qubit rings

PHYSICAL REVIEW A, 66, 034302 ~2002!

Thermal and ground-state entanglement in HeisenbergXX qubit rings

Xiaoguang Wang*Institute for Scientific Interchange (ISI) Foundation, Viale Settimio Severo 65, I-10133 Torino, Italy

~Received 2 April 2002; published 12 September 2002!

We study the entanglement of thermal and ground states in the HeisenbergXX qubit rings with a magneticfield. A general result is found that for even-number rings, pairwise entanglement between nearest-neighborqubits is independent of both the sign of exchange interaction constants and the sign of magnetic fields. As anexample we study the entanglement in the four-qubit model and find that the ground state of this model withoutmagnetic fields is shown to be a four-body maximally entangled state measured by the squareN-bit concur-rence.

DOI: 10.1103/PhysRevA.66.034302 PACS number~s!: 03.67.Hk, 03.65.Ud, 75.10.Jm

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Quantum entanglement is an important predictionquantum mechanics and constitutes indeed a valuablesource in quantum information processing@1#. Much effortshave been devoted to the study and characterization oVery recently one kind of natural entanglement, the thermentanglement@2–7#, has been proposed and investigatThe investigations of this type of entanglement, ground-sentanglement@8,9# in quantum spin models, and relation@10,11# between quantum phase transition@12# and entangle-ment provide a bridge between the quantum informattheory and condensed-matter physics.

Consider a thermal equilibrium state in a canonicalsemble. In this situation the system state is described byGibb’s density operator rT5exp(2H/kT)/Z, where Z5tr@exp(2H/kT)# is the partition function,H is the systemHamiltonian,k is Boltzmann’s constant that we henceforwill take equal to 1, andT is the temperature. AsrT repre-sents a thermal state, the entanglement in the state is cthermal entanglement@2#. In a recent paper@6# we showedthat in the isotropic HeisenbergXXX model the thermal entanglement is completely determined by the partition fution and is directly related to the internal energy. In genethe entanglement cannot be determined only by the partifunction.

In this Brief Report, we consider a physical HeisenbeXX N-qubit ring with a magnetic field and aim to obtasome general results about the thermal and ground-statetanglement. We consider a four-qubit model as an examand examine in detail the properties of entanglement. Bthe pairwise and many-body entanglements are conside

In our model the qubits interact via the following Hamtonian @13#:

H~J,B!5J(i 51

N

~s ixs i 11x1s iys i 11y!1B(i 51

N

s iz , ~1!

wheresW i5(s ix ,s iy ,s iz) is the vector of Pauli matrices,J isthe exchange constant, andB is the magnetic field. The positive and negativeJ correspond to the antiferromagnet~AFM! and ferromagnetic~FM! cases, respectively. We a

*Present address: Department of Physics, Macquarie UniverSidney, New South Wales 2109, Australia.

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sume periodic boundary conditions, i.e.,N11[1. Thereforethe model has the symmetry of translational invariance.

It is easy to check that the commutator@H,Sz#50, ~rota-tion symmetry about thez axis! which guarantees that reduced density matrixr125tr3,4, . . . ,N(rT) of two nearest-neighbor qubits, say qubit 1 and 2, for the thermal staterThas the form@8#

r125S u1 0 0 0

0 w z 0

0 z w 0

0 0 0 u2

D ~2!

in the standard basis$u00&,u01&,u10&,u11&%. Here u i j &[u i &^ u j &( i , j 50,1), u0& (u1&) denotes the state of spin-u~-down!, andSz5( i 51

N s iz/2 are the collective spin operatorThe Hamiltonian is invariant under the transformati)n51

N/2 Sn,N2n11 , whereSi j is the swap operator for qubitsiand j. This is reflection symmetry which implies that thmatrix elementz is a real number. For any operatorA12 act-ing on qubits 1 and 2, we have the relation

tr12~A12r12!5tr1,2, . . . ,N~A12rT!. ~3!

Then the reduced density matrixr12 is directly related tovarious correlation functions Gab5^s1as2b&5tr1, . . . ,N(s1as2brT) (a5x,y,z). Precisely the matrix ele-ments can be written in terms of the correlation functioand the magnetizationM5tr(( i 51

N s izrT) as

u15tr1, . . . ,N~ u00&^00u!5 14 ~112M1Gzz!,

u25tr1, . . . ,N~ u11&^11u!5 14 ~122M1Gzz!, ~4!

z5tr1, . . . ,N~ u01&^10u!5 14 ~Gxx1Gyy!,

whereM5M /N is the magnetization per site. In deriving thabove equation, we have used the translational invariancthe Hamiltonian.

Due to the fact@H,Sz#50 one hasGxx5Gyy . Then, theconcurrence@14# quantifying the entanglement of two qubiis readily obtained as@8#

ty,

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BRIEF REPORTS PHYSICAL REVIEW A66, 034302 ~2002!

