Anisotropy and Magnetic Field Effects on the Genuine Multipartite
  Entanglement of Multi-Qubit Heisenberg {\it XY} Chains
Anisotropy and Magnetic Field Effects on the Genuine
Multipartite Entanglement of Multi-Qubit Heisenberg XY Chains
Chang Chi Kwong and Ye Yeo
Department of Physics, National University of Singapore,
10 Kent Ridge Crescent, Singapore 119260, Singapore
Abstract
It has been shown that, for the two-qubit Heisenberg XY model, anisotropy and magnetic field
may together be used to produce entanglement for any finite temperature by adjusting the external
magnetic field beyond some finite critical strength. This interesting result arises from an analysis
employing the Wootters concurrence, a computable measure of entanglement for two-qubit states.
Recently, Mintert et al. proposed generalizations of Wootters concurrence for multipartite states.
These MKB concurrences possess a mathematical property that enables one to understand the
origin of this characteristic behavior. Here, we first study the effect of anisotropy and magnetic
field on the multipartite thermal entanglement of a four-qubit HeisenbergXY chain using the MKB
concurrences. We show that this model exhibits characteristic behavior similar to that of the two-
qubit model. In addition, we show that this can again be understood using the same mathematical
property. Next, we show that the six-qubit Heisenberg XY chain possesses properties necessary for
it to have the characteristic behavior too. Most importantly, it is possible to directly measure the
multipartite MKB concurrences of pure states. This may provide an experimental verification of
our conjecture that for a Heisenberg XY chain of any even number of qubits, it is always possible
to obtain non-zero genuine multipartite entanglement at any finite temperature by applying a
sufficiently large magnetic field.
PACS numbers: 03.67.Mn, 03.65.Ud
http://arxiv.org/abs/0704.1368v2
I. INTRODUCTION
The one-dimensional Heisenberg models have been extensively studied in solid state
physics (see references in [1]). Interest in these models has been revived lately by sev-
eral proposals for realizing quantum computation [2] and information processing [3] using
quantum dots (localized electron spins) as qubits. An intriguing phenomenon in such quan-
tum systems with more than one component is entanglement. It refers to the non-classical
correlations that exist among these components. Due to this nature, quantum entanglement
has recently been recognized as an indispensable physical resource for performing classically
impossible information processing tasks; such as quantum computation, teleportation [4],
key distribution [5] and superdense coding [6]. In any physically realistic consideration of
quantum-information-processing device, it is important to take into account the fact that it
would be in thermal contact with some heat bath. Consequently, entanglement in interacting
Heisenberg spin systems at finite temperatures has been investigated by a number of authors
(see, e.g., Ref.[7] and references therein). The state of a typical solid state system at thermal
equilibrium (temperature T ) is χ = e−βH/Z, where H is the Hamiltonian, Z = tr e−βH is the
partition function, and β = 1/kT , where k is the Boltzmann’s constant. The entanglement
associated with the thermal state χ is referred to as the thermal entanglement [1].
To achieve a quantitative understanding of the role of entanglement in the field of quan-
tum information science, it is necessary to quantify the amount of entanglement that is
associated with a given state. Several entanglement measures have been proposed. One
famous example is the Wootters concurrence [8]. Consider a two-qubit state ρ, the Wootters
concurrence
CW [ρ] ≡ max{λ1 − λ2 − λ3 − λ4, 0}, (1)
where λk (k = 1, 2, 3, 4) are the square roots of the eigenvalues in decreasing order of
magnitude of the spin-flipped density-matrix operator R = ρ(σy⊗σy)ρ∗(σy⊗σy), the asterisk
indicates complex conjugation. Wootters concurrence is closely related to the entanglement
of formation [9]. It appears as an auxillary function that has to be evaluated when computing
the entanglement of formation. Recently, Mintert, Kús and Buchleitner (MKB) [10, 11]
proposed generalizations of Wootters concurrence for multipartite quantum systems, which
can be evaluated efficiently for arbitrary mixed states. This is an extremely important
development in our investigation of multipartite entanglement, a subject we have yet to
achieve a complete understanding.
With the availability of such a good and computable measure of entanglement for systems
of two qubits, many thoroughly analyzed the thermal entanglement in two-qubit Heisenberg
models in terms of thermal concurrence [1, 12, 13, 14, 15, 16, 17]. The recent breakthroughs
in the experimental physics of double quantum dot (see, e.g., Ref.[18]) shows that these
studies are worthwhile pursuits. Interestingly, Kamta et al. [15] showed that the anisotropy
and the magnetic field may together be used to control the extent of thermal concurrence
in a two-qubit Heisenberg XY model and, especially, to produce entanglement for any
finitely large T , by adjusting the external magnetic field beyond some finitely large critical
strength. Such robustness is absent in the case of two-qubit Heisenberg XX model. A
natural question is whether thermal entanglement of a Heisenberg XY chain with more
qubits has similar behavior. This question is not only of practical importance but also of
fundamental importance (see, for instance, Ref.[19]). In this paper, we provide answers for
Heisenberg XY chains with any even number of qubits by analyzing the MKB concurrences
associated with their thermal states.
