Multipartite-Entanglement Tomography of a Quantum Simulator

Multipartite entanglement tomography, namely the quantum Fisher information (QFI) calculated with respect to different collective operators, allows to fully characterize the phase diagram of the quantum Ising chain in a transverse field with variable-range coupling. In particular, it recognizes the phase stemming from long-range antiferromagnetic coupling, a capability also shared by the spin squeezing. Furthermore, the QFI locates the quantum critical points, both with vanishing and nonvanishing mass gap. In this case, we also relate the finite-size power-law exponent of the QFI to the critical exponents of the model, finding a signal for the breakdown of conformal invariance in the deep long-range regime. Finally, the effect of a finite temperature on the multipartite entanglement, and ultimately on the phase stability, is considered. In light of the current realizations of the model with trapped ions and of the potential measurability of the QFI, our approach yields a promising strategy to probe long-range physics in controllable quantum systems.


Introduction
The experimental realization of quantum simulators [1,2] has made a significant progress in the recent years [3][4][5][6][7][8][9][10]: systems of trapped ions [11,12], ultracold atoms and molecules [13][14][15] and superconducting circuits [16] are currently able to simulate important models of quantum physics. A notable example is the long-range (LR) quantum Ising chain in a transverse field, which has been realized with up to ∼50 spins [8,9]. The experiments are rapidly approaching the point where the outcomes cannot be efficiently computed on a classical machine. We thus need methods for the reliable benchmarking of quantum simulators [17,18]. These might be given, for instance, by detecting specific properties of the ground state of the system that can be accessed without full state tomography.
The measurement of a local order parameter is a standard example of such benchmarking: it signals the onset of a dominant order in the system when tuning a control parameter that rules the competition between non-commuting terms in a many-body Hamiltonian. It has thus been used to detect a variety of quantum phase transitions (QPTs), in analogy to the detection of thermal phase transitions. This approach, however, provides no information about quantum correlations in the considered system. Moreover, a local order parameter cannot distinguish between topologically trivial and nontrivial phases [19].
Another approach, which has emerged in the last decades [20][21][22], is to characterize the system via the bipartite entanglement (BE) properties of the ground state. Entanglement between two parts of a many-body system is a pivotal figure of merit and it is analyzed typically via the Von Neumann entropy [20][21][22][23][24][25] or the entanglement spectrum [26][27][28][29][30]. An alternative approach to BE is the study of the two-body reduced density matrix [31][32][33], also quoted as pairwise entanglement. BE has attracted large attention because it can be efficiently computed [20] and it is a resource required for classical simulations of many-body systems with numerical methods [22,34]. It has been shown that in several short-range (SR) one-dimensional models BE diverges logarithmically with the system size at criticality, whereas it does not scale in any gapped phase [20][21][22]. Instead, for LR models such a violation of the area law is found also in gapped phases [35][36][37][38]. Yet, not only it is difficult to experimentally extract BE in large systems [18] but, furthermore, a logarithmic scaling might be hard to distinguish from a constant behavior in systems of relatively small size.
Here we consider a further possible approach to benchmark a quantum simulator. This is based on the susceptibility of the ground state gs y ñ | to unitary transformations e O if -ˆg enerated by some operator Ô and parametrized by the real value f, as given by the quantum Fisher information (QFI) [39][40][41]. (ˆ) for pure states. Importantly, the QFI is a witness of multipartite entanglement (ME) [43,44] for local operators Ô , as in the case of this manuscript, detects k-partite entanglement among N spins [42,45,46]. In particular, ME is able to capture the richness of multiparticle correlations of many-body states beyond BE. The QFI of a quantum states calculated with respect to different operators Ô provides a 'multipartite-entanglement tomography' that gives information not only about ME, but also about global properties of the correlation functions [47][48][49][50][51]. The QFI is thus able to recognize different phases and QPTs of a many-body model.
In the present paper, we illustrate these ideas for the Ising chain with variable-range interaction in a transverse field. We show how multipartite-entanglement tomography based on the QFI can give information about-and distinguish-the paramagnetic (PM), ferromagnetic (FM) and antiferromagnetic (AFM) phases of the model. For ordered phases, the optimal choice of operator Ô is given by the order parameter of the transition, characterized by diverging fluctuations, and giving a Heisenberg scaling of the QFI, . For disordered phases there is an important difference between the SR and LR regimes: while in the SR case the QFI is extensive, , in the LR case the QFI is superextensive, . This scaling law is directly related to the presence of power-law decaying correlation functions, where Ô here is a suitable collective operator-generally different from the order parameter-that maximizes the QFI in this regime. Interestingly, the LR disordered phase is also recognized by the spin-squeezing parameter. We discuss the change of scaling of the QFI at the critical points when interactions change from SR to LR, suggesting the breakdown of conformal invariance and capturing the meanfield limit of the model. We finally extend our analysis to finite temperature [52] and show that the large entanglement found in the ground state of the LR disordered phase is robust against temperature being protected by a finite energy gap. Our results can be readily tested in current experimental systems. In particular, the finite-size power-law scaling of the QFI is thus able-even at experimentally available sizes (N 50 » )-to detect the appearance of LR phases and to characterize QPTs beyond nearest-neighbor interaction. It is indeed worth pointing out that the QFI can be experimentally addressed: it is related to dynamical susceptibilities [49] and Loschmidt echo [53], and lower bounds can be obtained from the variation of statistical distributions of a measured observable [45,54], squeezing parameters [55][56][57][58], quantum coherence [59] and fidelity measures [60].

