Triple Spin Interaction and Entanglement

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Triple Spin Interaction and Entanglement

Willi-Hans Steeb

International School for Scientific Computing, University of Johannesburg, Auckland Park 2006, South Africa

Reprint requests to W.-H. S.; E-mail:steebwilli@gmail.com

Z. Naturforsch.68a,172 – 177 (2013) / DOI: 10.5560/ZNA.2012-0087 Received September 20, 2012 / published online February 15, 2013

We study a Hamilton operator ˆH for spin-1/2 with triple spin interactions. The eigenvalues and eigenvectors are determined and the unitary matrices exp(−i ˆHt/¯h)are found. Entanglement of the eigenvectors is investigated. A Hamilton operator ˆK for spin-1 and triple spin interaction is also discussed.

Key words:Spin Interactions; Hamilton Operators; Entanglement.

1. Introduction

In quantum theory Hamilton operators with spin- interactions have a long history [1–5]. Triple spin interaction have been studied by several authors [6–19].

Igl´oi [6] investigated an Ising model with three-spin interaction by finite-size scaling and applying free boundary conditions. Vanderzande and Igl´oi [7] studied the critical behaviour and logarithmic corrections of a quantum model with three-spin interaction. Al- caraz and Barber [8] and Wittlich [9] studied a one- dimensional Ising quantum chain with 3N sites with staggered three-spin coupling and periodic boundary conditions. Somma et al. [10] studied the unitary oper- atorU(t) =exp(iωtσ₁⊗σ₃⊗σ₂). Hereσ₁,σ₂,σ₃are the Pauli spin matrices

σ1= 0 1

1 0

, σ2=

0 −i

i 0

, σ3=

1 0 0 −1

. Pachos and Plenio [11] studied three-spin interactions in optical lattices. Three qubit Hamilton operators and Riemannian geometry has been discussed by Brandt [12]. Using the mean field method Jiang and Kong [13] studied a spin-1 quantum Ising model with three-spin interaction. Wang et al. [14] investigated the bifurcation in ground-state fidelity for a one- dimensional spin model with competing two-spin and three-spin interactions. Lanyon et al. [15] considered among others the triple-spin operatorσ3⊗σ1⊗σ1for universal digital quantum simulation with trapped ions.

Topilko et al. [16] considered magnetocaloric effects in

spin-1/2 XX chains with three-spin interactions. Zhang et al. [17] investigated the geometric phase of a qubit symmetrically coupled to a XY-spin chain with three spin interaction in a transverse magnetic field. The Greenberger–Horne–Zeilinger (GHZ) state and triple- spin operatorsσ₁⊗σ₂⊗σ₂,σ₂⊗σ₁⊗σ₂,σ₂⊗σ₂⊗ σ1,σ1⊗σ1⊗σ1have been discussed by Aravind [18]

in connection with Bell’s theorem without inequalities.

A nonlinear eigenvalue problem with triple-spin interaction has been solved by Steeb and Hardy [19]. Here we consider triple-spin interaction and entanglement for spin-1/2 systems. Spin-1 systems will also be discussed.

We consider the Hamilton operator with triple-spin interaction of spin-1/2,

Hˆ =hω¯ ₁(σ₁⊗I₂⊗I₂+I₂⊗σ₂⊗I₂+I₂⊗I₂⊗σ3) +hω¯ ₂(σ₁⊗σ₂⊗I₂+σ₁⊗I₂⊗σ₃+I₂⊗σ2⊗σ3) +hω¯ 3(σ₁⊗σ₂⊗σ₃),

whereσ1, σ2,σ3 are the Pauli spin matrices, and I2

is the 2×2 unit matrix. Thus the Hamilton operator Hˆ acts in the Hilbert spaceC⁸. The Pauli matrices are hermitian and unitary withσ₁²=σ₂²=σ₃²=I₂. The eigenvalues are+1 and−1 with the corresponding normalized eigenvectors

u₁= 1

√ 2

1 1

, u−1= 1

√ 2

1

−1

, v₁= 1

√2 1

i

, v−1= 1

√2 1

−i

,

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w₁= 1

0

, w₋₁= 0

1

.