C5maxF0,UGxxU2 1

2A~11Gzz!

224M2G , ~5!

which is determined by the correlation functionGxx ,Gzz,and the magnetization. In the literature there are a lotresults on correlation functions@15# in various quantum spinmodels, and they may be used for the calculations oftanglement.

By using the translational invariance of the Hamiltoniand the rotational symmetry about thez axis, we find a usefulrelation

Gxx5~U2BM!/~2J!, ~6!

whereU5U/N is the internal energy per site andU is theinternal energy. From the partition function we can obtaininternal energy and the magnetization through the wknown relations

U521

Z

]Z

]b,M52

1

Zb

]Z

]B, ~7!

where b51/T. As a combination of Eqs.~6! and ~7! thecorrelation functionGxx is solely determined by the partitiofunction. Therefore the concurrence can be obtained frompartition function and the correlation functionGzz. We seethat the partition function itself is not sufficient for determiing the entanglement.

Except for the symmetries used in the above discussthere are also other symmetries in our model. Consideroperator Lx[s1x^ s2x^ •••^ sNx satisfing Lx

251. Wehave the commutator@s ias i 11a ,Lx#50 and anticommuta-tor @s iz ,Lx#150. Then we immediately obtain

Gaa5tr$Lxs1as2aexp@2bH~J,B!#Lx%/Z

5tr$s1as2aexp@2bH~J,2B!#%/Z,

M5tr$Lxs1zexp@2bH~J,B!#Lx%/Z

52tr$s1zexp@2bH~J,2B!#%/Z.

These equations tell us that the correlation functionGaa andthe square of the magnetization is invariant under the traformationB→2B. From Eq.~5! and the invariance ofGaa

andM2 we find the following Proposition.Proposition 1. The concurrence is invariant under t

transformation B→2B.The proposition shows that the pairwise thermal entan

ment of the nearest-neighbor qubits is independent ofsign of the magnetic field. And it is valid for both the eveand the odd number of qubits.

In Ref. @3#, we observe an interesting result that the pawise thermal entanglement for two-qubitXX model with amagnetic field is independent of the sign of the exchaconstantJ. Now we generalize this result to the case ofbitrary even-number qubits.

For the case of even-number qubits we have

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H~2J,B!5LzH~J,B!Lz ,

Lz5s1z^ s3z^ •••^ sN21z .

The transformationLz changes the sign of the exchange costantJ. Note that the definition ofLz is different from that ofLx . It is straightforward to prove

Gxx5tr~Lzs1xs2xe2bH(J,B)Lz!52tr~s1xs2xe

2bH(2J,B)!,

Gzz5tr~Lzs1zs2ze2bH(J,B)Lz!5tr~s1zs2ze

2bH(2J,B)!,

M5tr~s1ze2bH(2J,B)!.

From these equations we see that the absolute valueuGxxu,the correlation functionGzz, and the magnetization are invariant under the transformationJ→2J. Hence from Eq.~5!and Proposition 1 we arrive at the following Proposition.

Proposition 2. For the even-number XX model withmagnetic field the pairwise thermal entanglement of nearneighbor qubits is independent of the sign of exchange cstant J and the sign of magnetic field B.

The proposition shows that the entanglement of AFM qbit rings is the same as that of FM rings.

Now we consider the case of no magnetic fields. Thenmagnetization will be zero and Eq.~5! reduces toC

5 12 max(0,uU/Ju2Gzz21). We can prove that the interna

energy is always negative. First, from the traceless propof the Hamiltonian, it is immediate to check that

limT→`

U5 limT→`

(n

Ene2En /T

(n

e2En /T

50, ~8!

where En is the eigenvalue of the HamiltonianH. In thislimit the correlation functionGxx and the magnetizationMare also zero. Further from the fact that

]U/]T5~^H2&2^H&2!/T25~DH !2/T2.0 ~9!

we conclude thatU is always negative. Then we arrive at thfollowing Proposition.

Proposition 3. The concurrence of the nearest-neighbqubits in the Heisenberg XX model without a magnetic fiis given by

C5H 1

2max@0,2U/J2Gzz21# for AFM,

1

2max@0,U/J2Gzz21# for FM.

~10!

From the proposition we know that even for the caseno magnetic fields the partition function itself cannot detmine the entanglement. In the limit ofT→0, the above equation reduces to

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C5H 1

2maxF0,2

E(0)

NJ2Gzz

(0)21G for AFM,

1

2maxF0,

E(0)

NJ2Gzz

(0)21G for FM,

~11!

whereE(0) is the ground-state energy andGzz(0) is the corre-

lation function on the ground state. Therefore the groustate pairwise entanglement of the system is determinedboth the ground-state energy and the correlation funcGzz

(0) . Next we consider a four-qubitXX model as an ex-ample.