Our paper is organized as follows. Since the results of MKB [10, 11] is crucial to our
analysis, we will present them in the following section. However, we do so explicitly in the
context of direct relevance to our paper, i.e., n-qubits. Specifically, when n = 2, the MKB
concurrence coincides with the Wootters concurrence. It thus allows us to understand the
special property of the thermal entanglement of the two-qubit Heisenberg XY model, in
the light of a mathematical property of the MKB concurrences (see Eq.(11)). This together
with a discussion of the two-qubit Heisenberg XY model [15] will be given in Section III.
We show, in Section IV, that the thermal entanglement of the four-qubit Heisenberg XY
chain has behavior similar to the two-qubit case. Namely, anisotropy and magnetic field
may together be used to produce genuine multipartite entanglement [20, 21] for any finitely
large T by adjusting the external magnetic field beyond some finitely large critical strength.
This can again be understood in the light of the same mathematical property of MKB con-
currences. Our results agree with those in Ref.[22], which employed bipartite entanglement
measure. In general, it is not sufficient to only consider bipartite entanglement. For instance,
the Greenberger-Horne-Zeilinger (GHZ) state [23] is a state with genuine multipartite en-
tanglement but yields zero entanglement between one particle and any other particle. On
the other hand, the W state [24] is one where every particle is entangled with every other
particle, but it has no genuine multipartite entanglement [10, 20]. Multipartite entangle-
ment surely is more interesting. Therefore, our study complements theirs. However, we
must emphasize that our analysis more importantly demonstrates that the thermal entan-
glement of the Heisenberg XY chains have the characteristic behavior due to two reasons.
First, in the presence of a finitely large external magnetic field, the ground state of the
model has non-zero genuine multipartite entanglement. Second, for non-zero temperatures,
this ground state is the highest weight state (with weight ≈ 1, i.e., the thermal state is
almost a pure state) when a finitely large enough magnetic field is applied. In Section V,
we show that this is indeed the case for the six-qubit Heisenberg XY chain. We thus have
firm mathematical basis (Eq.(11)) to establish that the thermal entanglement in this case
will exhibit similar characteristic behavior. Most important of all, the MKB concurrences
may be directly measured for multipartite pure states [25, 26]. It is thus possible to experi-
mentally verify our conjecture that for any even n, it is always possible to obtain non-zero
genuine multipartite entanglement at any finite temperature by applying a sufficiently large
magnetic field. Further discussions of this possibility and a summary of our results will be
presented in the concluding Section VI.
II. THE MKB CONCURRENCES
Consider an n-qubit pure state |ψ〉 ∈ H1⊗H2⊗· · ·⊗Hn, the MKB concurrence is defined
as the expectation value of a Hermitean operator A that acts on two copies of the state [10]:
C[|ψ〉〈ψ|] ≡
〈ψ| ⊗ 〈ψ|A|ψ〉 ⊗ |ψ〉. (2)
In general, A could have the form
{sji=±}
ps1is2i ···sniP
⊗ P (2)s2i ⊗ · · · ⊗ P
, (3)
where ps1is2i ···sni ≥ 0,
(Π+0 +Π
Π−1 (4)
are projectors onto the symmetric and antisymmetric subspaces of Hj ⊗ Hj. Here, Π±0 ≡
(|00〉 ± |11〉)(〈00| ± 〈11|), Π±1 ≡ (|01〉 ± |10〉)(〈01| ± 〈10|) and {|0〉, |1〉} is an orthonormal
basis of Hj. The summation in Eq.(3) is performed over the set {sji = ±}+, which contains
all n-long strings of +’s and −’s with even number of −’s. The superscript + indicates that
the string with n +’s is not included in the sum. This is because for separable states, its
expectation value in the symmetric twofold copy is non-zero. Terms with odd number of
− ’s are naturally excluded in the sum since their expectation values in the twofold copy
states is always zero.