We calculate the unitary matrixU(t) =exp(−i ˆHt/h)¯ to solve the Schr¨odinger and Heisenberg equation of motion. These unitary operators are applied to entangled and unentangled states. Entangled and unentangled states can be found depending on the pa- rameters ¯hω₁, ¯hω₂, ¯hω₃. As entanglement measure for the Hamilton operator ˆH, we consider the three- tangle.

2. A Theorem

For the calculation of the eigenvalues and eigenvectors of the Hamilton operator ˆH, we utilize the follow- ing theorem [20,21]. LetA₁,A₂,A₃ben×nmatrices over CandI_nbe then×nunit matrix. Consider the matrix

M=c₁(A₁⊗I_n⊗I_n+I_n⊗A₂⊗I_n+I_n⊗I_n⊗A₃) +c₂(A₁⊗A₂⊗I_n+A₁⊗I_n⊗A₃+I_n⊗A₂⊗A₃) +c₃(A₁⊗A₂⊗A₃),

wherec₁, c₂,c₃are constants. Since the terms in M commutate pairwise, we can write exp(M)as

e^M=e^c¹Â¹^⊗Iⁿ^⊗Iⁿe^c¹Îⁿ^⊗A²^⊗Iⁿe^c¹Îⁿ^⊗Iⁿ^⊗A³e^c²Â¹^⊗A²^⊗Iⁿ

·e^c²^A¹^⊗Iⁿ^⊗A³e^c²^Iⁿ^⊗A²^⊗A³e^c³^(A¹^⊗A²^⊗A³⁾.

If|ui,|vi,|wiare eigenvectors ofA₁,A₂,A₃, respec- tively, with eigenvaluesλ,µ,ν, we find the eigenvector|ui ⊗ |vi ⊗ |wiofMwith the eigenvalue

c₁(λ+µ+ν) +c₂(λ µ+λ ν+µ ν) +c₃(λ µ ν).

Then|ui ⊗ |vi ⊗ |wiis also an eigenvector of e^Mwith the corresponding eigenvalues

c₁(λ+µ+ν) +c₂(λ µ+λ ν+µ ν) +c₃(λ µ ν).

If the matricesA₁,A₂,A₃have the additional properties that A²₁=A²₂=A²₃=I_n(spin-¹₂ case withn=2), we obtain

e^c³^A¹^⊗A²^⊗A³= (I_n⊗I_n⊗I_n)cosh(c₃) + (A₁⊗A₂⊗A₃)cosh(c₃).

If the matricesA1,A2,A3have the additional properties thatA³_j=A_j with j=1,2,3 (spin-1 case withn=3), we obtain

e^c³^A¹^⊗A²^⊗A³=In⊗In⊗In+ (A₁⊗A₂⊗A₃)sinh(c₃) + (A²₁⊗A²₂⊗A²₃)(cosh(c₃)−1). 3. Spin-1/2 Case

The eight normalized eigenvectors of ˆHcan be con- structed from the normalized eigenvectors of σ₁,σ₂, σ3and the Kronecker products

e₁₁₁=u₁⊗v₁⊗w₁, e11−1=u₁⊗v₁⊗w−1, e1−11=u₁⊗v−1⊗w₁, e1−1−1=u₁⊗v₂⊗w₂, e−111=u−1⊗v₁⊗w₁, e−11−1=u−1⊗v₁⊗w−1, e−1−11=u−1⊗v−1⊗w₁, e−1−1−1=u−1⊗v−1⊗w−1

