Entanglement and quantum correlations in many-body systems: a unified approach via local unitary operations
M. Cianciaruso, S. M. Giampaolo, W. Roga, G. Zonzo, M. Blasone, F. Illuminati
EEntanglement and quantum correlations in many-body systems:a unified approach via local unitary operations
M. Cianciaruso,
1, 2
S. M. Giampaolo,
3, 4
W. Roga,
5, 4
G. Zonzo, M. Blasone,
1, 2 and F. Illuminati
4, 2, ∗ Dipartimento di Fisica “E. R. Caianiello”, Universit`a degli Studi di Salerno,Via Giovanni Paolo II 132, I-84084 Fisciano (SA), Italy INFN Sezione di Napoli, Gruppo collegato di Salerno, Italy International Institute of Physics, UFRN, Av. Odilon Gomes de Lima 1722, 59078-400 Natal, Brazil Dipartimento di Ingegneria Industriale, Universit`a degli Studi di Salerno,Via Giovanni Paolo II 132, I-84084 Fisciano (SA), Italy Department of Physics, University of Strathclyde, John Anderson Building, 107 Rottenrow, Glasgow G4 0NG, United Kingdom (Dated: October 22, 2015)Local unitary operations allow for a unifying approach to the quantification of quantum correlations amongthe constituents of a bipartite quantum system. For pure states, the distance between a given state and its imageunder least-perturbing local unitary operations is a bona fide measure of quantum entanglement, the so-calledentanglement of response, which can be extended to mixed states via the convex roof construction. On the otherhand, when defined directly on mixed states perturbed by local unitary operations, such a distance turns outto be a bona fide measure of quantum correlations, the so-called discord of response. Exploiting this unifiedframework, we perform a detailed comparison between two-body entanglement and two-body quantum discordin infinite XY quantum spin chains both in symmetry-preserving and symmetry-breaking ground states as wellas in thermal states at finite temperature. The results of the investigation show that in symmetry-preservingground states the two-point quantum discord dominates over the two-point entanglement, while in symmetry-breaking ground states the two-point quantum discord is strongly suppressed and the two-point entanglementis essentially unchanged. In thermal states, for certain regimes of Hamiltonian parameters, we show that thepairwise quantum discord and the pairwise entanglement can increase with increasing thermal fluctuations. PACS numbers: 03.67.Mn, 03.65.Ud, 75.10.Pq, 05.30.Rt
I. INTRODUCTION
Quantum correlations arise from the combination of the su-perposition principle and the tensor product structure of theHilbert space associated with a composite quantum system.For pure states, they are entirely captured by entanglement.On the other hand, in the case of mixed states, the situation be-comes more involved, as there can exist mixed separable (i.e.non-entangled) states that nevertheless can display non classi-cal features [1–3]. The existence of such states suggests thatthe total amount of quantum correlations is not, in general,quantified only by the entanglement but needs to be charac-terized also in terms of another quantity, related to quantumstate distinguishability, the so-called quantum discord.Entanglement and discord are fundamental resources forquantum information and quantum metrology [4–7], as wellas quite useful tools for the characterization of quantumphases in many-body systems [8–10]. For instance, topolog-ically ordered phases cannot be characterized by the Landau-Ginzburg paradigm based on symmetry breaking and localorder parameters, but rather by the long-range entanglementproperties featured by the ground state of the system [11–14].Within this generalized framework, quantum correlations (en-tanglement and discord) in many-body ground states allowfor the most fundamental characterization of complex quan-tum systems. In fact, even for systems that do not feature ex- ∗ Corresponding author: [email protected] otic phases and nonlocal quantum orders, the investigation ofground-state patterns of entanglement and discord can providea deeper understanding of locally ordered phases associated tospontaneous symmetry breaking [15–26].In spite of the ongoing efforts to characterize quantumground states by analyzing their quantum correlations, a sys-tematic comparative study of the behavior of entanglementand discord in quantum many-body systems is still lacking. Inthe present work we carry out such direct comparison withinthe powerful unified framework to the quantification of entan-glement and quantum correlations, based on the formalism oflocal unitary operations, introduced in Refs. [27–31]. In theabove-mentioned works, it has been shown that the distancebetween a given state and the state obtained from it by