By choosing the value of all the ps1is2i ···sni ’s in Eq.(3) to be 4, an entanglement monotone
Cn can be obtained. The resulting operator An can equivalently be written as 4(I − P (1)+ ⊗
+ ⊗ · · ·⊗P
+ ) [26]. The concurrence Cn of an n-qubit pure state |ψ〉 can then be written
Cn[|ψ〉〈ψ|] = 21−n/2
(2n − 2)〈ψ|ψ〉 −
Trρ2i . (5)
The above summation runs over all (2n − 2) reduced density operators ρi of the state |ψ〉.
Cn accounts for all possible types of entanglement in a state and takes the value zero if and
only if the state is fully separable.
For even number n of qubits, it is possible to define an MKB concurrence C(n) that detects
multipartite entanglement, by choosing the operator A = A(n) ≡ 2nP (1)− ⊗ P
− ⊗ · · · ⊗ P
We note that when n = 2,
C(2)[|ψ〉〈ψ|] = |〈ψ∗|σy ⊗ σy|ψ〉| = CW [|ψ〉〈ψ|]. (6)
That is, the MKB concurrence C(2) coincides with the Wootters concurrence. And, for n = 4,
we have
C(4)[|ψ〉〈ψ|] = |〈ψ∗|σy ⊗ σy ⊗ σy ⊗ σy|ψ〉|, (7)
which is also an entanglement monotone [11]. There is obviously no equivalent definition
for the case of odd number of qubits since the expectation value of A(n) are always zero for
odd n.
The MKB concurrence for a mixed state ρ of n qubits can be obtained via the convex
roof construction:
C[ρ] ≡ inf
piC[|ψi〉〈ψi|], ρ =
pi|ψi〉〈ψi|
, (8)
where the infimum is taken over all possible pure state decompositions of ρ. To evaluate
C[ρ], consider the spectral decompositions ρ =
i |φ̃i〉〈φ̃i| and A =
α |χ̃α〉〈χ̃α|, where the
eigenstates |φ̃i〉 and |χ̃α〉 are subnormalized such that their norms squared are the eigenvalues
corresponding to the states. If r is the rank of the operator A, it is possible to define r
complex symmetric matrices T α with elements T αjk ≡ 〈φ̃j| ⊗ 〈φ̃k|χ̃α〉. And, Eq.(8) becomes
C[ρ] = inf
|[V T αV T ]ii|2 (9)
where the infimum is now taken over the set of left unitary matrices V .
It can be shown that the following inequality holds [10]:
C[ρ] ≥ inf
|[V τV T ]ii|. (10)
Here, the matrix τ is defined to be
α zαT
α in terms of arbitrary complex numbers zα
satisfying only the condition that
α |zα|2 = 1. An algebraic solution of the inequality
Eq.(10) is given in Ref.[10] to be max{0, λ1 −
j>1 λj} where λi’s are singular values of τ
written in decreasing order. For C(n), when A = A(n) is of rank 1, T 1 = τ and the lower
bound in Eq.(10) turns out to be the exact value of C(n).
In general, an optimization over zα is also necessary to obtain the optimal lower bound
for C[ρ]. However, it is possible to obtain a good approximation to C[ρ] by approximating τ
with a matrix whose elements [10]
τij ≈
〈φ̃1| ⊗ 〈φ̃1|A|φ̃i〉 ⊗ |φ̃j〉
〈φ̃1| ⊗ 〈φ̃1|A|φ̃1〉 ⊗ |φ̃1〉
, (11)
where |φ̃1〉 is the eigenstate of ρ with the largest eigenvalue. This is the mathematical
property of MKB concurrences that will play a critical role in our understanding of the
characteristic behavior of the thermal entanglement of Heisenberg XY models. We will first
illustrate this explicitly with the two-qubit Heisenberg XY chain in the next section.
III. TWO-QUBIT HEISENBERG XY MODEL
The Hamiltonian H2 for the anisotropic two-qubit Heisenberg XY model in an external
magnetic field Bm ≡ ηJ (η is a real number) along the z axis is
(1 + γ)Jσx1 ⊗ σx2 +
(1− γ)Jσy1 ⊗ σ
1 ⊗ I2 + I1 ⊗ σz2), (12)
where Ij is the identity matrix and σ
j , σ
j , σ
j are the Pauli matrices at site j = 1, 2. The
parameter −1 ≤ γ ≤ 1 measures the anisotropy of the system and equals 0 for the isotropic
XX model [12] and ±1 for the Ising model [14]. (1 + γ)J and (1 − γ)J are real coupling
constants for the spin interaction. The model is said to be antiferromagnetic for J > 0
and ferromagnetic for J < 0. The thermal concurrence associated with the thermal state
χ2, Eq.(16), can be derived from Eq.(17). It is invariant under the substitutions η −→ −η,
γ −→ −γ, and J −→ −J . Therefore, we restrict our considerations to η ≥ 0, 0 ≤ γ ≤ 1,
and J > 0.
