{'fraction_non_alphanumeric': 0.04650388918930161, 'fraction_numerical': 0.01638866935651595, 'mean_word_length': 4.089331075359865, 'pattern_counts': {'":': 0, '<': 15, '<?xml version=': 0, '>': 6, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 2, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Quantum discord in finite XY chains

We examine the quantum discord between two spins in the exact ground state of finite spin 1/2 arrays with anisotropic XY couplings in a transverse field B. It is shown that in the vicinity of the factorizing field B_s, the discord approaches a common finite non-negligible limit which is independent of the pair separation and the coupling range. An analytic expression of this limit is provided. The discord of a mixture of aligned pairs in two different directions, crucial for the previous results, is analyzed in detail, including the evaluation of coherence effects, relevant in small samples and responsible for a parity splitting at B_s. Exact results for finite chains with first neighbor and full range couplings and their interpretation in terms of such mixtures are provided.


I. INTRODUCTION
The great interest on quantum entanglement in recent years was triggered by the key role it played in certain quantum processing tasks like quantum teleportation and superdense coding [1][2][3]. It was also shown to be essential for achieving an exponential speedup over classical computation in pure-state based quantum computation [4]. However, it was recently recognized that in the case of mixed-state based quantum computation, like the "deterministic quantum computation with one qubit" model introduced by Knill and Laflamme [5], an exponential speed up can take place without a substantial presence of entanglement [6]. This has turned the attention to alternative measures of quantum correlations in mixed states, like the quantum discord, introduced by Ollivier and Zurek [7], which can detect those quantum correlations present in certain separable mixed states [8] and hence not captured by the entanglement of formation [9], but still useful and crucial for certain quantum tasks. It was in fact recently shown by Datta, Shaji and Caves [10] that the circuit of [5] does exhibit a finite non-negligible value of the quantum discord between the control qubit and the remaining mixed qubits. Since then, interest on the quantum discord and other alternative measures has grown considerably [11][12][13][14][15] and several studies of their behavior in spin pairs immersed in a spin chain have been performed [16][17][18][19].
The aim of this work is to analyze the quantum discord of spin pairs in the exact ground state of finite spin 1/2 arrays with anisotropic XY or XY Z type couplings in a uniform transverse magnetic field B. The exact ground state of a finite chain will have a definite spin parity and this property will be seen to deeply affect the discord for fields lower than the critical field B c , where we will show that the main results can be interpreted in terms of the discord of mixtures of aligned pairs. Our interest in these systems is motivated in particular by the remarkable factorization phenomenon they can exhibit at a particular finite value B s < B c of the magnetic field [20][21][22][23][24][25][26][27]: At such field, they have an exactly separable (i.e., factorized) ground state. This feature was first discovered in [20] in one-dimensional (1-d) XY chains with first neighbor couplings, and later shown to occur also in more general systems, like 2-d arrays [22], cyclic [24] and general [25,26] arrays with arbitrary range couplings with a common anisotropy and also systems in non-uniform fields [26]. For transverse fields, these separable ground states actually break parity symmetry and are hence degenerate, coinciding B s in a finite system with the last crossing of the two lowest opposite parity levels [24]. A most remarkable related feature is that in the immediate vicinity of B s , pairwise entanglement, though weak, reaches full range [23,24], regardless of the coupling range [24], changing at B s from antiparallel (B < B s ) to parallel (B > B s ) type [23], indicating an "entanglement transition". This suggests the possibility of a non-zero discord between distant pairs at least in the vicinity of B s , with universal features (independence of separation and coupling range). Here we will show that this is indeed the case, and derive also its analytic limits at this field.
Moreover, distributed pairwise entanglement is necessarily weak due to the well-known monogamy property [28,29] associated with entanglement sharing: If in a system of n spins or qubits one spin is strongly entangled with a second spin, it cannot be strongly entangled with any of the remaining spins. This fundamental feature follows from the bound [28,29] j C 2 ij ≤ C 2 i ≤ 1 satisfied by the pairwise concurrences [30] C ij measuring the entanglement between spins i and j, where C i represents the concurrence between i and the rest of the chain. If all pairs are equally entangled, the maximum value that can be reached by C ij is in fact just 2/n [31,32]. In contrast, the quantum discord is not affected by such bound and can simultaneously reach non-negligible finite values between any two spins, as will be seen to occur in the vicinity of B s , leading to a quite different behavior with the applied field and separation in the whole region |B| < B c . The properties of the ground state discord in the vicinity of the separability field in anisotropic XY chains were not discussed in previous references. We will also analyze here finite size effects, which lead to a finite step of the discord at the factorizing field B s and other parity transitions, visible in small chains.
