Quantum discord in the ground state of spin chains

Quantum discord in the ground state of spin chains

Abstract

The ground state of a quantum spin chain is a natural playground for investigating correlations. Nevertheless, not all correlations are genuinely of quantum nature. Here we review the recent progress to quantify the ’quantumness’ of the correlations throughout the phase diagram of quantum spin systems. Focusing to one spatial dimension, we discuss the behavior of quantum discord close to quantum phase transitions. In contrast to the two-spin entanglement, pairwise discord is effectively long-ranged in critical regimes. Besides the features of quantum phase transitions, quantum discord is especially feasible to explore the factorization phenomenon, giving rise to nontrivial ground classical states in quantum systems. The effects of spontaneous symmetry breaking are also discussed as well as the identification of quantum critical points through correlation witnesses.

Quantum Information; Quantum Phase Transitions; Quantum Discord
\catchline

1 Introduction

Quantum mechanics relies on properties of complex vector spaces, whose elements are associated with wavefunctions. Therefore any correlation exploiting the quantum superposition principle (expressing the notion of sum in vector spaces) is indeed of quantum nature. This simple evidence has been exploited as a guiding principle to quantify quantum correlations beyond the generic notion of ’correlation’ in a condensed matter system. In this context, quantum spin systems naturally provide quantum correlations, especially at , where the system is in its ground state. Such correlations are diversely important throughout the phase diagram of the system. For example, entanglement, the most famous quantum correlation, turned out strong enough close to quantum phase transitions (QPT) [1, 2], but it can be very small deep in the phase of a highly correlated quantum spin system where quantum fluctuations provide a factorized ground state  [3, 4, 5] (see also Ref. \refciteAmico:08 for a review on entanglement in many-body systems).

A recently devised measure of quantum correlation is the quantum discord (QD) [7], providing a ’quantum/classical sieve’ explicitely based on the superposition principle of quantum mechanics [7, 8, 9]. With QD, it was evidenced that indeed there are quantum correlations in separable mixed states. This, in turn, may disclose new scenarios for the phase diagram of many-body quantum systems. The first computations of QD for spin systems in the thermodynamical limit have been carried out by Dillenschneider [10] and Sarandy [11], where QD between two spins in a mixed state realized by the rest of the system was considered. It turned out that the non-analyticities of energy derivatives at QPTs were reflected not only in the two-spin entanglement [12] and QD [10, 11] but also in the classical correlations, establishing all them as useful tools to analyze quantum critical phenomena. Further investigations were then performed in a number of many-body systems (see, e.g., Refs. [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 90]), where entanglement, QD and classical correlations were compared. The bulk of evidence we can rely on so far suggests that genuine quantum correlations, including entanglement and QD, exhibit the behavior expected by scaling theory for the long-range physics close to QPTs. For the short-range physics, in contrast, a rather complex situation emerges because entanglement, QD and standard correlation functions may vanish for different ranges [23, 24] 1. This is clearly evidenced at factorizing fields occurring in the ordered phase of the system (characterized by a nonvanishing spectroscopic gap). Close to factorization, the entanglement range diverges despite the ordinary correlation functions decay exponentially[31].

In this article we review the properties of QD throughout the phase diagram of spin systems in a one-dimensional lattice, with nearest-neighbor exchange interactions. For these systems, different ground states may be achieved through order-disorder QPTs occurring via the spontaneous symmetry breaking (SSB) mechanism. We consider both the so called thermal states and the symmetry broken ground states, enjoying and breaking the symmetries of the Hamiltonian, respectively. As long as the system is studied through its energy spectrum, all thermodynamical observables stay unaltered irrespective whether the thermal ground state or the state with broken symmetry are considered. Nevertheless, the difference comes in entanglement and QD (the same argument can be applied to any other measure of quantum correlation based on reduced density matrix). Entanglement was analyzed in the SSB scenario in the Refs. [32, 33, 34, 35]. For the analysis of QD, SSB effects have been first accounted in Refs. [36, 37]. The article is organized as follows. In section 2 we introduce QD. The models we deal with are introduced in the section 3; in there we provide a summary of the symmetry breaking mechanism to make the article self contained. In the sections 4 and 5 we review the properties of the QD for thermal and symmetry broken ground states respectively. As illustration, in section 6 we discuss a witness for multipartite quantum correlations [38], for both thermal and symmetry broken ground states. In section 7 we draw our conclusions.

