Introduction

The measurement of temperature is a key aspect in science, technology and in our daily life. Many ingenious solutions have been designed to approach different situations and required accuracies1. What is the ultimate limit to the precision at which the temperature of a macroscopic state can be determined? An elegant answer to this question is offered by estimation theory2,3,4: The precision is related to the heat capacity of the system5,6.

In view of the groundbreaking potentialities offered by present-day nanotechnologies7,8,9,10,11,12 and the need to control the temperature at the nano-scale, it is highly relevant to question whether the heat capacity is still the relevant (fundamental) precision limit to small-scale thermometry. The extensivity of the heat capacity is a consequence of the growing volume-to-surface ratio with the size13. However, at a microscopic level such construction may present some limitations14,15. Moreover a series of theoretical efforts recently concentrated on a self-consistent generalization of the classical thermodynamics to small-scale physics, where quantum effects become predominant16,17,18,19,20,21,22. In particular, a lot of attention has been devoted to the search for novel methods of precision nanothermometry that could exploit the essence of quantum correlations23,24,25,26,27,28. In this context, the possibility to correctly define the thermodynamical limit, and therefore the existence of the temperature in the quantum regime, has been thoroughly investigated. It has been shown that the minimal subset of an interacting quantum system, which can be described as a canonical ensemble, with the same temperature of the global system, depends not only on the strength of the correlations within the system, but also on the temperature itself29,30,31. Using a quantum information-oriented point of view, this phenomenon has also been highlighted in Gaussian fermionic and bosonic states, by exploiting quantum fidelity as the figure of merit32,33. Furthermore, the significant role played by quantum correlations has been recently discussed with specific attention to spin- and fermonic-lattice systems with short-range interactions34.

In this paper, we propose a quantum-metrology approach to thermometry, through the analysis of the local sensitivity of generic quantum systems to their global temperature. Our approach does not assume any constraint neither on the structure of the local quantum state, nor on the presence of strong quantum fluctuations within the system itself. It is motivated by the observation that the temperature is a parameter that can be addressed only via indirect measurements, as it labels the state of the considered systems. Specifically, we introduce a new quantity that we dub local quantum thermal susceptibility (LQTS), according to the following scheme: Given a quantum system in a thermal equilibrium state, the LQTS is a response functional, which quantifies the highest achievable accuracy for estimating the system temperature T through local measurements performed on a selected subsystem of (see Fig. 1).

Figure 1: Temperature estimation via local measures.
figure 1

We propose the following operationally grounded strategy, which is embodied by the local quantum thermal susceptibility (LQTS) functional. A composite quantum system is in thermal equilibrium with a bath at temperature T. The LQTS measures the highest achievable accuracy in the estimation of T under the hypothesis to perform only local measurements on a given subsystem of .

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The LQTS is in general not extensive with respect to the size of , yet it is an increasing function of the latter, and it reduces to the system heat capacity in the limit where the probed part coincides with the whole system . In the low-temperature limit, we shall also see that the LQTS is sensitive to the local distinguishability between the ground state and the first excited subspace of the composite system Hamiltonian. In this regime, even for a tiny size of the probed subsystem, our functional is able to predict the behaviour of the heat capacity and in particular to reveal the presence of critical regions. This naturally suggests the interpretation of as a sort of mesoscopic version of the heat capacity, which replaces the latter in those regimes where extensivity breaks down.

Results

The functional

Let us consider a bipartite quantum system at thermal equilibrium, composed of two subsystems and , and described by the canonical Gibbs ensemble . Here is the system Hamiltonian, which in the general case will include both local (that is, and ) and interaction (that is, ) terms, while denotes the associated partition function (β=1/kBT is the inverse temperature of the system, kB the Boltzmann constant, and {Ei} the eigenvalues of H). In this scenario, we are interested in characterizing how the actual temperature T is perceived locally on .