The eigenvalues and eigenkets of H2 are given by [15]
H2|Φ0〉 = B|Φ0〉,
H2|Φ1〉 = J |Φ1〉,
H2|Φ2〉 = −J |Φ2〉,
H2|Φ3〉 = −B|Φ3〉, (13)
where B ≡
B2m + γ
2J2 =
η2 + γ2J ,
|Φ0〉 =
(B +Bm)2 + γ2J2
[(B +Bm)|00〉+ γJ |11〉],
|Φ1〉 = 1√
[|01〉+ |10〉],
|Φ2〉 =
[|01〉 − |10〉],
|Φ3〉 = 1√
(B − Bm)2 + γ2J2
[(B − Bm)|00〉 − γJ |11〉]. (14)
The Wootters concurrence associated with the eigenkets, |Φ0〉 and |Φ3〉, are given by γ√
η2+γ2
Hence, they represent entangled states when γ 6= 0. We note that when η = 0, |Φ0〉 and
|Φ3〉 reduce to (|00〉 + |11〉)/
2 and (|00〉 − |11〉)/
2 respectively, so that the eigenstates
are the four maximally entangled Bell states: |Ψ0Bell〉, |Ψ1Bell〉, |Ψ2Bell〉, and |Ψ3Bell〉. And, in
the limit of large η,
CW [|Φ0〉〈Φ0|] = CW [|Φ3〉〈Φ3|] ≈ γη−1, (15)
only going to zero asympotically when η is infinitely large. In contrast, when γ = 0,
|Φ0〉 = |00〉 and |Φ3〉 = |11〉 are product states with eigenvalues ηJ and −ηJ respectively,
though |Φ1〉 and |Φ2〉 remain the same [12].
For the above system in thermal equilibrium at temperature T , its state is described by
the density operator
wi|Φi〉〈Φi|
[e−βB|Φ0〉〈Φ0|+ e−βJ |Φ1〉〈Φ1|+ eβJ |Φ2〉〈Φ2|+ eβB|Φ3〉〈Φ3|], (16)
where the partition function Z2 = 2 cosh βB + 2 cosh βJ , the Boltzmann’s constant k ≡ 1
from hereon, and β = 1/T . After some straightforward algebra, we obtain
eβJ ,
e−βJ ,
2γ2J2
sinh2 βB + 2γJ
sinh2 βB sinh βB,
2γ2J2
sinh2 βB − 2γJ
sinh2 βB sinh βB. (17)
In the zero-temperature limit, i.e., β −→ ∞, at which the system is in its ground state,
Eq.(16) reduces to the following three possibilities.
(a) 0 ≤ η <
1− γ2:
[eβJ |Φ2〉〈Φ2|+ eβB|Φ3〉〈Φ3|] −→ |Φ2〉〈Φ2|, (18)
with Z2 = e
βJ+eβB. Equation (17) gives CW [χ2] = 1, its maximum value, in agreement
with the fact that |Φ2〉 is a maximally entangled Bell state.
(b) η =
1− γ2:
χ2 −→
[|Φ2〉〈Φ2|+ |Φ3〉〈Φ3|]. (19)
From Eq.(17), the above equally weighted mixture has
CW [χ2] =
(1− γ). (20)
(c) η >
1− γ2:
χ2 −→ |Φ3〉〈Φ3|, (21)
and Eq.(17) yields accordingly
CW [χ2] =
η2 + γ2
. (22)
Therefore, for a given γ, ηcritical =
1− γ2 marks the point of quantum phase transition
(phase transition taking place at zero temperature due to variation of interaction terms in
the Hamiltonian of a system [1]). For values of γ other than γ = 1
, there is a sudden increase
or decrease in C[χ2] at ηcritical, depending on whether γ > 13 or γ <
, before decreasing to
zero asymptotically, as η is increased beyond the critical value ηcritical [15]. Here, we focus
on the behavior of the model at non-zero temperatures and subject to magnetic field of
appropriate strengths.
At non-zero temperatures, due to mixing, CW [χ2] decreases to zero as the temperature T
is increased beyond some critical value. In fact, as T → ∞, the statistical weights wi → 1/4
for all i and CW [χ2] → 0. However, for a large but finite T ,
1 + e−2βB + e−β(B−J) + e−β(B+J)
can always be made as close to unity as possible by increasing the strength η of the external
magnetic field. That is, when η is large enough, only |Φ3〉 contributes significantly to the
thermal state χ2. The η required for this to occur depends on T , larger η for higher T . This
is always possible for finite T .
