Discord and information deficit in the XX chain

We examine the quantum correlations of spin pairs in the cyclic $XX$ spin 1/2 chain in a transverse field, through the analysis of the quantum discord, the geometric discord and the information deficit. It is shown that while these quantities provide the same qualitative information, being non-zero for all temperatures and separations and exhibiting the same type of asymptotic behavior for large temperatures or separations, important differences arise in the minimizing local measurement that defines them. Whereas the quantum discord prefers a spin measurement perpendicular to the transverse field, the geometric discord and information deficit exhibit a perpendicular to parallel transition as the field increases, which subsists at all temperatures and for all separations. Moreover, it is shown that such transition signals the change from a Bell state to an aligned separable state of the dominant eigenstate of the reduced density matrix of the pair. Full exact results for both the thermodynamic limit and the finite chain are provided, through the Jordan-Wigner fermionization.


I. INTRODUCTION
The investigation of quantum correlations in mixed states is presently attracting strong attention [1]. While in bipartite pure states such correlations can be identified with entanglement, it was recently recognized that separable (non-entangled) bipartite mixed states, defined as states which can be created by local operations and classical communication, and which are therefore convex mixtures of product states [2], may still exhibit useful quantum correlations, stemming from the non-commutativity of the different products. The mixed state based quantum algorithm introduced by Knill and Laflamme (KL) [3] has shown that an exponential speed-up over classical algorithms can in fact be achieved without entanglement [4], in contrast with the case of pure states [5].
This has turned the attention to alternative measures of quantum correlations for mixed states, such as the quantum discord [1,[6][7][8], which are able to capture the quantumness of such mixed states, vanishing just for states diagonal in a product basis and coinciding with entanglement in the pure state limit. A finite discord between the control qubit and the remaining maximally mixed qubits was in fact found in the KL algorithm [9], renewing the interest on this measure [10][11][12][13][14]. Other measures with similar properties include the closely related one-way information deficit [1,15,16], the geometric discord [17], which allows an easier evaluation, and the generalized entropic measures of ref. [18], which include the previous ones as particular cases. Various applications and operational interpretations of the quantum discord and related measures were recently provided [1,10,16,[19][20][21][22][23][24]. We remark that all these measures require a minimization over local measurements in one of the constituents (which can be viewed as the determination of the least disturbing local measurement [25]), which makes their evaluation difficult in systems with high dimensionality.
Spin chains provide an interesting scenario for studying these measures and their relation with criticality [1,[26][27][28][29][30][31][32][33][34][35][36]. In particular, the state of a spin pair in an entangled ground state (GS) is in general a mixed state, entailing that differences between the entanglement and the quantum discord of a spin pair will arise already at zero temperature [26,28,30]. These differences become significant in the exact ground states of finite XY chains for transverse fields lower than the critical field B c [30], with the quantum discord reaching full range in this region.
The aim of the present work is to analyze in detail the behavior of the quantum discord, the geometric discord and the one-way information deficit of spin pairs in chains with XXtype first neighbor couplings in a transverse field, at both zero and finite temperature.
Such model is particularly interesting for both quantum information and condensed matter physics, exhibiting distinct features such as eigenstates with definite magnetization along the field axis and a special critical behavior [37]. It is first shown that in contrast with entanglement [38][39][40], discord-type measures exhibit common features such as a non-zero value for all separations L at all temperatures T > 0. Exact asymptotic expressions for the decay with L and T will be provided, on the basis of the exact treatment based on the Jordan-Wigner fermionization [39][40][41][42]. Nonetheless, we will also show that important differences between the quantum discord on the one side, and the geometric discord and information deficit on the other side, do arise in the minimizing local spin measurement. While in the quantum discord the latter is always orthogonal to the transverse field (even at strong fields if T > 0), in the geometric discord and information deficit it exhibits a perpendicular to parallel transition as the field increases, at a field lower than the T = 0 critical field B c . Such transition in the minimizing measurement is present at all temperatures and separations, and as will be shown, is a signature of the transition from a Bell state to a separable aligned state of the dominant eigenstate of the reduced density matrix of the pair. This difference indicates a distinct response of the minimizing measurement in these quantities to the onset of quantum correlations.