For this purpose, we resort to quantum metrology35 and define the LQTS of subsystem as

where is the fidelity between two generic quantum states ρ and σ (refs 36, 37). The quantity (1) corresponds to the quantum Fisher information (QFI; refs 3, 4) for the estimation of β, computed on the reduced state . It detects how modifications on the global system temperature are affecting , the larger being the more sensitive being the subsystem response. More precisely, through the quantum Cramér–Rao inequality, quantifies the ultimate precision limit to estimate the temperature T, by means of any possible local (quantum) measurement on subsystem . In the specific, it defines an asymptotically achievable lower bound,

on the root-mean-square error of a generic local estimation strategy, where Test is the estimated value of T, is the expectation value for a random variable x and N is the number of times the local measurement is repeated.

By construction, is a positive quantity that diminishes as the size of is reduced, the smaller being the portion of the system we have access to, the worse being the accuracy we can achieve. More precisely, given a proper subset of , we have . In particular, when coincides with the whole system , equation (1) reaches its maximum value and becomes equal to the variance of the energy,

which depends only on the spectral properties of the system and which coincides with the system heat capacity5,6 (note that, rigorously speaking, the LQTS quantifies the sensitivity of the system to its inverse temperature β; the corresponding susceptibility to T=1/(kBβ) gets a correction term, which also enters the standard definition of the heat capacity).

An explicit evaluation of the limit in equation (1) can be obtained via the Uhlmann’s theorem38 (see the ‘Methods’ section for details). A convenient way to express the final result can be obtained by introducing an ancillary system isomorphic to and the purification of ρβ defined as

where is the spectral decomposition of the system Hamiltonian. It can then be proved that

where {|ej〉} are the eigenvectors of the reduced density matrix living on , obtained by taking the partial trace of |ρβ〉 with respect to the ancillary system , while {λj} are the corresponding eigenvalues (which, by construction, coincide with the eigenvalues of ).

Equation (5) makes it explicit the ordering between and : the latter is always greater than the former due to the negativity of the second contribution appearing on the right hand side. Furthermore, if H does not include interaction terms (that is, ), one can easily verify that reduces to the variance of the local Hamiltonian of , and is given by the heat capacity of the Gibbs state which, in this special case, represents , that is, . Finally we observe that in the high-temperature regime (β→0) the expression (5) simplifies yielding

where and denote the Hilbert space dimensions of and respectively, and we defined having set, without loss of generality, Tr[H]=0.

A measure of state distinguishability

In the low-temperature regime, the LQTS can be used to characterize how much the ground state of the system differs from the first excited subspaces when observing it locally on . This is a direct consequence of the fact that the QFI (which we used to define our functional) accounts for the degree of statistical distinguishability between two quantum states (in our case the reduced density matrices and ) differing by an infinitesimal change in the investigated parameter (in our case the inverse temperature β). Therefore for β→∞, the LQTS can be thought as a quantifier of the local distinguishability among the lowest energy levels in which the system is frozen.

To clarify this point, let us consider the general scenario depicted in Fig. 2, where we only discuss the physics of the ground state (with energy E0=0) and of the lowest excited levels with energy Ei bounded by twice the energy of the first excited level, Ei≤2E1. The degeneracy of each considered energy eigenstate is denoted by ni. From equation (5), it then follows that up to first order in the parameter we get

Figure 2: Energy spectrum of a quantum system.
figure 2

The figure provides a schematic representation of the low-energy spectrum for a generic many-body quantum system. For simplicity the ground-state (gs) energy E0 is set to zero. Here Πi denotes the normalized projector on the eigenspaces of energy Ei, which can be ni-fold degenerate.

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Here , where Πi is the normalized projector on the degenerate subspace of energy Ei. Moreover, is the span of the local subspace associated to the ground state, that is, with and being the number of non-zero eigenvalues (pj>0) of .

