Separability conditions and limit temperatures for entanglement detection in two qubit Heisenberg XYZ models

We examine the entanglement of general mixed states of a two qubit Heisenberg XYZ chain in the presence of a magnetic field, and its detection by means of different criteria. Both the exact separability conditions and the weaker conditions implied by the disorder and the von Neumann entropic criteria are analyzed. The ensuing limit temperatures for entanglement in thermal states of different XYZ models are then examined and compared with the limit temperature of the symmetry-breaking solution in a mean field type approximation. The latter, though generally lower, can also be higher than the exact limit temperature for entanglement in certain cases, indicating that symmetry-breaking does not necessarily entail entanglement. The reentry of entanglement for increasing temperatures is also discussed.


I. INTRODUCTION
Entanglement is one of the most distinctive features of quantum mechanics, representing the ability of composite quantum systems to exhibit correlations which have no classical analogue. Recognized already by Schrödinger [1], it has recently become the object of intensive research due to the key role it plays in the field of quantum information [2][3][4][5][6]. Rigorously, a mixed state ρ of a bipartite system is said to be separable or classically correlated [7] if it can be expressed as a convex combination of uncorrelated densities, i.e., ρ = ν q ν ρ ν A ⊗ ρ ν B , where ρ ν A , ρ ν B are mixed states of each subsystem and q ν are non-negative numbers. Otherwise, ρ is entangled or inseparable. When separable, ρ satisfies all Bell inequalities as well as other properties characteristic of classical systems.
A pure state ρ = |Φ Φ| is separable just for tensor product states |Φ = |φ A |φ B , but in the case of mixed states, like thermal states ρ ∝ exp[−H/T ], with H the system Hamiltonian, it is in general much more difficult to determine whether ρ is separable or not. Only in special cases, like a two-qubit or qubit+qutrit system, simple necessary and sufficient conditions for separability are known [8,9]. Moreover, the entanglement of formation of a mixed state [10] has been explicitly quantified only for a two-qubit system [11]. Nonetheless, it is known that any mixed state becomes separable if it is sufficiently close to the fully mixed state [12,13]. For thermal states of finite systems, this implies that a finite limit temperature for entanglement [14], T e , will always exist such that ρ becomes separable ∀ T ≥ T e . It is then interesting to analyze if it is possible to estimate this temperature with simple separability criteria, and how it is related to the critical temperature T c of the symmetry-breaking solution in a mean field type approximation, which is the conventional starting point for describing interacting manybody systems. Such solutions (i.e., like deformed or superconducting) normally reflect the presence of strong correlations and collective behavior.
The aim of this work is to examine these issues in a sim-ple yet non-trivial model where the exact entanglement conditions and quantification can be easily obtained. For this purpose, we will consider a system of two qubits interacting through a Heisenberg XY Z Hamiltonian [15] in the presence of an external magnetic field. Interest in this model stems from the potential use of Heisenberg spin chains for gate operations in solid state quantum computers [16,17]. The pairwise entanglement of thermal states of isotropic [18,19] and anisotropic XY [20][21][22] Heisenberg models have accordingly been recently studied, and several interesting features have appeared already in the two-qubit case [21], like the possibility of entanglement reentry for increasing temperatures or magnetic fields.
We will first review the exact separability conditions for general mixtures of the eigenstates of arbitrary XY Z Hamiltonians, examining in particular thermal states and the possibility of entanglement reentry. We will also analyze the weaker conditions provided by the disorder criterion [23], which is the strongest one based just on the spectrum of ρ and one of its reductions, and is hence more easy to implement in general than other criteria. Violation of the disorder conditions also ensures distillability [24]. These conditions are here exact in the absence of a magnetic field. Although the disorder criterion admits a generalized entropic formulation [25], it is stronger than the von Neumann entropic criterion [26], based on the same information, whose predictions will also be analyzed. The ensuing exact and approximate limit temperatures for entanglement in thermal states of different XYZ models will then be examined.
Finally, we will discuss the mean field (i.e., independent qubit) approximation for thermal states, with the aim of comparing the previous limit temperatures with the corresponding mean field critical temperature T c . It will be shown, remarkably, that for T > 0, symmetrybreaking is not necessarily a signature of entanglement, so that T c may be higher than T e , although it is usually lower. The model and methods are described in sec. II, while three different examples are analyzed in detail in sec. III. Conclusions are finally drawn in IV.