2 Quantum discord

Let us begin by considering a composite system in a bipartite Hilbert space . The system is characterized by quantum states described by density operators , where is the set of bound, positive-semidefinite operators acting on with trace given by . Then, denoting by the density matrix of the composite system and by and the density matrices of parts and , respectively, the total (classical + quantum) correlations between parts and can be provided by the quantum mutual information [39]

 I(ρA:ρB)=S(ρA)−S(ρA|ρB), (1)

where is the von Neumann entropy for subsystem (with the symbol log denoting logarithm at base 2) and

 S(ρA|ρB)=S(ρ)−S(ρB) (2)

is the quantum conditional entropy for given the part . A remarkable observation realized in Ref. \refciteOllivier:01 is that the conditional entropy can be introduced by a different approach, which yields a result in the quantum case that differs from Eq. (2). Indeed, let us consider a measurement performed locally only on part . This measurement can be described by a set of projectors . The state of the quantum system, conditioned on the measurement of the outcome labelled by , becomes

 ρk=1pk(I⊗Bk)ρ(I⊗Bk), (3)

where denotes the probability of obtaining the outcome and denotes the identity operator for the subsystem . The conditional density operator given by Eq. (3) allows for the following measurement-based definition of the quantum conditional entropy:

 S(ρ|{Bk})=∑kpkS(ρk). (4)

Therefore, the quantum mutual information can also be alternatively defined by

 J(ρ:{Bk})=S(ρA)−S(ρ|{Bk}). (5)

Eqs. (1) and (5) are classically equivalent but they are different in the quantum case. The difference between them is due to quantum effects on the correlation between parts and and provides a measure for the quantumness of the correlation. Indeed, by generalizing the measure set to a positive-operator valued measure (POVM) and maximizing over such POVMs, we can define the classical correlation between parts and as [8]

 C(ρ)=max{Bk}J(ρ:{Bk}), (6)

Then, quantum correlations can be accounted by subtracting Eq. (6) from Eq. (1), which yields

 Q(ρ)=min{Bk}[I(ρA:ρB)−J(ρ:{Bk})]. (7)

Quantum correlations as measured by define QD. As provided by Eq. (7), it is an asymmetric quantity that measures the quantum correlations between subsystems and revealed by measurements on part . Then, if QD vanishes, the system is in a quantum-classical state. In order to define classical-classical states, we can symmetrize QD with respect to measurements on each subsystem [40, 41], requiring the vanishing of QD for having total classicality.

Here we apply the above notions to the situation where and are individual spins of a quantum spin chain. To compute QD, we will adopt the specific case of von Neumann local measurements, which are provided by the orthogonal projectors , such that

 ^Bk=V^ΠkV†,k=0,1 (8)

where define the computational basis and is a rotation operator. Then, can be parametrized as

 Misplaced & (9)

where and are respectively the azimuthal and polar axes of a qubit over in the Bloch sphere [10, 11]. Hence, QD as given by Eq. (7) can be directly obtained by minimizing over all angles and . For two spin-1/2 systems, it is shown that the optimal POVM in Eq. \eqrefqcorrel is a projective measurement (4-projector-elements POVMs) [42, 43]. Remarkably (see Refs. [44, 45, 46]), for the particular case of Bell-diagonal states (marginal states are maximally mixed), such optimal POVM can be achieved through von Neumann measurements, while for the less restricted case of -symmetric states (X-states), there are examples where optimization over POVMs may be inequivalent to von Neumann measurements. In such a situation and also for generic nonsymmetric states, which is the case of states with SSB, orthogonal measurements are not more effective than 4-elements POVMs [47], with von Neumann measurements in Eq. \eqrefqcorrel providing just upper bounds for QD (as an illustration, see Ref. \refciteAmico:12).

3 The models

Magnetic interactions constitute a simple mechanism to produce classical and quantum correlations. In this context, we consider the chain, which is composed by spin- particles localized in a one dimensional lattice, interacting anisotropically along the three spatial directions with Heisenberg interaction strengths , and subjected to a uniform external field , which is governed by the Hamiltonian

 H=12∑i,j[JxSxiSxj+JySyiSyj+JzSziSzj]−h∑iSzi, (10)

where () are spin- operators defined on the -th site of the chain. Factorization of the ground state and QPTs characterize the phase diagram of the system. The bulk of evidence at disposal so far indicates that the two phenomena are interconnected, with the factorization being a precursor of QPTs [6].

