to the quantum fluctuations the QFI F ðβÞ associated with the inverse temperature β. in energy given by the average WYD information Q π ^ ; E , According to the quantum Cramér-Rao bound this functional which only vanishes in the classical limit where quantifies the minimum extent to which the inverse temperature E ðTÞ; π ^ ðTÞ ¼ 0. The second correction is a dissipation term fluctuates from the perspective of S, and we denote these S S stemming from the temperature dependence of the internal fluctuations by Δβ . Given that the state of S takes the form energy operator Eq. (11). Notably this term can still be present in βH ðTÞ π ^ ðTÞ :¼ e =Z we can immediately apply Theorem 1 by the classical limit where the energy operator may depend on ^ ^ identifying B ¼ E ðTÞ with θ = β, leading to θ S temperature if the coupling is non-negligible. As expected both F ðβÞ K π ; E . Applying Eq. (7) for the single-shot case S S S terms can be dropped in the limit of vanishing coupling and the ^ ^ (n = 1) and using the fact that K π ^ ; E = ΔU  Q π ^ ; E ,we standard FDR is recovered. S S S S S obtain the following thermodynamic uncertainty relation: Bound on signal-to-noise ratio for temperature estimates. Let 1 1 Δβ  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  : us denote the uncertainty in the temperature from a given ð12Þ ΔU ^ S ΔU  Q π ^ ; E unbiased estimation scheme by ΔT , with measurements per- S S S S 11,12,14,55 formed on S alone. It is known that in the weak- coupling limit, the optimal signal-to-noise ratio for estimating T from a single measurement is bounded by C ðTÞ: This is the main result of the paper and represents the strong- S coupling generalisation of Eq. (1). It can be seen that the bound on the uncertainty in the inverse temperature is increased ð15Þ C ðTÞ: ΔT whenever quantum energy fluctuations are present. These additional fluctuations are quantified by the non-negative Q π ^ ; E , increasing which implies a larger lower bound on S S This bound is tight for a single measurement of T and implies Δβ . One recovers the usual uncertainty relation when Q π ; E S S S that precise measurements of the temperature require a large heat can be neglected, which is the case when the interaction capacity. The result follows straightforwardly from the quantum commutes with the bare Hamiltonians of S and R or when the ^ Cramér-Rao inequality and the standard FDR. interaction is sufficiently weak. We note that Q π ^ ; E vanishes Using our modified FDR Eq. (14), we here give the strong- for classical systems and Eq. (12) reduces to the original coupling generalisation of the bound Eq. (15). Considering uncertainty relation Eq. (1), but with energy fluctuations ^ ^ estimates of T rather than the inverse temperature β, a simple quantified by E instead of the bare Hamiltonian H . S S change of variables reveals that the QFI with respect to T is If one repeats the experiment n times, then the uncertainty in pffiffiffi related to that of β, F ðβÞ¼ T F ðTÞ. From Theorem 1 we again the estimate can be improved by a factor of 1= n . We remark S S ^ ^ have T F ðTÞ K π ; E , and combining this with Eqs. (14) that in the weak coupling limit, where H ðTÞ’ H , the state of S S S S S S and (7) we obtain: belongs to the exponential family, and hence the bound on Δβ becomes tight for a single measurement in agreement with Eq. 2 ^ ^ ð16Þ (1). However, when V is non-negligible the Hamiltonian of  C ðTÞ ∂ E : S T S ∪ R S ΔT mean force cannot generally be expressed in the linear form ^ ^ ^ βH ðTÞ = βX þ Y . This means in general it is necessary to take S S 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 volume of the system, the coupling limits. In these regimes the POVM saturating Eq. (16)is heat capacity is defined as the temperature derivative of the given by the maximum-likelihood estimator measured in the 31,32 internal energy U ðTÞ , i.e. basis of the relevant symmetric logarithmic derivative .We stress that Eq. (16) is valid in the classical limit, in which case it is ∂U ð13Þ C ðTÞ :¼ : always tight. We remark that the RHS of Eq. (16) can alternatively ∂T be expressed in terms of the skew information, in which case 2 2  2 ðT=ΔT Þ  ΔU =T  Q π ^ ; E =T . S S S S 4 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 Application to damped harmonic oscillator. While the bound operator, n ^ ¼ ^a ^a , with annihilation operator ^a ¼ T T T T Eq. (16) is tight in the high temperature limit, for general open qffiffiffiffiffi quantum systems the accuracy of the bound is not known. We i ^ ^ x þ p with A = M ω . T T T 2h A show that the bound is very good for the example of a damped The internal energy operator is now obtained by straightfor- harmonic oscillator linearly coupled to N harmonic oscillators in ward differentiation, see Eq. (11), and given by 34,56,57 the reservoir . Experimentally, such a model describes the 58 2 behaviour of nano-mechanical resonators and BEC impu- ^ ^ a þ a ^ 2 2 T T 59 Mω ^x ð19Þ rities . Here the system Hamiltonian is H ¼ þ , while ^ ^ E ðTÞ¼ α H ðTÞ g ; 2M 2 S T S T 2 2 2 P 2 ^p M ω ^x N j j j j the reservoir Hamiltonian is H ¼ þ and the R j¼1 2M 2 ′ ′ ω A interaction term is given by T T where α ¼ 1  T and g ¼ hω T . Using this operator we T T T ω A ! T T X λ obtain analytic expressions for C ðTÞ, F ðTÞ, Q π ^ ; E and S S S S V ¼ λ ^x  ^x þ ^x : ð17Þ S ∪ R j j 2 ^ ∂ E in Supplementary Note 3. 