A. Model and separability conditions
We will consider a Heisenberg XY Z chain [15] for two qubits in an external magnetic field b along the z axis. Denoting with S = s A + s B the total spin of the system, the corresponding Hamiltonian can be written as The ferromagnetic (antiferromagnetic) case corresponds to v i ≥ 0 (≤ 0), and the standard XY model to v z = 0. Its normalized eigenstates H|Φ j = E j |Φ j are given by with ∆ = v − 1 + b 2 /v 2 − , u ± = 1 ± b/∆ and |± ± ≡ |± |± the separable eigenstates of S z (standard basis). The states |Φ 0,3 are maximally entangled, whereas |Φ 1,2 are entangled for v − = 0, with concurrence v − /∆ (see Appendix). They become maximally entangled for b = 0, in which case the set of states (2) is just the Bell basis.
We will first consider general statistical mixtures of the previous eigenstates, which can be written as where p j ≥ 0, 3 j=0 p j = 1 and with O ≡ TrρO. Eqs. (3) comprise standard thermal states as well as those arising in more general statistical descriptions [27,28], and represent the most general two-qubit state with good permutational and phase flip symmetry U = −e iπSz real in the standard basis. The two-site density matrix of an N qubit XY Z chain with cyclic boundary conditions is in fact also of this form [22].
Exact separability conditions. For the state (3), they can be most easily determined with the Peres criterion [8], sufficient for two qubits [9], and can be cast as or, in terms of the averages imposing bounds on the averages of the last two terms in (1b). If ρ is entangled, only one of Eqs. (5) is violated, and its concurrence is given precisely by the difference between the left and right hand sides of the broken inequality (see Appendix). The entanglement arises essentially from one of the states , if ρ is sufficiently close to the fully mixed state. If b = 0, ρ is diagonal in the Bell basis and Eqs. (5) reduce accordingly to p j ≤ 1/2 ∀ j [26], while Eqs. (6) to 1 ≤ S 2 ≤ 1 + 2 S 2 i for i = x, y, z, as S z = 0. Disorder and entropic separability conditions. The disorder criterion [23] states that if ρ is separable, ρ is majorized by the reduced densities ρ A,B ≡ Tr B,A ρ, which means that ρ is more mixed (i.e., disordered) than ρ A , ρ B . In a two qubit system, this implies that the largest eigenvalue of ρ should not exceed that of ρ A and ρ B , which is in general a necessary condition that becomes sufficient when ρ is pure or diagonal in the Bell basis [23,25].
For the state (3), ρ α = 1/2 + S z s α z for α = A, B, and the disorder criterion leads to the inequalities which in terms of total spin averages can be recast as Eqs. (7) or (8) are clearly less stringent in general than Eqs. (5) or (6), but become exact for b = 0 ( S z = 0), i.e., when ρ is diagonal in the Bell basis.
The standard entropic criterion [26], based on the von Neumann entropy S 2 (ρ) = −Tr ρ log 2 ρ, states that if ρ is separable, S 2 (ρ) ≥ S 2 (ρ α ) for α = A, B. Although exact for pure states (in which case S 2 (ρ) = 0 and S 2 (ρ A ) = S 2 (ρ B ) is just the entanglement of ρ [11]), for mixed states it is in general weaker than the disorder criterion [25], except when both ρ and ρ α have rank two. Fig. 1 depicts, for p 1 = 0 and b/v − = 1, the regions where the state (3) is entangled and where entanglement is detected by the disorder and the standard entropic criteria.