3.1 QPTs and symmetry breaking

QPTs occur when an external perturbation (like pressure, magnetic field, etc.) causes quantum fluctuations that are effective enough to lead to a qualitative change of the ground state of the system. In the present article, we consider order-disorder QPTs within the symmetry breaking mechanism. Accordingly, the ordered phase of the spin system is characterized by a finite magnetization, that is a local order parameter breaking the symmetry of the Hamiltonian. In the case of Eq. (10), the symmetry is a global symmetry generated by rotations of around the direction of all spins: , where . This indicates that the ground states corresponding to opposite are indeed degenerate and span a ground-state-manifold where any state is a possible ground state of the system. Because of such a symmetry (together with the fact that ), the magnetization in the vanishes identically in the ground state:

 ⟨gs|σx,yj|gs⟩=⟨gs|Π†σx,yjΠ|gs⟩=−⟨gs|σx,yj|gs⟩. (11)

The so called ’thermal ground state’ is the ground state that is an equal mixture of the two degenerate ground states; it can be thought as the limit of the thermodynamical state. Thermal states enjoy the same symmetry of the Hamiltonian. Superpositions of degenerate ground states may also preserve the symmetry. However, they are demonstrated to be not stable to small perturbations because of the lack of clustering property [49]. Therefore the symmetry is spontaneously broken in physical systems (in the thermodynamical limit) because any small in-plane local magnetic field lifts the degeneracy and there is no observable connecting the two ground states (super-selection rule [50]). For finite systems, for example, we could have a superposition of all the spins in opposed directions like . These two states are orthogonal and preserve the symmetry, but they are not stable against small perturbations either (see also Ref. \refciteRossignoli). Here we will consider both the thermal ground states and the state with explicit symmetry breaking of the symmetry. We observe that thermal ground state and the state with broken symmetry provide the same thermodynamics of the system (see [52] for a recent reference); the corresponding density matrices, however, differ sensibly.

3.2 Factorization of the ground state

The ground state of Eq.(10) is, in general, a highly entangled state. Nonetheless, there may exist points where it is indeed a product state, which is factorized as the tensor product of individual spin states [3]. For the model Hamiltonian in Eq. (10) the factorization occurs at 2

 hf=2√(Jx−Jz)(Jy−Jz). (13)

Indeed, a qualitative change of the entanglement, called Entanglement Transition (ET), occurs at , involving two gapped phases and with a divergent entanglement range [53, 54]. We also mention that analysis of the ground state for finite size systems demonstrated that the factorization can be viewed as transition between ground states of different parities (see Ref. [55] and and a related article in this special issue [56]). Close to the factorization, the system is characterized by the following pattern of correlations functions:

 gα,β(r)=Aα,β+Bα,β(r)(h−hf)+O[(h−hf)2], (14)

where we remark that is the same constant . Eq. (14) was obtained for the quantum XY model (case II below) in Ref. \refcitebaroni, but it holds for the generic model in Eq.(10[58].

We will discuss Eq. \eqrefgeneral-spin in the following cases.

(I)

The non-integrable antiferromagnetic model (), with (this is the case experimentally realized with  [59]). At zero field the system is critical in the same universality class of the isotropic XY model (see Fig. 1 for ). At finite the system acquires a finite gap, vanishing at were a QPT occurs of the Ising type. The order parameter is . The factorization is displayed at .

(Ii)

For , the (quantum anisotropic ) Hamiltonian can be diagonalized by first applying the Jordan-Wigner transformation and then performing a Bogoliubov transformation [60, 61, 62]. The quantum Ising model corresponds to while the (isotropic) -model is recovered for . In the latter (isotropic) case the model enjoys an additional symmetry resulting in the conservation of the total magnetization along the -axis. The properties of the Hamiltonian are governed by the dimensionless coupling constant (where is set as the energy scale and working in units such that ). The phase diagram is sketched in Fig. 2 [63]. In the interval the system undergoes a second order quantum phase transition at the critical value . The order parameter is the magnetization in -direction, , different from zero for . In the phase with broken symmetry the ground state has a two-fold degeneracy reflecting a global phase flip symmetry of the system. The magnetization along the -direction, , is different from zero for any value of , but has a singular behavior in its first derivative at the transition point. In the whole interval the transition belongs to the Ising universality class. For the QPT is of the Berezinskii-Kosterlitz-Thouless type. The ET transition, occurring at the the factorization field is also associated with a change in the way the correlations decay with distance: for , the decay is oscillatory while, for , the decay is monotonic [62]. The quantum XY model can be experimentally realized with the magnetic compound  [64].