To study the pairwise entanglement we need to solveeigenvalue problems of the four-qubit Hamiltonian. We woin the invariant subspace spanned by vectors of fixed numr of reversed spins. The subspacer 50 (r 54) is triviallycontaining only one eigenstateu0000& (u1111&) with eigen-value 4B (24B). The subspacer 51 is four dimensional.The corresponding eigenvectors and eigenvalues are gby

uk&51

2 (n51

4

exp~2 inkp/2!un&0~k50, . . . ,3!, ~12!

and 4J cos(kp/2)12B, respectively. Here the ‘‘numbestate’’ un&05T n21u1&0 , u1&05u1000&, and T is the cyclicright shift operator that commutes with the HamiltonianH@16#. Due to the fact thatH commutesLx , uk&85Lxuk& isalso a eigenstate with eigenvalue 4J cos(kp/2)22B. Thestatesuk& and uk&8 are the so-calledW states@17# whosecorresponding concurrence is 1/2. We have diagonalizedHamiltonian in the subspaces ofr 50,1,3,4. Now we diago-nalize the Hamiltonian in the subspacer 52 and use thefollowing notations:

u1&15u1100&,un&15T n21u1&1~n51,2,3,4!,~13!

u1&25u1010&,um&25T m21u1&2~m51,2!.

The action of the Hamiltonian is then described by

Hun&152J (m51

2

um&2 ,Hum&252J(n51

4

un&1 . ~14!

By diagonalizing the corresponding 636 matrix the eigen-values and eigenstates are given by

E6564JA2,uC6&51

2A2S (

n51

4

un&16A2 (m51

2

um&2D ,

E150,uC1&51

A2~ u1010&2u0101&),

E250,uC2&51

A2~ u1100&2u0011&), ~15!

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E350,uC3&51

A2~ u1001&2u0110&),

E450,uC4&51

2~ u1100&1u0011&2u1001&2u0110&).

From Eqs.~12! and ~15! all the eigenvalues are obtained64B (1),62B (2), 4J62B (1),24J62B (1), 64JA2(1),0 (4), where the numbers in the parentheses denotedegeneracy. Then the partition function simply follows as

Z5412cosh~4A2bJ!12 cosh~4bB!

14@11cosh~4bJ!#cosh~2bB!. ~16!

From Eqs.~7!, ~12!, ~15!, and ~16!, we get the internal en-ergy, magnetization, and correlation functionGzz,

2ZU52JA2sinh~4A2bJ!12B sinh~4bB!

12B@11cosh~4bJ!#sinh~2bB!

14J sinh~4bJ!cosh~2bB!, ~17!

2ZM52 sinh~4bB!12@11cosh~4bJ!#sinh~2bB!,

ZGzz52 cosh~4bB!2cosh~4A2bJ!21.

Then according to the relation~6! the correlation functionGxx is obtained as

GxxZ52A2sinh~4A2bJ!22 sinh~4bJ!cosh~2bB!.~18!

The combination of Eqs.~17!, ~18!, and ~5! gives the exactexpression of the concurrence. It is easy to see that thecurrence is independent of the sign ofJ and B, which isconsistent with the general result given by Proposition 2even-number qubits.

Figure 1 gives a three-dimensional plot of the concurreagainst the temperature and magnetic field. The exchaconstantJ is chosen to be 1 in the following. We observe

FIG. 1. The concurrence against the temperature and magfield. The parameterJ51.

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threshold temperatureTc52.363 38 after which the entanglement disappears. It is interesting to see that the threstemperature is independent of the magnetic fieldB for ourfour-qubit model. For two-qubitXX model the thresholdtemperature is also independent ofB @3#. We further observethat near the zero temperature there exists a dip whenmagnetic field increases from zero. The dip is due toenergy level crossing at pointBc152(A221)50.828 43.WhenB increases formB52 near the zero temperature thentanglement disappears quickly since there is another lcrossing pointBc252 after which the ground state becomu1111&. We also see that it is possible to increase entanment by increasing the temperature in the range of magnfield B.2.

Now we discuss the ground-state entanglement (T50).WhenB,Bc1, the ground state isuC2& with the eigenvalueE(0)524A2J. It is direct to check thatGzz

(0)521/2. Thenaccording to Eq.~11!, the concurrence is obtained asC5A2/221/450.45711. WhenBc1,B,Bc2 the ground stateis uk52&8 and the corresponding concurrence is 1/2. WhB.Bc2 the ground state becomesu1111&, so there exists noentanglement.