ap-plying a least perturbing local unitary operation is a bona fide measure of entanglement, the so-called entanglement of re-sponse (or unitary entanglement ) [27–29], in the case of purestates. In the case of mixed states, it is a bona fide measureof quantum correlations (discord), the so-called discord of re-sponse [30, 31].We apply these distance-based measures to investigate theground-state behavior of pairwise entanglement and discordin the one-dimensional XY models in a transverse magneticfield with periodic boundary conditions. Within the set ofwell-behaved distances, featuring the correct properties ofmonotonicity under completely positive and trace preserving(CPTP) dynamical maps, we pick the trace distance (otherpossible relevant choices would include, e.g., the relative en-tropy, the Bures, and the Hellinger distance). We show thatthe pairwise discord of response is always larger than the a r X i v : . [ qu a n t - ph ] O c t pairwise entanglement of response, and strongly dominatesit in symmetry-preserving ground states, particularly for largeinter-particle distances and at the factorizing field (this lastfact being trivial, since at the factorizing field the pairwise en-tanglement always vanishes identically).For symmetry-breaking ground states, we observe thatwhile the hierarchy between discord and entanglement con-tinues to hold, nevertheless, compared to the symmetry-preserving case, the pairwise discord is strongly suppressedwhile the pairwise entanglement remains either unchanged orincreases slightly for decreasing values of the external fieldbelow the factorization point. Moving from the trace distance,which is monotonically non-increasing under CPTP maps, tothe Hilbert-Schmidt distance, that does not share such a prop-erty, we show that in symmetry-breaking ground states thephysically correct hierarchy is reversed: the pairwise discordof response is dominated by the entanglement of response.This unphysical result thus provides an important illustrationof the fact that the Hilbert-Schmidt metric does not yield aproper and correct quantification of quantum correlations (afirst well-known example was provided earlier by Piani inRef. [32]).The paper is organized as follows. In Section II we reviewthe unifying approach to the quantification of quantum cor-relations based on local unitaries, by recalling the definitionsof the entanglement and discord of response. In Section IIIwe recall the main features of the one-dimensional XY mod-els in transverse field with periodic boundary conditions. InSections IV, V and VI we perform the comparison betweenthe entanglement of response and the discord of responsefor spin pairs in infinite XY chains (thermodynamic limit),respectively in symmetry-preserving and symmetry-breakingground states, as well as in thermal states at finite temperature.Conclusions and outlook are discussed in Section VII. II. ENTANGLEMENT AND DISCORD OF RESPONSE
We start by briefly reviewing some basic definitions andresults concerning the quantification of entanglement andquantum correlations via local unitary operations. Through-out the present work, we will focus on a bipartite quan-tum system AB composed of two distinguishable subsys-tems A and B . Such quantum system is associated withan Hilbert space H = H A ⊗ H B which is the tensor productof the Hilbert spaces pertaining to each subsystem, so that d ≡ dim H = d A d B . Moreover, the space of states of AB ischaracterized by the convex set of density operators (i.e. semi-positive definite and trace-class operators with unit trace) on H , whose extremal points are the unit-trace projectors over H that represent pure states.Let us denote by ρ AB Φ ≡ | Φ AB (cid:105)(cid:104) Φ AB | and Λ , respectively,a generic pure state of the bipartite quantum system AB andthe set of local unitary operators U A ≡ U A ⊗ I B such that I B is the identity operator on H B and U A is any unitary op-erator on H A whose spectrum is given by the d A -th complexroots of unity. The entanglement of response [27, 28] of ρ AB Φ , E (cid:0) | Φ AB (cid:105) (cid:1) , is defined by: E (cid:0) | Φ AB (cid:105) (cid:1) ≡ min U A ∈ Λ D T r (cid:0) ρ AB Φ , ˜ ρ AB Φ (cid:1) , (1)where ˜ ρ AB Φ ≡ U A ρ AB Φ U † A and D T r ( ρ, σ ) ≡ T r | ρ − σ | isthe trace distance between the states ρ and σ . In other words,when the whole quantum system AB is in a pure state ρ AB Φ ,the entanglement of response quantifies the quantum correla-tions between parts A and B in terms of the distinguishabilitybetween the state ρ AB Φ and the state ˜ ρ AB Φ obtained from ρ AB Φ by applying to it a minimally perturbing local unitary trans-formation.There are at least two distinct ways to extend