The model
We study the one-dimensional quantum Ising chain in a transverse field, with variable-range interaction and open boundary conditions. The corresponding Hamiltonian is where Nis the number of spins (in the following we assume even N), i n ŝ ( ) is the Pauli matrix for the ith spin (i=1, 2, K,N) along the direction n, and 0  > sets the energy scale. The parameter θä[−π/2, π/2] rules the competition between the transverse external field of magnitude cos  q and the spin-spin interaction of strength sin  q. The decay power 0  a specifies the range of the spin-spin interaction, which is FM for θ<0 and AFM for θ>0. For a  ¥, equation (1) reduces to the well-known quantum Ising model with nearest-neighbor interaction [61,62].
For α=0, equation (1) corresponds to a chain with infinite-range interaction, formally equivalent to the Lipkin-Meshkov-Glick model [63]. For finite values of α, equation (1) is a paradigmatic model to study the physical effects induced by LR interaction. Indeed, various theoretical works pointed out that this model displays many interesting and peculiar features [64,65], ultimately connected to the effective violation of locality [66,67], including the semialgebraic decay for correlations in gapped regimes [35,37,68], the related violation of the area law for the Von Neumann entropy [35,36] and anomalous distribution for the entanglement spectrum [35,66], and the breakdown of conformal invariance at criticality [69,70]. Moreover, new phases displaying these features, but not belonging to the classification schemes for SR systems, have been identified theoretically in this model [35,37,71]. The interesting physics associated to LR interaction concerns also fermionic lattice systems, characterized by nontrivial topological invariants [36-38, 69, 72-77]. Notably, for these systems, BE is known to characterize only partially the LR regimes, not being able to distinguish in general the different LR phases [38,74], while ME appears to be more indicative [50][51][52].

Phase diagram
The phase diagram of the model shown in figure 1 is determined by the competition between the two noncommuting terms in equation (1): the longitudinal exchange interaction and the transverse magnetic field.