In Section II we summarize the main features of the previous measures, including the equations that determine the stationary local measurements. The application to the spin 1/2 XX chain is made in section III, where we first discuss some general properties of these measures in this model and show that spin measurements parallel and perpendicular to the field are always stationary. We then consider in detail the thermodynamic limit and the finite case. Details of the exact calculation are provided in the Appendix. Conclusions are finally given in IV.

II. DISCORD AND GENERALIZED INFORMATION DEFICIT
Let us consider a bipartite quantum system A+B initially in a state ρ AB . A local complete projective measurement M B on system B is defined by a set of orthogonal projectors Π B j = The state of the total system after an unread measurement of this type becomes In [18,25] we considered the minimum generalized information loss by such measurement, where S f (ρ) = Tr f (ρ) denotes a general entropic form, with f a smooth strictly concave function for p ∈ [0, 1], satisfying f (0) = f (1) = 0 [43]. Eq. (2) satisfies I B f ≥ 0 for any such f , becoming the generalized entanglement entropy S f (ρ B ) = S f (ρ A ) in the case of pure states (ρ 2 AB = ρ AB ). However, it can be non-zero in separable mixed states, vanishing just for states which are already of the form (1) [18], i.e., states which remain unchanged after the local measurement M B and are hence diagonal in a product basis {|i j A ⊗ |j B }.
If f (ρ) = −ρ log 2 ρ, S f (ρ) becomes the von Neumann entropy S(ρ) and Eq. (2) becomes the one way information deficit [1,15,16], which we will denote as I B 1 . It can be rewritten in terms of the relative entropy [44,45] S(ρ||ρ ′ ) = −Tr ρ(log 2 ρ ′ − log 2 ρ) as [18] For a pure state, I B 1 becomes the standard entanglement entropy S(ρ A ) = S(ρ B ). If f (ρ) = 2ρ(1 − ρ), S f (ρ) becomes the so called linear entropy S 2 (ρ) = 2(1 − Tr ρ 2 ) and Eq. (2) becomes where ||O|| 2 = Tr O † O and the last minimization can be extended to any state of the general form (1). It is then seen that (4) is proportional to the geometric discord introduced in [17], defined as the square of the minimum Hilbert-Schmidt distance from ρ AB to a classically correlated state with respect to B. For pure states I B 2 becomes the squared concurrence C 2 AB [46], which for such states is just the linear entropy of any of the subsystems [47]. Both measures (3)-(4) can then be regarded as particular cases of the generalized information deficit (2). We may similarly define [25] , q > 0 [48]. I B q reduces to (4) for q = 2 and to (3) for q → 1 (S q (ρ) → S(ρ) in this limit). Normalization of f (ρ) was chosen such that I B f = 1 ∀ S f for a two-qubit Bell state On the other hand, the quantum discord [6,7] for a measurement in B can be written as where S(A|M B ) denotes the conditional von Neumann entropy of A given a measurement  [25,49] Tr which leads to d B (d B − 1) real equations [25]. In the quantum discord (5), an additional is to be added in (6), with f (ρ) = −ρ log 2 ρ [25] (see also [50,51]). Nevertheless, in the case of the geometric discord I 2 , the final equations can be simplified considerably. In particular, for a general mixed state of two qubits where σ = 2s are the Pauli matrices, where λ 1 is the largest eigenvalue of the positive semi-definite 3×3 matrix M 2 = r B r t B +J t J. The minimizing M B is a spin measurement along the direction of the associated eigenvector can also be obtained for this case [25].