the entangle-ment of response to mixed states: the convex roof extension,which then identifies the entanglement of response of mixedstates, and the discord of response defined directly as the dis-tance between a given mixed state and the one obtained fromit through the action of the least perturbing local unitary oper-ation [30, 31]. More precisely, the entanglement of response of a bipartite mixed state ρ AB , E (cid:0) ρ AB (cid:1) , is defined as: E (cid:0) ρ AB (cid:1) ≡ min { | Φ ABi (cid:105) ,p i } (cid:88) i p i E (cid:0) | Φ ABi (cid:105) (cid:1) , (2)where the minimization is performed over all the decompo-sitions of ρ AB in pure states (cid:80) i p i | Φ ABi (cid:105)(cid:104) Φ ABi | = ρ AB , p i ≥ , (cid:80) i p i = 1 . On the other hand, the discord of re-sponse of a bipartite state ρ AB , Q ( ρ AB ) , is defined as [31](see also Ref. [30] for earlier related work): Q ( ρ AB ) ≡ min U A ∈ Λ D T r (cid:0) ρ AB , ˜ ρ AB (cid:1) , (3)where, as in the case of pure states, ˜ ρ AB ≡ U A ρ AB U † A .Therefore, the entanglement and the discord of responsequantify different aspects of bipartite quantum correlations viatwo different uses of local unitary operations. The discord ofresponse arises by applying local unitaries directly to the gen-erally mixed state ρ AB , while the entanglement of responsestems from the application of local unitaries to pure states. Byvirtue of their common origin, it is thus possible to perform adirect comparison between these two quantities.In terms of the trace distance, the two-qubit entanglementof response is simply given by the squared concurrence [31,33], whereas the two-qubit discord of response relates nicelyto the geometric discord [34], whose closed formula is knownonly for a particular class of two-qubit states [35], although itcan be computed for a more general class of two-qubit statesthrough a very efficient numerical optimization. III. XY MODELS
In this section we recall some key aspects of the quan-tum many body systems we shall focus on, that is the one-dimensional anti-ferromagnetic XY models in transversefield with periodic boundary conditions [36–40]. Such quan-tum spin models consist of a periodic chain of N -spins, withanisotropic nearest-neighbor spin-spin interactions competingwith a transverse magnetic field, whose dynamics is governedby the following Hamiltonian: H = N (cid:88) i =1 (cid:20)(cid:18) γ (cid:19) σ xi σ xi +1 + (cid:18) − γ (cid:19) σ yi σ yi +1 − hσ zi (cid:21) . (4)Here σ αi , α = x, y, z , are the Pauli matrices on site i , γ is theanisotropy in the xy plane, h is the strength of the transversemagnetic field, and the periodic boundary conditions implythat σ αN +1 ≡ σ α . The XY models reduce to the isotropic XX model and to the Ising model for γ = 0 and γ = 1 ,respectively.Regardless of the value of γ , in the thermodynamic limit,these models feature a quantum phase transition at h = h c =1 . For h > h c = 1 and for any value of γ , the ground statespace is non-degenerate and there is a finite gap in the energyspectrum between the ground state and the first excited state.On the other hand, for h < h c , two different cases arise: for γ = 0 the ground state space remains non-degenerate whilethe energy spectrum becomes gapless, whereas for γ > theground state space becomes two-fold degenerate, the energyspectrum is gapped, and the system can be characterized by anon vanishing local order parameter m x = ( − i (cid:104) σ xi (cid:105) (spon-taneous on-site magnetization). Besides the quantum criti-cal point, there exists another relevant value of the externalmagnetic field, that is h f = (cid:112) − γ , the factorizing field .Indeed, at this value of h , the system admits a two-fold de-generate, completely factorized ground state [17–19, 41, 42].The two degenerate factorized states collapse onto a singlestate for γ = 0 . This corresponds t the isotropic, gapless XX model, for which the factorizing field and the critical field co-incide: h f = h c = 1 .Since our goal is to compare the two-spin entanglementof response and the two-spin discord of response, we needto determine the pairwise reduced density matrix ρ ij both insymmetry-preserving and symmetry-breaking ground states,as well as thermal states at finite temperature. The pairwisereduced density matrix ρ ij is defined as the partial trace onthe state of the whole chain with respect to all spins exceptthose at sites i and j . While the ground state of the entirechain is a pure state, the reduced state of a pair of spins is ingeneral mixed. The two-site density matrix can be expandedas follows [43]: ρ ij = 14 (cid:88) α,β =0 (cid:104) σ αi σ βj (cid:105) σ αi σ βj (5)where σ i = I i is the identity