Long range quantum coherence, quantum & classical correlations in Heisenberg XX chain
aa r X i v : . [ qu a n t - ph ] J a n Long Range Quantum Coherence, Quantum & Classical Correlationsin Heisenberg XX Chain
Zakaria Mzaouali a, ∗ , Morad El Baz a a ESMaR, Faculty of Sciences, Mohammed V University, Av. Ibn Battouta, B.P. 1014, Agdal, Rabat, Morocco
Abstract
A comparative study of pairwise quantum coherence, quantum and classical correlations is addressed for non-nearestspin pairs of the 1D Heisenberg spin- XX chain. Following the Jordan-Wigner mapping, we diagonalise the hamil-tonian of the chain and we check this procedure numerically as well. Using the “Pauli basis expansion” formalismwe get the pairwise quantities studied in this work at any distance. We then, show the role of quantum correlationsin revealing quantum phase transitions, the robustness of quantum discord to the temperature and the dominance ofquantum correlations over their classical counterpart in the magnetic and thermal interval in quantum spin chains.We conclude the paper by shedding light from a resource-driven point of view on the new born quantity “quantumcoherence” where we discuss its role in detecting quantum phase transitions being a long-range quantity, and how itoutclasses the usual quantum correlations measures in the robustness against the temperature, which indicates poten-tial uses in the framework of quantum information processing.
Keywords:
Entanglement, Quantum Discord, Classical Correlations, Quantum Coherence, Spin Chains, QuantumPhase Transitions
1. Introduction
Correlations, the key elements to understand many-body systems can be classified into classical and quantumtypes. The existence of quantum correlations were first pointed out in 1935 by E. Schr¨odinger [1] in the context ofnon-separable (i.e entangled) states. Since then, much e ff ort was devoted in studying the nature of entanglement [2, 3]especially in the framework of resource theory in quantum information processing, through quantum teleportation [4],quantum communication [5] and quantum computation [6].The quantum information approach to condensed matter physics has been very fruitful at giving new a perspectiveto understand the collective phenomena in many-body systems. Indeed, since the early 2000s various measures ofquantum entanglement have been employed to characterize the features of the ground and excited states of quantummatter [7, 8]. In this sense, quantum spin systems play an essential role in these developments as they describe thee ff ective interactions in a collection of physical systems [9] like quantum Hall systems, high-temperature supercon-ductors, heavy fermions and magnetic compounds. Furthermore, systems that can be described by interacting spinsare interesting because they manifest quantum fluctuations and can be realised by a variety of physical approaches[10, 11].The study of spin entanglement in quantum spin chains began in 2000s. Since then, many types of entanglementat both zero and finite temperatures have been widely studied in various spin models [7]. However, entanglementmeasures fails at capturing other forms of non-classical correlations [12, 13] that emerge for free in the ground andthermal states of condensed matter models and that can be exploited as resources for quantum technologies. As amatter of fact, quantum discord [14] is the most e ff ective measure of quantum correlations beyond entanglement, andit has been heavily studied in spin chains at both zero [15, 16, 17] and finite temperature [17, 18, 19]. As entanglement ∗ Corresponding author
Email address: [email protected] (Zakaria Mzaouali)
Preprint submitted to Physica A January 7, 2019 lays an important role in identifying quantum phase transitions [8, 20], quantum discord has received much attentionin this regard as well [21, 22]. Moreover, it was applied in several contexts like open quantum systems [23], quantumdynamics [24] and even biophysics [25].Recently, the concept of quantum coherence has received much attention in the quantum information communityas it plays an essential role in phenomena like quantum interference, bipartite and multipartite entanglement [26].Various schemes were proposed for detecting coherence [27, 28], but it was never quantified in the language ofquantum information theory until the seminal work of Baumgratz, Cramer and Plenio [29] in which they constructeda quantitative theory that captures the resource character of coherence in a mathematically rigorous fashion. Suchdevelopments led to number of applications using coherence as a basic ingredient in various fields such as quantumcommunication [30] and in farther other arenas, such as thermodynamics [31] and even certain branches of biology[32].Few works were dedicated to study quantum coherence in condensed matter systems [33, 34, 35, 36]. In fact, theinvestigations that were carried out in spin chains like the XY model had the sole purpose of revealing the connectionbetween quantum coherence and quantum phase transitions. These studies has shown the role played by coherencein detecting important features like critical points, but it is still early to say how e ffi cient quantum coherence is indetecting quantum phase transitions as the field needs more models and measures to investigate these connections.Motivated by these developments, the aim of this paper is to study and compare the behavior of non-nearest quan-tum coherence, quantum and classical correlations in an infinite 1D spin- XX chain in the presence of a magneticfield. This is an analytically solvable model by means of the Jordan-Wigner transformation which we check numer-ically as well. Also this model has well-known physical properties hence it is a suitable ground for studying theinterface between quantum information theory and condensed matter physics.This article is organised as follows. In the next section we introduced the model, the analytical and numericaldiagonalisation process by the Jordan-Wigner mapping. In section 3, we introduce the two-sites density matrix andthe various measures of correlations considered in this paper. In section 5, we discuss our results on non-nearestquantum coherence, quantum and classical correlations in the XX chain. A conclusion and the summary of the resultsare presented in 6.