III. APPLICATION TO THE XX MODEL
We now consider a chain of N spins s i with first neighbor XX couplings in a uniform transverse magnetic field. The Hamiltonian reads and is obviously invariant under rotations around the z axis, satisfying [H, S z ] = 0, with S z = i s iz the z-component of total spin. Its eigenstates can then be characterized by the total magnetization M along z. The sign of the field B and the coupling strength J can be changed by local rotations e iπs jz at all and even spins j respectively (assuming N even in the cyclic case N + 1 ≡ 1), so that we will set in what follows B ≥ 0, J ≥ 0.
We will examine the spin 1/2 case, where exact results for finite N as well as the thermodynamic limit N → ∞ can be obtained via the Jordan-Wigner fermionization (see Appendix).
We will focus on the cyclic case N + 1 ≡ 1, where pair correlations between spins i and j in the ground state or in the thermal state ρ ∝ exp[−βH] will depend just on the separation For any global state ρ satisfying [ρ, S z ] = 0, the reduced state ρ ij = Tr ij ρ of any pair i = j will commute with s iz + s jz . In the cyclic case, ρ L ≡ ρ ij will then have the form where (10) is the representation in the standard basis and (11) the eigenvector expansion, and The eigenvectors of ρ L in the ground or thermal state will not depend then on the field or separation. For B ≥ 0 and J ≥ 0 in (9), p − L ≥ p + L and α L ≥ 0. The largest eigenvalue of ρ L will then correspond to the Bell state |Ψ + if and to the aligned separable state | ↓↓ if α L < α c L . Hence, in the ground state we may expect as the field decreases a transition from |↓↓ to |Ψ + in the dominant eigenstate of We will see such crossing reflected in the transition exhibited by the geometric discord and the information deficit (but not the quantum discord). We will also find the same effect at finite temperatures.

A. Parallel and perpendicular geometric discord and information deficit
We first discuss the general properties of the discord and information deficit of the states (10). Due to the permutation symmetry of ρ ij , we will omit in what follows the superscript B (i.e., j) in I f and D, as For α L = 0, ρ L is diagonal in the standard basis and will then have zero entanglement and discord: E = D = I f = 0 ∀ S f . It will be, however, classically correlated, being a product state ρ i ⊗ ρ j only when . Quantum correlations will then be driven solely by α L , and will lead to a finite value of D and I f ∀ α L = 0. The geometric discord (4) for such state can be evaluated immediately and reads and the superscript in I 2 indicates the direction of the minimizing local spin measurement (along z if |α L | < α t L and along any orthogonal direction k if |α L | > α t L ). Hence, I 2 increases first quadratically with α L and exhibits then a parallel → perpendicular transition at α L = α t L , corresponding to a transition field B L t . For p − L > p L such transition correlates with that exhibited by the dominant eigenstate of ρ L (Eq. (14)). In (15) is to be contrasted with the concurrence of ρ L , which requires a finite threshold value of α L : Hence, discord-type quantum correlations with zero entanglement will arise for 0 < |α L | ≤ The behavior of the generalized information deficit (2) is similar to that of the geometric discord. For a spin measurement along a vector k forming an angle γ with the z axis, the eigenvalues of the post-measurement state ρ ′ L are, setting δ = s z = (p + L − p − L )/2 and µ, ν = ±1, It is then verified that ∂I γ f /∂γ = 0 at γ = 0 and γ = π/2: Both parallel (γ = 0) and perpendicular (γ = π/2) measurements are always stationary, in agreement with the general considerations of [25]. Intermediate minima may also arise for a general S f , but the essential f . For small α L and δ = 0, the minimum I γ f for any S f will be obtained for γ = 0, with where k f = |f ′′ (p L )| (we assumed here p L = 0). Hence, as α L increases from 0, all I f will exhibit an initial quadratic increase with α L , like the geometric discord.
On the other hand, if δ = 0 (p + L = p − L ), as in the case of zero field in the ground or thermal state, the minimum I γ f for any S f is attained for (15). Hence, all I f 's will in this case exhibit, like the geometric discord, a parallel → perpendicular transition at the same value of α L . Moreover, for p − L > p L , α t L coincides in this case exactly with α c L , i.e, with the value where the dominant eigenstate of ρ L becomes a Bell state.