on site i , and (cid:104) σ αi σ βj (cid:105) denotesthe two-body correlation function between σ αi and σ βj . Theoperator expansion in eq. (5) depends on different corre-lation functions. However, this number can be reduced byresorting to the symmetries of the Hamiltonian. Translationalinvariance of the lattice implies that the reduced density ma-trix depends only on the inter-spin distance r = | i − j | . Also,since the Hamiltonian is real, ρ ij = ρ ∗ ij . Finally, except for thesymmetry-breaking ground states, the global phase-flip sym-metry implies that [ σ zi σ zj , ρ ij ] = 0 . Therefore, for both the h Q h E FIG. 1: Nearest-neighbor discord of response (upper panel)and nearest-neighbor entanglement of response (lower panel) forsymmetry-preserving ground states, in the thermodynamic limit, asfunctions of the external field h , and for different values of theanisotropy γ . Solid blue curve: γ = 0 ; dashed red curve: γ = 0 . ;dot-dashed green curve: γ = 0 . ; double-dot-dashed black curve: γ = 0 . ; dotted orange curve: γ = 1 . In the lower panel, to eachof these curves, there corresponds a vertical line denoting the asso-ciated factorizing field h f . In the upper panel, the solid vertical linedenotes the critical field h c = 1 . symmetry-preserving ground states and the thermal states, theonly correlation functions different from zero are (cid:104) σ zi (cid:105) and (cid:104) σ αi σ αj (cid:105) for α = x, y, z . Such correlation functions are read-ily obtained by generalizing the approach of Ref. [36] at nonvanishing external field for the finite size system, or directlyfrom Refs. [38, 39] in the thermodynamic limit.On the other hand, when dealing with the symmetry-breaking ground states, i.e. when the system is in the thermo-dynamic limit, the external field is below the quantum criticalpoint, and γ > , the set of nonvanishing correlation func-tions includes also (cid:104) σ xi (cid:105) and (cid:104) σ xi σ zj (cid:105) . The explicit expressionof the former was first derived in Ref. [39] while (cid:104) σ xi σ zj (cid:105) canbe evaluated by a simple generalization of the same procedure. IV. SYMMETRY-PRESERVING GROUND STATES
We first compare the two-body entanglement of responseand the two-body discord of response in symmetry-preservingground states. For two neighboring spins, these two quantitiesare plotted in Fig. 1 as functions of the external field h andfor different values of the anisotropy γ . For any intermediatevalue of γ , the nearest-neighbor entanglement of response E exhibits the following behavior. If h < h f , E decreasesuntil it vanishes at the factorizing field h = h f . Otherwise,if h > h f , E first increases until it reaches a maximum atsome value of h higher than the critical point h c = 1 , then itdecreases again until it vanishes asymptotically for very largevalues of h in the paramagnetic phase(saturation). Overall, E features two maxima at h = 0 and h > h c and two minima at h = h f (factorization) and h → ∞ (saturation). In the Isingmodel ( γ = 1 ) the point h = 0 corresponds to a minimum,since it coincides with the factorizing field h f = (cid:112) − γ ,while in the isotropic XX model ( γ = 0 ) there is no secondmaximum for large fields h > h c , since the ground state isalways completely factorized as soon as h ≥ h c .On the other hand, regardless of the value of γ , the nearest-neighbor discord of response Q always features a single max-imum. Depending on the value of γ such maximum can beeither in the ordered phase h < h c or in the disordered (para-magnetic) phase h > h c , moving towards higher values of h with increasing γ . Remarkably, Q never vanishes at the fac-torizing field, except in the two extreme cases of γ = 0 , .Indeed, at the factorizing field h = h f and for any γ (cid:54) = 0 , ,the symmetry-preserving ground state is not completely fac-torized but rather is a coherent superposition of the two com-pletely factorized symmetry-breaking ground states. Conse-quently, while the two-body entanglement of response mustvanish in accordance with the convex roof extension, the two-body discord of response remains always finite.When increasing the inter-spin distance r , the pairwise en-tanglement of response E r and discord of response Q r be-have even more differently (see Fig. 2). E r dramatically dropsto zero as r increases, except in a small region around thefactorizing field h = h f that gets smaller and smaller as r increases, in agreement with the findings of Ref. [44]. Onthe other hand, while in the disordered and critical phases Q r vanishes as r increases, in the ordered phase Q r sur-vives even in the limit of infinite r . Indeed, in both the dis-ordered and