Critical lines
For any fixed α, the Ising chain hosts two QPTs driven by the control parameterθ. Each QPT separates a magnetically disordered phase from an ordered one, according to the spontaneous symmetry breaking of the spin-flip Z 2 invariance of the Hamiltonian (1) in the thermodynamic limit. This behavior results in two lines of critical points 0 c  q a -( ) and 0 c q a > + ( ) , where transitions from a PM phase to FM and AFM phases take place, respectively. For α>0, both the critical lines signal second-order QPTs. The model is analytically solvable in two limit cases: for nearest-neighbor interaction (a = ¥) within a Jordan-Wigner transformation [78]; and for infinite-range interaction (α=0), within a Bethe ansatz [79,80] and in the thermodynamic limit [81]. In the case a = ¥ the exact location of the critical points is well known [62]: . For α=0, instead, the fully-connected chain has a second-order FM transition at θ=0 [82,83] and a first-order AFM transition at θ=π/2 [84]. For any finite value of α, the emerging QPTs at finite N are signaled by a minimum of the mass gap Δ N (α, θ), as a function of θ. In order to locate the transitions, we determine min , For θ<0, the LR interaction enforces the FM order, even at strong magnetic fields: at fixed α, the PM phase progressively shrinks when increasing N, and it disappears in the large-N limit if  and θ<0 (hatched region), the thermodynamic limit is not defined. The solid black lines, separating the ordered phases from the disordered one, mark a vanishing mass gap in the thermodynamic limit, they interpolate the numerical data c q -(blue dots) and c q + (orange dots). Triangles are known results in the literature for the FM transition (blue triangles, [85]) and AFM transition (red triangles, [35]), see also [86]. The red lines show the position of the FM critical points as calculated by a perturbative expansion at the first order (solid) and at the second order (dashed) in θ→0. The horizontal dashed line denotes a massive critical line at α=1, separating the short-range paramagnetic (PM) phase from the long-range (LRPM) one.
For θ>0, instead, the LR interaction strongly frustrates the AFM order: frustration entails a preference for the system to endure in the disordered phase, even at low magnetic fields. Consequently, the AFM critical point shifts towards larger values of θ as α decreases. In particular, the fully-connected chain α=0 becomes completely frustrated and the corresponding AFM phase has a vanishing extension, reducing to the single point 0 2 c q p = + ( ) . Finally, we notice that many studies, based on different numerical methods, have investigated the AFM and FM critical lines [35,37,[85][86][87][88][89][90][91]. Our numerical results agree well with the literature. In particular, in figure 1, for comparison, we report the the location of the PM-to-FM QPT based on scaled exact diagonalization in the FM regime (for α1) [85], and the location of the PM-to-AFM QPT based on maxima of the half-chain Von Neumann entropy (for α0.5) [35].

Quantum phases
The characterization of the different phases bounded by c q a -( )and c q a + ( )is primarily done in terms of suitable order parameters that recognize the onset of the dominant FM and AFM order. We can distinguish three phases, see figure 1.

FM phase
For sufficiently strong FM interaction, 2 c  p q q a -< -( ), the system exhibits an ordered FM phase, where the Z 2 symmetry is spontaneously broken in the limit N  ¥. The order is detected by the longitudinal magnetization J z z gs gs perturbation is not added, in the limit θ→π/2 the ground state is the Greenberger-Horne-Zeilinger (GHZ) for all values of α, while in the N  ¥ limit this state becomes degenerate with 2 . Here and in the following, n ñ | and n ñ | denote the eigenstates of n ŝ .
The FM phase for 1  a (hatched region in figure 1) deserves a comment since, here, the energy is superextensive. In this case, the thermodynamic limit is not well definite. Yet, we do not encounter special difficulties in characterizing this regime within our numerical studies at finite N. In particular, the ground state for θ=−π/2 is the same for every value of 0  a . Furthermore, as discussed below, the QFI is superextensive in the FM phase (above and below α=1) with the same scaling exponent.

AFM phase
acts as the order parameter, with In particular, at θ→π/2, the ground state of a finite-size chain is the Néel state for any α>0. For α=0, instead, each spin is coupled with all the others via the same strength, regardless of their mutual distance: the ground state at θ=π/2 becomes the symmetric Dicke state (often also indicated as twin-Fock state) given by the equally weighted superposition of all possible permutational symmetric combinations of N/2 spin-up and N/2 spin-down particles (for an even number of spins). It should be noticed that for θ>0, the energy of the ground state is extensive for all values of 0  a , even for α=0 and θ=π/2. This fact allows for a proper definition of the quantum phases even at 1  a .

