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Entropy 2015,17, 1634-1659; doi:10.3390/e17041634
entropy
ISSN 1099-4300
www.mdpi.com/journal/entropy
Article
Quantum Discord and Information Deficit in Spin Chains
Norma Canosa, Leonardo Ciliberti and Raúl Rossignoli *
Departamento de Física-IFLP, Universidad Nacional de La Plata, C.C.67, La Plata 1900, Argentina; E-Mails: canosa@fisica.unlp.edu.ar (N.C.); leocili@gmail.com (L.C.)
* Author to whom correspondence should be addressed; E-Mail: rossigno@fisica.unlp.edu.ar; Tel.:+54-221-4247-201; Fax:+54-221-4252-006.
Academic Editor: Demosthenes Ellinas, Giorgio Kaniadakis, Jiannis Pachos and Antonio M. Scarfone
Received: 14 January 2015 /Accepted: 19 March 2015 /Published: 26 March 2015
Abstract: We examine the behavior of quantum correlations of spin pairs in a finite anisotropic XY spin chain immersed in a transverse magnetic field, through the analysis of the quantum discord and the conventional and quadratic one-way information deficits. We first provide a brief review of these measures, showing that the last ones can be obtained as particular cases of a generalized information deficit based on general entropic forms. All of these measures coincide with an entanglement entropy in the case of pure states, but can be non-zero in separable mixed states, vanishing just for classically correlated states. It is then shown that their behavior in the exact ground state of the chain exhibits similar features, deviating significantly from that of the pair entanglement below the critical field. In contrast with entanglement, they reach full range in this region, becoming independent of the pair separation and coupling range in the immediate vicinity of the factorizing field. It is also shown, however, that significant differences between the quantum discord and the information deficits arise in the local minimizing measurement that defines them. Both analytical and numerical results are provided.
Keywords: quantum correlations; entropic measures; quantum discord; information deficit; spin systems
1. Introduction
The investigation of non-classical correlations in mixed states of composite quantum systems has attracted strong attention in recent years. While in pure states, such correlations can be identified with entanglement [1-7], in the case of mixed states, separable (unentangled) states, defined in general as convex mixtures of product states [8], i.e., as states that can be generated by local operations and classical communication (LOCC), may still exhibit non-classical features. The latter emerge from the possible non-commutativity of the different products and lead, for instance, to a finite value of the quantum discord [9-13] and other recently introduced related quantifiers of quantum correlations [14,15]. These quantifiers include the one-way information deficit [12,13,16-18], the geometric discord [19], generalized entropic measures [20,21] and, more recently, the local quantum uncertainty [22,23] and the trace distance discord [24-27]. While entanglement is certainly necessary for quantum teleportation [28] and for an exponential speed-up in pure state-based quantum computation [29,30], interest in these new measures has been triggered by the existence of mixed state-based quantum algorithms, like that of [31], able to achieve an exponential speedup over the best classical algorithms for a certain task, with vanishing entanglement [32], but finite quantum discord [33]. Additionally, various operational interpretations of the quantum discord and other related measures have been provided [15,22,26,34-39].
In this article, we will concentrate on the quantum discord [9-13] and the generalized entropic measures of [20], which include as particular cases the von Neumann-based one-way information deficit [12,13,16-18] and the geometric discord [19] and which represent a generalized information deficit. The quantum discord, as well as all other related measures require a rather complex minimization over a local measurement or operation, which has limited their applicability to small systems or special states. The optimization problem for the quantum discord was in fact recently shown to be NP complete [40]. The advantage of the generalized entropic formalism is, first, the possibility of using simpler entropic forms, like the linear entropy, which, as will be discussed in Section 2, enables an easier evaluation (it does not require the diagonalization of the density matrix) and a more direct experimental access (it can be determined without a full state tomography). This entails that an explicit solution of the associated optimization problem for certain states can be achieved. The generalized formalism also allows one to identify some universal properties, i.e., valid for any entropic form (and not just for a particular choice of entropy) satisfied by the post-measurement state.
We first provide in Section 2 an overview of the main concepts and properties associated with these measures. We then apply these measures to examine the quantum correlations of spin pairs in the exact ground state of finite spin 1/2 chains with XY-type couplings in a transverse magnetic field, through their entanglement, quantum discord and information deficit. All separations between the pairs are considered. Several important studies of the quantum discord in spins chains have been made [41-52], but the relation to the generalized information deficit and the differences between their optimizing measurements in these spin pairs have not yet been analyzed in detail. We have recently investigated these aspects for an XX spin chain in [51] and will here extend this analysis to the anisotropic XY case. It is first shown that in contrast with the pair entanglement, the quantum discord and the information deficit exhibit, for the exact ground state of these chains, common features, such as
an appreciable finite value below the critical field, for all separations. Moreover, they approach a finite common non-zero value [44] at the remarkable factorizing field [49,50,53-60] that these chains can exhibit in the anisotropic case. On the other hand, we will also show that important differences between the quantum discord on the one side and the standard and generalized information deficit on the other side do arise in the minimizing local spin measurement that defines them. While in the quantum discord, the direction of the latter is always orthogonal to the transverse field, in the other measures, it exhibits a perpendicular to parallel transition as the field increases, which is present for all separations and which reflects significant qualitative changes in the reduced state of the pair. This difference indicates a distinct response of the minimizing measurement of these quantities to the onset of quantum correlations.
2. Measures of Quantum Correlations
2.1. Quantum Entanglement
We start by providing a brief overview of the basic notions. A pure state |^AB) of a bipartite system A + B is separable iff (if and only if) it is a product state |^A)|^B). Otherwise, it is entangled. The Schmidt decomposition [5]:
\*ab) = Y,V^\ka)\kb) (1)
where |kA(B)) denote orthonormal states for subsystem A(B) and > 0, EU= 1 Pfc = 1, allows one to easily distinguish separable pure states (ns = 1) from entangled states (ns > 2). Here, ns is the Schmidt rank of |^AB) (ns < Min[dA, dB], with dA(B) the Hilbert space dimensions of A(B)). Pure sate entanglement can be measured by the entanglement entropy [3,4]:
E(A,B) = S(pa) = S(pb) = Pk logPk (2)
where pa(b) = TTb(a) Pab = EU= 1 Pk |^a(b) )(&a(b)|, with pab = |^ab)(^ab |, are the reduced states of A(B) and S(p) = —Tr p log p is the von Neumann entropy. We will set in what follows logp = log2 p, such that E(A, B) = 1 for a maximally entangled two-qubit state (ns = 2, p1 = p2 = 1/2).