critical phases, and when r goes to infinity, theonly nonvanishing one-body and two-body correlation func-tions in the symmetry-preserving ground states are (cid:104) σ zi (cid:105) and (cid:104) σ zi σ zi + r (cid:105) , so that the two-body reduced state can be writ-ten as a classical mixture of eigenvectors of σ zi σ zi + r . On theother hand, in the ordered phase, also the two-body correlationfunction (cid:104) σ xi σ xi + r (cid:105) appears, while (cid:104) σ xi (cid:105) vanishes due to sym-metry preservation, thus preventing the two-body marginal ofthe symmetry-preserving ground state from being a mixtureof classical states.The long range nature of the pairwise discord of responsenot only tells us that quantum correlations beyond entangle-ment cannot be monogamous, at variance with entanglementitself [45, 46], but also reveals the unavoidable quantum na-ture of the symmetry-preserving ground states in the orderedphase. Indeed, symmetry-preserving ground states are, ingeneral, entangled coherent superpositions of the symmetry-breaking ordered ground states, and the latter are the mostclassical ones among all possible ground states, as recently h Q r h Q r h E r h E r FIG. 2: Two-body discord of response (upper panel) and two-bodyentanglement of response (lower panel) for symmetry-preservingground states, in the thermodynamic limit, as functions of the exter-nal field h , in the case of γ = 0 . , for different inter-spin distances r .Solid blue curve: r = 2 ; dashed red curve: r = 3 ; dot-dashed greencurve: r = 8 ; dotted black curve: r = ∞ . In both panels, the twosolid vertical lines correspond, respectively, to the factorizing field(left) and to the critical field (right). Inset (both panels): same, butwith γ = 0 ; the solid vertical line corresponds to the critical point. proven in Refs. [25, 26]. At the factorizing fields, thesymmetry-preserving ground states are maximally entangledcoherent superpositions (Schroedinger cats).Let us now analyze the behavior of the nearest-neighborentanglement of response E and discord of response Q inclose proximity to the quantum critical point h c = 1 . Fig. 3shows that both ∂ h E and ∂ h Q manifest a logarithmic di-vergence at the critical point, for any non zero anisotropy γ .However, while in the entanglement case such logarithmic di-vergence is always positive, in the discord case it can be eitherpositive or negative, depending on the anisotropy γ .By performing finite size scaling analysis, one can showthat both ∂ h E and ∂ h Q allow for an accurate description ofthe quantum phase transition occurring at h c = 1 , providingus with the corresponding critical exponent ν [21, 47–49]. Inthe following, for the sake of illustration, we obtain the criticalexponent ν for the XY -model with intermediate anisotropy γ = 0 . , although the same result applies to any non zeroanisotropy. To do this, we need to suitably compare the twofollowing behaviors [47]. (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) h (cid:182) h Q (cid:45) (cid:45) (cid:45) (cid:45) ln (cid:72) N (cid:76) (cid:72) (cid:182) h Q (cid:76) h (cid:61) h m (cid:45) (cid:45) (cid:45) ln (cid:72) N (cid:76) l n (cid:72) (cid:160) h c (cid:45) h m (cid:164) (cid:76) h (cid:182) h E ln (cid:72) N (cid:76) (cid:72) (cid:182) h E (cid:76) h (cid:61) h m (cid:45) (cid:45) (cid:45) (cid:45) ln (cid:72) N (cid:76) l n (cid:72) (cid:160) h c (cid:45) h m (cid:164) (cid:76) FIG. 3: First derivative of the nearest-neighbor discord of response(upper panel) and of the nearest-neighbor entanglement of response(lower panel) in symmetry-preserving ground states, as functions ofthe external field h in proximity of the critical point, in the case of γ = 0 . , for different chain lengths N . Blue curve: N = 30 ; redcurve: N = 40 ; green curve: N = 60 ; black curve: N = 90 ; orangecurve: N = 120 ; magenta curve: N = 180 ; brown curve: N = ∞ .In both panels, the solid vertical line represents the critical point.In both panels, the upper inset shows the dependence on the chainsize N of the renormalized critical point h m . The lower inset in theupper panel displays the dependence on the chain size N of the valueattained at the renormalized critical point h m by the first derivativeof the nearest-neighbor discord of response. The lower inset in thelower panel displays the same dependence for the nearest-neighborentanglement of response. On the one hand, we have the dependence on the chain size N of the value attained at the renormalized critical point h m by ∂ h E (resp., ∂ h Q ), where h m is its maximum (resp., min-imum) in the close proximity of the critical point h c = 1 (seeFig. 3): ∂ h E ( N )1 (cid:12)(cid:12)(cid:12) h m = 0 .