PM SR and LR phases
A disordered PM phase is displayed by the system for weak spin-spin interaction, both in the FM and in the AFM regime, c c q a q q a < < -+ ( ) ( ). The polarization provided by the transverse external magnetic field dominates over the spin-spin interaction and determines the structure of the ground state. In particular, at θ=0, the ground state is given by the coherent spin state 2 polarized along the −x direction by the magnetic field. In the following, we distinguish a PM SR phase from a LR one. This distinction is not based on an order parameter since the spin-flip Z 2 symmetry is preserved: 0 , in the full PM phase. Instead, for 0 c q q a < < + ( )and 1  a , a logarithmic violation of the area law for the Von Neumann entropy has been found in [35], and shown not to originate from finite-size effects. The analogy with critical gapless systems motivated the introduction of an effective central charge [35], that has also been used as a tool for probing the phase diagram [37]. Finally, by means of a Jordan-Wigner transformation, the LR Ising chain can be mapped into a LR interacting fermionic chain [37], that, only in the PM regime at α1, turns out to be characterized by the appearance of massive edge modes [37], similar to the ones found in the LR Kitaev chain [36]. All these peculiar features induce to conjecture the existence of a new PM phase at α1 [37], bounded from above by a transition with nonvanishing mass gap at α≈1. This PM gapped phase, still preserving the Z 2 symmetry, will be quoted here and in the following as long-range paramagnetic (LRPM) phase, to distinguish it from the ordinary PM phase occurring at α1.
In spite of the above theoretical clues, no valid observable for the experimental detection of the conjectured LRPM phase has been identified so far, mainly because BE is challenging to be observed in extended systems (see e.g. [14, 92]). A similar open question holds for the nature of the AFM transitions at c q a + ( ): from the scaling of the Von Neumann entropy, the breakdown of conformal invariance induced by the LR interaction has been suggested [37]. However, no detection criterion for observing the spontaneous breakdown of the conformal symmetry has been available so far to our knowledge. A promising method based on the inspection of the finitesize scaling of the ground-state energy density was suggested [36], but its reliable use is currently forbidden by the limited size in experimental realizations of the LR Ising chain [8,9].