The same behavior occurs when p ± L = 1 4 ± δ (implying p L = 1 4 ) with α L , δ small, a typical situation to be encountered at high temperatures or large separations. A series expansion Hence we obtain in this case a universal parallel → transverse transition at |α L | = |δ| ∀ S f and L. In other words, all I f behave like the geometric discord in this limit.
In contrast, the minimizing projective spin measurement of the quantum discord D will not exhibit such transition for the present Hamiltonian. We obtain, setting now f (p) = −p log 2 p, Hence, D z ≡ D 0 = I z 1 , but D γ < I γ 1 if | cos γ| < 1 and δ = 0 (however, at zero field, δ = 0 and D γ = I γ 1 ∀ γ, implying D = I 1 ). While both γ = 0 and γ = π/2 are again always stationary, the minimum D γ will be always obtained for γ = π/2 (D = D ⊥ ) for the actual reduced states derived from the ground or thermal state determined by H, directly reflecting the spin-spin coupling in (9) (which involves the spin components perpendicular to the field axis). This will also occur for small α L , since in this limit the actual values of p ± L will correspond to a product state, entailing no preferred direction in D γ for α L = 0. In fact, for small α L and γ = π/2, Eq. (19) leads, for p L = p + L p − L > 0, to which is always smaller than D z = I z f ≈ α 2 L p L ln 2 . Nonetheless, a quadratic increase with α L is also present.

B. The thermodynamic limit
We will first examine the previous quantities in the ground and thermal state of (9) in the large N limit, where we may express all elements of ρ L in terms of the integrals (see appendix) where β = 1/k B T and L = 0, 1, . . ., with g 0 = 1/2 + s z the intensive magnetization. We then obtain with A L the first L × L block of the matrix of elements A ij (i, j = 1, . . . , L). Thus, α 1 = g 1 , with g 0 = ω/π.
Results for I 2 , I 1 , D and the eigenvalues of ρ L are shown in Figs. 1-2 for L = 1 and 3.
It is first verified that while the minimum quantum discord corresponds to D ⊥ ∀ |B| < J, the minimum geometric discord I 2 exhibits, for decreasing B, the expected sharp I z 2 → I ⊥ at this point. In such a case p 1 − p + 1 = p − 1 − p 1 = α 1 , so that α c 1 = α t 1 (Eq. (15)) and hence B L t = B L c for L = 1. At T = 0 we have, explicitly, and this transition occurs at B t ≈ 0.67J, i.e., sin ω = π/2 − ω, corresponding to an intensive magnetization s z ≈ −0.235. It is also seen that I 2 ≥ C 2 ∀ B, i.e., the geometric discord remains larger than the corresponding entanglement monotone.
The behavior of the information deficit I 1 is similar, except that the previous transition is smoothed through a small crossover region 0.55 B/J 0.67 where an intermediate measurement (0 < γ < π/2) provides the actual minimum: As B decreases, the minimizing angle γ increases smoothly from 0 to π/2 in this interval.
For higher separations, the behavior is similar except that values of I f and D are lower and the transition field B L t is shifted towards lower fields, in agreement with the decrease of the field B L c where |Ψ + becomes dominant, as seen in Fig. 2 for L = 3. B L t remains close to B L c but the agreement is not exact. The quantum discord continues to be minimized by a perpendicular measurement ∀ |B| < J. Notice that in this case the concurrence is very Results for large separations L 3 can be fully understood with the small α L , δ expressions (17), (18) and (20). For large L we may neglect g L in p ± L and p L , in which case with η L approaching a finite value as L increases (η L → 0.294 at B = 0, decreasing with increasing B). For sufficiently large L, Eq. (17) then leads to with k f = |f ′′ (p L )| (k f = 4 in I 2 and 1 p L ln 2 in I 1 ). Hence, all I f 's decrease as L −1 for increasing separations L.