Equation (7) can be interpreted as follows. Our capability of measuring β relies on the distinguishability between the states and , with ɛ<<β. In the zero-temperature limit, the system lies in the ground state and locally reads as , while at small temperatures, the lowest energy levels start to get populated. If their reduced projectors (i≥1) are not completely contained in the span of , that is , there exist some local states whose population is null for T=0 and greater than zero at infinitesimal temperatures. Such difference implies that the first order in does not vanish. On the contrary, if the reduced projectors are completely contained in the span of , that is , we can distinguish from only thanks to infinitesimal corrections to the finite-valued populations of the lowest energy levels (see the ‘Methods’ section for an explicit evaluation of the latter). In conclusion, the quantity acts as a thermodynamical indicator of the degree of distinguishability between the ground-state eigenspace and the lowest energy levels in the system Hamiltonian.

LQTS and phase estimation

A rather stimulating way to interpret equation (5) comes from the observation that, in the extended scenario where we have purified as in equation (4), the global variance (3) formally coincides with the QFI associated with the estimation of a phase which, for given β, has been imprinted into the system by a unitary transformation , with H′ being the analogous of H on the ancillary system , that is, where (refs 4, 35, 39). Interestingly enough, a similar connection can be also established with the second term appearing in the right hand side of equation (5): indeed the latter coincides with the QFI that defines the Cramér–Rao bound for the estimation of the phase of , under the constraint of having access only on the subsystem (that is, that part of the global system which is complementary to ). Accordingly, we can thus express the LQTS as the difference between these two QFI phase estimation terms, the global one versus the local one or, by a simple rearrangement of the various contributions, construct the following identity

that establishes a complementarity relation between the temperature estimation on and the phase estimation on its complementary counterpart , by forcing their corresponding accuracies to sum up to the energy variance 〈ΔH2〉β of the global system (3).

Local thermometry in many-body systems

We have tested the behaviour of our functional on two models of quantum spin chains, with a low-energy physics characterized by the emergence of quantum phase transitions (QPTs) belonging to various universality classes40.

Specifically, we consider the quantum spin-1/2 Ising and Heisenberg chains, in a transverse magnetic field h and with a z axis anisotropy Δ respectively,

Here denotes the usual Pauli matrices (α=x, y, z) on the i-th site, and periodic boundary conditions have been assumed. We set J=1 as the system’s energy scale. At zero temperature, the model (9) presents a -symmetry breaking phase transition at |hc|=1 belonging to the Ising universality class. The Hamiltonian (10) has a critical behaviour for −1≤Δ≤1, while it presents a ferromagnetic or antiferromagnetic ordering elsewhere. In the latter case, the system exhibits a first-order QPT in correspondence to the ferromagnetic point Δf=−1, and a continuous QPT of the Kosterlitz–Thouless type at the antiferromagnetic point Δaf=1.

Figure 3 displays the small-temperature limit of for the two models above, numerically computed by exploiting expression (21) in the ‘Methods’ section. We first observe that, as expected, for all the values of h and Δ, the LQTS monotonically increases with increasing the number of contiguous spins belonging to the measured subsystem . More interestingly, we find that even when reduces to two or three sites, its thermal behaviour qualitatively reproduces the same features of the global system (represented in both models by the uppermost curve). In particular, even at finite temperatures and for systems composed of 12 sites, the LQTS is sensitive to the presence of critical regions where quantum fluctuations overwhelm thermal ones. The reminiscence of QPTs at finite temperatures has been already discussed via a quantum-metrology approach, through the analysis of the Bures metric tensor in the parameter space associated with the temperature and the external parameters41. The diagonal element of such tensor referring to infinitesimal variations in temperature, corresponds to the thermal susceptibility of the whole system. The latter quantity has been recently studied for the XY model28, showing its sensitivity to critical points of Ising universality class.

Figure 3: The LQTS in the Ising and the Heisenberg models at low temperature.
figure 3

We numerically computed the LQTS of equation (1) in the low-temperature limit for a chain with L=12 sites in the following two cases: (a) the Ising model as a function of the adimensional transverse field h; (b) the Heisenberg XXZ chain as a function of the anisotropy Δ. The uppermost (red) curve corresponds to the global quantum thermal susceptibility, that is the heat capacity. The other curves stand for different sizes of the measured subsystem of ( increases along the direction of the arrow). The inset in b magnifies the data around Δ=−1. In the XXZ model, the LQTS with =1 can be proved to rigorously vanish. The inverse temperature has been fixed in both cases at β=9.