(Iii)

The antiferromagnetic anisotropic chain can be obtained from Eq. (10) by setting , , and . At zero field it presents a critical phase with quasi-long range order (quasi-lro) for ; this is separated by two classical phases with QPTs at . For the phase is a strip in the phase diagram, eventually turning into polarized phases for sufficiently strong magnetic field. Here we remark that the factorization phenomenon degenerates in the saturation occurring as a first order transition. Despite the similarities the factorization and saturation are quite distinct phenomena. In the case in which and for any value of the model is referred to as the model. The two isotropic points and describe the antiferromagnetic and ferromagnetic chains respectively (see the phase diagram in Fig. 3 [63]). The one-dimensional model can be solved exactly by Bethe Ansatz technique (see e.g. [63]) and the correlation functions can be expressed in terms of certain determinants (see [65] for a review). Correlation functions, especially for intermediate distances, are in general difficult to evaluate, although important steps in this direction have been made [66, 67].

The zero temperature phase diagram of the model at zero magnetic field shows a gapless phase in the interval with power law decaying correlation functions [68, 69]. Outside this interval the excitations are gapped. The two phases are separated by a Berezinskii-Kosterlitz-Thouless phase transition at while at the transition is of the first order. In the presence of the external magnetic field a finite energy gap appears in the spectrum.

3.3 The reduced density matrix

In order to compute the QD between any two spins and at distance along the chain, the key ingredients are the single-site density matrices and the two-site density matrix of the composite subsystem [see Eq.\eqrefqcorrel]. Due to translational invariance along the chain, single-site density matrices are the same for any spin and they are given by

 ^ρA=^ρB=12(1+gzgxgx1−gz), (15)

where are the local expectation values of the magnetization along the three different axes.

The two-site reduced density matrix for a Hamiltonian model Eq.(10) reads

 ^ρr=14⎛⎜ ⎜ ⎜ ⎜⎝Aaa% FaBCbaCBbFbbD⎞⎟ ⎟ ⎟ ⎟⎠ (16)

in the basis , where and are eigenstates of (because of translational invariance, this density matrix depends only on the distance between the two spins: ). The various entries in Eq. \eqrefrhoAB are related to the two-point correlators and to the local magnetizations, according to the following:

 Misplaced & (17)

The entries above express the parity coefficients, while

 Misplaced & (18)

As long as the state displays the -symmetry of the Hamiltonian (i.e., the thermal ground state) the matrix elements and vanish. In this case, the reduced density matrix correspond to an state. On the other hand, the requirement characterizes the state with symmetry breaking [34, 35].

4 Quantum discord in thermal ground states

In this section we summarize the properties of QD in the thermal ground states of spin models with -symmetry. In this case the density matrices are given by requiring into Eq. (16).

4.1 XY model

We begin with the transverse field XY chain (case (I) of Sec. 3). Here, we will be interested in the correlations between arbitrary distant spin pairs (not only nearest-nighbor pairs). Then, the reduced state for spin pairs at a distance reads

 ρ0r=14{I0n+⟨σz⟩(σz0+σzr)+∑i=x,y,z⟨σi0σir⟩σi0σir}, (19)

where is the identity operator acting on the joint state space of the spins and . The magnetization density as well as the two-point correlation functions can be directly obtained from the exact solution of the model [62]. Then, the total information shared by the spins in the state (19) is given by