or higher for L ≥ 2. Hence, in this limit we obtain, at leading non-zero order, implying that we may directly apply Eqs. (18) and (20). Therefore, I f and D will vanish exponentially with increasing L, i.e., I f , D ∝ (T /J) −2L . Nonetheless, for all I f 's there is still a transition field B L t ∀ T such that I ⊥ f < I z f for |B| < B L t , with B L t decreasing with increasing T and approaching the field B L c for the onset of |Ψ + as the dominant eigenstate of ρ L . The final result for high T derived from Eq. (18) is where k f = |f ′′ (p L )| ≈ |f ′′ (1/4)| and as determined from the condition I ⊥ f = I z f , which coincides in this limit with that derived from the crossing condition (14). Hence, for first neighbors (L = 1) B L t approaches for high T the finite limit J/2, whereas for L ≥ 2 it decreases as (J/T ) L−1 , as verified in the right panel of Fig. 3 for I 2 . In this limit the transition fields B L t approach B L c ∀ I f . For lower T they remain quite close. It is also seen in Fig. 3 that in the case of I 2 , B L t = B L c ∀ T for L = 1, as previously demonstrated.
for high T , where k D ≈ 2 ln 2 . Again, D ⊥ ≈ I ⊥ 1 for B → 0. We finally notice that for T > 0 and strong fields B ≫ J, T , we have g L ≈ e −βB π π 0 e βJ cos ω cos(Lω)dω = e −βB I L (βJ) , where I L (x) denotes the modified Bessel function of the first kind (I L (x) ≈ e x / √ 2πx for x → ∞ while I L (x) ≈ (x/2) L /L! for x → 0). Hence, in this limit g L decreases exponentially with the field, with p L ≈ g 0 and α L ≈ g L . The geometric discord then becomes decreasing as e −2B/T for strong fields and also quite fast with separation if B ≫ T ≫ J On the other hand, I 1 and D will decrease for strong fields as α L (∝ e −B/T ).

C. The finite case
In a finite chain, the exact ground state has a definite discrete magnetization M. Therefore, it will exhibit N transitions M → M + 1 as the field decreases from above B c = J, This state leads to a L-independent rank 2 reduced state ρ L , with p + L = 0, p − L = 1 − 2/N and p L = α L = 1/N in (10). For such state we obtain, if N ≥ 4, in agreement with the thermodynamic limit result (32) (for large N the second critical field ). Note that for where α L > α c , a perpendicular measurement is preferred in both I 2 and I 1 ). In contrast, D is minimized by a perpendicular measurement ∀ N, with for large N (though D ⊥ ≈ D z = I z 1 at leading order). For lower fields, it is seen that for small L ≥ 2, I 2 is maximum at the parallel- with the fields B L c where dominant eigenstate changes from the Bell state to an aligned state, for all separations L = 1, . . . , N/2. For L = 1 there is again almost exact coincidence between both fields for all T , since the deviation from the thermodynamic limit condition (25) is small. For larger L the agreement is not exact for low T , but they become again coincident for high temperatures ∀ L, where deviations from the thermodynamic limit results become small.

IV. CONCLUSIONS
We have examined the behavior of discord-type measures of quantum correlations for the case of spin pairs in the cyclic XX chain. Their behavior is substantially different from that of the pair entanglement, acquiring at T = 0 non-zero values for all pair separations L if B < B c and decaying only as L −1 for large L. Moreover, they remain non-zero for all temperatures, decaying as T −2L for sufficiently high T . Thus, they all exhibit the same "universal" asymptotics, independently of the particular choice of entropic function in I f .
It can then be most easily accessed through the geometric discord, which offers the simplest evaluation. The ensuing picture is, consequently, quite different from that exhibited by the pair entanglement [39], which, although reaching full range in the immediate vicinity of B c , is appreciable just for the first few neighbors, as seen in Fig. 5, and strictly vanishes beyond a low limit temperature. Hence, critical systems like the XX chain seem to offer vast possibilities for discord-type quantum correlations between close or distant pairs.