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In the low-temperature regime, such global sensitivity can be understood within the Landau–Zener (LZ) formalism42. This consists of a two-level system, whose energy gap ΔE varies with respect to an external control parameter Γ, and presents a minimum ΔEmin in correspondence to some specific value Γc. Conversely, the global heat capacity (3) may exhibit a maximum or a local minimum at Γc, according to whether ΔEmin is greater or lower than the value of ΔE* at which the expression is maximum in ΔE, respectively. Indeed it can be shown that 〈ΔH2〉β for a two-level system exhibits a non-monotonic behaviour as a function of ΔE, at fixed β (see the ‘Methods’ section). Quite recently, an analogous mechanism has also been pointed out for the global heat capacity in the Lipkin–Meshkov–Glick model27. The LZ formalism represents a simplified picture of the mechanism underlying QPTs in many-body systems. However, by definition, the temperature triggers the level statistics and the equilibrium properties of physical systems. Therefore, both the heat capacity of the global system5,6 and the LQTS of its subsystems are expected to be extremely sensitive to the presence of critical regions in the Hamiltonian parameter space.

In the Supplementary Note 1, we performed a finite-size scaling analysis of as a function of the size of the measured subsystem. For slightly interacting systems, one expects the LQTS to be well approximated by the heat capacity of (at least when this subsystem is large enough). The latter quantity should scale linearly with its size . This is indeed the case for the Ising model (9), where a direct calculation of 〈ΔH2〉β can be easily performed28. Our data for the scaling of the stationary points of close to QPTs suggest that significant deviations from a linear growth with are present (see the Supplementary Fig. 1). This indicates that correlations cannot be neglected for the sizes and the systems considered here. A similar behaviour has been detected for the XXZ model, as shown in the Supplementary Fig. 2.

Discussion

We have proposed a theoretical approach to temperature locality based on quantum estimation theory. Our method deals with the construction of the local quantum thermal susceptibility, which operationally highlights the degree at which the thermal equilibrium of the global system is perceived locally, avoiding any additional hypothesis on the local structure of the system. This functional corresponds to the highest achievable accuracy up to which it is possible to recover the system temperature at thermal equilibrium via local measurements. Let us remark that, even if in principle, the Cramér–Rao bound is achievable, from a practical perspective it represents a quite demanding scenario, as it requires the precise knowledge of the Hamiltonian, the possibility to identify and perform the optimal measurements on its subsystems, and eventually a large number of copies of the system. However, in this manuscript, we have adopted a more theoretical perspective, and focused on the geometrical structure of the quantum statistical model underlying local thermalization.

In the low-temperature regime, our functional admits an interpretation as a measure of the local state distinguishability between the spaces spanned by the Hamiltonian ground state and its first energy levels. Furthermore, we established a complementarity relation between the highest achievable accuracy in the local estimation of temperature and of a global phase, by showing that the corresponding accuracies associated with complementarity subsystems sum up to heat capacity of the global system. Finally, we considered two prototypical many-body systems featuring quantum phase transitions, and studied their thermal response at low temperatures. On one hand, we found that optimal measurements on local systems provide reliable predictions on the global heat capacity. On the other hand, our functional is sensitive to the presence of critical regions, even though the total system may reduce to a dozen of components and the measured subsystem to one or two sites.

Let us remark that most of the results presented herewith do not refer to any specific choice of the interaction Hamiltonian, between and . As an interesting implementation of our scheme, we foresee the case of non-thermalizing interactions43,44, whose potentialities for precision thermometry have been already unveiled.

We conclude by noticing that, while in this article we focused on temperature, the presented approach can be extended to other thermodynamic variables (like entropy, pressure, chemical potential and so on), or functionals45. In the latter case, quantum-estimation-based strategies, not explicitly referred to a specific quantum observable, but rather bearing the geometrical traits of the Hilbert space associated to the explored systems, may provide an effective route.