Sec. II discusses the quantum discord and its evaluation in typical reduced states of a spin pair in such chains, describing in detail the case of a mixture of two aligned states, which represents the exact reduced state in the vicinity of B s . Coherence effects in these mixtures, relevant for small chains, are also examined. Sec. III discusses the behavior of the discord of spin pairs with the applied field and separation in finite cyclic XY chains with first neighbor as well as full range couplings, including its interpretation and the differences with the pair entanglement. The appendix discusses the details of the exact definite parity solution of the finite cyclic XY chain. Conclusions are finally drawn in IV.

A. Quantum Discord
The quantum discord (D) is a measure of quantum correlations based on the difference between two distinct quantum generalizations of the classical mutual information, or equivalently, the classical conditional entropy [7,10]. Given a bipartite system A + B in a mixed state ρ AB , and denoting with ρ A = Tr B ρ AB , ρ B = Tr A ρ AB the reduced density operators of each subsystem, D can be expressed as [7]  Here represents the joint state after such measurement if the result is unknown, with ρ ′ is over all sets of local projectors.
The last bracket in (1b) is the direct quantum extension of the classical conditional entropy [3,33] for a system with joint probability distribution p ij and marginal distribution p B j = i p ij ), which, in contrast with the classical case, is not necessarily positive. On the other hand, the first bracket is the conditional entropy S {P B j } (A|B) = j p j S(ρ j ) after a local measurement in system B, where p j = Tr P B j ρ AB is the probability of outcome j and ρ j = P B j ρ AB P B j /p j the state after such outcome. It represents the average lack of information about A after such measurement and is a non-negative quantity. The second term in (1a) is considered a measure of the classical correlations between A and B in [34], the discord measuring then the quantum part of these correlations.
In the case of a pure state (ρ 2 AB = ρ AB ), S(ρ AB ) = 0 while the first bracket in (1b) vanishes for any choice of local projectors, and the discord reduces then to the entanglement entropy [35] In the case of mixed states, however, it does not coincide in general with the entanglement of formation [9] (the convex roof extension of the entanglement entropy). While the latter vanishes for any separable state, i.e., for a convex superposition of product states , the discord can be non-zero for these states. The discord vanishes in the case of densities diagonal in an orthogonal product basis with the orthogonal set {|i jA } depending on |j B ), but will not vanish in general for a mixture of non-commuting product states, which is still a separable state. The difference (1) can be shown to be non-negative [7] due to the subtle concavity property of the conditional von Neumann entropy S(ρ AB ) − S(ρ B ) [33].

B. Quantum discord and entanglement of spin pairs in definite parity states
Let us now describe the basic elements to evaluate the discord and entanglement of a spin pair in a typical eigenstate of a finite chain of n spins, where the rest of the spins will play the role of an environment. We will consider spin 1/2 chains with XY Z couplings of arbitrary range in a uniform transverse magnetic field B along the z axis, such that the chain Hamiltonian has the form where s iµ denotes the spin components at site i (in units of ). H commutes with the spin parity operator where σ iz = 2s iz , which changes s iµ to −s iµ ∀ i and µ = x, y. Non-degenerate eigenstates |Ψ ν of H will then have a definite spin parity P z = ±1, a symmetry which will play a fundamental role in our discussion. The reduced density matrix of an arbitrary pair i, j in such eigenstate, will then contain no elements connecting states of opposite parity, commuting therefore with the pair parity P ij z = σ iz σ jz . In the standard basis {|00 , |01 , |10 , |11 } (|kl ≡ |k i |l j , with s iz |k i = 1 2 e iπki |k i , k i = 0, 1), ρ ij will then be of the form where, setting O ≡ tr ρ ij O and s i± = s x ± is iy , with a + b + c + c ′ = 1. We will here consider translationally invariant systems such that s iz = s jz and hence . Moreover, α and β will be real since H can be represented by a real matrix in the standard product basis of the full space. Non-negativity of ρ ij implies |α| ≤ √ ab, |β| ≤ c, with a, b, c non-negative. The internal entanglement of the pair can be measured through the entanglement of formation, which for the case of two qubits can be explicitly calculated as [30] where C is the concurrence [30]. It is given here by The entanglement of the pair is of parallel (antiparallel) type if the first (second) entry is positive [23,24]. Just one of these entries can be positive for a non-negative ρ ij . On the other hand, in order to evaluate the discord of the pair, we need first the eigenvalues of ρ ij , given for , and those of the single spin density matrix, which are obviously We also need to consider a measurement of the spin at site j along an arbitrary axis z ′ defined by the angles γ and φ. The state of the pair after such measurement (Eq. (2)) is where such that s jz ′ |k ′ = 1 2 e iπk |k ′ for s jz ′ = s jz cos γ + s jx sin γ cos φ + s jy sin γ sin φ. For real α, β, the eigenvalues of ρ ′ ij are where ν = ±1, µ = ±1 and |α + β| 2 ≡ α 2 + β 2 + 2αβ cos 2φ, corresponding ν = 1 (−1) to the eigenvalues of the first (second) term in (13). The eigenvalues of We have then all elements to evaluate the difference whose minimum (with respect to θ, φ) is the discord D (Eq. (1)). For αβ ≥ 0, minimization with respect to φ yields cos 2φ = 1 and just the minimization over γ is finally required, which can be restricted to the interval [0, π/2]. The minimum for the pair densities used in sec.
becomes independent of ν and hence degenerate. A general evaluation of the discord for states of the form (6) was recently provided [14].
C. The case of a mixture of two aligned states A particular case of (6) of exceptional interest is that of a statistical mixture of two aligned states along arbitrary directions, not necessarily opposite, such that the local states involved are non-orthogonal. Choosing the z axis as the bisector of the angle between the two directions, we can write this state as where |θ = exp[iθs iy ]|0 = cos 1 2 θ|0 + sin 1 2 θ|1 and (19) is again the standard basis representation. Eq. (18) is the exact reduced state of any two spins in the immediate vicinity of the factorizing field (see sec. III) if coherence terms are neglected. It also provides the basic approximate picture of the pair state in the region |B| < B c .
The response of D and E to the coherence term is shown in Fig. 3. For sufficiently small ε, the correction to D is linear in ε for θ not close to π/2, with the discord increasing (decreasing) for ε < 0 (> 0), while at θ = π/2 the correction is quadratic and positive (at θ = π/2, D = 1 − µ=±1 h( 1+µε 2 ) ≈ 1 2 ε 2 log 2 e). Entanglement, on the other hand, becomes finite as soon as |ε| increases, becoming even larger than the discord for θ close to π/2 (where C = |ε| and E ∝ −ε 2 log ε 2 for small ε). As seen in the bottom panel, at an intermediate θ entanglement remains smaller than the discord just in an interval around ε = 0, becoming slightly larger before reaching the pure limit ε = ±1, where D and E coincide.

III. QUANTUM DISCORD IN XY CHAINS
We have now all the elements to evaluate and understand the behavior of the discord of an arbitrary spin pair in the ground state of a finite chain. We first consider a finite 1-d XY cyclic chain of n spins with first neighbor couplings, where J ij z = 0 and J ij µ = δ j,i±1 J µ , µ = x, y , with n + 1 ≡ 1. The exact solution for finite n can be obtained through the Jordan-Wigner fermionization [36] (see Appendix). The exact ground state will have a definite (field dependent) spin parity and the reduced density of an arbitrary pair will be of the form (6), where the elements (7)-(9) can be evaluated with the expressions of the Appendix. Exact results for the discord and entanglement of pairs are shown in Figs. 4-5 for χ = J y /J x = 0.5 and two different sizes n. For a first neighbor coupling and even n, the sign of J x can be changed by a local transformation s 2i,µ → −s 2i,µ for µ = x, y, so that both the ferromagnetic (J x > 0) and antiferromagnetic (J x < 0) cases at fixed χ will exhibit exactly the same entanglement and discord. They are also independent of the sign of B.