 I(ρ0r)=S(ρ0)+S(ρr)−S(ρr), (20)

with

 S(ρ0)=S(ρr)=−1∑i=01+(−1)i⟨σz⟩2log21+(−1)i⟨σz⟩2 (21)

and

 S(ρ0r)=1∑i=0(ξilog2ξi+ηilog2ηi), (22)

where

 ξi = 1+⟨σz0σzr⟩4+(−1)i4√(⟨σx0σxr⟩−⟨σy0σyr⟩)2+4⟨σz⟩2, ηi = 14[1−⟨σz0σzr⟩+(−1)i(⟨σx0σxr⟩+⟨σy0σyr⟩)]. (23)

Following [11, 24, 14], we obtain that the minimum in Eq. (7) is attained, for all values of , , and , by the following set of projectors: , with , where are the eigenstates of . Thus one obtains

 J(ρr)=Hbin(p1)+Hbin(p2), (24)

where is the binary entropy

 Hbin(x)=−xlog2x−(1−x)log2(1−x) (25)

and

 p1=12(1+⟨σz⟩),p2=12(1+√⟨σx0σxr⟩2+⟨σz⟩2). (26)

This provides therefore an analytical expression for evaluating QD. We can then use this expression to investigate QD at quantum criticality. By fixing (Ising model) and considering pairwise QD between first neighbors, we can identify the ferromagnetic-paramagnetic QPT by looking at either classical or quantum correlations, as shown in Figs. 4 and 5.

As easily observed, derivatives of both and exhibit a non-analytical behavior at the quantum critical point as we approach to the limit of an infinite system.

Remarkably, besides the non-analytical behavior of pairwise QD derivatives, we can show that QD exhibits a long-range decay [24]. This is in contrast with the two-spin entanglement behavior, which is typically very short-ranged. In order to illustrate the long-range behavior of QD, we plot in Fig. 6 pairwise QD as a function of distance . The curves in Fig. 6 are the exponential fits of QD. We observe that, for both examples of anisotropies considered, and , the decay of QD with distance can be well fitted by an exponential function , where , , and are constants. Nevertheless, we notice that, while for QD vanish exponentially, in the cases where , we obtain a constant long-distance value for QD that depends only on and . It is also remarkable to observe that the factorization phenomenon can be traced already for the thermal ground state: it is the unique value of the field where the same quantum correlations are present at any length scale (left inset of Fig. 8). This is due by the peculiar pattern of correlation functions close to the factorizing field in Eq.(14).

4.2 XXZ model

As a second example, let us consider now spin-1/2 chain with (the case (III) of Sec.3). In this case the Hamiltonian exhibits invariance, which ensures that the element of the reduced density matrix given by Eq. (16) vanishes. Moreover, the ground state has magnetization density (), which implies that . For convenience, we consider correlations between nearest-neighbor spin pairs, i.e. , and then expand in terms of Pauli operators, which reads

 ρ=14[I⊗I+3∑i=1(ciσi⊗σi)], (27)

with

 Missing dimension or its units for \hskip (28)

By taking into account those notations, classical correlations in Eq. (6) can be analytically worked out [70], yielding

 C(ρ)=(1−c)2log(1−c)+(1+c)2log(1+c), (29)

with . For the mutual information we obtain

 I(ρ)=2+3∑i=0λilogλi, (30)

where

 λ0 = 14(1−c1−c2−c3), λ1 = λ2=14(1+c3), λ3 = 14(1+c1+c2−c3). (31)

In order to compute and we write , , and in terms of the ground state energy density. By using the Hellmann-Feynman theorem [71, 72] for the XXZ Hamiltonian , we obtain

 c1 = c2=12(Gxx+Gyy)=Δ∂εxxz∂Δ−εxxz, c3 = Gzz=−2∂εxxz∂Δ, (32)

where is the ground state energy density

 εxxz=⟨ψ0|HXXZ|ψ0⟩L=−12(Gxx+Gyy+ΔGzz), (33)

with denoting the ground state of . Eqs. (32) and (33) hold for a chain with an arbitrary number of sites, allowing the discussion of correlations either for finite or infinite chains. Indeed, ground state energy as well as its derivatives can be exactly determined by Bethe Ansatz technique [73], which allows us to obtain the correlation functions , , and . In Fig. 7, we plot classical and quantum correlations between nearest-neighbor pairs for an infinite XXZ spin chain.