On the other hand, a general mixed state pAB (pAB > 0, Tr pAB = 1) of a bipartite system A + B is separable iff it can be expressed as a convex mixture of product states [8]:
Pab separable ^ pab = ^ Pa Pa ® Pb Pa > 0 (3)
where EaPa = 1 and pA(B) denote mixed states for subsystem A (B). Otherwise, it is entangled. The meaning is that a separable state can be created by LOCC, i.e., Alice prepares a state pA with probability Pa and tells Bob to prepare a partner state pB.
For pure states pAB = |^AB)(^AB|, Equation (3) is equivalent to the previous definition (|^AB) = |^A)|^B)), but in the case of mixed states, product states pAB = pA ® pB are just a very particular case of separable states. The latter also include: (a) classically-correlated states, i.e., states diagonal in a standard product basis {^j) = |iA)|jB)},
Pab = Pj|iA)(u|®|jB)(jB| , Pij > 0 (4)
where Y1 i jPj = 1 and |iA(B)) are orthonormal states of A(B); (b) classically-correlated states from one of the subsystems, say B, which are of the form:
PAB = Pj PA/j ® |jB)(jB1 , Pj > 0 , (5)
where j Pj = 1 and pA/j are states of A, which are then diagonal in a conditional product basis {|ijj) = |iA/j)|jB)} with |iA/j) the eigenstates of pA/j (the case (4) recovered when all pA/j commute); and (c) convex mixtures of product states, which are not of the previous forms of (a) or (b). The latter typically possess entangled eigenstates. For this reason, it is much more difficult to determine whether a mixed state is separable or entangled. The well-known positive partial transpose criterion [61,62] (pAbb > 0, with ptB kl = pil,kj for pilkj = (i1|pAB|kj)) provides a necessary criterion for separability, which is sufficient for two-qubit or qubit-qutrit states.
For mixed states, the marginal entropies S(pA), S(pB) no longer provide a measure of entanglement. Instead, it is possible to use the entanglement of formation [63], defined through the convex roof extension of the pure state definition:
E (A, B) = Min S (pA) (6)
where the minimization is over all decompositions of pAB as convex mixtures of pure states (pa > 0, aPa = 1) and S(pA) = S(pB) is the entanglement entropy of the pure state |^AB). Equation (6) vanishes iff pAB is separable and reduces to the entanglement entropy (2) for pure states. It is an entanglement monotone [64], i.e., it does not increase by LOCC, staying unaltered under local unitary operations pAB ^ UA ® UB pAB UA ® UB. Its evaluation is, however, difficult in general. A general analytic expression has been derived just for the two-qubit case [65,66], which will be specified in Section 3.
While the marginal entropies are no longer entanglement indicators, it can still be shown [67] that if
S(pa) > S(pAB) or S(pB) > S(pAB), pAB is entangled, i.e.,
pAB separable ^ S(pa) < S(pab) , S(pB) < S(pab). (7)
Equation (7) provides an entropic criterion for separability [67] (necessary, but not sufficient in general), which can be also extended to more general entropic forms [68-70] and which will be invoked in Section 2.3.
2.2. Quantum Discord
For the classically-correlated states (4) or in general (5), there is a complete local measurement on B that leaves the state unaltered. This is not the case for entangled states, nor for separable states not of the form of (4) or (5). Let us recall here that a general positive operator valued measurement (POVM) [5] on system A + B is defined by a set of operators {Mj} satisfying MjMj = IAB = IA ® IB, such that the probability of outcome j and the joint state after such an outcome are:
Pj = Tr pab Mj , pAB/j = Mj pabMj/Pj (8)
The post-measurement state if the outcome is unknown is then:
pAB = ^ Pj pAB/j = Y1 Mj pAB Mj . (9)
Standard projective measurements correspond to the case where the Mj are orthogonal projectors (MkMj = jMj), while a local measurement on B corresponds to Mj = IA ® MB. By a complete local measurement on B, we will mean one based on rank one orthogonal projectors MB = Pj8. It is then apparent that the states (4) and (5) remain unchanged (pAB = pAB) after a local measurement on B based on the projectors Pj8 = |jB)(jB |. For the states (4) (but not necessarily (5)), there is also a local measurement on A (that based on the projectors |iA) (iA|), which leaves them unchanged.
The quantum discord [9-13] is a measure of quantum correlations, which, unlike the entanglement of formation, can distinguish the classically-correlated states (5) from the rest of separable states: It vanishes iff pAB is of the form of (4) or (5), being positive in the other separable states (c), and reduces to the entanglement entropy (2) in the case of pure states. It can be defined as the minimum difference between two distinct quantum versions of the mutual information, or equivalently, of the conditional entropy:
D(A|B) = Min[I(A, B) - I(A,Bmb)] = Min S(A|Bmb) - S(A|B) (10)
where the minimization is over all local measurements MB on B and:
I (A, B) = S (pA) - S (A|B), S (A|B) = S (pAB ) - S (pB ) (11)
are, respectively, the standard quantum mutual information and conditional entropy, while:
I(A,BMb) = S(pA) - S(A|BMb), S(A|BMb) = ^PjS(pA/;) (12)
are the mutual information and conditional entropy after the local measurement MB, with pA/j = TrB pAB/j the reduced state of A after outcome j. Equation (10) is always non-negative [9-11], a property that arises from the concavity of the conditional von Neumann entropy [71]. In the case of complete local projective measurements MB, we have:
S(A|BMb) = S(pAB) - S(p'B) (13)
where p'B = TrA pAB and pAB is the post-measurement state (9). It is then apparent that if the state is of the form (4) or (5), a measurement MB based on the projectors Pj8 = |jB )(jB | leads to S (A|BMb ) = S(A|B) and, hence, D(A|B) = 0. For all other states (i.e., entangled states or separable states not of the form of (4) or (5)), D(A|B) > 0. In the case of pure states, S(pAB) = 0, while S(A|BMb) = 0 if MB is any complete local measurement, entailing D(A|B) = S(pB) = E(A, B). For mixed states, the quantum discord can be related to the entanglement of formation E(A, C) with a third system C purifying the whole system [35-38].