15 ln N + const ,∂ h Q ( N )1 (cid:12)(cid:12)(cid:12) h m = − .
59 ln N + const . (6)On the other hand, we have the dependence on the proxim-ity to the critical point h c = 1 of ∂ h E (resp., ∂ h Q ) in the (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) N (cid:72) h (cid:45) h m (cid:76) (cid:72) (cid:45) e x p (cid:72) (cid:182) h Q (cid:45) (cid:72) (cid:182) h Q (cid:76) h m (cid:76)(cid:76) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) ln (cid:72)(cid:160) h c (cid:45) h (cid:164)(cid:76) (cid:182) h Q (cid:72) (cid:165) (cid:76) (cid:45) (cid:45) N (cid:72) h (cid:45) h m (cid:76) (cid:72) (cid:45) e x p (cid:72) (cid:182) h E (cid:45) (cid:72) (cid:182) h E (cid:76) h m (cid:76)(cid:76) (cid:45) (cid:45) (cid:45) (cid:45) ln (cid:72)(cid:160) h c (cid:45) h (cid:164)(cid:76) (cid:182) h E (cid:72) (cid:165) (cid:76) FIG. 4: Finite size scaling of the first derivative of the nearest-neighbor discord of response (upper panel) and of the nearest-neighbor entanglement of response (lower panel) in symmetry-preserving ground states, for γ = 0 . . The inset in the upper panelshows the first derivative of the nearest-neighbor discord of responsein the thermodynamic limit as a function of the proximity to the criti-cal point. The same in the inset in the lower panel, but for the nearest-neighbor entanglement of response. thermodynamic limit (see insets of Fig. 4): ∂ h E ( ∞ )1 = − .
15 ln | h c − h | + const ,∂ h Q ( ∞ )1 = 0 .
59 ln | h c − h | + const . (7)According to the scaling ansatz relative to the case of alogarithmic divergence [47], the critical exponent ν is sim-ply given by the opposite of the ratio between the pre-factorsof the logarithms in Eq. (7) and Eq. (6) , respectively. We thusobtain ν = 1 , in agreement with the known fact that all theanisotropic XY models belong to the Ising universality class.Moreover, Fig. 4 highlights the precision of the above finitesize scaling analysis by showing that, via a proper scaling ofthe functions ∂ h E and ∂ h Q [47], it is possible to make allthe data for different sizes N collapse onto a single curve.We conclude this section by studying how the nearest-neighbor entanglement of response E and the nearest-neighbor discord of response Q scale with the system size N in close proximity to the factorizing field h = h f = (cid:112) − γ .In spite of the significant difference between E and Q atthe factorizing field in the thermodynamic limit, as shown in
12 14 16 18 20 22 24 26 (cid:45) (cid:45) (cid:45) (cid:45) N . l n (cid:160) E h f N (cid:45) E h f (cid:165) (cid:164) , l n (cid:160) Q h f N (cid:45) Q h f (cid:165) (cid:164) FIG. 5: Dependence on the chain size N of the value attained at thefactorizing field by the nearest-neighbor discord of response (solidline) and nearest-neighbor entanglement of response (dashed line)for symmetry-preserving ground states, at different values of γ . Bluelines: γ = 0 . ; red lines: γ = 0 . ; black lines: γ = 0 . . Fig. 1, their finite size scalings in the proximity of h f are ex-tremely similar, as shown in Fig. 5. Indeed, both the entangle-ment and the discord scale with N according to an exponentialdecay that is independent of the inter-spin distance r and getsfaster and faster as the anisotropy γ increases. Interestingly,for any fixed value of the anisotropy γ , the decay rate of theentanglement of response is twice that of the correspondingrate for the discord of response. V. SYMMETRY-BREAKING GROUND STATES
In this section we move the focus of the comparison be-tween two-body entanglement of response and discord ofresponse from symmetry-preserving to symmetry-breakingground states. Spontaneous symmetry breaking manifests it-self in the thermodynamic limit, in the ordered phase h So far we have focused our analysis on the ground states ofthe XY models, be them symmetry-preserving or symmetry-breaking. We will now consider XY models in thermal equi-librium with a bath at finite temperature.The behavior of the nearest-neighbor entanglement of re-sponse E and of the nearest-neighbor discord of response Q , as well as that of their first derivatives, as functions of