The second important result is that in spite of the similar behavior, these measures exhibit substantial differences in the minimizing local spin measurement that defines them. The quantum discord, which minimizes a conditional entropy, always prefers here measurements along a direction orthogonal to the transverse field, even if correlations are weak (i.e., large L, high T or strong fields B if T > 0). The information deficit-type measures, which evaluate the minimum global information loss due to such measurement and include the geometric discord and the one-way information deficit, exhibit instead a transition in the optimum measurement, from perpendicular to parallel to the field as the latter increases, present for all pair separations and at all temperatures. Such difference was previously observed in certain two-qubit and two-qutrit states [25,49].
In the present model such behavior is a signature of the transition exhibited by the dominant eigenstate of the reduced state of the pair, which changes from a maximally entangled state to a separable state in the immediate vicinity of the measurement transition.
Hence, the latter reveals an actual relevant change in the structure of the reduced state.
Moreover, for contiguous pairs and in the case of the geometric discord, both transitions occur exactly at the same field, at all temperatures. For general separations there is also exact agreement between both fields at high T , for all measures I f . In the finite chain the T = 0 measurement transition coincides of course with one of the ground state magnetization transitions M → M +1. These results indicate that the "least disturbing" local measurement optimizing these quantities can be significantly different from that minimizing the quantum discord, even though they coincide exactly in some regimes, being essentially affected by the main component of the reduced state. Its changes can then be used to characterize different quantum regimes, even when entanglement is absent.
The authors acknowledge support from CONICET (LC, NC) and CIC (RR) of Argentina.

Appendix A: Exact solution of the cyclic chain
We give here a brief summary of the method employed for obtaining the exact solution of the cyclic XX chain for both finite N and the thermodynamic limit, at both 0 and finite T [39]. Through the Jordan-Wigner transformation [41], and for each value σ = ±1 of the S z parity P z = exp[iπ(S z +N/2)], the XX Hamiltonian can be mapped exactly to the fermionic where N + 1 ≡ 1 and c j , c † j denote fermion annihilation and creation operators. Eq. (A1) can be solved exactly through a discrete Fourier transform c † j = 1 √ N k∈Kσ e iω k j c ′ † k to fermion operators c ′ k , where ω k = 2πk/N and k is half-integer (integer) for σ = 1 (−1), i.e., K σ = {−[ N 2 ] + δ σ , . . . , [ N −1 2 ] + δ σ } with [. . .] the integer part and δ 1 = 1 2 , δ −1 = 0. We then obtain The 2 N energies are then k∈Kσ λ k (N k − 1/2), where N k = 0, 1 and σ = (−1) k N k . The single fermion energies λ k depend on the global parity σ and are degenerate (λ k = λ −k ) for |k| = 0, N/2. A careful comparison of the ensuing levels leads to the critical fields (38).
The exact partition function Z of the spin system corresponds to the full grand-canonical ensemble of the fermionic representation. However, due to the parity dependence of the latter, this requires a (fermion) number parity projected statistics [39]. Z can then be written as a sum of partition functions for each parity, where P σ = 1 2 (1+σP z ) is the projector onto parity σ and Z σ ν = e βBN/2 k∈Kσ (1+(−1) ν e −βλ k ) for ν = 0, 1. The thermal average of an operator O can then be written as where s j± = s jx ± is jy and A L is the L × L matrix of elements (A L ) ij = 2g i−j+1 − δ i,j−1 . All elements in (10) can then be analytically evaluated.
For N → ∞ and finite separations L, we can ignore parity effects and directly employ Wick's theorem in terms of the final averages g L = c † i c j , with sums over k replaced by integrals over ω ≡ ω k . This leads to Eqs. (21)- (23). When the ground state is non-degenerate, Eqs. (22)-(23) can also be applied for finite N in the T → 0 limit, using the exact contractions g L ≡ c † i c j 0 = 1 N k∈Kσ N k cos(Lω k ), with N k = 0, 1 the occupation of level k.