and determine a finite limit temperature for entanglement T e , such that they are satisfied ∀ T ≥ T e . Nonetheless, entanglement, as measured by the entanglement of formation or concurrence, may not be a decreasing function of T for T < T e when the ground state is less entangled than the first excited state [18][19][20][21], and even entanglement vanishing plus reentry may occur [21], as discussed below. Note that the spectrum of H, and hence Eqs.  Let us consider for instance a mixture of |Φ 2 and |Φ 3 (p 0 = p 1 = 0 in (3a)), which corresponds to the outer border in fig. 1. This state is separable just for (9) will approximately be of this form for low T if E 2 and E 3 are sufficiently close and well below the remaining levels. Hence, if E 2 < E 3 , C(ρ) will initially decrease as T increases from zero, vanishing at the temperature where p 2 = p c , but will exhibit a reentry for T > T r , with Eq. (10a) [(10b)] broken for T < T r (T r < T < T e ). Due to the remaining levels, C(ρ) will actually vanish in a small but finite temperature interval around T r (see case 3 in sec. III). When |Φ 2 becomes separable (v − /b → 0), p c → 1 and T r → 0, whereas when it becomes maximally entangled (b/v − → 0), p c → 1/2 and T r → ∞, so that no reentry takes place in this limit. In constrast, if E 3 < E 2 , p 2 < 1/2 ≤ p c ∀ T ≥ 0 and no reentry or enhancement of C(ρ) can take place. Nor can it occur for a mixture of |Φ 0 and |Φ 3 or |Φ 1 and |Φ 2 , since they are separable just for equal weights, as seen from Eqs. (5). For the state (9), the disorder conditions (7) become e −βvz sinh |βv + | ≤ cosh(β∆) + | b ∆ | sinh |β∆| , (12b) and lead to a lower limit temperature for entanglement detection, T d e ≤ T e , with T d e = T e just for b = 0. The entropic criterion leads to an even lower limit temperature T s e ≤ T d e . The reentry effect cannot be detected by the disorder (and hence by the entropic) criterion. Violation of Eqs. (7) requires p j > 1/2 for some j, so that in the thermal case just the entanglement arising from the ground state can be detected. For a mixture of |Φ 2 and |Φ 3 , Eqs. (7) are broken just for p 2 > p d = (2 − |b/∆|) −1 or p 2 < p ′ d = (2 + |b/∆|) −1 (see Fig. 1), which does not allow to detect the reentry when E 2 < E 3 since p ′ d ≤ 1/2.

B. Symmetry-breaking mean field approximation
The thermal state (9) represents the density operator that minimizes the free energy In a finite temperature mean field or independent qubit approximation, Eq. (13) is minimized among the subset of uncorrelated trial densities, given in this case by with arbitrary ρ A , ρ B , obtaining thus an upper bound to the minimum free energy. The only way such an approximation can reflect entanglement is through symmetry breaking: the optimum density that minimizes F (ρ mf ) may break some of the symmetries present in the Hamiltonian H, and become degenerate. In these cases a critical temperature T c will exist such that the optimum density becomes symmetry conserving for T ≥ T c . At T = 0, symmetry breaking implies entanglement if the ground state of H is non-degenerate, since for pure states separability corresponds to an uncorrelated density. However, this is not necessarily the case for T > 0, where symmetry-breaking just indicates, in principle, that the true thermal state is not uncorrelated. On the other hand, entanglement does not necessarily imply symmetry-breaking either, both at T = 0 or T > 0, as correlations need to be in general sufficiently strong to induce a symmetry-breaking mean field [29]. The densities ρ α , α = A, B, can be parameterized as s α = Trρ α s α = − 1 2 λ α tanh[ 1 2 β|λ α |]/|λ α | , so that Eq. (14) corresponds to an approximate independent qubit Hamiltonian h = α λ α · s α . Minimization of F (ρ mf ) with respect to λ α leads then to the selfconsistent equations (see for instance [30]) where H mf = Tr ρ mf H. A similar equation obviously holds for the n qubit case. In the case (1), Permutational symmetry will be broken if λ A = λ B , and phase flip symmetry if λ α x = 0 or λ α y = 0. The latter has to be broken in order to see any effect from the last two interaction terms in (1b) at the mean field level, since otherwise their mean field averages vanish. In such a case the sign of one of the λ α x (or λ α y ) remains undetermined, giving rise at least to a two-fold degeneracy.
For instance, in the ferromagnetic case v i ≥ 0, H mf is minimum for s A = s B and permutational symmetry needs not be broken. Hence, λ A,B = λ. Defining T c decreases as χ increases, with T c → 0 for χ → 1 and This solution is insensitive to v m . As discussed in sec. III, T c is usually lower than T e , but can also be higher. For example, if b = 0 and v x > v y = v z > 0, T c = v x /2, but the ensuing exact thermal state, diagonal in the Bell basis, is separable ∀ T > 0 (T e = 0), as the ground state is degenerate (E 2 = E 3 − v x /2) and hence p j ≤ 1/2 ∀ j, T .

