Quantum and classical correlations in the anisotropic XY model with a single defect

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Quantum and classical correlations in the anisotropic XY model with a single defect

W.W. Cheng

^a,b,n

, C.J. Shan

^b

, Y.B. Sheng

^a

, L.Y. Gong

^a

, S.M. Zhao

^a

, J.-M. Liu

^c

aInstitute of Signal Processing & Transmission, Nanjing University of Posts and Telecommunication, Nanjing 210093, China

bDepartment of Physics, Hubei Normal University, Huangshi 435002, China

cNational Laboratory of Solid State Microstructures, Nanjing University, Nanjing 210093, China

H I G H L I G H T S

Quantum and classical correlations in theXYmodel with a defect are studied.

Quantum discord depends on the positions of the lattices, defect and the quantum phase where the system resides in.

Quantum discord between long-range sites can be modiﬁed by a local control of defect under ferromagnetic phase.

a r t i c l e i n f o

Article history:

Received 20 November 2012 Received in revised form 1 April 2013

Accepted 31 May 2013 Available online 17 June 2013 Keywords:

Quantum correlation XYmodel

Defect

a b s t r a c t

We investigate the quantum and classical pairwise correlations in the one-dimensional anisotropicXY spin chain with a single defect. The rigorous analysis and numerical results reveal that the quantum discord and classical correlation in different quantum phases present different responses to the defect. In particular, we show that the non-local long-range quantum discord can be modiﬁed by a local control of the defect under ferromagnetic phase in such system.

1. Introduction

The concept of correlation, i.e. information of one subsystem to another in a composite system, is an important element in many- body theory of condensed matter physics. For a quantum system, the correlations are often involved with the classical and quantum sources. Usually, the existence of quantum correlation as the fundamental characteristic of quantum mechanics can be char- acterized by entanglement, which was pointed out by Schrödinger in 1935. Since then, the entanglement has been regarded as the only kind of quantum correlation. In the past decades, intensive researches indicated that this quantum correlation as a potential resource in quantum computation and quantum information processing (QIP) plays a key role, and thus attracts much attention in many branches of physics both in theoretical aspects and experimental ones[1–3]. However, it has been also shown that

some separable states (non-entangled state) may speed up certain tasks over their classical counterparts [4–6], which means that there are other quantum correlations which cannot be captured by entanglement. Recently, a so-called quantum discord concept has been proposed to measure and characterize these non-classical correlations[7], inspiring much attention to this quantum correlation as a source in QIP[8–20]. For example, a recent experimental work demonstrated that the separable state can achieve higher ﬁdelity in remote state preparation than the entangled state, underlining that quantum discord quantiﬁes the non-classical correlations required for the task[21].

On the other hand, the ability to modify and manipulate the quantum correlation in quantum systems is a very important issue in QIP. In this aspect, spin chain systems provide natural models to explore the quantum correlation behaviors. In contrast to other systems in which local operator can hardly affect non-local physical quantities, here, due to the coupling interaction among spins, local control of certain parameters in a spin system may modify and manipulate the non-local quantum correlation. Along this line, many pioneer schemes have been proposed to control the behaviors of entanglement in spin chain systems by means of Contents lists available atSciVerse ScienceDirect

journal homepage:www.elsevier.com/locate/physe

Physica E

http://dx.doi.org/10.1016/j.physe.2013.05.026

nCorresponding author at: Institute of Signal Processing & Transmission, Nanjing University of Posts and Telecommunication, Nanjing 210093, China.

Tel.:+86 2583492417.

E-mail address:[email protected] (W.W. Cheng).

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defect or disorder[22,23], e.g., Apollaro et al. have shown that a single magnetic defect can produce a state with localized pairwise entanglement[22]. As a more fundamental quantum correlation, the manipulation of quantum discord in spin chain systems by the local control has been rarely considered and may be of more signiﬁcance for practical operations. Moreover, a recent work showed that the power of quantum computation associated with the Rényi entropy depends on the different quantum critical phases [24], implying that the ability to manipulate quantum correlation may depend on different quantum phases too. Conse- quently, in this work, we investigate the quantum discord and classical correlation in different quantum phases in the anisotropic XYspins chain with a single defect. Specially, the response of the long-range quantum discord to the defect in different quantum critical phases in such a system will be investigated carefully.

This work is organized as follows. InSection 2, we give a brief introduction to the anisotropic XY model and the correlations (quantum discord and classical correlation). In Section 3, the response of the quantum discord and classical correlation to the defect are investigated carefully. Finally, we give a summary for our main results.