theexternal field, are plotted in Figs. 8 and 9 for different valuesof the temperature T . The appearance of thermal effects has arounding off effect that removes all singularities in correspon-dence of the critical point h c . Indeed, a sharp quantum phasetransition can occur only at zero temperature. Specifically, assoon as the temperature T increases from zero to some finitevalue, the zero temperature singularity of ∂ h E at the criticalpoint h c is smoothed into a maximum of ∂ h E localized ata value of the external field h higher than h c . Moreover, themore the temperature increases, the more this maximum isshifted away from the critical point h c and the correspondingvalue of ∂ h E decreases. Similarly, the divergence of ∂ h Q at the critical point h c is replaced by either a minimum or amaximum of ∂ h Q (depending on γ ) at a value of the externalfield h lower than h c . Furthermore, the higher T , the morethis extremal point moves away from the critical point h c andthe corresponding absolute value of ∂ h Q decreases.Obviously, thermal effects also remove and/or distort theground-state factorization phenomenon, that occurs at zerotemperature for h f = (cid:112) − γ . Indeed, Fig. 8 shows that,as the temperature varies, h f either belongs to a region where E vanishes identically, or is a regular point at which E isstrictly nonzero. Accordingly, as soon as the temperature T h (cid:182) h Q h (cid:182) h E FIG. 9: First derivative of the nearest-neighbor discord of response(upper panel) and of the nearest-neighbor entanglement of response(lower panel) for thermal states, in the thermodynamic limit, as func-tions of the external field h , with γ = 0 . , for different values ofthe temperature T . Solid blue curve: T = 0 ; dashed red curve: T = 0 . ; dot-dashed green curve: T = 0 . ; double-dot-dashedblack curve: T = 0 . ; dotted orange curve: T = 0 . . In bothpanels, the solid vertical line represents the critical point. increases from zero to some finite value, the discords evalu-ated for different inter-spin distances do not coincide anymorewhen h = h f (see Fig. 10).From Fig. 8 it also emerges that, as the temperature T in-creases, the peaks corresponding to E and Q tend to flattento zero quite differently, with the pairwise discord being morerobust than the pairwise entanglement against thermal effects.Interestingly, Figs. 11 and 12 show that, for any anisotropy γ , there exist some values of the external field h such thateither the pairwise entanglement or the pairwise discord in-crease with the temperature. This contrasts the common in-tuition for which thermal effects can only be detrimental toquantum features. Furthermore, this surprising behavior ap-pears mostly in the case of the pairwise discord. Indeed, thelatter increases with the temperature for many values of theexternal field h and any sufficiently small inter-spin distance r , whereas the pairwise entanglement displays such behavioronly for particular values of h and, essentially, in the case ofpairs of nearest-neighboring spins.More precisely, for sufficiently large anisotropy γ , Q in-creases with the temperature for all values of the external field h except for those belonging to a small interval around and h Q r h E r FIG. 10: Two-body discord of response (upper panel) and two-bodyentanglement of response (lower panel) for thermal states at tem-perature T = 0 . , in the thermodynamic limit, as functions of theexternal field h , with γ = 0 . , for different values of the inter-spindistance r . Solid blue curve: r = 2 ; dashed red curve: r = 3 ; dot-dashed green curve: r = 4 ; dotted black curve: r = ∞ . In bothpanels, the two solid vertical lines correspond, respectively, to thefactorizing field (left) and to the critical field (right). including the critical point h c = 1 . For sufficiently small γ , Q increases with the temperature for all values of the externalfield h except for those belonging to an interval that lies belowthe critical point h c = 1 and shifts towards lower values of h as γ decreases. Consequently, there is no correspondence be-tween the growth of the pairwise discord with the temperatureand