III. EXAMPLES
We now examine in detail the previous limit temperatures in three different cases. We set in what follows b ≥ 0, v ± ≥ 0, since the concurrence and limit temperatures are independent of their signs.
1) v − = 0, v + > 0 (XXZ model). The states |Φ 1,2 are in this case separable, with ∆ = b in Eq. (2). Entanglement can then only arise through the violation of Eq. (5b), i.e., Eq. (10b) in the thermal case, which is now independent of the magnetic field b. If v + > v z , the thermal state (9) will then be entangled for any b if T > 0, up to a limit temperature T e that is independent of b. However, the ground state is |Φ 3 if b < b 0 ≡ v + − v z and |Φ 2 if b > b 0 , so that for b > b 0 , ρ becomes entangled only at finite temperature T > 0, in agreement with Eq. (11) (T r → 0 for v − /b → 0). On the other hand, if v + < v z , no entanglement occurs at any temperature. These features can be appreciated in Fig. 2 for v z = 0 (XX model), where b 0 = v + and [20] T e = αv + , α = 1/ ln[1 + The disorder criterion can now detect entanglement just through the violation of Eq. (7) for j = 0, 3, i.e., Eq. (12b) in the thermal case, which can occur only for b < b 0 , i.e., when |Φ 3 is the ground state. The entanglement arising for T > 0 when b > b 0 cannot be detected. In addition, the limit temperature T d e determined by (12b) will depend on b, decreasing as b increases and vanishing for b → b 0 . Its behavior for v z = 0 is shown in the central panel of fig. 2 The limit temperature T s e of the entropic criterion is still lower. For v z = 0 and b → 0, T s e → 0.478v + , with C[ρ(T s e )] → 0.584. In this case the ground state critical field b 0 coincides with the mean field critical field b c . Hence, a stable symmetry breaking mean field solution is here feasible just The corresponding exact limit temperature for entanglement (constant) and the limit temperatures below which entanglement is detected by the disorder and by the von Neumann entropic criterion. The critical temperature for the symmetry-breaking mean field approximation is also shown. Bottom: The concurrence at the previous limit temperatures.
for b < b 0 , with T c given by Eq. (18) with v M = v + . Since now [H, S z ] = 0, this "deformed" solution breaks the rotational invariance around the z axis and possesses, accordingly, a continuous degeneracy. As seen in Fig. 2, for v z = 0 T c is much lower than T e , lying actually quite close to the entropic limit temperature T s e . For b → 0, T c → v + /2, with C[ρ(T c )] → 0.55. Note, however, that for b → b 0 , T c > T d e due to the logarithmic vanishing of T c in this limit, where C[ρ(T c )] → 1/2.
2) v − > 0, v + = 0. This case of maximum anisotropy (v x = −v y ) represents, for v z = 0, the two-qubit version of the standard Lipkin model, widely employed in nuclear physics to test symmetry-breaking mean field based descriptions [29]. It describes the interplay between a single particle term bS z and a monopole interaction that induces a deformed mean field. The states |Φ 1,2 are now entangled, whereas the states |Φ 0,3 become degenerate. Hence, in the thermal case p 3 = p 0 , and entanglement can only arise from the states |Φ 1,2 , i.e., through the violation of Eq. (5a) (Eq. (10a) in the thermal case). This requires ∆ > −v z , i.e., that |Φ 2 be the ground state.
The limit temperature T e determined by (10a) depends now on the field b, with ρ entangled for 0 ≤ T < T e and C(ρ) a decreasing function of T , as seen in Fig. 3 for with T e ≈ αv − [1 + 1 A remarkable feature is that as b increases, T e now increases, even though the entanglement of |Φ 2 decreases, since the energy gap ∆ between the ground and the first excited states increases. Moreover, for b → ∞, T e ≈ b/ ln(2b/v − ) → ∞, being then possible to make ρ entangled at any temperature by increasing the field.