2. The Model
The Hamiltonian of the Heisenberg XX chain describing a set of localized spin- particles interacting with nearest-neighbors exchanging coupling on a 1D lattice in an external magnetic field is given by : H XX = J N X i = (cid:0) S xi S xi + + S yi S yi + (cid:1) − h N X i = S zi , (1)where S i is the spin- operator on site i . A positive (negative) exchange coupling J favors anti-ferromagnetic (ferro-magnetic) ordering of the spins, and h is the external magnetic field which interacts only with the z -component of thespins.The pecularity of the XX model is the possibility of exact diagonalisation by using a method that is very di ff erentfrom the Bethe Ansatz [37], and can be verified numerically as well. This method consists in mapping each spinoperator to a fermion operator, following the Jordan-Wigner transformation. For simplicity, we first introduce the spinladder operators : S ± = S x ± iS y with S + = ! , S − = ! . (2)In 1928, Jordan and Wigner showed [38] that the spin ladder operators can be represented exactly by the fermionoperators ( c i ’s) with the following mapping : S + i = c + i e − j π i − X l = c + l c l , S − i = e j π i − X l = c + l c l c i , S zi = c + i c i − , (3)2here j is the imaginary unit.Rewriting the Hamiltonian (1) in terms of the ladder operators (2) and by performing the transformation (3), theHamiltonian is readily shown to take the form : H XX = J N X i = (cid:16) c + i c i + + c + i + c i (cid:17) − h N X i = c + i c i − ! . (4)Since The XX model is translation invariant, we can introduce the Fourier transform of the fermion operators : d k = √ N N X i = e − jikc i and d + k = √ N N X i = e jikc + i . (5)The final diagonalised form of the Hamiltonian of the XX-chain written in terms of the spinless fermion creationand annihilation operators is : H XX = X k ǫ ( k ) d + k d k with ǫ ( k ) = J cos( k ) − h . (6)In the rest of this paper the coupling constant J will be equal to unity. The Jordan-Wigner transformation can beverified using “QuSpin” [39, 40] a Python package for dynamics and exact diagonalisation of quantum many-bodysystems. E n e r g y FermionSpin
Figure 1: The Spectrum of the XX-model in the spin and fermion representation
In Figure.1 we plot for a chain of 15 spins, J = h = solidline corresponds to the eigenenergies of the usual spin representation of the XX model (1), while the blue crosses arethe eigenvalues of the fermionic Hamiltonian (4) obtained by the Jordan-Wigner mapping. We see that for each spinstate there is a corresponding fermion state, which means that the two representations match exactly, as predicted.