Thus far we have discussed thepairwise entanglementinboth the ground state and the thermal state. Now we disthe many-body entanglement in the ground state. ReceCoffmanet al. @18# used concurrence to examine three-qusystems, and introduced the concept of the square 3-bitcurrence as a way to quantify the amount of three waytanglement in three-qubit systems. Later Wong and Chtensen@19# generalize the square 3-bit concurrence toeven-number squareN-bit concurrence. The squareN-bitconcurrence is defined as

t1,2, . . . ,N[u^cus1y^ s2y^ •••^ sNyuc* &u2, ~19!

ur

ntra

24

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with uc& a multiqubit pure state. The squareN-bit concur-rence works only for an even number of qubits.

For the ground stateuC2& we haveuC2* &5uC2&, and

s1y^ s2y^ s3y^ s4yuC2&5uC2&, ~20!

i.e., the ground state is an eigenstate of the operators1y^ s2y^ s3y^ s4y . Therefore we findt1,2, . . . ,451, whichmeans that the ground state has four-body maximal entanment. For another two ground states when varying the mnetic field it is easy to check thatt1,2, . . . ,450, which meansthat the ground states have no genuine four-body entanment. Note that the ground stateuk52&8 has pairwise en-tanglement, but no four-body entanglement.

In conclusion, we have found that the pairwise thermentanglement of nearest-neighbor qubits is independenthe sign of exchange constants and the sign of magnfields in theXX even-number qubit ring with a magnetfield. For determining the concurrence we need to knowonly the partition function, but also one correlation functioGzz. For the four-qubit model we observe that there existthreshold temperature that is independent of the magnfield. The effects of level crossing on the thermal entangment and ground-state entanglement are also discussednally we find that the ground state is a four-body maximaentangled state according to the potential many-bodytanglement measure, the squareN-bit concurrence.

The authors thank Paolo Zanardi, Irene D’AmicHongchen Fu, Professor Allan I. Solomon, ProfessJoachim Stolze, Professor Guenter Mahler, and ProfeVladimir Korepin for helpful discussions. This work wasupported by the European Community through Grant NIST-1999-10596~Q-ACTA!.

M.

@1# See, for example, C.H. Bennett and D.P. DiVincenzo, Nat~London! 404, 247 ~2000!.

@2# M.A. Nielsen, Ph.D thesis, University of Mexico, 1998, e-priquant-ph/0011036; M.C. Arnesen, S. Bose, and V. VedPhys. Rev. Lett.87, 017901~2001!.

@3# X. Wang, Phys. Rev. A64, 012313~2001!; Phys. Lett. A281,101 ~2001!.

@4# X. Wang, H. Fu, and A.I. Solomon, J. Phys. A34, 11 307~2001!; X. Wang and Klaus Mo” lmer, Eur. Phys. J. D18, 385~2002!.

@5# P. Zanardi and X. Wang, e-print quant-ph/0201028.@6# X. Wang and P. Zanardi, Phys. Lett. A301, 1 ~2002!.@7# G.L. Kamta and A.F. Starace, Phys. Rev. Lett.88, 107901

~2002!.@8# K.M. O’Connor and W.K. Wootters, Phys. Rev. A63, 052302

~2001!.@9# D.A. Meyer and N.R. Wallach, e-print quant-ph/0108104.

@10# T.J. Osborne and M.A. Nielsen, e-print quant-ph/01090e-print quant-ph/0202162.

@11# A. Osterloh, L. Amico, G. Falci, and R. Fazio, Nature~Lon-

e

l,

;

don! 416, 608 ~2002!.@12# S. Sachdev,Quantum Phase Transitions~Cambridge Univer-

sity Press, Cambridge, 2000!.@13# E.H. Lieb, T.D. Schultz, and D.C. Mattis, Ann. Phys.~N.Y.!

16, 407 ~1961!.@14# W.K. Wootters, Phys. Rev. Lett.80, 2245~1998!.@15# For examples, H.E. Boos, V.E. Korepin, Y. Nishiyama, and

Shiroishi, e-print cond-mat/0202346; K. Fabricius, U. Lo¨w,and J. Stolze, Phys. Rev. B55, 5833~1997!; V.S. Viswanath, S.Zhang, J. Stolze, and G. Mu¨ller, ibid. 49, 9702~1994!.

@16# D. Kouzoudis, J. Magn. Magn. Mater.173, 259 ~1997!; K.Barwinkel, H.J. Schmidt, and J. Schnack,ibid. 220, 227~2000!.

@17# W. Dur, G. Vidal, and J.I. Cirac, Phys. Rev. A62, 062314~2000!; W. Dur, ibid. 63, 020303~2001!; M. Koashi, V. Buzek,and N. Imoto,ibid. 62, 050302~2000!.

@18# V. Coffman, J. Kundu, and W.K. Wootters, Phys. Rev. A61,052306~2000!.

@19# A. Wong and N. Christensen, Phys. Rev. A63, 044301~2001!.

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qubit rings