Methods

Derivation of useful analytical expressions for the LQTS

Let us recall the definition of the LQTS for a given subsystem of a global system at thermal equilibrium:

where is the fidelity between two generic quantum states ρ and σ. According to the Uhlmann’s theorem38, we can compute as

where the maximization involves all the possible purifications and of and , respectively through an ancillary system a. A convenient choice is to set a=, with isomorphic to . We then observe that, by construction, the vector |ρβ〉 of equation (4), besides being a purification of ρβ, is also a particular purification of . We can now express the most generic purification of the latter as

where V belongs to the set of unitary transformations on a, where represents the identity operator on a given system , and where in the last equality we introduced the vector , being the eigenvectors of H. We can thus write the fidelity (12) as

Since we are interested in the small-ɛ limit, without loss of generality we set V=exp(i ɛ Ω), with Ω being an Hermitian operator on the ancillary system a. It comes out that, up to corrections of order , the LQTS reads

where we have defined . By differentiating the trace with respect to Ω, we determine the minimization condition for it, yielding

with and , H′ being the analogous of H which acts on (by construction H |ρβ〉=H′|ρβ〉). Equation (16) explicitly implies that Ω does not depend on ω, which, without loss of generality, can be set to zero. Moreover, it enables to rewrite the LQTS in equation (15) as

The solution of the operatorial equation (16) can be found by applying Lemma 1 presented at the end of this section, yielding

with Ω0 being an operator which anti-commutes with Ω, being the Moore–Penrose pseudoinverse of to the power n, R being the projector on kernel of , P=a−R being its complementary counterpart, and with h.c. denoting the hermitian conjugate term. By substituting this expression in equation (17), we finally get

where =∑iλi|ei〉〈ei| is the spectral decomposition of , sharing the same spectrum with . The expression above holds for both invertible and not invertible . To the latter scenario belongs the case in which H=+, where one can easily prove that the LQTS reduces to the variance of the local Hamiltonian , that is, (notice that the non-zero eigenvalues of are which correspond to , being , and the purification of through the ancillary system ). The expression above can also be rewritten as

which can be cast into equation (5) by simply exploiting the fact that the system is symmetric with respect to the exchange of with .

It is finally useful to observe that LQTS can be also expressed in terms of the eigenvectors of , =∑iλi|gi〉〈gi| as:

where we have used the Schmidt decomposition of |ρβ〉, with respect to bipartition ,

In particular, expression (21) can be exploited to numerically compute the LQTS, for instance when dealing with quantum many-body systems (see Fig. 3 and the discussion in the Supplementary Note 1).

Lemma 1: For any assigned operators X, Y satisfying the equation

the following solution holds

where is the Moore Penrose pseudoinverse of X, R is the projector on kernel of X, P=−R ( indicates the identity matrix) and W0 is a generic operator which anti-commutes with Y (see also ref. 46). Furthermore if X and Y are Hermitian, equation (23) admits solutions which are Hermitian too: the latter can be expressed as

where now W0 is an arbitrary Hermitian operator which anti-commutes with Y.

Proof: Since (23) is a linear equation, a generic solution can be expressed as the sum of a particular solution plus a solution W0 of the associated homogeneous equation, that is, an operator which anti-commute with X,

A particular solution W of equation (23) can be always decomposed as

Notice that by definition, RX=XR=O, where O identifies the null operator. Multiplying (23) on both sides by R, one gets the condition RYR=O. The operator W, solution of equation (23), is defined up to its projection on the kernel subspace, that is

Therefore, without loss of generality we can set

Multiplying equation (23) by on the right side and repeating the same operation on the left side, we get:

On the other hand, PWP satisfies the relation

This equation can be solved recursively in PWP and gives

thus concluding the first part of the proof. The second part of the proof follows simply by observing that, if X and Y are Hermitian, and if W solves equation (23), then also its adjoint counterpart does. Therefore, for each solution W of the problem, we can construct an Hermitian one by simply taking (W+W†)/2.

