Abstract
Thermodynamics relies on the possibility to describe systems composed of a large number of constituents in terms of few macroscopic variables. Its foundations are rooted into the paradigm of statistical mechanics, where thermal properties originate from averaging procedures which smoothen out local details. While undoubtedly successful, elegant and formally correct, this approach carries over an operational problem, namely determining the precision at which such variables are inferred, when technical/practical limitations restrict our capabilities to local probing. Here we introduce the local quantum thermal susceptibility, a quantifier for the best achievable accuracy for temperature estimation via local measurements. Our method relies on basic concepts of quantum estimation theory, providing an operative strategy to address the local thermal response of arbitrary quantum systems at equilibrium. At low temperatures, it highlights the local distinguishability of the ground state from the excited submanifolds, thus providing a method to locate quantum phase transitions.
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 accuracies^{1}. 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 theory^{2,3,4}: The precision is related to the heat capacity of the system^{5,6}.
In view of the groundbreaking potentialities offered by presentday nanotechnologies^{7,8,9,10,11,12} and the need to control the temperature at the nanoscale, it is highly relevant to question whether the heat capacity is still the relevant (fundamental) precision limit to smallscale thermometry. The extensivity of the heat capacity is a consequence of the growing volumetosurface ratio with the size^{13}. However, at a microscopic level such construction may present some limitations^{14,15}. Moreover a series of theoretical efforts recently concentrated on a selfconsistent generalization of the classical thermodynamics to smallscale physics, where quantum effects become predominant^{16,17,18,19,20,21,22}. 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 correlations^{23,24,25,26,27,28}. 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 itself^{29,30,31}. Using a quantum informationoriented point of view, this phenomenon has also been highlighted in Gaussian fermionic and bosonic states, by exploiting quantum fidelity as the figure of merit^{32,33}. Furthermore, the significant role played by quantum correlations has been recently discussed with specific attention to spin and fermoniclattice systems with shortrange interactions^{34}.
In this paper, we propose a quantummetrology 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).
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 lowtemperature 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/k_{B}T is the inverse temperature of the system, k_{B} the Boltzmann constant, and {E_{i}} 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 metrology^{35} 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 rootmeansquare error of a generic local estimation strategy, where T^{est} 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 capacity^{5,6} (note that, rigorously speaking, the LQTS quantifies the sensitivity of the system to its inverse temperature β; the corresponding susceptibility to T=1/(k_{B}β) 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 theorem^{38} (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 ρ_{β} defined as
where is the spectral decomposition of the system Hamiltonian. It can then be proved that
where {e_{j}〉} are the eigenvectors of the reduced density matrix living on , obtained by taking the partial trace of ρ_{β}〉 with respect to the ancillary system , while {λ_{j}} 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 hightemperature 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 lowtemperature 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 E_{0}=0) and of the lowest excited levels with energy E_{i} bounded by twice the energy of the first excited level, E_{i}≤2E_{1}. The degeneracy of each considered energy eigenstate is denoted by n_{i}. From equation (5), it then follows that up to first order in the parameter we get
Here , where Π_{i} is the normalized projector on the degenerate subspace of energy E_{i}. Moreover, is the span of the local subspace associated to the ground state, that is, with and being the number of nonzero eigenvalues (p_{j}>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 zerotemperature 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 finitevalued 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 groundstate 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 〈ΔH^{2}〉_{β} of the global system (3).
Local thermometry in manybody systems
We have tested the behaviour of our functional on two models of quantum spin chains, with a lowenergy physics characterized by the emergence of quantum phase transitions (QPTs) belonging to various universality classes^{40}.
Specifically, we consider the quantum spin1/2 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 ith 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 h_{c}=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 firstorder QPT in correspondence to the ferromagnetic point Δ_{f}=−1, and a continuous QPT of the Kosterlitz–Thouless type at the antiferromagnetic point Δ_{af}=1.
Figure 3 displays the smalltemperature 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 quantummetrology approach, through the analysis of the Bures metric tensor in the parameter space associated with the temperature and the external parameters^{41}. 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 model^{28}, showing its sensitivity to critical points of Ising universality class.