3. Correlation Measures in Spin Chains
Quantifying correlations that emerge in a given quantum system requires the knowledge of the density matrix ofthe state characterizing the system. Here we are interested in studying the bipartite correlations between two spins i and j distant by some lattice spacing m , where i = j + m . Using Pauli basis expansion [41], the two sites densitymatrix ρ i , i + m is given by : ρ i , i + m = X α,β = p αβ σ α i ⊗ σ β i + m , (7) color on the online version p αβ = h σ α i σ β i + m i , and σ α i are the Pauli matrices, α = , , ,
3. In terms of Pauli ladder operators σ ± = ( σ x ± i σ y ),we can write in the computational basis {|↑↑i , |↑↓i , |↓↑i , |↓↓i} , the density matrix of two spins as follows : ρ i , i + m = h P ↑ i P ↑ i + m i h P ↑ i σ − i + m i h σ − i P ↑ i + m i h σ − i σ − i + m ih P ↑ i σ + i + m i h P ↑ i P ↓ i + m i h σ − i σ + i + m i h σ − i P ↓ i + m ih σ + i P ↑ i + m i h σ + i σ − i + m i h P ↓ i P ↑ i + m i h P ↑ i σ − i + m ih σ + i σ + i + m i h σ + i P ↓ i + m i h P ↑ i σ + i + m i h P ↓ i P ↓ i + m i , (8)where P ↑ = (1 + σ z ) and P ↓ = (1 − σ z ). The brackets denote the ground-state and thermodynamic averagevalues at zero and finite temperatures.The Hamiltonian (1) has several symmetries that reduces the number of non-zero elements of the density matrix(8). For instance, translation invariance means that ρ i , i + m = ρ i + m , i , therefore we have p αβ = p βα . The global phase-flipsymmetry implies that the commutator h σ zi σ zi + m , ρ i , i + m i =
0, then the coe ffi cients p = p , p = p , p = p and p = p must be zero. Furthermore, the z -component of the total magnetization commutes with the Hamiltonian (1) hP i σ zi , H XX i = |↑↑i and |↓↓i can not be possible for anytwo spins. Then the coe ffi cient p and p corresponding to the matrix elements |↑↑i h↓↓| and |↓↓i h↑↑| are zero. Thedensity matrix (8) reduces to : ρ i , i + m = X + i , i + m Y + i , i + m Z ∗ i , i + m Z i , i + m Y − i , i + m
00 0 0 X − i , i + m . (9)In terms of the occupation number operator n i = c + i c i , the elements of the density matrix are given by : X + i , i + m = h n i n i + m i , X − i , i + m = − h n i i − h n i + m i + h n i n i + m i , Y + i , i + m = h n i i − h n i n i + m i , Y − i , i + m = h n i + m i − h n i + m n i i , Z i , i + m = h c + i (cid:16) i + m − Y k = i { − c + k c k } (cid:17) c i + m i . (10)In the coming sections we investigate various contributions of correlations at long distances of m in the XX chain.We study the correlations between second nearest (2 N ) m =
2, third nearest (3 N ) m = N ) m =
4. Then, by using Wick’s theorem [42] the set of equations (10) reduces to : X + i , i + m = f − f m , X − i , i + m = − f + f − f m , Y + i , i + m = Y − i , i + m = f − f + f m , (11)while for the element Z i , i + m we havefor m = Z i , i + = f − f f + f , for m = Z i , i + = f − f f f + f f + f f − f f + f f − f f ) + f , for m = Z i , i + = f − f f f + f f + f f f + f f − f − f f f f + f f f − f f + f f − f f f − f f + f f f − f f f + f f f ) + f f − f f − f f f + f f f + f f − f f + f f − f f ) + f f − f f + f ) + f . (12)For the non-negative integer number m : f m = π Z π cos( km ) g ( k ) dk , (13)where g ( k ) = + e βǫ ( k ) is the Fermi-Dirac distribution, β = k B T and the Boltzmann constant is taken equal to unity.4 .1. Entanglement One candidate to study the bipartite properties of non-local quantum correlations of a quantum state is by meansof the concurrence [43, 44] defined as follows : C ( ρ ) = max { , λ − λ − λ − λ } , (14)where the λ i ’s are the square roots of the eigenvalues of the matrix ρ ˜ ρ in decreasing order, with ˜ ρ being a transformedmatrix of ρ i.e., ˜ ρ = ( σ y ⊗ σ y ) ρ ∗ ( σ y ⊗ σ y ). In the case of the two sites density matrix ρ i , i + m (9) considered in thispaper, the concurrence then can be calculated from the local-density’s, hopping term, and site-site correlations of thefermions. Thus, Eq.