For p 0 = p 3 , entanglement will be detected by the disorder criterion through the violation of Eq. (7) for j = 1, 2, i.e., of Eq. (12a). For v z = 0, this will occur for any value of b but below the lower limit temperature , becoming then also infinite in this limit. Hence, the emergence of entanglement for large fields is also detected (since it is a ground state effect) but above a higher threshold. Note also that T d e /T e ≥ 1/2 ∀ b, with C[ρ(T d e )] ≤ 0.33. The limit temperature of the entropic criterion lies very close to T d e for b → ∞ (as T d e /∆ → 0 in this limit) but becomes smaller as b decreases, with T s e → 0.478v − for b → 0. For v z = 0, a phase flip symmetry breaking mean field solution becomes here feasible only for b < b c = v − . For b > b c , ground state correlations, though non-vanishing, are not strong enough to induce a symmetry-breaking mean field, so that the entanglement effect for large fields cannot be captured by the mean field. The permutationally invariant solution corresponds to λ x = 0 and λ y = 0, so that the critical temperature is given again by Eq. (18) with v M = v − . Hence, T c → v − /2 for b → 0, lying again very close to T s e in this limit, while . This is the case with finite anisotropy γ = v − /v + > 0, where entanglement vanishing plus reentry may occur as T increases. For v + > v z ≥ 0, the two lowest states are |Φ 2 and |Φ 3 , with For b above but close to b 0 , Eq. (10a) [(10b)] will be broken for 0 ≤ T < T − r (T + r < T < T e ), with T − r < T + r . Hence, as T increases from zero, the concurrence will first decrease, vanishing for T ∈ [T − r , T + r ], but will exhibit a reentry for T > T + r , vanishing finally for T ≥ T e . This behavior is depicted in Fig.4 for v z = 0 and γ = 0.7, where b 0 ≈ 0.71v + and the reentry occurs for b 0 < b < b r ≈ 1.1v + . For b close to b 0 , T − r and T + r are practically coincident and equal to the value given by Eq.
T r e −2v+/Tr ). For b > b r the reentry disappears and T e becomes the continuation of T − r , undergoing then a sharp drop at b = b r . For b → ∞, T e → ∞, as in case 2, while for b → 0, T e → 0.93v + . At fixed T < 0.93v + , entanglement vanishing plus reentry will then also occur as b increases.
As discussed in sec. II, the disorder criterion cannot detect the reentry for increasing T . Instead, the limit temperature T d e vanishes for b → b 0 , as seen in Fig. 4, with (12a) broken for b > b 0 and (12b) for b < b 0 . Nevertheless, T d e → T e /2 for b → ∞, so that the entanglement effect for large fields will be detected, whereas T d e → T e for b → 0, with T e − T d e ∝ b for b → 0. Note also that T s e lies very close to T d e for b 0.55v + , but becomes lower as b decreases, with T s e → 0.39v The ratio T c /v + is then larger than in case 1. For γ = 0.7, T c lies close to T e for b v + , with T c → 0.85v + and C[ρ(T c )] → 0.034 for b → 0. However, the most striking effect is that T c > T e for 1.1 b/v + 1.33, i.e., for b just above the reentry interval. In this region, ρ becomes separable at a low temperature, yet correlations remain strong to induce a symmetry-breaking mean field. On the other hand, for b > b c the ground state remains entangled but correlations are not strong enough to induce symmetry-breaking, as occurs in case 2.

IV. CONCLUSIONS
We have examined the exact and the disorder separability conditions for general mixed states of two qubits interacting through a general XY Z Heisenberg Hamiltonian, which can be succinctly expressed in terms of total spin expectation values. The disorder conditions are exact in the absence of a magnetic field, but become weaker as the field increases and are unable to detect the reentry of entanglement for increasing temperatures in thermal states, an effect which may here arise when the ground state is less entangled than the first excited state. The von Neumann entropic criterion leads to still lower limit temperatures and is not exact even for zero field. Nonetheless, both the disorder and entropic criteria do predict the increase in the limit temperature for large fields occurring in anisotropic models.
The critical temperature for the symmetry-breaking mean field solution is normally also lower than the exact limit temperature for entanglement in the examples considered and always vanishes for sufficiently large fields. However, it can also be higher, particularly when the lowest energy levels are close and entangled, implying that such solutions, normally regarded as signatures of the presence of strong correlations in the system, are not rigorous indicators of entanglement for T > 0. It is well known that in small systems, the sharp thermal mean field transitions are to be interpreted just as rough indicators of a smooth crossover between two regimes. The concept of entanglement allows, however, to formulate a crossover precisely. Finite systems regain in this sense a critical-like behavior for increasing T , becoming classically correlated (but not uncorrelated) for T ≥ T e , and with an entanglement undetectable through the eigenvalues of ρ and one of its reductions for T d e ≤ T < T e .