2. XYmodel and correlations

We consider the Hamiltonian for the anisotropicXYmodel with an externalﬁeld as follows:

H¼−1þγ 2 ∑^L

i

J_i;iþ1s^xis^xiþ1−1−γ 2 ∑^L

i

J_i;iþ1s^yis^yiþ1− ∑^L

i

his^zi; ð1Þ where γ is the anisotropy parameter, J_i_;_iþ1¼Jð1þαⁱ;iþ1Þ, αⁱ;iþ1

denotes the system defect parameter and it equals to zero for all sites except the ones nearest to the defect site located atim, i.e., αi_m−1;im¼αim;imþ1¼α.h¼hð1þβiÞ, whereβithus denotes the external magneticﬁeld defect parameter on sitei, andβim¼βonly on siteimand otherβiequals to zero.s^x;y;zi denote the Pauli matrices.

By using the Jordan–Wigner transformation, the Hamiltonian can be mapped onto quasi-free spinless fermions, taking the following form:

H¼ ∑^L

i;j¼1

c^†_iMijcjþ1 2 ∑^L

i;j¼1

ðc^†_iNijc^†_jþcjNijciÞ; ð2Þ

where M and N are symmetric and antisymmetric real LL matrices, respectively. Explicitly,Mij¼−ðð1þβiÞδijþλδi;jþ1ð1þαijÞþ λδiþ1;jð1þαijÞÞ, M1L¼ML1¼−λð1þα1;LÞ and Nij¼γðλð1þαijÞδi;jþ1− λð1þαijÞδiþ1;jÞ,N_1;L¼γð1þαL;1Þ ¼−N_L;1, whereλ¼J=h.

The quadratic Hamiltonian can be diagonalized by making a linear transformation of the fermionic operators[23,25]. Here, we conﬁne our attentions to the quantum discord and classical correlation between two lattices at positionsiandjon the chain.

In the quantum physics, the total correlations between two subsystemsAandB are described by quantum mutual information,

LðρABÞ ¼SðρAÞ þSðρBÞ−SðρABÞ; ð3Þ where SðρÞ ¼Trðρlog2ρÞ is the von Neumann entropy. And the classical correlations for such a system can be deﬁned as following:

CðρABÞ ¼max

fB_kg LðρABjfBkgÞ; ð4Þ

where LðρABjfBkgÞ is a variant of quantum mutual information based on a given measurement basis fBkg on the subsystem B, which takes

LðρABjfBkgÞ ¼SðρAÞ−∑

k

p_kSðρkÞ; ð5Þ

where ρk¼ ðIA⊗BkÞρABðIA⊗BkÞ is the post-measurement state of subsystemAafter obtaining the outcomekon subsystemBwith the probabilitypk¼Tr½ðIA⊗BkÞρABðIA⊗BkÞ. Then the pure quantum correlation described by the quantum discord can be deﬁned as[7], DðρABÞ ¼LðρABÞ−CðρABÞ: ð6Þ The maximum in Eq.(4)is taken over a complete set of projective measuresfBkgon the partitionB.

Therefore, we need to consider the two-site reduced density matrixρijobtained by tracing out all spins except those at sites i andj. If there is no symmetry breaking, by taking the thermal ground state, the reduced density operator for the sites i and j reads

ρij¼

ρ11 0 0 ρ14

0 ρ22 ρ23 0 0 ρ32 ρ33 0 ρ41 0 0 ρ44

0 BB BB

@

1 CC CC

A: ð7Þ

Here, the elements of the matrixρijcan be expressed by the spin– spin correlation functions [23,25]. For such a non-symmetry X states, we can obtain the classical correlation explicitly as following[12,13],

CðρijÞ ¼maxðC¹;C²Þ; ð8Þ where Cj¼Hðρ11þρ22Þ−Dj withD1¼−∑⁴i¼1ρiilog₂ρii−Hðρ11þρ33Þ andD2¼HðτÞwithτ¼ ð1þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

½1−2ðρ33þρ44Þ²þ4ðjρ14j þ jρ23jÞ²

q Þ=2.

And the quantum discord can be derived as,

DðρijÞ ¼minðD1;D2Þ; ð9Þ where Dj¼Hðρ11þρ33Þ þDjþ∑⁴i¼1λⁱijlog₂λⁱij, and λⁱijði¼1;2;3;4Þ are the eigenvalues of the reduced density matrixρij. In the above expressions,H(x) denotes the functionHðxÞ≡−xlog2x−ð1−xÞlog2ð1−xÞ.

Subsequently, we determine how the defect parametersαandβ affect the quantum discord and classical correlation in different quantum critical phases.