the occurrence of a gap in the energy spectrum betweenthe ground state and the first excited state. VII. CONCLUSIONS AND OUTLOOK In this paper, by resorting to a unifying approach to thequantification of bipartite quantum correlations based on lo-cal unitary operations, we have performed the first, direct andcomprehensive, comparison between the two-body entangle-ment and two-body quantum discord in infinite XY quan-tum spin chains, both in symmetry-preserving and symmetry-breaking ground states as well as in thermal states at finitetemperature.For symmetry-preserving ground states, we have shownthat the pairwise entanglement captures only a modest por- T Q T (cid:69) (cid:45) (cid:45) T (cid:182) T Q (cid:45) (cid:45) (cid:45) T (cid:182) T E FIG. 11: Nearest-neighbor discord of response (leftmost upperpanel) and its first derivative (leftmost lower panel) for thermal states,in the thermodynamic limit, as functions of the temperature T , with γ = 0 . , for different values of the external field h . Solid blue line: h = 0 ; dashed red line: h = 0 . ; dot-dashed magenta line: h = h f ;double-dot-dashed green line: h = 0 . ; triple-dot-dashed black line: h = h c = 1 ; dotted orange line: h = 1 . . The same in the rightmostpanels, but for the nearest-neighbor entanglement of response. T Q T E (cid:45) (cid:45) (cid:45) T (cid:182) T Q (cid:45) (cid:45) (cid:45) T (cid:182) T E FIG. 12: Next-nearest-neighbor discord of response (leftmost upperpanel) and its first derivative (leftmost lower panel) for thermal states,in the thermodynamic limit, as functions of the temperature T , in thecase of γ = 0 . , for different values of the external field h . Solidblue line: h = 0 ; dashed red line: h = 0 . ; dot-dashed magenta line: h = h f ; double-dot-dashed green line: h = 0 . ; triple-dot-dashedblack line: h = h c = 1 ; dotted orange line: h = 1 . . The same inthe rightmost panels, but for the next-nearest-neighbor entanglementof response. tion of the total pairwise quantum correlations. This fact istrivially obvious at the factorizing field and quite intuitive forlong-range inter-spin distances: in both cases, the pairwise entanglement vanishes.Conversely, for symmetry-breaking ground states, we haveshown that the pairwise quantum correlations are stronglysuppressed in the whole ordered phase h < h c , while the pair-wise entanglement is either unchanged or undergoes a slightenhancement, thus contributing the largest amount to the totalpairwise quantum correlations. When adopting the Hilbert-Schmidt distance, we have also found that the two-body dis-cord of response can be even smaller than the correspondingentanglement, thus providing a fundamental physical illustra-tion of the fact that the Hilbert-Schmidt distance, being noncontractive under CPTP maps, does not allow for a properquantification of quantum correlations.For thermal states at finite temperature, we have shown thatthe pairwise discord of response is in general more robust thanthe pairwise entanglement against thermal effects. Moreover,we have also shown that a surprising resilience to thermal ef-fects can occur both for the pairwise discord and the pairwiseentanglement, whereby these quantum features can, in someregions of the Hamiltonian parameters, increase with the tem-perature, although this behavior appears most enhanced in thecase of the pairwise discord.The fact that pairwise quantum correlations and pairwiseentanglement can increase with the temperature in someregimes of the Hamiltonian parameters, together with thecomplex behavior of the maximum pairwise entanglement anddiscord as functions of the anisotropy are two puzzling fea-tures whose physical origin is at present not fully understoodand thus deserves further investigation.It is finally worth remarking that in order to directly com-pare entanglement and quantum discord on equal footing, onemight resort to other unifying approaches to the quantifica-tion of entanglement and quantum correlation besides the onebased on the formalism of local unitary operations that wehave used in the present work. 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