It is immediately seen that pair entanglement and discord exhibit significant differences for fields |B| < B c = 1 2 (1 + χ)J x (the critical field of the thermodynamic limit n → ∞). While in this case entanglement practically vanishes at the factorizing field [20,21,[23][24][25] where the chain possesses an exactly separable and degenerate parity-breaking ground state [24] (see Eq. (28)), the discord remains non-zero, reaching in fact its maximum in its vicinity. Besides, entanglement rapidly decreases as the separation L between the spins increases, being nonzero for L > 2 only in the immediate vicinity of B s , where it is very small. In contrast, the discord decreases only slightly with separation for |B| < B c , reaching a saturation value for large L. Moreover, it is strictly independent of L at the factorizing field B s . In order to understand these results, we recall that at B = B s , the uniform parity-breaking separable state where s iz |0 i = − 1 2 |0 i , is an exact ground state (we have here assumed J x > 0; for J x < 0, θ i → (−1) i θ). It is obviously a state with spins fully aligned along an axis forming an angle θ with the original −z axis. Due to parity symmetry, the partner state | − Θ = P z |Θ is also an exact ground state at B s . They can therefore be exact eigenstates of H only when levels of opposite parity cross [24]. The ground state of the present chain undergoes actually n/2 spin parity transitions as the field B increases from 0 (reminiscent of the m/2 S z transitions of the XX limit [37]), the last one precisely at B s . Outside these transitions, the energy gap between the lowest states of each parity is small for |B| B c , but nonetheless finite in a finite chain.
In the immediate vicinity of B s , the exact ground state will therefore possess a definite parity and the correct side-limits at B s will be determined by the states where −Θ|Θ = cos n θ [24]. The reduced state of any two spins derived from (30) will be given precisely by the same mixture (19) if the complementary overlap −Θ n−2 |Θ n−2 = cos n−2 θ is neglected. We can then immediately understand the qualitative difference between entanglement and discord for |B| < B c . As B approaches B s , the discord between any two spins will approach a common finite limit for any separation L, given by Eq. (21) with θ obtained from (29) (D ≈ 0.145 in the case of fig. 4, where θ = π/4 at B s ). In contrast, the pair entanglement will vanish for B → B s (for a negligible complementary overlap) as the state (18) is separable. In Figs. 4-5 we have taken the exact ground state with its correct parity. The non-negligible discord between any two spins for |B| < B c can then be understood in a similar way, as in this region the ground state can be seen, approximately, as a definite parity combination (30) of "mean field" states with broken-symmetry (28), with cos θ = B/J x , plus additional corrections. The reduced state of a spin pair will then be again given essentially by the mixture ρ(θ) (Eq. (18)) plus smaller corrections, with the discord arising principally from ρ(θ) (although corrections are non-negligible; see Fig. 6). Let us remark that the same reduced density (18) arises from the statistical mixture ρ 0 = 1 2 (|Θ + Θ + | + |Θ − Θ − |) if the overlap is discarded. ρ 0 represents the T → 0 + limit of the thermal state of the system at B s . The limit (21) of the discord at B s remains then also valid at sufficiently low T .
On the other hand, for strong fields B ≫ J x , the ground state is essentially the state with all spins fully aligned along the −z axis plus small perturbative corrections. As seen in Fig. 4, the discord in this region is rather small and decreases rapidly with separation, since the previous superposition effects are no longer present. Moreover, the discord between first neighbors is very close to the entanglement of formation, as verified by a perturbative expansion: For |B| ≫ J x and L = 1 in the case (26) we obtain, setting η = (J x − J y )/(8B), Thus, E is in this region slightly greater than D, as verified in the top panel of Figs. 4-5.
Results for a small chain of n = 10 spins are shown in Fig. 5. Although the behavior is similar to that for n = 100, finite size effects become important and the overlap −Θ|Θ can no longer be neglected. The effects on the discord and entanglement of the ground state parity transitions taking place for |B| < B s are now visible, giving rise to small steps in these quantities. The final step takes place at B s , where D exhibits now a finite discontinuity due to the parity splitting arising from the coherence term (sec. II D), no longer negligible: the actual reduced state of a spin pair derived from the states (30) is given by Eq. (22) with where the − (+) sign corresponds to the left (right) side at B s , i.e., negative (positive) spin parity. The side-limits of D at B s are then given by Eq. (24) for the values (29)-(33) of θ and ε (leading to D − ≈ 0.153, D + ≈ 0.137 in the case of Fig. 5). Small but non-zero common side-limits at B s of the entanglement between any two spins also arise [24], as determined by Eqs. (10), (25). In contrast with the discord, Eq. (25) satisfies of course the monogamy inequality [28,29], reaching its maximum value 2/n in the XX limit J y → J x (where θ → 0 and C(ε) → 0 but C(−ε) → 2/n, as |Θ − approaches the W -state [24]).
