N-tangle, Entangled Orthonormal Basis, and a Hierarchy of Spin Hamilton Operators
Willi-Hans Steeb and Yorick Hardy
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.66a,615 – 619 (2011) / DOI: 10.5560/ZNA.2011-0025 Received April 30, 2010 / revised June 29, 2011
AnN-tangle can be defined for the finite dimensional Hilbert spaceH=C2
N, withN=3 orN even. We give an orthonormal basis which is fully entangled with respect to this measure. We provide a spin Hamilton operator which has this entangled basis as normalized eigenvectors ifNis even. From these normalized entangled states a Bell matrix is constructed and the cosine–sine decomposition is calculated. IfNis odd the normalized eigenvectors can be entangled or unentangled depending on the parameters.
Key words:Entanglement; Spin Hamilton Operators; Orthonormal Basis; Cosine–Sine Decomposition.
Two level and higher level quantum systems and their physical realization have been studied by many authors (see [1] and references therein). We consider a spin Hamilton operator acting in the finite-dimensional Hilbert spaceH=C2
Nand the normalized states
|ψi=
1
∑
j1,j2,...,jN=0
cj1,j2,...,jN|j1i ⊗ |j2i ⊗ · · · ⊗ |jNi
in this Hilbert space. Here|0i,|1idenotes the standard basis. Letεjk(j,k=0,1)be defined byε00=ε11=0, ε01=1,ε10=−1.
Letσx,σy,σz be the Pauli spin matrices. We con- sider entanglement for the eigenvectors of the hierar- chy of spin Hamilton operators
HˆN=hω¯ (
N-factors z }| { σz⊗σz⊗ · · · ⊗σz)+∆1(
N-factors z }| { σx⊗σx⊗ · · · ⊗σx) +∆2(
N-factors z }| { σy⊗σy⊗ · · · ⊗σy)
with N ≥2. Here ⊗ denotes the Kronecker prod- uct [2–4],ω >0,∆1,∆2≥0, andσx⊗ · · · ⊗σx,σy⊗
· · · ⊗σy,σz⊗ · · · ⊗σzare elements of the Pauli group PN. TheN-qubit Pauli group [5] is defined by
PN:={I2,σx,σy,σz}⊗N⊗ { ±1,±i},
whereI2is the 2×2 identity matrix. TheN-qubit Clif- ford groupCN is the normalizer of the Pauli group – a unitary matrixU acting onN-qubits is contained in CNif
U MU−1∈ PN for each M∈ PN.
Thus the Hamilton operator ˆHN acts in the finite- dimensional Hilbert space H=C2N. The eigenvalue problem for the case with∆2=0 has been studied by Steeb and Hardy [6,7].
Many authors developed methods for the detec- tion and classification of entangled states in finite di- mensional Hilbert spaces for mixed states and pure N-qubit states [8–23]. A pure N-partite state is sep- arable if and only if all the reduced density matri- ces of the elementary subsystems describe pure states.
In a bipartite case, separability can be determined by calculating the Schmidt decomposition of the state.
The concept of the Schmidt decomposition cannot be straightforwardly generalized to the case ofNseparate subsystems [12]. Besides these two well-known meth- ods, a separability condition based on comparing the amplitudes and phases of the components of the state has been presented. There are some other approaches to detect the separability of pure states [10,11]. Here we select the entanglement measure given by Wong
c
2011 Verlag der Zeitschrift f¨ur Naturforschung, T¨ubingen·http://znaturforsch.com
and Christensen [18]. The tangle of Wong and Chris- tensen [18] can only be used to detect entanglement, since there are entangled states on which it vanishes.
We note that theW state
|Wi=
√1
3(|0i ⊗ |0i ⊗ |1i+|0i ⊗ |1i ⊗ |0i+|1i ⊗ |0i ⊗ |0i) has vanishing Wong–Christensen tangle and yet is not separable. D¨ur et al. [19] showed that three qubits can be entangled in two inequivalent ways. Ac´ın et al. [20]
described the classification of mixed three-qubit states.
Verstraete et al. [21] showed that four qubits can be entangled in nine different ways. Osterloh and Sie- wert [22] constructedN-qubit entanglement from an- tilinear operators. Entanglement witness in spin mod- els has been studied by T´oth [23]. A symbolic C++
program to calculate the tangle of Wong and Chris- tensen [18] has been given by Steeb and Hardy [24].
LetNbe even orN=3. Wong and Christensen [18]
introduced anN-tangle by
τ1...N=2
1
∑
α1,...,αN=0 ...