(14) takes the following form : C ( ρ i , i + m ) = max n , (cid:16) | Z i , i + m | − q X + i , i + m X − i , i + m (cid:17)o , (15)where Z i , i + m , X + i , i + m and X − i , i + m are given by the set of equations (11) and (12). & Classical Correlations
A bipartite quantum state can carry quantum and classical correlations [12], entanglement quantifiers like theconcurrence fails at detecting quantum correlations other than the non-local ones. To measure the total quantumcorrelations in a given bipartite quantum system ρ AB we use Quantum Discord (QD) a measure introduced by Olivier &Zurek [14], and independently by Henderson & Vedral [12]. It is based on the di ff erence between the total correlationsand classical correlations (CC): QD ( ρ AB ) = I ( ρ A : B ) − CC ( ρ AB ) , (16)where the total correlations are quantified by the quantum mutual information : I ( ρ A : B ) = S ( ρ A ) + S ( ρ B ) − S ( ρ AB ) , (17)with S ( ρ ) = − tr( ρ log ρ ) being the von-Neumann entropy.Classical correlations are defined as the maximization of I ( ρ A : B |{ Q Bk } ) over the set of positive operator valuedmeasurements (POVM) { Q Bk } on subsystem B, and is given by : CC ( ρ AB ) = sup { Q Bk } I ( ρ A : B |{ B Y k } ) . (18)It reduces to : CC ( ρ AB ) = S ( ρ A ) − min Q Bk S ( ρ B | B Y k ) , (19)where S ( ρ AB | Q Bk ) = P k p k S ( ρ Bk ) is the conditional entropy based on the measurement Q Bk . p k = Tr (cid:2) ( Q Bk ⊗ I B ) ρ AB ( Q Bk ⊗ I B ) (cid:3) ,and ρ kB = ( Q Bk ⊗ I B ) ρ AB ( Q Bk ⊗ I B ) p k are the probability and the state for a measurement outcome k .From Eq.(17) and Eq.(19) quantum discord takes the form : QD ( ρ AB ) = S ( ρ A ) − S ( ρ AB ) + min Q Bk S ( ρ B | B Y k ) . (20)The di ffi culty in evaluating QD rises from the minimization procedure of the conditional entropy, and obtaining ananalytical expression of CC ( ρ AB ) is not an easy task for general states. However, following the method of C.Z Wangand al in [45] an explicit expression for CC ( ρ AB ) and QD ( ρ AB ) is available for X-states described by density matricesof the form : ρ AB = ρ ρ ρ ρ ρ ρ ρ ρ . (21)5We obtain CC ( ρ AB ) and QD ( ρ AB ) as follows : CC ( ρ AB ) = max { CC , CC } , QD ( ρ AB ) = min { QD , QD } , (22)where CC j = H ( ρ + ρ ) − D j , QD j = H ( ρ + ρ ) + X k = λ k log ( λ k ) + D j , (23)and D = H ( w ) , D = − X j = ρ j j log ( ρ j j ) − H ( ρ + ρ ) , w = + p [1 − ρ + ρ )] + | ρ | + | ρ | ) , (24) H ( x ) : = − x log ( x ) − (1 − x ) log (1 − x ) being the binary Shannon entropy and λ k are the eigenvalues of the matrix ρ AB .Then, for the two sites density matrix (9) considered in this paper, the analytical expression for CC ( ρ i , i + m ) and QD ( ρ i , i + m ) are given by : CC ( i , i + m ) = H ( X + i , i + m + Y + i , i + m ) − H + q [1 − Y − i , i + m + X − i , i + m )] + | Z i , i + m | ! , CC ( i , i + m ) = H ( X + i , i + m + Y + i , i + m ) + { X + i , i + m log ( X + i , i + m ) + Y + i , i + m log ( Y + i , i + m ) + Y − i , i + m log ( Y − i , i + m ) + X − i , i + m log ( X − i , i + m ) } + H ( X + i , i + m + Y − i , i + m ) , QD ( i , i + m ) = H ( X + i , i + m + Y − i , i + m ) + { X + i , i + m log X + i , i + m + X − i , i + m log X − i , i + m + ( Y + i , i + m − | Z i , i + m | ) log ( Y + i , i + m − | Z i , i + m | ) + ( Y − i , i + m + | Z i , i + m | ) log ( Y − i , i + m + | Z i , i + m | ) } + H + q [1 − Y − i , i + m + X − i , i + m )] + | Z i , i + m | ! , QD ( i , i + m ) = H ( X + i , i + m + Y − i , i + m ) + { X + i , i + m log X + i , i + m + X − i , i + m log X − i , i + m + ( Y + i , i + m − | Z i , i + m | ) log ( Y + i , i + m − | Z i , i + m | ) + ( Y − i , i + m + | Z i , i + m | ) log ( Y − i , i + m + | Z i , i + m | ) }− { X + i , i + m log ( X + i , i + m ) + Y + i , i + m log ( Y + i , i + m ) + Y − i , i + m log ( Y − i , i + m ) + X − i , i + m log ( X − i , i + m ) }− H ( X + i , i + m + Y − i , i + m ) , (25)where X + i , i + m , X − i , i + m , Y + i , i + m , Y − i , i + m and Z i , i + m are given by the set of equations (11) and (12).
4. Quantum Coherence
Arising from quantum superposition, quantum coherence is a fundamental feature of quantum mechanics andplays a central role in the field of quantum information science. In the framework of resource theory, Baumgratz etal. [29] proposed a formalism for measuring quantum coherence which triggered the interest of many researchersand a variety of coherence measures emerged since then. In general, all these coherence measures can be classifiedinto either the entropic or the geometric class of measures [46]. The classification depends on whether the measureis based on the entropy functional or has a metric nature which implies a geometric structure. Interestingly, there is a6easure which combines both entropic and metric properties introduced in [46], and is based on the quantum versionof the Jensen-Shannon divergence [47] J ( ρ, ρ d ) defined as : J ( ρ, ρ d ) = S (cid:16) ρ + ρ d (cid:17) − S ( ρ )2 − S ( ρ d )2 . (26)The quantum coherence is then simply the square root of the quantum Jensen-Shannon divergence : QC = p J ( ρ, ρ d ) = r S (cid:16) ρ + ρ d (cid:17) − S ( ρ )2 − S ( ρ d )2 , (27)Where ρ is the density matrix of the state and ρ d that of the closest incoherent state under the distance measure and S ( . )is the von Neumann entropy. The closest incoherent state is taken to be the denstiy matrix where all the o ff -diagonalelements of ρ are zero [46].Then, in terms of Y + i , i + m , Y − i , i + m and Z i , i + m given by the set of equations (11) and (12), the analytical expression ofquantum coherence for the XX chain described by the density matrix (9) is : QC ( ρ i , i + m ) = "n − (cid:16) Y + i , i + m − | Z i , i + m | (cid:17) log (cid:16) Y + i , i + m − | Z i , i + m | (cid:17) − (cid:16) Y − i , i + m + | Z i , i + m | (cid:17) log (cid:16) Y − i , i + m + | Z i , i + m | (cid:17)o − n − ( Y + i , i + m − | Z i , i + m | ) log ( Y + i , i + m − | Z i , i + m | ) − ( Y − i , i + m + | Z i , i + m | ) log ( Y − i , i + m + | Z i , i + m | ) o − n − Y + i , i + m log Y + i , i + m − Y − i , i + m log Y − i , i + m o . (28)As the function of quantum coherence depend only on three terms of the density matrix (9), this will make it easier tostudy its behavior numerically.
5. Results
In this section, we present our numerical treatment based on the analytical approach of the entanglement, quantumdiscord, classical correlations and quantum coherence between second nearest (2N), third nearest (3N) and fourthnearest (4N) neighbors.