Note that, in the classical Ising limit , we have a fully polarized ferromagnet. The ground state is then a doublet given by the vectors and , yielding the mixed state

 ρ=12|↑↑⋯↑⟩⟨↑↑⋯↑|+12|↓↓⋯↓⟩⟨↓↓⋯↓|. (34)

Indeed, this is simply a classical probability mixing, with and . The same applies for the antiferromagnetic Ising limit , where a doubly degenerate ground state arises. Moreover, observe that the classical (quantum) correlation is a minimum (maximum) at the infinite-order QCP . On the other hand, both correlations are discontinuous at the first-order QCP . The behavior of QD at the first-order QPT is in agreement with the expected behavior for entanglement, as shown by Wu et al. in Ref. \refciteWu:04. In fact, the approach used by Wu et al. for entanglement also applies for any function of the reduced density matrix of two-spins. It is only based on the fact that the non-analyticities in the derivatives of the ground state energy typically come from the elements of the reduced density matrix and, as such, may be inherited by entanglement or other functions of the density matrix elements. There is a caveat in the fact that entanglement and QD definitions involve a maximization or minimization processes, which may create accidental non-analyticities or even hide ones. In fact, the behavior of the entanglement as measured by concurrence [74] or negativity [75] at , is not the one expected for first-order QPTs, but rather for second order QPTs, since it is not discontinuous at but its derivative is. Such a misidentification of the order of the QPT is due to the maximization process involved [76]. Remarkably, although involving an optimization, QD does indicate the correct order of the QPT, exhibiting in this case a behavior that is superior to entanglement. For , the result for QD is in agreement with the entanglement behavior (maximum or minimum at the infinity-order QPT), even though there is no proof of the generality of this behavior for arbitrary infinite-order QPTs.

5 Quantum discord in ground states with symmetry breaking

In this section, we discuss the main features of QD for given by Eq.(16) with non vanishing given by (18). This correspond to the ground state of the Hamiltonian (10) (with nearest neighbor couplings), with broken symmetry. In this case the reduced density matrix is accessed via DMRG for finite systems with open boundaries [77], by adding a small symmetry-breaking longitudinal field (of typical magnitude ) to the Hamiltonian 3. In Fig. 8, it is displayed QD for the quantum XY model. Similarly, QD for XYX model in the ground state with symmetry breaking is obtained in Fig.(9).

QD is substantially affected by SSB 4. We observe that quantum correlations are typically much smaller deep in the ordered ferromagnetic phase , rather than in the paramagnetic one . Nonetheless, as we shall see, they play a fundamental role to drive the order-disorder transition at the QPT, where exhibits a maximum, as well as the correlation transition at , where is rigorously zero. The QPT is marked by a divergent derivative of the QD (see also [10, 11, 14]). Such divergence is present at every , for the symmetry broken state; on the other hand, for the thermal ground state, it is not present at . A thorough finite-size scaling analysis is shown in Fig. (10(b)) proving that . For the thermal ground state (in the thermodynamic limit), we found that diverge logarithmically as , within the Ising universality class.

At the factorizing field , all the correlation measures are zero in the state with broken symmetry (see symbols in Fig. 8); in particular, we numerically found a dependence

 Qr∼(h−hf)2×(1−γ1+γ)r (35)

close to it. Such behavior is consistent with the expression of correlation functions close to the factorizing line Eq.(14), and here appears incorporating the effect arising from the non vanishing spontaneous magnetization.

It is found that exponentially tends to the asymptotic value (see Fig. 10(b)). In [54, 53], it was shown that marks the transition between two different patterns of entanglement. The factorization is thus a new kind of zero-temperature transition of collective nature, not accompanied by a change of symmetry. We emphasize, though, that this transition does not correspond to any non analyticity in the ground state as a function of . 5. At the level of spectral properties of the system, we interpret this result as an effect of certain competition between states belonging to different parity sectors for finite  [55]; as these states intersect, the ground-state energy density is diverging for all finite (such divergence, though, vanishes in the thermodynamic limit).

6 Witnessing quantum correlated states

Computation of QD involves an extremization procedure. However, if we just want to find out whether a state exhibits quantum correlations, such a procedure can be avoided by introducing witness operators, namely, observables able to detect the presence of nonclassical states. A witness is defined here as a Hermitian operator whose norm is such that, for all nonclassical states, , with a convenient norm measure adopted. Therefore, is a sufficient condition for nonclassicality (or, equivalently, a necessary condition for classicality). Let us begin by defining a classical state. We will concetrate in von Neumann meansurements, but our approach can be generalized for POVMs.