δ1,...,δN=0
cα1...αNcβ1...βNcγ1...γNcδ1...δN
×εα1β1εα2β2· · ·εαN−1βN−1εγ1δ1εγ2δ2· · · εγN−1δN−1εαNγNεβNδN
.
This includes the definition for the 3-tangle [8]. Let N =4. Consider the two states with c0000 =1/√
2, c1111=1/√
2 (all other coefficients are 0), andc0000= 1/√
2, c1111 =−1/√
2 (all other coefficients are 0).
Using this measure of entanglement, we find for both cases that the states are fully entangled, i.e.τ1234=1.
Fourteen more states can be constructed with cj1j2j3j4=1/√
2, cj¯1j¯2j¯3j¯4=±1/√ 2, where ¯jdenotes the NOT-operation, i.e. ¯0=1 and ¯1= 0. These sixteen states form an orthonormal basis in the Hilbert spaceC16.
This result can be extended forN≥4 andNeven.
The orthonormal basis would be given by
|φj1...jNi=
√1
2(|j1i ⊗ |j2i ⊗ · · · ⊗ |jNi ± |j¯1i ⊗ |j¯2i ⊗ · · · ⊗ |j¯Ni).
These states are also fully entangled using the measure given above. These states are also related to a Hamilton operator described below.
Let us now find the spectrum of ˆHN and the unitary matrixUN(t) =exp(−i ˆHNt/¯h). Since tr ˆHN =0 for all N, we obtain
2N
∑
j=1Ej=0,
whereEjare the eigenvalues of ˆHN. Consider the her- mitian and unitary operators
Σz,N :=σz⊗σz⊗ · · · ⊗σz, Σx,N :=σx⊗σx⊗ · · · ⊗σx, Σy,N :=σy⊗σy⊗ · · · ⊗σy.
We have to distinguish between the caseNeven and the caseNodd. IfNis even then the commutators vanish, i.e.
[Σx,N,Σy,N] =0, [Σy,N,Σz,N] =0, [Σz,N,Σx,N] =0. IfNis odd then the anti-commutators vanish, i.e.
[Σz,N,Σx,N]+=0, [Σz,N,Σx,N]+=0, [Σz,N,Σx,N]+=0. Note thatΣx,N,Σy,N, andΣz,Nare elements of the Pauli groupPNdescribed above.
Thus setting ˆHN =HˆN0+HˆN1+HˆN2with HˆN0=hω(σ¯ z⊗σz⊗ · · · ⊗σz), HˆN1=∆1(σx⊗σx⊗ · · · ⊗σx), HˆN2=∆2(σy⊗σy⊗ · · · ⊗σy),
we find that for N even, owing to the result given above,
[HˆN0,HˆN1] =0,[HˆN1,HˆN2] =0,[HˆN0,HˆN2] =0. Then the unitary operatorUN(t) =exp(−i ˆHNt/h)¯ for Neven can easily be calculated since
UN(t) =exp(−i ˆHN0t/h)exp(−i ˆ¯ HN1t/¯h)
·exp(−i ˆHN2t/h).¯
IfNis odd, owing to the result given above, we have [HˆN0,HˆN1]+=0, [HˆN1,HˆN2]+=0, [HˆN0,HˆN2]+=0. Here too the time evolutionUN(t) =exp(−i ˆHNt/h)¯ can easily be calculated. We use the abbreviation
E:=
q
¯
h2ω2+∆21+∆22.
W.-H. Steeb and Y. Hardy·N-tangle, Entangled Orthonormal Basis 617 Consider now the general cases. IfNis odd the Hamil-
ton operator has only two eigenvalues, namelyE and
−E. Both are 2N−1times degenerate. The unnormal- ized eigenvectors for+Eare given by
E+hω¯ 0 0 ... 0 0
∆1−(−i)N∆2
,
0 E−hω¯
0 ... 0
∆1+ (−i)N∆2
0
, . . . ,
0
... 0 E+hω¯
∆1−(−i)N∆2
0 ... 0
.
The unnormalized eigenvectors for−Eare given by
E−hω¯ 0 0 ... 0 0
−(∆1−(−i)N∆2)
,
0 E−hω¯
0 ... 0
−(∆1−(−i)N∆2) 0
, . . .
0 ... 0 E−hω¯
−(∆1−(−i)N∆2) 0
... 0
.