We start by comparing the behavior of the 2N, 3N and 4N spin pairs entanglement and quantum discord when themagnetic field is switched on, at T =
0. 7 .0 0.2 0.4 0.6 0.8 1.0 1.2Magnetic Field0.000.020.040.060.080.100.120.14 E n t a n g l e m e n t (a) Entanglement at zero temperature Q u a n t u m D i s c o r d (b) Quantum discord at zero temperatureFigure 2: The e ff ect of the magnetic field on (a) entanglement and (b) quantum discord at T =
0, for the 2N, 3N and 4N spin pairs.
Figure. 2a shows that when the magnetic field “ h ” is switched o ff none of the spin pairs are entangled, and as wegradually increase the magnetic field we see creation of entanglement when h reaches a critical value h Ec of for 2 N , for 3 N and for 4 N . Increasing the magnetic field beyond the critical value h Ec , enhances entanglement betweenthe spin pairs until its saturation at a critical value close to h Ed = h Ec for entanglement creation and the critical field h Ed at whichentanglement dies was proposed. A relation that is in agreement with the results from the finite size systems [49].For quantum discord however, the comportment is di ff erent. In Figure. 2b we see that quantum discord persists evenwhen h = h < h Ed ). In general, quantum discord originates from thecoherence that arises from quantum superposition, which exists in the subsystems of a quantum system and persistseven if the system is in a product state, while the amount of entanglement in a seperable state is always zero. Q u a n t u m D i s c o r d Figure 3: Quantum discord versus temperature at h =
0, for the 2N, 3N and 4N spin pairs.
Next, we analyze the e ff ect of the temperature on entanglement and quantum discord when the magnetic fieldis switched o ff . As depicted in Figure. 3, in all three cases, quantum discord starts with a maximum value anddecreases as T increases, and tends asymptotically to zero. This is in contrast to the case of entanglement wherefrom Figure. 2a it is clear that entanglement is absent when h =
0. This behavior is expected as when one increasesthe temperature, the quantum fluctuations are dominated by the thermal ones, but this does not kill the quantumnessof the system. The process that make it vanish is decoherence, and since absence of entanglement does not implyclassicality and the fact that quantum discord is more robust under decoherence, explains the asymptotic behavior ofthe total quantum correlations as dipected in Figure. 3. This shows that product states are not negligible at all [50]and it also confirms the robustness of quantum discord against temperature and its potential usefulness as resource for8uantum technologies.
Magnetic Field T e m p e r a t u r e (a) 2N Magnetic Field T e m p e r a t u r e (b) 3N Magnetic Field T e m p e r a t u r e (c) 4NFigure 4: T − h Phase diagram of entanglement for the 2N, 3N and 4N spin pairs.
The T − h phase diagram of the entanglement and quantum discord of the 2N, 3N, 4N spin pairs is depicted inFigure. 4 and Figure. 5. It is clear from the figures that as the distance between the pairs increases, the quantumcorrelations magnitude decreases which is expected in the sense that as the distance increase the strength of thecorrelations decreases. Furthermore, from Figure. 4 entanglement is seen to be easily destroyed by the temperatureas the distance between pairs increase, while quantum discord is more robust as shown in Fig. 5. Such behavior isdue to the dominance of thermal fluctuation over quantum correlations when one increases the temperature. However,the thermal and magnetic interval of entanglement in the 3N and 4N spin pairs is smaller than the 2N pairs, while forquantum discord this does not change, it is defined in the entire interval. Magnetic Field T e m p e r a t u r e (a) 2N Magnetic Field T e m p e r a t u r e (b) 3N Magnetic Field T e m p e r a t u r e (c) 4NFigure 5: T − h Phase diagram of quantum discord, for the 2N, 3N and 4N spin pairs. .2. Classical Correlations The behavior of classical correlations in the 1D XX chain is addressed in this subsection. We plot in Figure. 6 a 3Dpanorama of classical correlations for the 2N, 3N and 4N spin pairs. As seen for entanglement and quantum discordin the previous subsection, classical correlations also decreases with distance between spins and classicality persistsin the overall thermal and magnetic interval. Furthermore, in comparison with the behavior of quantum discord inFigure. 