{definition}

If there exists any measurement such that then describes a classical state under von Neumann local measurements.

Therefore, it is always possible to find out a local measurement basis such that a classical state is kept undisturbed. In this case, we will denote , where is the set of -partite classical states. Observe that such a definition of classicality coincides with the vanishing of global QD [41]. Note also that by itself is a classical state for any , namely, . Hence, can be interpreted as a decohered version of induced by measurement. A witness for nonclassical states can be directly obtained from the observation that the elements of the set are eigenprojectors of . This has been shown in Ref. \refciteSaguia:11, being stated by Theorem 2 below (see also Ref. \refciteLuo:08).

{theorem}

, with and denoting the index string . {proof} If then . Then, similarly as in Ref. \refciteLuo:08, a direct evaluation of yields . However, also implies that . Therefore,

 ∑j[ρ,Πj][ρ,Πj]†=0. (36)

Hence, . On the other hand, if then provides a basis of eigenprojectors of . Then, from the spectral decomposition, we obtain

 ρ=∑jpjΠj=∑i1,⋯,iNpi1,⋯,iNΠi1A1⊗⋯ΠiNAN, (37)

which immediately implies that . Hence .

We can now propose a necessary condition to be obeyed for arbitrary multipartite classical states.

{theorem}

Let be a classical state and the reduced density operator for the subsystem . Then . {proof} From theorem 6, if then the spectral decomposition of yields Eq. (37). Therefore,

 ρAi=∑rpnΠnAi. (38)

Hence, by direct evaluation, we obtain that .

Observe that, given a composite multipartite state , it is rather simple to evaluate the commutator , with no extremization procedure as usually required by QD computation. From the necessary condition for classical states above, we can define a witness for nonclassicality by the norm of the operator . Indeed, if , where

 W=[ρ,ρA1⊗⋯⊗ρAN]. (39)

Therefore, is sufficient for nonclassicality.

For concreteness, we will take as defined by the trace norm, namely,

 ∥W∥=Tr√WW†. (40)

In order to apply the witness in the XY model, we will first consider the case of the thermal (-symmetric) ground state. For this case, a plot of as a function of is provided in Fig. 11 for . Observe that this plot shows that the only possible classical state appears for . Indeed, this point corresponds to an infinite transverse magnetic field applied, which leads the system to a product state, with all spins individually pointing in the direction.

However, besides large , the ground state with SSB exhibits a further nontrivial factorization (product state) point at This point identifies a change in the behavior of the correlation functions decay, which pass from monotonically to an oscillatory decay [62]. Besides, it has been realized that, at this point, the product ground state is two-fold degenerated even for finite systems [79, 80, 36].

On the other hand, if SSB is taken into account, which must be the case in the thermodynamic limit, this picture dramatically changes. Indeed, in the broken case, the two-spin reduced density matrix is more complicated (see Eqs. (16), (18)). Tight lower and upper bounds for and are obtained in Ref. \refciteOliveira. In our plots, either of such bounds essentially yields the same curves, which led us to keep the results produced with the lower bound. Concerning the results for , we can compare the plots for the thermal ground state and the ground state with SSB also in Fig. 11. Observe that, in the case with SSB, the only point for which is . Indeed, this corresponds to a classical state, as can be confirmed by the exact computation of QD. Since our witness identifies classical-classical states, we adopted the symmetric version of QD [40, 41]. Remarkably, this classical state is associated with one of two doubly-degenerated factorized (fully product) ground state, which appears as a consequence of the SSB. This unique classical point for the XY model is then clearly revealed by the witness evaluation. Another aspect of the witness observed from Fig. 11 is that it displays nonanalyticity in its first derivative at the QPT . This phenomenon occurs both for the symmetric and broken ground states, which promotes to a simple tool also to detect QPTs.