The mutual information I (A, B) is a measure of all correlations between A and B, being non-negative and vanishing just for product states pAB = pA ® pB. The bracket in (10) can then be interpreted as the difference between all correlations (classical + quantum) present in the original state minus the
classical correlations left after the local measurement on B, which leaves then the quantum correlations. The evaluation of Equation (10) is, nevertheless, difficult in the general case, being in fact an NP complete problem [40] due to the minimization over all possible local measurements MB. Nonetheless, the minimum is always attained for measurements based on rank one projectors P.B, not necessarily orthogonal [15,72,73].
2.3. Information Deficit
The one-way information deficit can be considered as an alternative measure of quantum correlations, with basic properties similar to those of the quantum discord. It can be defined as [12,13,16-18,20]:
I (A|B) = M;n S (pAB ) — S (pAB ) (14)
where pAB is the post-measurement state (9) and MB is here restricted to complete local projective measurements on B, such that pAB is of the form of (5). Like the quantum discord, Equation (14) is a non-negative quantity, which also vanishes just for the states (4) or (5) and which also reduces to the entanglement entropy (2) in the case of pure states. These properties will be shown below in a more general context, although they are also apparent from the alternative expression:
I (A|B) = M;n S (pab ||pAb ) (15)
where S (p||a) = Tr p(log p—log a) is the relative entropy [71,74] and a quantity satisfying S (p||a) > 0, with S(p||a) = 0 iff p = a. Equation (15) can be shown by noting that pAB is the diagonal part of pAB in the basis defined by the projective measurement (the minimization in (15) can in fact be extended to all pAB of the form of (5) [20]). Nevertheless, differences with the quantum discord may arise in the minimizing measurement, as discussed in the next section. We also note that if the minimizing measurement of D(A|B) is projective and in the basis of eigenstates of pB, then p'B = pB and Equations (10)-(13) lead to D(A|B) = I(A|B). Otherwise, D(A|B) < I(A|B), since for projective measurements, Equations (10)-(13) imply D(A|B) < S(pAB) — S(pAB) — [S(p'B) — S(pB)] < S (pAB ) — S (pAB ).
Equation (14) admits a simple interpretation in terms of the entanglement generated between the system and a measuring apparatus M performing the complete local measurement [18]. The measurement on the local basis {| iB)} can be represented through a unitary operator UBM satisfying UBM|jB0M) = |jBjM), where |0M) is the initial state of the apparatus and {|jM)} an orthogonal basis of M, such that:
pAB = TrM pABM , pABM = (IA 0 UBM)(pAB 0 |0M) (0M|)(1A 0 UBM) (16)
Since S(pAB) = S(pAB 0 |0M) (0M|) = S(pAbM), it is seen that Equation (14) is the difference between the entropy of the subsystem AB and that of the total system ABM after the measurement, and according to Equation (7), such a difference can be positive only if there is entanglement between AB and M. Thus, a positive I(A|B) indicates that entanglement between AB and M is generated by any complete local measurement MB. On the other hand, if I(A|B) = 0, then pAB is of the form of (5) and for a measurement in the basis {|jB)}, pAbM = Zj PjPA/j 0 |jBjM)(jBjM| is clearly separable, so that no
entanglement is generated by this measurement. It can be shown [18] that Equation (14) coincides in fact with the minimum distillable entanglement between AB and M generated by the complete local measurement on B. A similar interpretation for the quantum discord in terms of the minimum partial distillable entanglement can also be obtained [18]. Other operational interpretations can be found in [26,34,36-39].
2.4. Generalized Information Deficit
It is possible in principle to extend Equation (14) to more general entropic forms, since in contrast with the quantum discord (10), its positivity is not related to specific properties of the von Neumann entropy S(p), as shown below. We consider here generalized entropies of the form [75]:
Sf (p) = Tr f (p) (17)
where Tr f (p) = i f (Pi), withPi the eigenvalues of p and f (p) a smooth strictly concave real function defined for p e [0,1] and satisfying f (0) = f (1) = 0. These entropies fulfill the same basic properties as the von Neumann entropy, with the exception of additivity: we have Sf (p) > 0, with Sf (p) = 0 iff p is a pure state (p2 = p), while all Sf (p) are maximum for the maximally mixed state p = I/d, where d = Tr I is the Hilbert space dimension of the system. Moreover, they are strictly concave, i.e., Sf (E a PaPa) > E a Pa Sf (pa), for Pa > 0, E a Pa = 1, with equality iff all Pa are coincident. The von Neumann entropy is obviously recovered for f (p) = — p log p.
Concavity of Sf (p) implies the fundamental majorization property:
p' X p ^ Sf (p') > Sf (p) (18)
where p' X p indicates that p' is majorized by p [71,76,77] (also denoted as p' more mixed than p):
p' X P pj pj, i = 1,..., d — 1 (19)
j=i j=i
where pj, pj denote the eigenvalues of p and p' sorted in decreasing order (equality in (19) obviously holds for i = d). If the dimensions of p and p' differ, Equation (18) still holds (for f (0) = 0) after completing with zeros the smallest set of eigenvalues. Conversely, while the reverse of Equation (18) does not necessarily hold, indicating that majorization provides a more strict concept of mixedness or disorder than that defined by a single choice of entropy, it does hold if Sf (p') > Sf (p) V f of the previous form [69,70]:
Sf (p') > Sf (p) V Sf ^ p' X p (20)
Equation (18) remains actually valid for more general entropic forms (like increasing functions F(Sf) of Sf or, in general, Schur concave functions [76,77]), but Equation (20) indicates that the forms (17) are already sufficient to capture majorization. Among the various properties implied by majorization, we mention that for states with the same dimension, p' X p iff p' is a convex mixture of unitary transformations of p [71,76,77], i.e., iff p' = aPaUapU^, with Ua unitary andpa > 0.
Now, for any projective measurement (local or non-local) performed on the system A + B, it can be easily shown that Sf (pAB) > Sf (pAB) V Sf, i.e.,
pAb X Pab (21)
The reason is that the post measurement state pAB conserves just the diagonal elements p, = (v' | pAB | v') of pAB in a certain orthonormal basis {|v')} determined by the projectors, and hence, S/ (pAB) =
Evf ) = Evf (EM |(^|v')|2p^) > E^ |(^|v')|2/= S/(pab), where and ^ denote here the eigenvalues and eigenvectors of pAB. This relation is not restricted to rank one projectors (just choose an orthonormal basis {|v')} where pAB is diagonal), so that it holds for local projective measurements. Equation (21) remains actually valid for any measurement satisfying MMj = 1AB, i.e., which leaves the maximally-mixed state 1AB/dAB unchanged [20].