5, we see the dominance of quantum correlations over their classical counterpart while in the transverse Isingmodel the inverse phenomenon takes place [17]. More interestingly, as it can be seen from Figure. 5 in the presenceof the temperature quantum correlations can be increased in some regions by the presence of the magnetic field, whilefor classical correlations it decreases with respect to the magnetic field for zero and finite values of the temperature. M a g n e t i c F i e l d T e m p e r a t u r e C l a ss i c a l C o rr e l a t i o n s (a) 2N M a g n e t i c F i e l d T e m p e r a t u r e C l a ss i c a l C o rr e l a t i o n s (b) 3N M a g n e t i c F i e l d T e m p e r a t u r e C l a ss i c a l C o rr e l a t i o n s (c) 4NFigure 6: 3D panorama of classical correlations for the 2N, 3N and 4N spin pairs We conclude this section by a study of quantum coherence in the XX model. Figure. 7 shows the typical behaviorof quantum coherence with respect to the temperature and the magnetic field for the 2N, 3N and 4N spin pairs, inwhich we observe the similarity with quantum discord depicted previously in Fig. 2b and Fig. 3; this can be explainedas follows. Q u a n t u m C o h e r e n c e (a) Variation of Quantum Coherence with respect to the Temperature Q u a n t u m C o h e r e n c e (b) Variation of Quantum Coherence with respect to the magnetic fieldFigure 7: E ff ect of (a) the temperature and (b) the magnetic field on Quantum Coherence for several lattice spacing
10n general, coherence stems from individual sites (local coherence) or may be distributed along sites (intrinsiccoherence) [46] and, one can show that due to spin-flip symmetry of the XX chain, the one site density matrix isdiagonal which means that the local coherence as defined in [46] is zero for the XX model. So, the total coherenceobserved in the system will entirely originate from the correlations between the two spins, and since quantum discordrepresents total quantum correlations, the behavior of quantum coherence will be similar to that of quantum discord.It is worthwhile noting that in some models (e.g. Heisenberg spin models with Dzyaloshinsky-Moriya interactions[51]) the inverse behavior is observed, and the behavior of quantum coherence is reminiscent of entanglement notquantum discord.Moreover, in [52] it was found that quantum discord can be understood from the discrepancy between the relativequantum coherence for the total system and that of the subsystem chosen for the calculation of quantum discord. Thisset up a clear interplay between quantum correlations and quantum coherence.Finally, a representation of the complete spectrum of quantum coherence for the 2N, 3N and 4N spin pairs issketched in Figure. 8, in which we see the long-range property of quantum coherence. M a g n e t i c F i e l d T e m p e r a t u r e Q . C o h e r e n c e (a) 2N M a g n e t i c F i e l d T e m p e r a t u r e Q . C o h e r e n c e (b) 3N M a g n e t i c F i e l d T e m p e r a t u r e Q . C o h e r e n c e (c) 4NFigure 8: 3D plot of quantum coherence for the 2N, 3N and 4N spin pairs This characteristic makes it a candidate, like quantum discord, to characterize quantum phase transitions [36]. Wealso observe from the figure that like the other types of correlations studied in this paper the magnitude of quantumcoherence decreases with the distance between the spins.Furthermore, from Fig. 7a and Fig. 8 we see that quantum coherence decreases with the temperature which is asimilar treat with respect to other types of correlations studied in this paper. For quantum correlations, this is due ingeneral to the dominance of thermal fluctuation over quantum fluctuations, when we reach a threshold temperatureas we increase it, which forces quantum correlations to decay. However, what is more interesting about quantumcoherence is that it outperforms entanglement and quantum discord in the robustness to the temperature as it can beseen from Figure. 7a and Figure. 8 which is a direct consequence of the inclusion relation : Entanglement ⊂ Quantumdiscord ⊂ Quantum coherence. Knowing that entanglement is a form of coherence and given the non-local nature ofquantum coherence, this may make it an outstanding candidate from the perspective of resource theory to performquantum information processing task using quantum spin chains.