Multipartite-entanglement phase diagram
In order to characterize the phase diagram of the Hamiltonian (1) beyond the analysis of order parameters and BE, we study here the QFI and its lower bound given by the spin-squeezing parameter.
The QFI of a generic state p k k k k r = å ñá | |, relative to an arbitrary operator Ô , is given by (see the recent reviews [39][40][41] and references therein) [ˆˆ] ( ): the QFI thus quantifies the susceptibility of r to unitary parametric transformations. For pure states yñ | , equation (2) reduces to the variance : the QFI thus quantifies the 'spread' of the state over the eigenstates of Ô . Importantly, the QFI is a witness of ME [43,44]. Specifically, for collective operators O o i i = åˆ(i labeling the lattice sites) the violation of the inequality are genuinely N-partite entangled, f Q =N being the ultimate (Heisenberg) bound [42][43][44].
Here, we numerically study the QFI of the ground state gs y ñ  [49,50]: It should be noticed that different operators yield different values of the QFI. The calculation of the QFI for different operators provides a 'tomographic survey of ME' for the given quantum states [in particular of the ground state of the Hamiltonian (1)] that is able, as illustrated below, to fully characterize the phase diagram. We optimize the QFI by calculating the optimal eigenvalue of the 6×6 covariance matrix The optimal QFI is obtained by calculating the maximum eigenvalue of this matrix. We find 6 ME is quantified by the number of particles in the largest nonseparable subset [41,93]. A pure state of N particles is k-separable (also indicated as k-producible in the literature) if it can be written as ...
A mixed state is k-separable if it can be written as a mixture of k-separable pure states. A state that is k-separable but not (k−1)-separable is called k-particle entangled: it contains at least one state of k particles that does not factorize.
that, depending on the values of the parameters θ and α, see figure 2, the optimal operators are J ŷ , J ẑ or J z st( ) . We thus restrict our discussion below on these operators. Notice that J ẑ is the order parameter of the PM-to-FM QPT and J z is the order parameter of the PM-to-AFM transition, while J ŷ is not an order parameter. The J x operator is never optimal: in particular, we find f J , 0 at θ=0, since in this case the ground state is eigenstate of J x , and for θ=±π/2. We also analyze the Wineland spin-squeezing (WSS) parameter [41,55,56]  x < . This inequality is also a criterion for entanglement [94] and has been extended to witness ME [95]. The inverse of the spin-squeezing parameter (7) is a lower bound of the QFI [39][40][41]: for any state r we have where n ¢^i s a direction orthogonal to both n ⊥ and n P . Notice that for pure states the (ˆ) follows from the Heisenberg uncertainty relation. The spin squeezing is also related to the correlation function of collective spin operators and, for finite J n á ñ  , has the scaling properties of J n 2 D(ˆ) .
The investigation of ME in the ground state of the Ising chains, as witnessed by the QFI and the WSS, has been limited so far to the two extreme cases of nearest-neighbor a = ¥ [48,49,96] and infinite-range 0 a = interaction [47,96]. Several works have analyzed the QFI and the WSS in the ground state of the bosonic Josephson junction, which formally corresponds to the fully connected Ising model restricted to the Hilbert subspace of states that are symmetric under particle exchange [41,49,97]: see [41,[98][99][100] for experimental investigations in Bose-Einstein condensates. Notice that the ground state of the Hamiltonian(1) for α=0 is indeed given by symmetric states.
In the following we provide a study of the model (1) in the full range 0   a ¥. We find that the QFI witnesses ME, f Q >1, for any 0  a and 0 q ¹ , when calculated with respect to the optimal operators reported in figure 2(a). Instead, on the line θ=0, the ground state is separable (for any α), and the QFI does not overcome the bound f Q =1. In the PM phase for θ>0, ME is also witnessed by the spin-squeezing parameter, as shown in figure 2(b). We point out that, while the figure is obtained at N=50, we have checked the qualitative stability of the phase diagram as N increases up to N≈200: the change of behavior around α=1 becomes sharper.  is thus characterized by a pronounced maximum at c q q a = -( ), see figure 3, that diverges in the thermodynamic limit.

AFM regime
The AFM regime is richer than the FM one. In the AFM phase, for   , see appendix C, in very good agreement with the numerical calculations for large N.
The super-extensiveness of the QFI directly stems from the power-law tail in the algebraic decay of the correlation functions C yy i j , ( ) [35,37]. Interestingly, the behavior of the QFI in the PM phase is fully captured by the spin-squeezing parameter: we find [|ˆ], as shown in figures 2 and 3(b).

4.3.
Crossing the massive line 1 a = >Here we focus on the PM phase 0 c  q q < in the AFM regime. As discussed above, and shown in figure 2, when crossing the massive line α=1, the scaling of the QFI with N changes from extensive (for α>1) to superextensive (for 1  a ). This result can be taken as a strong indication for a gapped QPT occurring at α=1 from a SR phase to a LR phase. This is a consequence of the change of behavior of the correlation function that is captured by the QFI.   N (panel (b)). Both panels suggest (despite the system size limited to N=100) a sharp change of behavior around α=1: while To gain more insight into the behavior of   (9) is revealed by the fidelity susceptibility, again obtained from a perturbative calculation, see appendix B. It should be noticed, however, that the condition of validity of perturbation theory, 1 N  q a  ( ) , sets an upper limit for the validity of equation (9): for fixed 1 q  , the finite-size scaling for α<1 is only guaranteed when N ) . Namely, from equation (9) we cannot claim a superextensive scaling in the thermodynamic limit.
Summing up, our numerical and analytical results allow to locate the boundary between the SR and LR regimes at α=1, also improving the precision of previous studies [37].