with the corresponding eight eigenvalues E₁₁₁=h(3ω¯ ₁+3ω₂+ω₃), E11−1=h(ω¯ ₁−ω₂−ω₃), E1−11=h(ω¯ ₁−ω₂−ω₃), E1−1−1=h(−ω¯ ₁−ω₂+ω₃), E−111=h(ω¯ ₁−ω2−ω3), E−11−1=h(−ω¯ ₁−ω2+ω3), E−1−11=h(−ω¯ ₁−ω2+ω3), E−1−1−1=h(−3ω¯ ₁+3ω₂−ω3),

whereE11−1=E1−11=E−111andE1−1−1=E−11−1= E−1−11. The eigenvaluesE₁₁₁andE−1−1−1are not degenerate. Note that the eight normalized eigenvectors are pairwise orthogonal. Thus we have (spectral de- composition)

Hˆ=

∑

j,k,`∈{1,−1}

E_jk`e_jk`e^∗_jk`.

Now the unitary matrixU(t) = e^−it^H/¯^ˆ ^h can be con- structed from the normalized eigenvectors and eigenvalues of ˆHvia

e^−it^H/¯^ˆ ^h=

∑

j,k,`∈{1,−1}

e^−iE^jk`^t/¯^he_jk`e^∗_jk`.

We find

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U(t) =







u₁₁(t) 0 u₁₃(t) 0 u₁₅(t) 0 u₁₇(t) 0 0 u22(t) 0 u24(t) 0 u₂₆(t) 0 u28(t) u₃₁(t) 0 u₃₃(t) 0 u₃₅(t) 0 u₃₇(t) 0

0 u₄₂(t) 0 u₄₄(t) 0 u₄₆(t) 0 u₄₈(t) u₅₁(t) 0 u₅₃(t) 0 u₅₅(t) 0 u₅₇(t) 0

0 u₆₂(t) 0 u₆₄(t) 0 u₆₆(t) 0 u₆₆(t) u71(t) 0 u73(t) 0 u₇₅(t) 0 u77(t) 0

0 u₈₂(t) 0 u₈₄(t) 0 u₈₆(t) 0 u₈₈(t)







with

u₁₁(t) =u₃₃(t) =u₅₅(t) =u₇₇(t)

= e^−iE¹^t/¯^h/4+e^−iE²^t/¯^h/2+e^−iE⁴^t/¯^h/4, u₁₃(t) =−u₃₁(t) =−i e^−iE¹^t/¯^h/4+i e^−iE⁴^t/¯^h/4, u₁₅(t) =u₅₁(t) =e^−iE¹^t/¯^h/4−e^−iE⁴^t/¯^h/4, u₁₇(t) =−u₇₁(t)

=−i e^−iE¹^t/¯^h/4−i e^−iE⁴^t/¯^h/4+i e^−iE²^t/¯^h/2, u22(t) =u44(t) =u₆₆(t) =u88(t)

=e^−iE²^t/¯^h/4+e^−iE⁴^t/¯^h/2+e^−iE⁸^t/¯^h/4, u₂₄(t) =−u₄₂(t) =−i e^−iE²^t/¯^h/4+i e^−iE⁸^t/¯^h/4, u₂₆(t) =u₆₂=e^−iE²^t/¯^h/4−e^−iE⁸^t/¯^h/4,

u₂₈(t) =−u₈₂

=−i e^−iE²^t/¯^h/4+i e^−iE⁴^t/¯^h/2−i e^−iE⁸^t/¯^h/4, u35(t) =−u₅₃

=i e^−iE¹^t/¯^h4−i e^−iE²^t/¯^h/2+i e^−iE⁴^t/¯^h/4, u₃₇(t) =u₇₃=e^−iE¹^t/¯^h/4−e^−iE⁴^t/¯^h/4,

u₄₆(t) =−u₆₄

=i e^−iE²^t/¯^h/4−i e^−iE⁴^t/¯^h/2+i e^−iE⁸^t/¯^h, u₄₈(t) =u₈₄=e^−iE²^t/¯^h/4−i e^−iE⁸^t/¯^h/4, u₅₇(t) =−u₇₅=−i e^−iE¹^t/¯^h/4+i e^−iE⁴^t/¯^h/4, u68(t) =−u₈₆=−i e^−iE²^t/¯^h/4+i e^−iE⁸^t/¯^h/4

with 1↔111, 2↔11−1, 3↔1−11, 4↔1−1−1, 5↔ −111, 6↔ −11−1, 7↔ −1−11, 8↔ −1−1− 1.