The XX model is known to exhibit quantum phase transitions (QPT). At the thermodynamic limit, which is thecase studied here, the XX model undergoes only one continuous QPT at a critical field h c = J =
1, where the systemtransits from Mott-insulator phase to superfluid phase or vice versa. The ground state of the Mott-insulator phase isa separable state while the ground state of superfluid phase is sure to be an entangled state [53, 54]. In general, QPTare due to quantum fluctuations which stems from the Heisenberg uncertainty principle [53], and these fluctuationscan be captured by quantum correlations. Moreover, QPT is followed by a specific change in the nature of quantum11 .4 0.6 0.8 1.0 1.2 1.4 1.6h−6−5−4−3−2−10 ∂ Q E / ∂ h (a) Quantum entanglement (QE) ∂ Q D / ∂ h (b) Quantum discord (QD) ∂ Q C / ∂ h (c) Quantum coherence (QC)Figure 9: First derivative of entanglement, quantum discord and quantum coherence with respect to the magnetic field h , for the 2N, 3N and 4Nspin pairs correlations in the ground state of the system and this line of thoughts is what led at the beginning to the investigationsof QPT in the vicinity of entanglement [8].Studying the derivative of entanglement with respect to the magnetic field reveals the quantum critical point clearly.In Fig. 9a, the derivative of quantum entanglement shows a singular behavior for the 2N, 3N and 4N spin pairs, at h c = ffi cient in othermodels [18, 55], like in the case of the Ising model, where the entanglement vanishes for sites farther than 2N spinneighbors [17]. For this purpose, alternatives like multipartite entanglement, quantum discord and quantum coherenceare needed in the study of QPT in more general models.Finally, we conclude this section by studying the behavior of the derivative of quantum discord and quantum coherencewith respect to the magnetic field, where in Fig. 9b and Fig. 9c we see that both measures show a singular behaviorlike entanglement at the quantum critical point h c =
6. Conclusion
In summary, we presented in this paper a comparative study between various types of pairwise correlations thatmay emerge in a quantum spin system, such as : entanglement, quantum discord, classical correlations and quantumcoherence. The analytical expressions of the pairwise quantities were provided using the Jordan-Wigner transforma-tion which we also verified numerically, and the “Pauli basis expansion”. We then, showed the dominance of quantumcorrelations over their classical counterpart, the potential use of quantum discord in quantum information processingbeing robust against the temperature, and the role of quantum entanglement, quantum discord and quantum coher-ence in detecting QPT in the XX model. Finally, we focused on quantum coherence, an essential feature of quantummechanics which we proved to be a long range quantity and outperforming entanglement and quantum discord in therobustness to temperature. This motivates the generalization of the study of quantum coherence to more generalizedmodels in higher dimensions, and also to perform quantum information processing using quantum spin chains, withquantum coherence as a resource.
Acknowledgments
It is our pleasure to thank Prof. Abderrahmane Maaouni for his helpful advices on various numerical issuesencountered in this paper. Zakaria Mzaouali would like to thank ICTP-Trieste for its hospitality during his visit,where part of the work was done. 12 eferences [1] E. Schr¨odinger, Discussion of probability relations between separated systems, Math. Proc. Cambridge Philos. Soc. 31 (4) (1935) 555563.[2] R. Horodecki, P. Horodecki, M. Horodecki, K. Horodecki, Quantum entanglement, Rev. Mod. Phys. 81 (2009) 865–942.[3] A. Einstein, B. Podolsky, N. Rosen, Can quantum-mechanical description of physical reality be considered complete?, Phys. Rev. 47 (1935)777–780.[4] C. H. Bennett, G. Brassard, C. Cr´epeau, R. Jozsa, A. Peres, W. K. Wootters, Teleporting an unknown quantum state via dual classical andeinstein-podolsky-rosen channels, Phys. Rev. Lett. 70 (1993) 1895–1899.[5] H. Salih, Z.-H. Li, M. Al-Amri, M. S. Zubairy, Protocol for direct counterfactual quantum communication, Phys. 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