Note also that the strict concavity of S/ implies S/ (pAB) = S/ (pAB) iff pAB = PAB, as is apparent from the previous demonstration. In fact, if the off diagonal elements of pAB in the measured basis are sufficiently small, a second order expansion of S/ (pAB) leads to [20]:
SAp'ab) ~ sapab) (22)
where the fraction is always positive due to the strict concavity of / (and should be replaced by its limit —/''(p^) if pV ^ p^). Equation (22) is essentially the square of a weighted norm of the off-diagonal elements of pAB in the measured basis (i.e., of those lost in the measurement) and is therefore non-negative, vanishing (if /''(p) < 0 V p e (0,1)) only if all off-diagonal elements are zero. We may then define the quantity [20,21]:
I/ (A|B) = Min S/ (pAb ) — S/ (pab ) (23)
where the minimization is again over all complete local measurements on B. Equation (23) is non-negative, due to Equation (21), and vanishes iff pAB = pAB, i.e., iff pAB is already of the classically-correlated form of (4) or (5). It therefore vanishes only for the states with zero quantum discord. It obviously also remains invariant under local unitary operations.
In the case of pure states, it can be shown [20] that the minimum of Equation (23) is always attained for a measurement in the basis {|kB)} determined by the Schmidt decomposition (1), i.e., in the basis formed by the eigenstates of pB, which leads to:
I/(A|B) = S/(pa) = S/(PB) = Y, /(pk), (pab pure) (24)
It therefore reduces to the generalized entanglement entropy S/(pA) = S/(pB) of the pure state. The entanglement entropy can then be identified with the minimum information loss due to a local measurement [20]. It is apparent that for pure states, I/(A|B) = I/(B|A), a property which does not hold in the general case.
In the case of the von Neumann entropy, I/(A|B) becomes the standard information deficit (14), and Equation (24) implies that for pure states, it will coincide with the standard (von Neumann) entanglement entropy, like the quantum discord. Nevertheless, an important difference arises in the minimizing measurement, since that for the latter becomes undetermined in the case of pure states (it can be any measurement based on rank one projectors [72,73]), whereas all I/(A|B), including I(A|B), require a measurement in the basis {|kB)}, which is fully undetermined only in the case of maximally-mixed marginals.
Like the standard information deficit, If (A|B) is also an indicator of the minimum entanglement between the system and the measurement apparatus M generated by a complete local measurement. The von Neumann entropic criterion for separability (7) can actually be extended to any Sf [69,70]:
Pab separable ^ Sf (pa) < Sf (pab ), Sf (pB) < Sf (pab ) . (25)
The validity of Equation (25) for all Sf is stronger than the von Neumann-based criterion (23) [69,70] and equivalent to the disorder criterion of separability [68] (pAB separable pAB X pA(B)). By the same arguments given below, Equation (16), it follows that a positive If (A|B), i.e., Sf (pAB) > Sf (pAB) = Sf (pAbm ), is indicating the existence of entanglement between AB and M after any complete local projective measurement on B.
2.5. Minimizing Measurement
Equation (24) reflects a universal property exhibited by the local measurement minimizing If (A|B) for pure states: it is the same for all Sf. Such a measurement, i.e., a measurement in the basis {|kB)} determined by the Schmidt decomposition of the pure state, is also optimum, for all Sf, for the mixture of the pure state with the maximally-mixed state [20],
Pab = q|^AB)(^ab| + (1 — q)/AB/dAB , q e [0,1] (26)
These states exhibit then an unambiguous least disturbing local measurement, in the sense that it minimizes all If (A|B) and leads to a "least mixed" post-measurement state:
PAb = q^Pfc|kA)(kA| ® |kB)(kB| + (1 — q)lAB/dAB k=i
which majorizes any other post-measurement state emerging after a local measurement. This property does not hold for an arbitrary initial state pAB.
In the general case, the projective measurement MB = {|jB) (jB|} minimizing If (A|B) may depend on the choice of entropy Sf. It can be shown that it must satisfy the necessary stationary condition [21]:
TrA[f'(pAb), Pab] = 0 (27)
where f' denotes the derivative of f and pAB is the post-measurement state (9). Equation (27) implies,
explicitly^i[f'(pij)(ijj|Pab|ijk) — f'(pifc)(ikj|pAB|ikk)] = 0, wherepij = (ijj|Pabj) and |ijj) = |iA/j)|jB), with |ij/A) the eigenstates of pA/j. The minimizing measurement basis will not coincide in general with the eigenstates of pB, even though this holds for certain states, like pure states and the mixtures (26). Equation (27) shows that the eigenstates of pB will be stationary for any state pAB where the non-zero off-diagonal elements are of the form (ij |pAB |k1) with i = k and j = l, where |ij) = |iA)|jB) and |iA), |jB) are the eigenstates of pA and pB respectively [21].
In the case of the quantum discord and for MB restricted to complete local projective measurements, Equation (27) is to be replaced by (here f '(p) = — log p) [21]:
TrA[f'(pAb), PAB] — [f'(pAb), PB] = 0
More explicit expressions can be obtained for a two-qubit system, where we may write a general state
Pab = 1(1 ab + rA - vA + rB- Vb + vajvb) (29)
where aA = a ® I, aB = I ® a, with a* = (ax, , crz) the Pauli operators, and IAB = I ® I the identity. Since Tr = 0 and Tr c^ov = for v = x, y, z, we have ((O) = Tr pAB O):
rA(B) = (aA(B)), J = (aAaB) (30)
A complete projective measurement on B corresponds to a spin measurement along the direction of a unit vector k, represented by projectors / = ± k ■ cr). After this measurement, Equation (29) becomes:
9AB = \ [I + rA- uA + (rB ■ k)k ■ uB + ((T^Jfe)fe • crB] (31)
Equation (27) leads then to the explicit equation [21]:
axrB + a2J*rA + aJJk = Ak , (32)
where (cxi, a2, a3) = | E^=±i VlfjkV kl+Jfcf)'= K1 + ^ ' + + are
the eigenvalues of pAB, with v = ±1, and A is a proportionality factor. In the case of the quantum discord, Equation (28) leads to a similar equation, with f(p) —>• —plogp and cti —>• cti — | logp'./p^, where /4 = + rB ■ k) are the eigenvalues of p'B [21].