The normalization factors are 1
q
(E+hω¯ )2+∆21+∆22
, 1
q
(E−hω)¯ 2+∆21+∆22 ,
respectively. ForNodd the time evolution is given by
UN(t) =e−iωtΣz,N−i∆1tΣx,N/¯h−i∆2tΣy,N/¯h
=I2Ncos(Et/h)¯
−ihω Σ¯ z,N+∆1Σx,N+∆2Σy,N
E ·sin(Et/h).¯ ForNeven the four eigenvalues are given by
E1=hω¯ +∆1−∆2, E2=−¯hω−∆1+∆2, E3=−¯hω+∆1+∆2, E4=hω¯ −∆1−∆2. The eigenvalues are 2N−2times degenerate. The cor- responding 2N normalized eigenvectors for the caseN even are
√1 2
1 0 0 ... 0 0
±1
, 1
√2
0 1 0 ... 0
±1 0
, . . . , 1
√2
0
... 0 1
±1 0
... 0
.
They do not depend on∆and ¯hω. The first vector is the Greenberger–Horne–Zeilinger (GHZ)-state. It is well- known that these 2Neigenvectors form an orthonormal basis in the Hilbert spaceC2
N. As described above we apply the entanglement measure given by Wong and Christensen. It follows that these states are fully en- tangled. These states can also be generated from the GHZ-state by applying the unitary matrix
I2⊗ · · · ⊗I2⊗σx⊗I2⊗ · · · ⊗I2,
whereσx is at the jth position(j=1, . . .,N). Since these are local unitaries all states have the same entan- glement as the GHZ-state. Since forNeven we have
e−iωtΣz,N/¯h=I2Ncos(ωt)−iΣz,Nsin(ωt), e−i∆1tΣx,Nh¯=I2Ncos(∆1t/h)¯ −iΣx,Nsin(∆1t/h),¯ e−i∆2tΣy,N/¯h=I2Ncos(∆2t/h)¯ −iΣy,Nsin(∆2t/¯h), it follows that forNeven the unitary operatorUN(t)for the time evolution is given by
e−i ˆHNt/¯h
= e−iωtΣz,Ne−i∆1tΣx,N/¯he−i∆2tΣy,N/¯h
= I2Ncos(ωt)cos(∆1t/¯h)cos(∆2t/h)¯
−iΣz,Nsin(ωt)cos(∆1t/h)¯ cos(∆2t/h)¯
−iΣx,Ncos(ωt)sin(∆1t/h)cos(∆¯ 2t/h)¯
−iΣy,Ncos(ωt)cos(∆1t/h)sin(∆¯ 2t/h)¯
−Σz,NΣx,Nsin(ωt)sin(∆1t/h)cos(∆¯ 2t/h)¯
−Σz,NΣy,Nsin(ωt)cos(∆1t/h)sin(∆¯ 2t/¯h)
−Σx,NΣy,Ncos(ωt)sin(∆1t/h)¯ sin(∆2t/h)¯ +iΣz,NΣx,NΣy,Nsin(ωt)sin(∆1t/¯h)sin(∆2t/h).¯ For this basis we can form the 2N×2N(Neven) unitary matrix
B= 1
√ 2
1 0 0 . . . 0 0 . . . 0 0 1 0 1 0 . . . 0 0 . . . 0 1 0 0 0 1 . . . 0 0 . . . 1 0 0
... ... ...
0 0 0 0 0 0 0 0
0 0 0 1 1 0 0 0
0 0 0 1 −1 0 0 0
0 0 0 0 0 0 0 0
... ... ...
0 0 1 . . . 0 0 −1 0 0
0 1 0 0 0 0 −1 0
1 0 0 0 0 0 0 −1
.
For implementations ofBas quantum gates the cosine–
sine decomposition [2,4] is useful. This matrix has the cosine–sine decomposition
B=
U1 0 0 U2
C S
−S C
U3 0 0 U4
,
where the unitary matricesU1,U2,U3,U4are given by U2=I2N−1, U4=I2N−1,
U1=U3= 0 1
1 0
⊗ 0 1
1 0
⊗· · ·⊗
0 1 1 0
,
and the invertible matricesCandSare the 2N−1×2N−1 matrices
C=S= 1
√2I2N−1.
Thus the unitary matricesU1 andU3 are Kronecker products of the NOT-gate.
If N is odd then the eigenvectors are entangled if
¯
hω=0. If ¯hω→∞the eigenvectors become unentan- gled, i.e. can be written as product states.
We have provided a spin Hamilton operator acting in the finite dimensional Hilbert space H=C2
N that provides a fully entangled basis ifNis even. IfNis odd we vary the parameters ¯hω,∆1,∆2such that we can vary between entangled and unentangled states. Such Hamilton operators could also be investigated applying Riemannian geometry [25].
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