Second-order term corrections to LQTS

In the low-temperature regime (β→∞), we have computed the second-order correction term to the LQTS, that is of in equation (1), and found:

with Ek≥E1 and where the series in n is meant to converge to 1/2 when , that is, . To vanish, this second-order correction term requires a stronger condition with respect to one necessary to nullify the first-order term in the LQTS, equation (7). It is given by , and corresponds to the requirement that the system ground state must be locally indistinguishable from the first excited level.

Heat capacity in the two-level LZ scheme

Here we discuss the simplified case in which only the ground state (with energy E0) and the first excited level (with energy E1) of the global system Hamiltonian H play a role. In particular, we are interested in addressing a situation where the ground-state energy gap ΔE≡E1−E0 may become very small, as a function of some external control parameter Γ (for example, the magnetic field or the system anisotropy). A sketch is depicted in Fig. 4, and refers to the so-called LZ model42. This resembles the usual scenario when a given many-body system is adiabatically driven, at zero temperature, across a quantum phase transition point.

Figure 4: The two-level Landau–Zener model.
figure 4

A sketch of the behaviour of the two eigenenergies (E1, E2) as a function of some control parameter Γ. The gap ΔE=E2−E1 displays a pronounced minimum in correspondence of a given Γc value.

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In correspondence of some critical value Γc, the gap is minimum. For a typical quantum many-body system, such minimum value ΔEmin tends to close at the thermodynamic limit and a quantum phase transition occurs (notice that Γc may depend on the system size). Hereafter, without loss of generality, we will assume E0=0 and take E1=ΔE so that the system heat capacity (3) reduces to:

Here n0 and n1 are the degeneracy indexes associated to the levels E0 and E1, respectively. Notice that is always non-negative and exhibits a non-monotonic behaviour as a function of ΔE, at fixed β. Indeed it is immediate to see that in both limits ΔE→0 and ΔE→+∞. For fixed β, n0 and n1, the heat capacity displays a maximum in correspondence of the solution of the transcendental equation

In particular, for n0=n1=1, the latter relation is fulfilled for , while for n0=2, n1=1, it is fulfilled for .

It turns out that the behaviour of the heat capacity as a function of increasing Γ in a two-level LZ scheme depends on the relative sizes of ΔE* and ΔEmin, as pictorially shown in Fig. 5: (a) if ΔEmin>ΔE*, then will exhibit a maximum in correspondence of Γc; (b) if ΔEmin<ΔE*, a maximum at corresponding to ΔE=ΔE* will appear, followed by a local minimum at Γc and eventually by another maximum at where the former condition occurs again. Since ΔE* is a function of β, and ΔEmin depends on the system size, the point of minimum gap can be signalled by a maximum or by a local minimum depending on the way the two limits L→+∞ (thermodynamic limit) and β→+∞ (zero-temperature limit) are performed. In the Supplementary Note 2, we explicitly address the two many-body Hamiltonians considered in the last subsection of the ‘Results’ section, namely the Ising and the XXZ model (see the Supplementary Figs 3 and 4). Here in particular, we discussed the possible emergence of corrections to the low-temperature energy variance (34) when one takes into account the presence of the low-lying energy levels beyond the first excited one.

Figure 5: Heat capacity in the Landau–Zener model.
figure 5

The emergence of extremal points in the behaviour of as a function of increasing control parameter Γ is associated to the relative size of ΔEmin with respect to ΔE*. The red arrows denote the changing of ΔE during a typical Landau–Zener protocol. One realizes that, if ΔEmin>ΔE* one peak will appear (left), whereas if ΔEmin<ΔE* two peaks will appear (right).

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Data availability

The data that support the findings of this study are available from the corresponding author upon request.

Additional information

How to cite this article: De Pasquale, A. et al. Local quantum thermal susceptibility. Nat. Commun. 7:12782 doi: 10.1038/ncomms12782 (2016).