QFI along the massless critical lines
The QFI is also useful to probe directly conformal invariance along the critical lines c q  (α), see figure 6. Indeed, the QFI density f O ,

Scaling along the FM critical line
Along c q a -( ), the conformal scaling holds for α  3, see figure 6(a). This is consistent with the results of [64] where it has been shown that for 3  a the phase transition is in the universality class of the SR Ising transition, see also [85,91]. Actually, taking into account corrections beyond mean-field, the threshold for the onset of the SR universality class can be put at α=2.75 [89,106,107].
For α  3, we find that b a ( )decreases down to 0 b = at α=1. The scaling of the QFI density for 1α3 can be obtained from a Landau-Ginzburg approach. We introduce the effective action  t) represents, at the low-energy effective level, the order parameter J z z gs gs , g is a coupling constant, and x ¶ g denotes the fractional derivative. This action can be justified by a renormalization group procedure, suitably modified for LR models [70,75], and it is known to be dominant, in the range 1 3   a [85,103], with respect to the conformal Landau-Ginzburg action [101,104]  , and using the mean-field critical exponents η mf =3−α and z mf =(α−1)/2 calculated in [85], giving c . This prediction is expected to be accurate for α<5/3 [85]. For larger values of α, the deviation from equation (15) is probably a clue that a more careful renormalization group treatment is required, such to properly account the interplay between (12) and (13), as well as the interaction terms O x t , 4 f (| ( )| ), able to change the dynamical exponents. In figure 6(a), we also show the scaling of the spin-squeezing parameter xdoes not scale at the transition point. is analytically known and recovered by our numerics. In figure 6(b) we also report the scaling of the spin-squeezing parameter . For comparison, we report the results of [35] (red triangles). In both the panels, the error bars are mainly due to the uncertainty on the location of the critical points.

Scaling along the AFM critical line
In [35] the scaling of has been analyzed on the AFM critical line c q a + ( ), as a function of N and for values α  0.3. The coefficient Δ z has been found to depart from the SR value Δ z =0.25 for α≈2.25, then decreasing and reaching Δ z =0.2 at α≈0.5. The scaling coefficient Δ z can be directly related to the scaling of the QFI: The results for 1−2Δ z found in [35] are reported as triangles in the inset of figure 6(b) and compared to the values obtained in our numerical calculations (orange regions, corresponding to the data of the main panel). They agree with our results except in the range 1α2 where they are systematically above our findings. As a check of our numerical calculations, the inset of figure 6( for N=120 (purple circles). We see that the values of β(α) extracted in both cases are consistent with c b a + ( )obtained via the analysis of the QFI. We thus conclude that the slight discrepancy within our numerical results and those of [35] is most likely due to the uncertainty in locating the critical point c q a + ( ). It should be noticed however, that the interesting regime, where c b + is notably different from the SR scaling, is found for values of α  0.5, that were not analyzed in [35].
The results reported in figure 6 strongly suggest the breakdown of conformal invariance along c q a + ( ) and c q a -( )due to the LR interaction in (1), at small-enough α. The same breakdown has been previously inferred in [37], based on the scaling of the Von Neumann entropy. Oppositely to this quantity, the QFI density can be measured experimentally, yielding a direct way to probe the breakdown of conformal invariance in critical quantum systems.