3. Results and discussion

First, we give a brief comment to quantum and classical correlations in the Hamiltonian. When λ-0, the ground state becomes a product of spins pointing in the position of external magnetic direction. Obviously, both the quantum correlation and the classical ones are zero. In the λ-∞, the ground state approaches a products of spins pointing in the positivexdirection.

However, the ground state is doubly degenerate under the global phase ﬂip associated with a products of spins pointing in the negative x direction. In our study, the thermal ground state is choose[26]. So the quantum correlation is zero and the classical correlation approached to one under the limit λ-∞. And as the critical region is approached, the correlation length begins to increase. That is to say, each site begins to develop quantum correlation with its neighbors.

Now, we investigate the behavior of the correlations for the nearest-neighbor (nn) lattices. InFig. 1(a)–(d), we plot the quantum discordDand classical correlationCbetween the sites 49th and 50th with respect toλ, given different defect parametersαand β. Without lose the generality, the defect lattice locates onim¼51.

The system size L¼101 is taken based on theﬁnite-size scaling analysis. The previous works have shown that the quantum discord are maximal in a region close to the critical point λc¼1 for a pureXYmodel[19,27]. This character remains for the defect system as long as we consider the behavior ofλ^p for which the maximal point would shift with the varying parametersαandβ. FromFig. 1(a), we canﬁnd that theD∼λcurves are depressed when

W.W. Cheng et al. / Physica E 54 (2013) 72–77 73

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the system defectαincreases over the whole regionλ∈ð0;∞Þ, which means that the quantum correlation is suppressed by the system defectα. For the case where the magnetic defectβis involved, the quantum correlation is enhanced with increasingβover the whole region ofλ, as shown inFig. 1(b). These characters of the quantum discord between the nn sites are very similar to the behaviors of pairwise entanglement in the XY model with an impurity [23].

Since we know that the single diagonal (magnetic) defectβ can induce localization of the quantum correlation. So, around the critical region, although the quantum correlation share among the lattices, the magnetic defect β enhance the quantum discord between defect lattice and the others owning to the effect of correlation localization. As the non-diagonal defectαemerge, this effect would be reduced.Fig. 1(c) and (d) plot the responses of the classical correlation to the defect parametersαandβ, respectively.

Obviously, for a given λ, the classical correlation is enhanced (reduced) with increasing system defect α (magnetic defect β).

At the same time, the quantum counterpart is descendant (ascen- dent), and vice verse. All these tendency become more obvious around the critical region.Fig. 1also shows that the response of the two kinds of correlations between the nn sites to the defect does not depend very much on the quantum critical phase of the system.

Now, we check the effect of the defect on the correlations between the next-nearest-neighbor (nnn) sites. In Fig. 2, we present the quantum discordDand classical correlationCagainst λgiven different defect parameters α and β. On the whole, the response of theDandCbetween the nnn lattices to the defectα(β) are similar with counterparts in the nn case. Detailed, theD is suppressed (enhanced) by the defectα(β). This property also own to the effect of correlation localization induced by the defect.

However, there are still some slight difference compare with the case of nn in different critical region. For a XY model without defect, Dillenschneider have showed that theﬁrst derivative of the quantum discord between the nnn lattices exhibits a singularity around the pseudo-critical pointλ^p [27], which is different with the case of the nn. For the XY chain with random disorder or defect, the system is critical when the average value of theﬁeldhi

equals the average value of the couplingJi;iþ1. In other words, this feature remains true for the defect system as long as we take account of the singularity point λp which would vary with the

defect parametersαandβ. So the curvesD∼λwould shift to left (right) with increasing defect parametersαðβ). These features are shown inFig. 2(a) and (b). InFig. 2(c) and (d) we plot the classical correlationCbetween the two next nearest-neighbor lattices 49th and 51th which both locate on one side of the defectim. From the C∼λcurves, one observes the similar trend of the classical correlation in response to the system defect and magnetic defect with respect to the correlations between the nearest-neighbor sites, as shown inFig. 1(c) and (d).

For a pure system without defect, the reduced density matrix ρij only depends on the separation between the two lattices n¼ ji−jj. When a defect is introduced, matrix ρij is related not only to the sites i andj, but also to the defect positionim. In other word, the distance between the defect and subsystem (sitesi andj) may affect the correlation directly. In the above discussion, we have studied the correlations for the case with the two sitesi andj on one side ofim. Here, we examine the effect of the location of the defect im on the next-nearest- neighbor correlations D and C. In Fig. 3 is presented the quantum discord D and classical correlations C between the sites i¼50 and j¼52 as a function of λ for different defect parametersαandβ. FromFig. 3(a) and (b), it is observed that the response of quantum discord to the defect also depends on which quantum critical phase the system resides in. However, in contrary to the situation shown in Fig. 2(a) and (b), for the present case, with increasing system defectαand the magnetic onesβ, the response of the correlation to the defect are relative slight for a givenλ. This difference also comes from the property of correlation localization. For the classical correlations on such two cases,Fig. 3(c) and (d) show that the responses of classical correlationCto the defect parametersαandβare very similar to the situation with the two sites residing on one side ofim, as shown inFig. 2(c) and (d).