2.6. Particular Cases
One of the advantages of the generalized information deficit (23) is the possibility of using simple entropic forms, which can be more easily evaluated (and measured) than the von Neumann entropy. For instance, if f (p) = 2(p — p2), Equation (17) becomes the so-called linear entropy:
S2(p) = 2(1 — Tr p2) (33)
which follows from the linear approximation ln p « p — I in the von Neumann entropy, but is actually a quadratic function of p, i.e., a linear function of the purity P(p) = Tr p2. It is the simplest entropic form, and its evaluation does not require the knowledge of the eigenvalues of p (see Equation (39) below). Moreover, purity and, hence, S2(p), can be experimentally determined without a full-state tomography [78-80]. Equation (33) is actually the q = 2 case of the Tsallis entropies [81,82], obtained for /(p) =
1 - Tr pq
5» = y—, Q > 0 (34)
Equation (34) approaches the von Neumann entropy S(p) for q ^ 1, being strictly concave for q > 0. We have normalized (33) and (34), such that Sq(p) = 1 for a maximally-mixed two-qubit state. In the case (33), it is first seen that for post-measurements states pAB:
S2(pAb) — S2(pab) = 2Tr (pAb — p'Ab) = 2||pAb — pab||2 (35)
where ||O||2 = Tr O^O. Hence, the local projective measurement minimizing S2(A|B), which is that maximizing the post-measurement purity P(pAB), leads to the post-measurement state with the minimum Hilbert-Schmidt distance to the original state. The associated deficit:
12(A|B) = Min S2(pAb ) — S2(Pab ) (36)
coincides, apart from a constant factor, with the geometric discord [15,19,20]. For pure states, 12(A|B) will then coincide with the linear marginal entropies:
12(A|B) = S2 (pa) = S2 (pb ) = 2(1 — J] p2) (37)
In two qubit systems, Equation (37) is just the squared concurrence [65,66] of the pure state pAB.
While as a measure, the geometric discord fails to satisfy some additional properties fulfilled by the quantum discord or the information deficit [83], it offers the enormous advantage of a simple analytic evaluation in qudit-qubit systems [19,21,84], as discussed below, also admitting through the purity a more direct experimental access. Moreover, Equation (22) shows that if pAB is close to the maximally-mixed state 1AB/dAB, all If (A|B) will become proportional to 12(A|B) [20], as in this case f {PlyZj/{Pl,) ~ -f"(d"b") is nearly constant. In fact, all Sf( p) are linearly related to S2(p) in this limit [72,73].
Any state of a general system A + B can be written in the form of (29), replacing the Pauli operators by a complete set of orthogonal operators a in A and B satisfying Tr = 0, Tr = dA(B)iMV:
?AB = J^(Iab + rA ' Ga + Tb ' Gb + uajvb) (38)
where rA(B) and J (now a dA x dB matrix) are again given by Equation (30). The S2 entropy can then be readily evaluated as:
S2(pAB) = 2[1 - 3^(1 + |rA|2 + |rB|2 + ||J||2)] (39)
where || J||2 = TrJ*J. If B is now a qubit, the state after a spin measurement along direction k on B will have the form of (31) with | —>• ^j-. We then obtain, using Equation (39),
S2(p'AB) = 2 - ^(|rA|2 + ktM2k) (40)
where M2 = rBrfB + J* J is a 3 x 3 positive semidefinite symmetric matrix. Hence, I2(k) = S2(pAB) — S2(pAB) = ¿¡"(Tr M2 — k'M2k). Its minimum I2(A\B) can then be evaluated analytically as [19,21]:
L2(A\B) = MinI2{k) = ¿(TrM2 - AO (41)
where Ai is the largest eigenvalue of M2, the minimizing spin measurement being along the direction of the corresponding eigenvector. Equation (41) is valid for an arbitrary qudit-qubit state pAB. Let us notice that the stationary condition (27) or (32) reduces, for the linear entropy, precisely to the eigenvalue equation M2k = Ak, as in this case f'(pAB) <x PAB, and hence, ai = rB ■ k, a2 = 0 and a3 = 1 [21]. This indicates that the stationary measurements are those along the direction of the eigenvectors of M2.
For arbitrary q > 0, we may similarly define the quantities (in what follows cq = 1 — 21 q):
Iq(A|B) = Min Sq(pAb) — Sq(pab) = Min c-1 Tr (pAb — p'Ab),
Min SR(pAb) - SR(pab) = Min
q "MB 1 - q
Cq Iq (A|B )
Tr pab
1 - CqSq(PAB)
where:
log Tr pq
log [1 - Cq Sq (p)] , q > 0
are the Renyi entropies [71], which are just increasing functions of the Tsallis entropies (34) (and also approach the von Neumann entropy for q — 1). Equations (42)-(43) are again non-negative, vanishing iff pAB is of the form of (4) or (5), and approach the von Neumann information deficit (14) for q — 1. Equation (44) is again just an increasing function of Iq(A|B) (for fixed pAB) and does not depend on the addition of an uncorrelated ancilla C to A (pAB — pC ® pAB), as Tr pC cancels out. An analytic expression for 13(A|B) valid for any two-qubit state can also be obtained [21].
3. Application: Quantum Correlations of Spin Pairs in XY Chains
3.1. Model and General Expressions
We consider a spin 1/2 system with XYZ couplings of arbitrary range, immersed in a transverse magnetic field B along the z axis. The Hamiltonian reads:
where are the (dimensionless) components of the local spin at site i and Jj the coupling strengths.
The Hamiltonian (46) commutes with the Sz spin parity operator Pz, irrespective of the coupling range, anisotropy, dimension or geometry of the system [55-57],
[H, Pz] = 0, Pz = exp[i^(Siz + 1/2)] = nt-ffiz)
where aiz = 2siz. The non-degenerate eigenstates of H will then have a definite Sz parity Pz = ±1. Consequently, the reduced density matrix of an arbitrary spin pair i, j in any non-degenerate eigenstate
), Pij [Pij ,Pzij ]
the form:
Tr—t will then commute with the Sz parity operator of the pair P%J = crizcrjz:
0. In the standard basis ||00), 101 >, |10), 111 >}, p.^ will therefore be an X-type state of
( a+ 0 0 ß \
0 c+ a 0
0 ä c- 0
V ß 0 0 a- J
where the coefficients are all real (since H is real in the full standard basis) and given by
a± = \ ± \{siz + sjz) + (sizsjz) , (49)
C± = \ ± - S^) - (sizsjz) , (£) = (Si_sjT) (50)
with a+ + c+ + c_ + a_ = 1. This corresponds to rA and rB along z in (29) (rA(B) = a+ + c+(_) — C—(+) — «_), with J diagonal, i.e., Jmv = 4(s^sjv) = Vj^,with j = 2(a± p), jz = «+ + «_ — c+ — c_. The positivity of p¿j implies a < ^/c+c., |(3| < ^/a+a., with a±, c± non-negative. The single spin density matrix is:
Pi = Trj Pij =( a+ + C+ + ) (51)
\ 0 a_ + c_ I
Both Pij and Pi will obviously be typically mixed due to the entanglement with the rest of the chain. In what follows, we will consider translational invariant systems, such that (siz) is site independent,
i.e., {siz) = {sjZ) V i,j, implying c± = c = 1~a+~a-. in this situation, pi:,- = pji and D(A\B) = D(B| A) = D, If (A|B) = If (B| A) = If V Sf.