7 Conclusion

We reviewed on the two-spin QD in the ground state of spin chains of the XYZ type in a transverse field. Close to QPTs, the anomalies of QD reflect the universality class of the Hamiltonian of the system. Therefore we can conclude that the scaling of QD and entanglement is identical with the known scaling of correlations of standard many body theory. In contrast, the range of quantum correlations at the critical point is non universal: entanglement is typically short-ranged (with scaling laws depending on the microscopic details of the system), while QD decays algebraically (similarly as the standard correlation functions) following the universality paradigm [24]. In particular, we note that QD is generically more robust than entanglement. This ultimately arises because QD catches quantum correlations in separable (mixed) state, where entanglement is vanishing. Therefore, at zero temperature, the QD can be sensible at distances were entanglement is vanishing. Along similar lines of reasoning, QD can survive at temperatures where the states are mixed enough to kill the entanglement (see Ref. \refciteWerlang:12 in this special issue). The investigation of multipartite measures of quantum correlations [41, 81] as well as their impact in the characterization of QPTs is certainly a future challenge. Entanglement is believed entering in its multipartite form close to the QPT; the two-spin entanglement is then a small amount of the entanglement at disposal, ultimately because of the monogamy property  [32, 83]. On the other hand, it has already been shown that QD does not obey the usual monogamy relationship [84, 85, 86]. However, a generalized monogamy constraint can be established [87]. The implications of such monogamy for the behavior of multipartite QD at QPTs is object of current research. For studies of multipartite QD in spin models, see Refs. \refciteRulli:11,Campbell:11. Moreover, the application of genuine multipartite measures of QD (e.g., as defined by Refs. \refciteBraga:12,Maziero-2:12,Giorgi:12) is also a further promising topic.

At the factorizing field, the correlations do not depend on the spatial distance. Therefore all the QDs for different ranges crosses at (see the inset of Fig.8). The actual value of the discords is finite for thermal ground states and it is vanishing for ground states with symmetry breaking. This results since the factorized thermal states are mixed; in the case of symmetry breaking the factorized two-spin state is pure, and therefore QD coincides with entanglement (they are both vanishing at ). Here we remark that QD is a smooth function close to the factorizing field (in contrast, entanglement displays certain discontinuity), and therefore it can be analyzed easly (see Fig.10b)[36, 37]. In fact the finite size scaling of the factorization phenomenon was achieved through the QD and not through the analysis of entanglement. We finally comment that the role of multipartite entanglement was demonstrated to be negligible close to the factorizing field [54]. Therefore the two-spin entanglement developed so far can provide a complete enough scenario of entanglement close to .

To conclude, we mention that QD is also interesting from the experimental perspective. Pairwise non-classical correlations in highly mixed states were experimentally measured in NMR systems [91]. In that experimental protocol non linear witness operators (see section 6) were employed to simplify the optimization encoded in the discord. Interestingly enough the QD can be easly evaluated for spin cluster. This may be important to measure the non-classical correlation via neutron scattering[92].

Acknowledgements

We thank A. Hamma, D. Rossini, B. Tomasello, J. Maziero, L. Céleri, R. M. Serra, A. Saguia, and C. C. Rulli for discussions. Financial support from the Brazilian agencies CNPq (to M. S. S. and T. R. O.) and FAPERJ (to M. S. S.) is acknowledged. This work was performed as part of the Brazilian National Institute for Science and Technology of Quantum Information (INCT-IQ). The research of L. A. was supported in part by Perimeter Institute for Theoretical Physics. Research at Perimeter Institute is supported by the Government of Canada through Industry Canada and by Province of Ontario through the Ministry of Economic Development Innovation.

Footnotes

1. This, in turn, might be important at low temperature, because indeed thermal fluctuations act as infrared cutoff in the system (the QD close to QPT at low temperature is reviewed in Ref. \refciteWerlang:12 of this focus issue; see also Refs. [26, 27, 28, 29, 30]).
2. It is remarkable that the factorization occurs for -dimensional spin system on a bipartite lattice and for finite range exchange interaction even in presence of frustration [4, 5] as the field is varied across the factorizing field , yielding (12) where , and , indicating the range of interaction among the spins.
3. For the quantum Ising model the reduced density operator with could be accessed analytically, in principle. Nevertheless, we resort to DMRG because the is given in terms of integral contour formulas [78].
4. In contrast, bipartite entanglement is not affected by SSB around the quantum critical point. SSB slightly influences bipartite entanglement only for . However, multipartite entanglement is highly affected [35].
5. Accordingly, the fidelity (which can detect both symmetry breaking and non-symmetry breaking QPTs), is a smooth function at .

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