2M ω j j S j¼1 Figure 1 shows the square root of the average skew information Q π ^ ; E in units of ħω as a function of temperature for different S S coupling strengths. As expected we see that the quantum To allow a fully analytical solution, the reservoir frequencies fluctuations in energy vanish in the high temperature limit, while are chosen equidistant, ω = jΔ and the continuum limit is taken fluctuations grow with increased coupling strengths due to so that Δ → 0 (and N → ∞). The coupling constants are chosen as rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi increased non-commutativity between E ðTÞ and the state π ^ of 2γM Mω Δ S S j j 34 D the Drude-Ullersma spectrum , λ ¼ , where γ 2 2 ^ j ^ π ω þω the oscillator. Interestingly we see that Q π ; E decays D j S S exponentially to zero in the low temperature limit, implying that is the damping coefficient controlling the interaction strength and the state of the oscillator commutes with the internal energy ω is a large cutoff frequency. operator in this regime. Whether this is a general feature or As shown by Grabert et al. , the resulting Hamiltonian of specific to the example here remains an open question. mean force for the oscillator can be parameterised by a Figure 2 shows the optimal signal-to-noise ratio for estimating temperature-dependent mass and frequency, T determined by the Cramér-Rao bound Eq. (7), 2 2 2 p M ω ^x 1 2 T T ^ ðT=ΔT Þ ¼ T F ðTÞ, as a function of temperature T and H ðTÞ¼ þ ¼ hω n ^ þ ; ð18Þ S opt S S T T 2M 2 2 coupling strength γ. The bound we derived in Eq. (16) given by the heat capacity and an additional dissipation term is also where M and ω are given through the expectation values of p T T plotted and shows very good agreement with the optimum 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 0.08 / = 0.5 √Q / = 0.2 0.06 0.6 / = 0.1 0.04 0.4 ΔT 0.02 0.2 0.00 0 5 10 15 20 10 T Fig. 1 Skew information for the damped oscillator. Plot of quantum Fig. 2 Bound on temperature signal-to-noise ratio. The coloured plot shows rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi hi the optimal signal-to-noise ratio ðT=ΔT Þ of an unbiased temperature ^ S energetic fluctuations Q π ; E =hω for the damped oscillator as a opt S S estimate for the damped oscillator, as a function of temperature T and hi function of T/ħω for different coupling strengths γ. Here Q π ^ ; E is the coupling strength γ. This optimal measurement is determined by the quantum Fisher information, which places an asymptotically achievable average Wigner-Yanase-Dyson skew information for the effective energy lower bound on the temperature fluctuations ΔT through the Cramér-Rao operator E . These fluctuations are present when the state of the oscillator S 2 inequality . The mesh plot shows the upper bound on ðT=ΔT Þ derived S opt π ^ ðTÞ is not diagonal in the basis of E due to the non-negligible interaction S S 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 coupling γ leads to an increase in the skew information. The quantum capacity of the system at arbitrary coupling strengths. It can be seen that fluctuations are most pronounced at low temperatures where the thermal the upper bound becomes tight in both the high temperature and weak energies become comparable to the oscillator spacing, T ’ h ω.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 38,68,69 T → 0, albeit with different rates of decay. Outside of these limits where correlations play a prominent role . The results establish a connection between abstract measures of quantum the difference between the bound and ðT=ΔT Þ has a opt information theory, such as the QFI and skew information, and a maximum, and at the temperature and coupling for which this material’s effective thermodynamic properties. This provides a maximum occurs the bound is roughly 30% greater than 2 starting point for future investigations into nanoscale thermo- ðT=ΔT Þ . S 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 Proof of Theorem 1. Here we provide a sketch for the proof of the bound Eq. (8) influence fluctuations 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. Let us first denote the spectral decomposition of the state ^ρ ¼ θ ^ e =Z by ^ρ ¼ p ψ ψ , where the eigenstates satisfy A ψ ¼ λ ψ . the well-known complementarity relation Eq. (1) between energy θ θ n n n n θ n n n From this decomposition the QFI can be expressed as follows : and temperature to all interaction strengths. This derivation is based on a bound on the QFI for exponential states which we ψ ∂ ^ρ ψ n θ θ m FðθÞ¼ 2 : ð20Þ prove in Theorem 1. As Theorem 1 is valid for any state of full- p