ME at finite temperature
The calculation of the QFI can be straightforwardly extended to finite-temperature states, using equation (2) and assuming thermal equilibrium e Tre where T is the temperature and the Boltzmann k B is set to k B =1 [49,52,108]. The QFI is obtained here by full numerical diagonalization of the Hamiltonian (1) for fixed system sizes N 20  . The decay of the QFI density with T characterizes the robustness of ME in the various phases. In [52] it has been shown that [ˆˆ] ( ) where μ and ν indicate the degeneracy of the ground state and the first excited state, respectively, and Δ is the mass gap, namely the finite difference between the ground-state energy and the energy of the first excited state in the thermodynamic limit. Equation (16) identifies a temperature regime below a crossover temperature of the order of Δ, where the QFI density is at least constant, lower bounded by its zero-temperature limit. Figures 7(a)-(c) show the QFI density in the θ-α phase diagram at different temperatures (color scale, where T is expressed in unit of the magnetic coupling  in equation (1)), with the white regions corresponding to f 1 Q  . In the FM and AFM ordered phases, the zero-temperature QFI is much larger than in the PM phase, see figure 7(a). Yet, this large value is lost abruptly for arbitrary small temperature T (in the thermodynamic limit), reaching [52] is due to the presence of a spontaneous symmetry breaking of the spin-flip Z 2 symmetry at T=0, resulting in a quasidegeneracy of the ground state (that becomes an actual degeneracy in the thermodynamic limit only). In figure 7(b) and (c) we see that the QFI in the FM and AFM phases at finite temperature is not high enough to witness ME. The QFI density remains high only close to the critical lines and, most interestingly, in the LRPM phase. Indeed, in the LRPM phase the ground state is nondegenerate also in the thermodynamic limit, as well as in the PM phase, so that, according to equation (16), the superextensive ME witnessed by the QFI at T=0, for 0 c  q q < + and 1  a , survives up to temperatures T≈Δ. The typical decay of the QFI density in the LRPM and PM phases, compared to the lower bound equation (16), is shown in figures 7(d) and (e), respectively.
In figure 8 we plot the thermal phase diagram θ-T of the QFI density for different values of α. The colored region corresponds to f Q >1, where the QFI witnesses ME, while f 1 Q  in the white region. We clearly distinguish two 'lobes' on the FM and AFM sides of the phase diagram. On the basis of the results above, we argue that the FM lobe (at θ<0) disappears in the thermodynamic limit for 1  a due to the disappearance of the PM phase. The boundary of the AFM lobe (at θ>0) for small values of θ is well reproduced by the condition 1 R 2 x = (dashed line) corresponding to the thermal loss of spin squeezing. Figure 8 also clearly shows the sharp decrease of the QFI in the FM and AFM phases at vanishingly small temperature T 0   [52].
We conclude this section noticing that, due to the absence of edge states, even in the open chains, the discussion about the phase stability against temperature, performed by the scaling of the QFI density and based on equation (16), does not suffer of deviations from edge contributions, similar to those hypothesized in [109].   x = encloses the area where the spin-squeezing parameter is able to detect entanglement. The black circles mark the position of the critical points. In all the panels, N=10.
The present study suggests the potential relevance of QFI for the study of the interplay of temperature and disorder [113][114][115].

Discussion and conclusions
The scaling of the QFI calculated with respect to different collective operators yields a characterization of the full phase diagram of the LR Ising model that can be probed in current quantum simulators, even for a limited (N50) number of spins. This approach provides a clear signature of many physical effects that characterize the model, as the presence of a LRPM phase at 1  a , the change of the scaling along the critical massless lines for small values of α and a probe of the mean-field regime. The LRPM phase is particularly interesting. Here, ME can be also captured by the spin-squeezing parameter, that largely simplifies experimental detection and characterization of the state. The large ME (namely, superextensive QFI and inverse spin-squeezing parameter) found in the ground state is robust against temperature, being protected by an energy gap that remains finite in the thermodynamic limit. Furthermore, the LPRM phase can be addressed by preparing the ground state at θ=0 and adiabatically increasing the coupling strength, without crossing any QPT. Finally, we recall that the QFI studied here and the spin-squeezing parameter are directly related to metrological usefulness of a quantum state, see [41] for a recent review. Therefore, we can conclude that the ground state of the LR Ising model, especially in the robust LRPM phase, can be a resource for entanglement-enhanced metrology. , this approximation breaks down in the thermodynamic limit.
B.1. Ground-state energy and FM critical line Using the nondegenerate perturbation theory it is also possible to evaluate the shift of the lowest energy levels due to the (small) interaction term. Up to the second order in θ, we find E N 4

B.3. Fidelity susceptibility
Let us now evaluate, at fixed N and θ (such that 1 N  q a  ( ) ) the fidelity between the ground states corresponding to two close interaction ranges α and α+δα: at leading order in θ, , , We find that χ α asymptotically scales as N N ln where the width σ is the sole variational parameter. We further assume N 1  and a sufficiently localized wavepacket so to neglect border effects. We minimize the energy E H , in agreement with our numerical calculations predicting a scaling N 1/2 of the QFI density, see . We can thus roughly locate a diverging derivative of the QFI (signaling the critical point) at N arctan q = , that is θ→π/2 for N  ¥.