4. Entanglement

Entangled quantum states are an important compo- nent of quantum computing techniques such as quantum error-correction, dense coding, and quantum tele- portation [21–27]. Entanglement is the characteristic

trait of quantum mechanics which enforces its entire departure from classical lines of thought. We consider entanglement of pure states. Entanglement of Hamil- ton operators with triple-spin interaction has not been considered so far.

The eigenvectors given above are product states and hence not entangled. However, some of the eigenvalues are degenerate(E₂=E3=E₅andE4=E₆=E7)and thus we can form linear combinations of these eigenvectors which are again eigenvectors and can be entangled.

An N-tangle can be defined for the finite dimen- sional Hilbert spaceH=C²

N, withN=3 orN even.

Two-level and higher-level quantum systems and their physical realization have been studied by many authors. We consider the finite-dimensional Hilbert space H=C²

N and the normalized states

|ψi=

1 j₁,j₂,...,

∑

j_N=0

c_j₁_,_j₂_,...,_j_N|j₁i ⊗ |j₂i ⊗ · · · ⊗ |j_Ni

in this Hilbert space. Here|0i,|1idenotes the standard basis. Letε_jk(j,k=0,1)be defined byε₀₀=ε₁₁=0, ε01=1,ε10=−1. LetN be even orN=3. Then an N-tangle can be introduced by [27]

τ1...N=2

1

∑

α₁,...,αN=0 ...

δ1,...,δ_N=0

c_α₁_...α_Nc_β₁_...β_Nc_γ₁_...γ_nc_δ₁_...δ_N

·ε_α₁_β₁ε_α₂_β₂· · ·ε_α_N−1_β_N−1ε_γ₁_δ₁ε_γ₂_δ₂

· · ·ε_γ_N−1_δ_N−1εα_Nγ_Nε_β_N_δ_N .

This includes the definition for the 3-tangle withN= 3. A computer algebra program to find theN-tangle is given by Steeb and Hardy [28].

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ConsiderN=3 and the GHZ-state and theW-state

|GHZi= 1

√ 2





 1 0 0 0 0 0 0 1







, |Wi= 1

√ 3





 0 1 1 0 1 0 0 0





 .

Then we find for the GHZ-state thatτ=1 andτ=0 for theW-state. Note that the|Wistate is not separable.

Consider the triple-spin interaction Hamilton operator

HˆT

¯

hω =σ1⊗σ2⊗σ3.

The eigenvalues are +1 (fourfold) and−1 (fourfold).

A set of normalized eigenvectors are the separable states (and thus not entangled) given above. Owing to the degeneracy of the eigenvalue+1 (and analogously for the eigenvalue−1), we can form linear combinations of these separable eigenstates that are fully entangled. From the four separable eigenstates with eigenvalue+1

1 1

⊗ 1

1

⊗ 1

1 1 1

⊗ 1

1

⊗ 1

1 1

1

⊗ 1

1

⊗ 1

1 1 1

⊗ 1

1

⊗ 1

1

,

we find by linear combinations the fully entangled eigenstates (using the three tangle described above) with eigenvalue+1

1 2





 1 1 0 0 0 0 i

−i





 , 1

2





 0 1 i 0 1 0 0

−i





 , 1

2





 0 0 i

−i 1 1 0 0





 , 1

2





 0 1 i 0 1 0 0 i





 .