In the lowtemperature regime, such global sensitivity can be understood within the Landau–Zener (LZ) formalism^{42}. This consists of a twolevel system, whose energy gap ΔE varies with respect to an external control parameter Γ, and presents a minimum ΔE_{min} in correspondence to some specific value Γ_{c}. Conversely, the global heat capacity (3) may exhibit a maximum or a local minimum at Γ_{c}, according to whether ΔE_{min} is greater or lower than the value of ΔE* at which the expression is maximum in ΔE, respectively. Indeed it can be shown that 〈ΔH^{2}〉_{β} for a twolevel system exhibits a nonmonotonic 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 model^{27}. The LZ formalism represents a simplified picture of the mechanism underlying QPTs in manybody 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 system^{5,6} 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 finitesize 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 〈ΔH^{2}〉_{β} can be easily performed^{28}. 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 lowtemperature 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 manybody 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 nonthermalizing interactions^{43,44}, 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 functionals^{45}. In the latter case, quantumestimationbased 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 theorem^{38}, 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 ρ_{β}〉 of equation (4), besides being a purification of ρ_{β}, 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 smallɛ 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 ρ_{β}〉=H′ρ_{β}〉). 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 Ω_{0} being an operator which anticommutes 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 =∑_{i}λ_{i}e_{i}〉〈e_{i} 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 nonzero 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 , =∑_{i}λ_{i}g_{i}〉〈g_{i} as:
where we have used the Schmidt decomposition of ρ_{β}〉, with respect to bipartition ,
In particular, expression (21) can be exploited to numerically compute the LQTS, for instance when dealing with quantum manybody 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 W_{0} is a generic operator which anticommutes 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 W_{0} is an arbitrary Hermitian operator which anticommutes 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 W_{0} of the associated homogeneous equation, that is, an operator which anticommute 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.
Secondorder term corrections to LQTS
In the lowtemperature regime (β→∞), we have computed the secondorder correction term to the LQTS, that is of in equation (1), and found:
with E_{k}≥E_{1} and where the series in n is meant to converge to 1/2 when , that is, . To vanish, this secondorder correction term requires a stronger condition with respect to one necessary to nullify the firstorder 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 twolevel LZ scheme
Here we discuss the simplified case in which only the ground state (with energy E_{0}) and the first excited level (with energy E_{1}) of the global system Hamiltonian H play a role. In particular, we are interested in addressing a situation where the groundstate energy gap ΔE≡E_{1}−E_{0} 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 socalled LZ model^{42}. This resembles the usual scenario when a given manybody system is adiabatically driven, at zero temperature, across a quantum phase transition point.
In correspondence of some critical value Γ_{c}, the gap is minimum. For a typical quantum manybody system, such minimum value ΔE_{min} tends to close at the thermodynamic limit and a quantum phase transition occurs (notice that Γ_{c} may depend on the system size). Hereafter, without loss of generality, we will assume E_{0}=0 and take E_{1}=ΔE so that the system heat capacity (3) reduces to:
Here n_{0} and n_{1} are the degeneracy indexes associated to the levels E_{0} and E_{1}, respectively. Notice that is always nonnegative and exhibits a nonmonotonic 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, n_{0} and n_{1}, the heat capacity displays a maximum in correspondence of the solution of the transcendental equation
In particular, for n_{0}=n_{1}=1, the latter relation is fulfilled for , while for n_{0}=2, n_{1}=1, it is fulfilled for .
It turns out that the behaviour of the heat capacity as a function of increasing Γ in a twolevel LZ scheme depends on the relative sizes of ΔE* and ΔE_{min}, as pictorially shown in Fig. 5: (a) if ΔE_{min}>ΔE*, then will exhibit a maximum in correspondence of Γ_{c}; (b) if ΔE_{min}<ΔE*, a maximum at corresponding to ΔE=ΔE* will appear, followed by a local minimum at Γ_{c} and eventually by another maximum at where the former condition occurs again. Since ΔE* is a function of β, and ΔE_{min} 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→+∞ (zerotemperature limit) are performed. In the Supplementary Note 2, we explicitly address the two manybody 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 lowtemperature energy variance (34) when one takes into account the presence of the lowlying energy levels beyond the first excited one.
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).
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Acknowledgements
We thank G. Benenti and G. De Chiara for useful discussions. This work has been supported by MIUR through FIRB Projects RBFR12NLNA and PRIN ‘Collective quantum phenomena: from strongly correlated systems to quantum simulators’, by the EU Collaborative Project TherMiQ (Grant agreement 618074), by the EU project COST Action MP1209 ‘Thermodynamics in the quantum regime’, by EUQUIC, EUIPSIQS and by CRP Award—QSYNC.
Author information
Affiliations
Contributions
All the authors conceived the work, agreed on the approach to pursue, analysed and discussed the results; A.D.P. performed the analytical calculations; D.R. performed the numerical calculations; V.G. and R.F. supervised the work.
Corresponding author
Correspondence to Antonella De Pasquale.
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Supplementary Figures 14 and Supplementary Notes 12. (PDF 416 kb)
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De Pasquale, A., Rossini, D., Fazio, R. et al. Local quantum thermal susceptibility. Nat Commun 7, 12782 (2016). https://doi.org/10.1038/ncomms12782
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