The entanglement of the pair can be measured by the entanglement of formation (6), which for two qubit states, can be evaluated as: [65,66]
£ = q± = l2(l±VTrC2) (52)
V = ±
where C is the concurrence [65,66]. For the states (48) with c± = c, the concurrence of the pair is given by:
Cij = 2Max[||31 - c, |a| - v^TaT,0] (53)
The pair entanglement is of the parallel type (as in the Bell states ) if the first entiy in (53) is
positive and antiparallel (as in ) if the second entry is positive [54] (just one of them can be
positive).
On the other hand, the quantum discord of the pair can be readily evaluated with the expressions (48) and (31) (see [44] for details). The ensuing minimization over the spin measurement direction k (we will consider here just projective measurements) will normally lead to the direction corresponding to maximum correlation, according to the general arguments of [72,73]. In the XY chains, which will be considered, i.e., JZj = 0, with | Jj| < Jx and JXj > 0, the quantum discord for the states (48) will always prefer a measurement along the x axis, irrespective of the field intensity [44].
The information deficit (14) can be evaluated in a similar way. In contrast with the quantum discord, the optimizing measurement direction will be affected by the field intensity, exhibiting a smooth transition from the x to the z direction as the field increases for the systems considered, as discussed below. The angle 7 between k and the z axis can be determined from Equation (32), which leads explicitly to:
a1rB + a2jz rA
cos 7 =-—-----(54)
MJ2 — jz2)
when y = 0 [21], which is a transcendental equation (as the ai depend on 7).
The quadratic information deficit (36) can, however, be analytically evaluated with Equation (41). Here, M2 is already diagonal, M2jUV = rB + j). Assuming |jx| > |jy |, as will occur in the cases
considered, we obtain:
h = fMin\jl+jljl + rl+jf\ = 4Min[a2 + |32, ^ + c3-(°+-°2)c+(a-p)3] (55)
with the minimizing measurement direction k along the z (x) axis if the first (second) entry is minimum:
k =( ez' + j (56)
{ ex, jX > rB + j2
This entails that as the field B increases from zero, a sharp x — z transition in the minimizing measurement direction will take place for 12, reflecting the change in the largest eigenvalue of the matrix M2. This transition becomes softened in the von Neumann information deficit (14), where k will evolve smoothly from the x to the z axis within a narrow field interval located in the vicinity of the 12 transition. A measurement transition also occurs for other values of q in the quantities (42)-(43) (see [21] for an example).
3.2. Results
In Figures 1 and 2, we show results for the exact ground state of a finite chain with n spins coupled through cyclic (n + 1 = 1) first neighbor anisotropic XY couplings (Jj = 0, J^ = ij)i±1 Ju for ^ = x, y), for which the reduced pair states (48) will depend just on the separation L = |i — j | between the spins of the pair. The exact values of the elements of the density matrix (48) can be obtained, for any size n or separation L, through the Jordan-Wigner fermionization of the model [85] and its analytic parity-dependent diagonalization [55,86,87] (see the Appendix).
We will set Jx > 0, with | Jy | < Jx. This involves no loss of generality as the sign of Jx can be changed by a local rotation of angle n around the z axis at even sites (assuming n even in cyclic chains), which will not affect the value of the correlation measures, and the x axis can be chosen along the direction of maximum coupling.
Figure 1 depicts the behavior with increasing field B of the one-way information deficits I1 = I Equation (14)) and 12 (Equations (36)-(55)) of spin pairs in the exact definite parity ground state for the anisotropic case Jy = JX/2, together with that of the quantum discord (14) and the concurrence (53). It is first seen that 11, 12 and D exhibit a similar qualitative behavior, acquiring appreciable finite values for any separation L in the interval |B | < Bc = (Jx + Jy)/2, in marked contrast with the concurrence, which is appreciable just for first and second neighbors (except for the immediate vicinity of the factorizing field; see below). The Sz parity symmetry is essential for this result. In fact, all measures converge to a finite common value, independent of the separation L, at the factorizing field [49,50,53,55-60]:
Bs = \J Jy Jx (57)
existing for 0 < Jy < Jx, where the system possesses a pair of degenerate completely separable exact ground states [49,50,55-57] given by |8) = |0,..., 0) and | — 6) = Pz|8) = | — 0,..., —0), where |0) = ||) is the single spin state forming an angle 0 with the —z direction and cos0 = Bs/Jx =
\J Jyj Jx. Actually, in the finite case, this field coincides with the last parity transition of the exact (and
hence, of definite parity) ground state [55], such that the latter approaches, as side limits at B definite parity combinations [55-57]:
l©±)
|e>±|-e> V2(i±(-e|e»
Bs, the
Here, |6+) (|6_)) is the ground state limit for B ^ B+ (B ^ B-). Discarding the overlap (—6|6) = cosn9, which is negligible if n and 9 are not too small (cosn9 « e"n02/2 for small 9), Equation (58) leads to a common reduced state for any pair i, j, given by [44,55]:
p* o IWI + I - o | -
This is a separable mixed state, and therefore, it leads to a zero concurrence for any pair, as seen in Figure 1 (where results at Bs correspond to the side limits (58)). However, it is not of the classically-correlated form of (4) or (5) if (—9|9) = cos 9 = 0 or 1, i.e., if | ± 9) are non-orthogonal and distinct, leading then to a common appreciable value of D, /i, /2 and, in fact, all //. We also notice that the same reduced state (59) is obtained from the mixture |(|6) (6| + | — 6) (—6|), which represents the low temperature limit of the thermal state p k exp[-H/ kT] at B =
N = 40
X = 0.5 0.2
l=2 20 l=1
0 0.5 1. 1.5
N = 40
X = 0.5
20 l=1
N = 40 X = 0.5
2 20 l=1
0.5 1. B/Jx (b) 1.5
l=1 N = 40 X = 0.5
0.5 1.