The present values of the discord at the factorizing field, determined by Eqs. (21) or in general (24), are actually much more general: Uniform chains or arrays with ferromagnetic (J ij x > 0) XY couplings of arbitrary range but common anisotropy χ = J ij y /J ij x ∈ (0, 1) will also exhibit a factorizing field, given for spin 1/2 by [24,25] where the chain will possess again the same degenerate separable ground states (28). Side-limits at B s will then be determined by the same definite parity states (30).
. As a result, the ground state pair discord in all these systems will be finite and independent of pair separation or coupling range in the vicinity of B s , and given by Eqs. (21) or (24), with the values (29)- (33). Similar arguments will apply in the vicinity of more general factorizing fields [26].
An example is provided in Fig. 6, where results for a fully and uniformly connected XY array (LMG model [38]), with the same anisotropy J y /J x = 0.5, are depicted. The exact result can be here obtained by direct diagonalization, as H can be expressed in terms of the total spin operators S µ = i s iµ . The reduced pair density will obviously be independent of separation at any field. The pairwise entanglement is then small for large n, with a O(n −1 ) concurrence [24].
The discord is, however, non-negligible and practically n-independent for large n. It is verified in Fig. 6 that at The result for first neighbor coupling at the same anisotropy and size n is also depicted (b), for large separation (L = n/2). The limits at the factorizing field Bs are exactly coincident. Bottom: Same details for n = 10 spins. Here (c) depicts the result (24) for the actual mixture (22), which includes the coherence term. Side limits at Bs are again coincident. B = B s , the same previous limit (21) (n = 100) and sidelimits (24)-(33) (n = 10) at B s are obtained. Moreover, the simple model (18) (large n) or (22) (small n) for the reduced density of a spin pair accurately describes the discord in the whole region |B| < B c = J x (and not just at B s ) if the mean field value cos θ = B/J x is employed for θ, as seen in Fig. 6. This indicates that the effects on D of correlations beyond the basic mean field description with parity restoration of the ground state given by Eq. (30), become in this system very small, being negligible for large n. In contrast, such mean field model cannot accurately describe the pair discord (either for small or large separations) in the nearest-neighbor chain away from B s .

IV. CONCLUSIONS
We have first discussed in detail the discord of a mixture of aligned pairs in two different directions. While exhibiting no entanglement if coherence effects between both directions are negligible, these mixtures do exhibit a finite and non-negligible quantum discord if the directions are not coincident nor opposite. In the presence of coherence terms, however, entanglement becomes finite and can be larger than the discord.
Such mixtures become of crucial importance for studying the pair discord in the exact (and hence of definite parity) ground state of finite XY and XY Z spin chains in a transverse field. They represent the actual reduced state of an arbitrary pair in the vicinity of the factorizing field B s . Previous results imply then a finite discord between any two spins in the vicinity of B s , irrespective of pair separation or coupling range. These mixtures are also the main term of the reduced pair state in the whole region |B| < B c , implying there a non-negligible pair discord even for pairs not linked by the couplings, as was seen in the nearest-neighbor case. Such mixtures do in fact accurately describe the pair discord ∀ |B| < B c in the fully connected XY model.
The behavior of the discord, which is free from the monogamy restriction, differs then considerably from that of the pairwise entanglement, whose limits at B s are small and determined by the coherence term. This term gives rise to a parity splitting and hence to a finite discontinuity of the discord at B s , visible in small chains.
A final remark is that the present results, together with those previously obtained for the entanglement [24], allow to identify the factorizing field as a quantum critical point for the ground state of an XY or XY Z chain of small size: At B s , the last ground state parity transition takes place and in its immediate vicinity, pair quantum correlations, as measured by the discord, become independent of both pair separation and coupling range.