þ p n m n;m 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 exponential operator : spread in energy, unbiased estimates of the underlying tempera- hi ture are limited to a greater extent due to coherences between ^ ^ ^ A ð1aÞ A a A θ θ ^ θ ð21Þ ∂ e :¼ dae ∂ ½A e : θ θ θ energy states. These coherences only arise for quantum systems beyond the weak coupling assumption. We found that these Combining Eqs. (20) and (21) eventually yields additional temperature fluctuations are quantified by the average f ðp ; p Þ WYD skew information, thereby establishing a link between 2 n m 2 FðθÞ¼ ΔB þ ðp þ p Þ B ; ð22Þ θ n m nm p þ p quantum and classical forms of statistical uncertainty in nanos- n m n<m cale thermodynamics. With coherence now understood to be an ^   ^ ^ where B ¼ ψ ∂ A ψ , ΔB ¼ Var ^ρ ; B and nm n θ θ m θ θ θ important resource in the performance of small-scale heat 60,61 2ðp  p Þ engines , our findings suggest that the skew information could n m fpðÞ ; p :¼ : ð23Þ n m lnðÞ p =p be used to unveil further non-classical aspects of quantum ther- n m modynamics. This complements previous results that connect Similarly, expanding in the basis ψ leads to an expression for the average 51 n skew information to both unitary phase estimation and quan- skew information with respect to the operator B : tum speed limits . X Q ^ρ ; B ¼ðÞ ðp þ p Þ f ðp ; p Þ B : θ n m n m θ nm ð24Þ Our second result Eq. (14) is a generalisation of the well- n<m known FDR to systems beyond the weak coupling regime. This further establishes a connection between the skew information Using the inequality and the system’s heat capacity C ðTÞ. Proving that the heat f ðp ; p Þ n m 1; ð25Þ capacity, with its strong coupling corrections, vanishes in the p þ p n m zero-temperature limit in accordance with the third law of ther- ^ ^ and comparing Eqs. (22) and (24) yields FðθÞ ΔB  Q ^ρ ; B . The theorem modynamics remains an open question. The appearance of the θ θ θ ^ ^ ^ then follows from the fact that ΔB ¼ K ^ρ ; B þ Q ^ρ ; B according to Eq. (2). skew information in Eq. (14) suggests that quantum coherences θ θ θ θ θ may play a role in ensuring its validity. Recent resource-theoretic 62,63 Derivation of the fluctuation-dissipation relation. Here we briefly outline the derivations of the third law could provide a possible avenue proof of the modified FDR Eq. (14). First note that according to the definition Eq. for exploring the impact of coherences. (5), the classical uncertainty in energy is given by By applying the FDR to temperature estimation we derive our third result, Eq. (16), an upper bound on the optimal signal-to-  1a  a ^ ^ ^ K π ^ ; E ¼ datr π ^ δE π ^ δE ; ð26Þ S S S S S S noise ratio expressed in terms of the system’s heat capacity. Notably the bound implies that when designing a probe to ^ ^ ^ where δE :¼ E ðTÞ E ðTÞ . By using the integral expression Eq. (21), we S S S measure T, its bare Hamiltonian and interaction with the sample show in Supplementary Note 2 that this equation for K π ; E is equivalent to S S should be chosen so as to both maximise C ðTÞ whilst mini- ^ ^ K π ^ ; E ¼ T tr E ∂ π ^ : ð27Þ ^ S S S T S mising the additional dissipation term ∂ E . It is an interesting T S open question to consider the form of Hamiltonians that achieve Using the product rule for the differential of an operator, along with the equation this optimisation in the strong coupling scenario. Furthermore, ^ ^ K π ^ ; E = ΔU  Q π ^ ; E given by Eq. (5), completes the derivation. S S S S S one expects that improvements to low-temperature thermometry resulting from strong interactions, such as those observed in , Data availability. The data that support the findings of this study are available from the corresponding author upon reasonable request. will be connected to the properties of the effective internal energy 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 relative scaling of C ðTÞ and ∂ E , and exploring this further S T S remains a promising direction of research. Advancements in nanotechnology now enable temperature sensing over micro- 64,65 scopic spatial resolutions , 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 constitution. J. Chem. Soc. 1, 349–384 (1932). The presented approach opens up opportunities for exploring 66,67 the intermediate regime between the limiting cases of 6 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 2. Mandelbrot, B. 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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 Attribution 4.0 International License, which permits use, sharing, Author contributions 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/. Competing interests: The authors declare no competing interests. © The Author(s) 2018 Reprints and permission information is available online at http://npg.nature.com/ reprintsandpermissions/ 8 NATURE COMMUNICATIONS (2018) 9:2203 DOI: 10.1038/s41467-018-04536-7 www.nature.com/naturecommunications | | |

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