1. B/Jx
Figure 1. (a,c) The one-way information deficits 12, Equation (36) (a) and /i, Equation (14) (c) as a function of the scaled magnetic field B/Jx, for spin pairs with separation L = 1, 2,..., n/2 in the exact ground state of a cyclic chain of n = 40 spins with first neighbor anisotropic XY couplings (x = / = 1/2). (b,d) The quantum discord D, Equation (10) (b) and the concurrence C, Equation (53) (d) for the same pairs. The results for different separations coincide exactly at the factorization field Bs = \J JyJx ~ 0.71./,..
/i N = 10 * = 0.5
L=5 L'
L=1 X = 0.5
1. b/Jx
Figure 2. The same quantities of Figure 1 for n = 10 spins. In this case, the parity transitions of the ground state lead to small, but appreciable discontinuities in all quantities, with the last transition (indicated by the vertical dotted line) taking place at the factorizing field Bs. For this size, the concurrence also presents small, but finite side limits at Bs.
It is then possible to obtain straightforward analytic expressions for the side limits of D [44], 12 and I
at the factorizing field through the state (59), which leads to a± = \ (l±cos 9)2 and cx = (3 = c = | sin2 9 in (48), with cos2 9 = Jy/ Jx. That for 12 is particularly clean and given by:
(!-x)2 y > 1/Q
WH xiik' x<i/z where x = (60)
with the minimizing measurement at Bs being along z if x > 1/3 and along x if x < 1/3. Equation (60) applies for all separations L.
For small chains, the results are similar, but the effects of the parity transitions of the ground state (it undergoes n/2 parity transitions as the field increases from zero, the last one at B = Bs [55]) are now appreciable trough the finite discontinuities exhibited by 12, I1 and D, as seen in Figure 2. At the factorizing field, these discontinuities arise from the overlap (—6|6), which now cannot be strictly neglected. This leads to an additional term a ± cosn-2 9(19) (—91 ® 19)(—91 + h.c.) in Equation (59), which originates slightly distinct side limits of D [44] and also 12 and I1 at Bs. Moreover, it also leads to small, but finite and distinct common side limits of the concurrence at B = Bs [55-57], which was known to reach full range in its vicinity [54]. All of these side limits are, nevertheless, still independent of the pair separation L. In the case of 12, they are given, for x ^ 1/3, by:
} - 2 (1 ± x"/2)2 (61)
which corrects the upper line in Equation (60) for finite n (or x ^ 1), and + (—) corresponds to
i xn/2 —1 (1_x)
the right (left) side limit. The side limits of the concurrence are C(Bf) = , as obtained
from (53) [55-57].
The behavior of the quantum discord for longer range ferromagnetic-type couplings is qualitatively similar [44]. Moreover, a factorizing field still exists for longer range couplings with a constant anisotropy x = /JX [56,57], in which case the reduced pair state at Bs is again given by Equation (59) with cos 9 = v/\\ and Equations (60)-(61) remain then valid.
In Figure 3, we compare the behavior of /2, /1 and D for first neighbors in the chains of Figures 1 and 2, with that of the associated entanglement monotone, i.e., the squared concurrence C2 for /2 and the entanglement of formation E for /1 and D, such that both quantities coincide for pure states. It is seen that for strong fields, differences are very small, in agreement with the weak entanglement of the pair with the rest of the chain in this regime (pi)i+1 is almost pure). The strong differences arise for B < Bc and especially in the vicinity of the factorizing field, due to the arguments exposed above. For |B| < Bc, the reduced pair state becomes appreciably mixed in the definite parity ground states, including the states (58) at the factorizing field, due to the entanglement with the rest of the chain. Significant differences between /f (and D) with the corresponding entanglement monotone become then feasible.
B/Jx B/JX
B/Jx B/JX
Figure 3. Plot of /2 (top), /1 (center) and D (bottom) together with the associated entanglement monotones for a first neighbor pair (L = 1) in the ground state of the chains of Figures 1 and 2.
It is also seen that /2 is in this case an upper bound of C2 for all fields, whereas /i is not an upper bound of E for low fields, while D is not an upper bound, even for strong fields, indicating the lack of an order relationship between D and E, even in this regime. In the case of /2, it is easy to show from Equations (53) and (55) that for X states, it is always an upper bound of C2 when the minimizing measurement is along z [21]. In, fact, for strong fields |B| » Jx, a perturbative expansion [20]
for the present chain leads to C
n2(log e — log n2 — 2), where:
2(n — n2), /2 ~ 4n2, /1 ~ n2(log e — log n2) and D
Hence, in this limit, /2 — C2 O(—n2) becomes negative.
O(n3) and /1 — E = O(—n3 logn2), both positive, whereas D — E
5.5. Minimizing Measurement
Although /1, /2 and D show a similar qualitative behavior, both measures /1 and /2 exhibit a more pronounced maximum, in comparison to that of the quantum discord, as appreciated in Figures 1-3. This reflects the transition in the orientation of their local minimizing spin measurements as the field increases, which, as mentioned above, is not present in the quantum discord. The latter prefers in the present system a measurement along the x axis, even for large fields and for any separation between the spins, following the strongest correlation [72,73]. As seen in Figure 4 and as previously stated, /2 exhibits instead a sharp transition from a direction parallel to the x axis (7 = n/2) to a direction parallel to the z axis (7 = 0), i.e., parallel to the field. This transition takes place, in the case shown in Figure 1, for all separations L at B & 0.65Jx. In the case of the information deficit /1, the transition becomes smooth, as the angle 7 takes all of the intermediate values between zero and n/2 (as determined by Equation (54)) for all separations in a narrow field interval centered at the /2 critical field, as also seen in Figure 4.
0 0.5 1
Figure 4. The angle 7 determining the direction of the minimizing local spin measurement for D, /1 and /2, as a function of the scaled transverse magnetic field, for a chain of n = 40 spins with Jy = Jx/2. Results for all separations L of the pair are shown.