The four vectors are also linearly independent. Anal- ogously we can make this construction for the eigenvalue −1 to find fully entangled states, i. e. the three tangle isτ= +1.

5. Spin-1 Case

Consider the Hamilton operator with triple-spin interaction with spin-1 system with the Hamilton operator

Kˆ=hω¯ ₁(s₁⊗I₃⊗I₃+I₃⊗s₂⊗I₃+I₃⊗I₃⊗s₃) +hω¯ ₂(s₁⊗s₂⊗I₃+s₁⊗I₃⊗s₃+I₃⊗s₂⊗s₃) +hω¯ ₃(s₁⊗s₂⊗s₃),

wheres₁,s₂,s₃are the trace-less and hermitian 3×3 matrices

s₁= 1

√2





0 1 0

1 0 1

0 1 0



, s₂= 1

√2





0 −i 0

i 0 −i

0 i 0



,

s₃=





1 0 0

0 0 0

0 0 −1



,

andI₃is the 3×3 identity matrix. The Hamilton operator ˆK acts in the Hilbert spaceC²⁷. Here one can investigate whether the state inC²⁷ can be written as a product state of two vectors inC⁹ andC³,C³ and C⁹, orC³,C³andC³. Note thats³_j=sjfor j=1,2,3 and thuss⁴_j=s²_j for j=1,2,3. The eigenvalues of the matricess1,s2,s3are given by+1, 0,−1. The normalized eigenvectors fors₁are

u₁=1 2





√1 2 1



, u₀= 1

√ 2



 1 0

−1



,

u−1=1 2



 1

−√ 2 1



.

The normalized eigenvectors fors₂are

v₁=1 2





√1 2i

−1



, v₀= 1

√ 2



 1 0 1



,

v−1=1 2



 1

−√ 2i

−1



.

The normalized eigenvectors fors₃is the standard basis denoted byw₁,w₀,w−1. Thus the 27 normalized eigenvectors (separable states) are given by

e_jk`=u_j⊗v_k⊗w_`, j,k, `=1,0,−1

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with the 27 eigenvalues

Ejk`=hω¯ 1(j+k+`) +hω¯ 2(jk+j`+k`) +hω¯ ₃(jk`).

Now for the unitary matrixV(t) =e^−it^K/¯^ˆ ^h, we find V(t) =

∑

j,k,`∈{1,0,−1}

e^−iE^jk`^t/¯^he_jk`e^∗_jk`. Note that forz∈C, we have

e^zs¹^⊗I³^⊗I³=I₃⊗I₃⊗I₃+ (s₁⊗I₃⊗I₃)sinh(z) + (s²₁⊗I₃⊗I₃)(cosh(z)−1), e^zs¹^⊗s²^⊗I³=I₃⊗I₃⊗I₃+ (s₁⊗s₂⊗I₃)sinh(z)

+ (s²₁⊗s²₂⊗I₃)(cosh(z)−1), e^zs¹^⊗s²^⊗s³=I3⊗I3⊗I₃+ (s₁⊗s2⊗s₃)sinh(z)

+ (s²₁⊗s²₂⊗s²₃)(cosh(z)−1).

Withz=−iωt, we arrive at

e^−iωts¹^⊗s²^⊗s³ =I₃⊗I₃⊗I₃−i(s₁⊗s₂⊗s₃)sin(ωt) + (s²₁⊗s²₂⊗s²₃)(cos(ωt)−1).

For ω1=0, ω2=0, the eigenvalues are highly degenerate, and we can form eigenvectors which are entangled.

6. Conclusion

We have studied spin Hamilton operators with triple-spin interaction. If the eigenvalues are degenerate then by linear combinations, we can construct entangled states from unentangled states.

Acknowledgement

The author is supported by the National Research Foundation (NRF), South Africa. This work is based upon research supported by the National Research Foundation. Any opinion, findings and conclusions or recommendations expressed in this material are those of the author(s) and therefore the NRF do not accept any liability in regard thereto.

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