Another interesting issue is the behavior of the long-range correlations in response to the system and magnetic defects in different quantum critical phases. For a system without defect, Fig. 4(a) shows that the quantum discord exhibits a qualitatively different behavior below and above the critical point λc¼1. In details, a slower decay of the quantum discord is exhibited in the ferromagnetic phase region (λ41), which means that the system presents a long-range quantum correlation, and it drops to zero Fig. 1.Quantum discordDand classical correlationCfor the nearest-neighbor spin site (49th and 50th) in the one-dimensional anisotropicXYmodel. Hereafter, parameters L¼101,γ¼1:0,im¼51. (a)D∼λfor differentα,β¼0:0. (b)D∼λfor differentβ,α¼0:0. (c)C∼λfor differentα,β¼0:0. (d)C∼λfor differentβ,α¼0:0.

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quickly with increasing separationn−1 between the two spins in the paramagnetic phase region (λo1). Around the critical point, the quantum discord as a function of the separation exhibits an exponential decay behavioraþbexpð−cnÞ, where a,b,care the constants depending only onλandγ[18]. Upon the introduction of the defect, the quantum discord between the long-distance spins in the paramagnetic phase region remains less affected no matter the system defectαor magnetic defectβvaries. However, in the ferromagnetic phase region, the quantum discord between the 1st spin and the sites near to the defectimcan be modiﬁed by both the system defect and magnetic defect, as shown inFig. 4(b) and (c), respectively. In facts, the spin-spin correlation 〈s^xis^xj〉 remains a relative value due to the spin coupling interactions^xis^xiþ1 in the

ferromagnetic phase region. These results also indicate that the effect of localization can enhance the quantum discord even the lattice is far away from the defect. The classical correlation presents similar responses to the defect with its quantum counterpart. The only difference is that an increasing classical correlation is always accompanied with a descendant quantum correlation, and the detailed results are shown inFig. 4(d)–(f).

4. Conclusion

In conclusion, we have studied the quantum discord and classical correlations in the anisotropic XY spin model with a Fig. 2.Quantum discordDand classical correlationCfor the next-nearest-neighbor spin sites (49th and 51th) in the one-dimensional anisotropyXYmodel. (a)D∼λfor differentα,β¼0:0. (b)D∼λfor differentβ,α¼0:0. (c)C∼λfor differentα,β¼0:0. (d)C∼λfor differentβ,α¼0:0.

Fig. 3.Quantum discordDand classical correlationCfor the next-nearest-neighbor spin site (50th and 52th) in one-dimensional anisotropicXYmodel. (a)D∼λfor different α,β¼0:0. (b)D∼λfor differentβ,α¼0:0. (c)C∼λfor differentα,β¼0:0. (d)C∼λfor differentβ,α¼0:0.

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single defect. In the different quantum phase regions, the responses of these correlations to the defect are investigated carefully. For the nearest-neighbor spin sites, the response of the quantum discord and classical correlations to the defect is inde- pendent of the quantum phase. However, the quantum discord for the next-nearest-neighbor sites depends not only on the positions of the latticesiandjand the location of defectim, but also on the quantum phase where the system resides in. Furthermore, the non-local long-range quantum discord can also be remotely manipulated by a local-control in the ferromagnetic phase. These results hint that for different quantum phases, not only the quantum system have different physical properties but also the ability to manipulate the quantum correlation is qualitatively different.

Acknowledgments

The authors acknowledge theﬁnancial support from the NSFC (Grant nos. 11105049, 61271238 and 11234005), the 973 Project of China (Grant no. 2011CB922101), the Natural Science Foundation of Nanjing University of Posts and Telecommunications (NY211034), and the Priority Academic Program Development of Jiangsu Higher Education Institutions, China.

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Fig. 4.Quantum discordDand classical correlationCbetween the long-range spin lattice sites (1st andnth) in different quantum phases. (a)D∼nfor differentλ, α¼0:0;β¼0:0. (b)D∼nfor differentλ,α¼1:0;β¼0:0. (c)D∼nfor differentλ,α¼0:0;β¼1:0. (d)C∼nfor differentλ,α¼0:0;β¼0:0. (e)C∼nfor differentλ,α¼1:0;β¼0:0.

(f)C∼nfor differentλ,α¼0:0;β¼1:0.

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