The value of the field where the transition in the optimizing local measurement for /2 occurs depends on the anisotropy, but only slightly on the separation L, except in the XX limit (Jy ^ Jx), as can be
seen in the left panel of Figure 5. The same holds for the field interval where the "transition" (actually the evolution from n/2 to zero of the measurement angle 7) in /1 takes place (Figure 5b). In the case of /2, if x = 1/3, the measurement transition for all separations L occur exactly at the factorizing field Bs = y/xJx, as follows from Equation (60).
The measurement transition reflects essentially the qualitative change experienced by the reduced state of the pair for increasing fields. Away from the XX limit, the dominant eigenstate of Pj (that with the largest eigenvalue) for not too low fields is the entangled state = u|H) + v|tt) with v/u =
-J and e = ,, . Above the measurement transition field (i.e., when the optimum measurement
is parallel to the field), v/u becomes small (< 0.25), indicating that the pair is approximately aligned with the field. Instead, below the transition field, v/u increases, approaching one for B ^ 0 (where |^+) becomes a parallel Bell state), and the least disturbing measurement is along x. For very low fields, the dominant eigenstate may shift to the antiparallel Bell state It-l-H-l-t) arjsjng from the central
block of (48), and in this case, the measurement along x is still preferred. On the other hand, in the XX limit, |3 = 0 in (48), and the dominant eigenstate is either at low fields or |^+) = 1for strong fields; and the measurement transition of /2 indicates essentially the field where the sharp transition in the dominant eigenstate (from maximally entangled to separable) takes place [51]. Such a measurement transition for increasing fields persists even at finite temperatures [51].
. i/V3 bs L=1
L=20 1/3 I2
Figure 5. (a) The field where the transition in the minimizing measurement of /2 takes place, as a function of the anisotropy x = / . A direction along the x (z) axis is preferred below (above) the transition field. The factorizing field Bs is also shown. All transition fields coincide with the factorizing field if x = 1/3, Equation (60). (b) The fields delimiting the interval where the smoothed transition in the minimizing measurement of the von Neumann information deficit /1 takes place.
4. Conclusions
We have examined the behavior of the quantum discord and the standard and quadratic one-way information deficit of spin pairs in the exact definite parity ground state of a finite anisotropic cyclic XY spin 1/2 chain in a transverse field. We have first provided a brief overview of the quantum discord, the standard von Neumann-based one-way information deficit and the generalized information deficit, which
contains the standard, as well as the quadratic deficit as particular cases and which can be interpreted as a measure of the minimum entanglement generated between the system and the measurement apparatus after a complete local projective measurement. The first important result is that the behavior of all of these measures is quite distinct from that of the pair entanglement for fields below the critical field, acquiring finite appreciable values for all separations of the spins of the pair. Moreover, they reach (as side limits) a common (independent of the separation) finite value at the factorizing field, which in a finite chain, is the field where the last ground state parity transition takes place. These finite limits can be evaluated analytically. The entanglement of pairs also reaches full range in its vicinity, although its value is much smaller and vanishes at this field, except for very small samples. Parity effects are of crucial importance for the proper description of these measures in finite systems below the critical field.
The second important result is that the behavior of the optimizing local spin measurement of both the standard and generalized information deficit is quite distinct from that optimizing the quantum discord, exhibiting a transition in the direction of the spin measurement, from that of maximum correlation to that parallel to the field. The details of this transition depend on the choice of entropy (it is sharp for /2 and smooth for /1). The quantum discord prefers instead that of maximum correlation, even for strong fields. Hence, the quantum discord, which is based on the minimization of a conditional entropy, "detects" in this way this direction [72,73], while the information deficits, based on the minimization of a total entropy, are more sensible to changes in the structure of the reduced state of the pair.
A final comment is that the generalized formalism permits the use of simple entropic forms involving just low powers of the density matrix, leading to measures of the form of (42) or (43), which can be more easily evaluated and optimized and which are also more easily accessible from the experimental side.
A. Appendix
We briefly discuss here the exact solution of the finite cyclic XY chain with first neighbor couplings, which requires one to take into account exactly the parity effects [55,85-87]. The Jordan-Wigner transformation [85] allows one to rewrite the Hamiltonian (46) in the XY case (Jj = 0) for J^ = ^ = x,y, and for each value ±1 of the parity Pz, as a quadratic form in fermion creation and annihilation operators c|, c defined by c| = exp[—in Ej=1 sj-], with the reverse transformation given by = c| exp[in j—1 c!cj]. This leads to:
H± = Y1~ 2) - ivt(J+4ci+1 + J-44+i + h-c•) i=1
= '/;//;, Tj) (62)
fceK±
where J± = \{JX ± Jy) and n + 1 = 1, </?" = 1, r/+ = 1 - 2 5m [85]. In (62), K+ = {\,..., n -K- = {0,..., n — 1} and:
Afc = \J(B — J+ cos ujk)2 + J- sin2 u)k , u)k = '2nk/n (63)
The last form of (62) is obtained through a parity-dependent discrete Fourier transform cj followed by a BCS-type Bogoliubov transformation c^ = u^ak + v^a„-fc,
fceK±
iii/i„ /, — to quasiparticle feraiionic operators ak, a\, with ( = ± (£? — J+ cosu^/A*.].
For B > 0, we may set Ak > 0 for k = 0 and A0 = J+ — B, in which case, the quasiparticle vacuum of H^1 has the right parity, and the lowest energy is E^ = — ^ J2keK± Ak. At the factorizing field (57), Ak = J+ — Bs cos and E± = —nJ+ /2 [55].
The reduced state of a spin pair in the exact ground state can then be obtained from the basic contractions (aj,a^) = 0, (aj,a^) = 0, leading to (c'kc'w) = , (c'kc'k) = «kvfc4 ,-k' and
(L = i — j):
= i E e~iUkL^k = h + - <CH>± = ¿ E e~ÜJkLUkVk = CJL
keK± keK±
The application of Wick's theorem then leads to [44,85]:
(siz) = /0, (siz ) = f0 — fL + gL
(Si-s^) = ¡[det(A+)Tdet(A~L)]
where (A±) are L x L matrices of elements = 2(/i-j±1 + gi-j±i)- These results, valid for any
finite n, were checked through direct diagonalization for small n.
Acknowledgments
Norma Canosa and Leonardo Ciliberti acknowledge support from CONICET and Raúl Rossignoli acknowledges the support from CIC of Argentina.
Author Contributions
All authors contributed equally to this paper. All authors have read and approved the final manuscript. Conflicts of Interest
The authors declare no conflict of interest. References
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