Next, we note that in the limit of large η,
λ1 ≈ λ2 ≈ 0 ≈ λ4, λ3 ≈ γη−1. (24)
It follows that
CW [χ2] ≈ γη−1 ≈ CW [|Φ3〉〈Φ3|]. (25)
Hence, for a finitely large T , the thermal concurrence of the system in a large enough
magnetic field is very well approximated by the concurrence of |Φ3〉 in the same magnetic
field. In other words, the entanglement associated with the thermal state χ2 in this case is
mainly due to that associated with the eigenstate |Φ3〉.
In summary, for any non-zero T and an appropriate η, |Φ3〉 is the highest weight eigen-
state. It follows from Eq.(11) that this is the state which will mainly contribute to the MKB
concurrence C(2)[χ2] or equivalently the Wootters concurrence (see Eq.(6)). An important
point to note here is that |Φ3〉 has non-zero concurrence as long as η is finite (see Eq.(15)).
Another is that γ 6= 0, otherwise |Φ3〉 will be a product state with no entanglement. These
clearly explain the characteristic robustness of the thermal entanglement of the two-qubit
Heisenberg XY model described in Ref[15] - the system at finite T can always be entangled
provided large enough magnetic field is applied. They also identify the necessary character-
istic features for a model to exhibit such behavior. We shall illustrate that this is indeed
the case for the four-qubit Heisenberg XY chain in the next section.
IV. FOUR-QUBIT HEISENBERG XY MODEL
The Hamiltonian Hn for an anisotropic n-qubit Heisenberg XY chain in an external
magnetic field Bm = ηJ along the z-axis is
[(1 + γ)σxj σ
j+1 + (1− γ)σ
j+1 + ησ
j ], (26)
where the periodic boundary condition σαn+1 = σ
1 (α = x, y, z) applies. Like in the two-qubit
model, we consider the case when η ≥ 0, 0 ≤ γ ≤ 1, and J > 0.
In this section, we consider n = 4. After some straightforward algebra, we obtain the
eigenvalues and eigenvectors of H4. Firstly, we present
H4|Φ15〉 = −ω+J |Φ15〉, (27)
where
[η2 + 2(1 + γ2)]±
[η2 + 2(1 + γ2)]2 − 8η2, (28)
|Φ15〉 = N−Ω (Ω
1 |0000〉+ Ω−2 |0011〉+ Ω−3 |0101〉+ Ω−2 |0110〉
+Ω−2 |1001〉+ Ω−3 |1010〉+ Ω−2 |1100〉+ |1111〉), (29)
with N±Ω ≡ 1/
1 + (Ω±1 )
2 + 4(Ω±2 )
2 + 2(Ω±3 )
Ω±1 =
(2η ± ω+)(ω+2 − 8)− 8γ2(η ± ω+)
Ω±2 =
2η ± ω+
Ω±3 = ±
2η ± ω+
. (30)
|Φ0〉, which can be obtained from |Φ15〉 by substituting Ω−i (i = 1, 2, 3) with Ω+i or Ω−i → Ω+i ,
satisfies H4|Φ0〉 = ω+J |Φ0〉. The MKB concurrences, as defined in Section II, for |Φ15〉 are
given by
C(4)[|Φ15〉〈Φ15|] = 2 Ω
1 + 2(Ω
2 + (Ω−3 )
1 + (Ω−1 )
2 + 4(Ω−2 )
2 + 2(Ω−3 )
C4[|Φ15〉〈Φ15|] =
7[2(Ω−2 )
2 + (Ω−3 )
2]2 + 2[4(Ω−1 )
2 − Ω−1 + 4][2(Ω−2 )2 + (Ω−3 )2] + 7(Ω−1 )2
1 + (Ω−1 )
2 + 4(Ω−2 )
2 + 2(Ω−3 )
In the limit of large η,
ω+ ≈ 2η + 2γ
4γ2 − γ4
Ω−1 ≈
2γ2 − γ4
Ω−2 ≈ −
γ3 − 4γ
Ω−3 ≈
4γ − 3γ3
. (32)
It follows that
C(4)[|Φ15〉〈Φ15|] ≈
8γ2 − 4γ4
, (33)
C4[|Φ15〉〈Φ15|] ≈
24γ − 9γ3
. (34)
We note that |Φ15〉 is a state with genuine four-partite entanglement [27], which remains
non-zero even for large η and going to zero only in the asymptotic limit of infinite magnetic
field.
Secondly, we have
H4|Φ14〉 = −[(α+ + α−)γ + 2]J |Φ14〉,
H4|Φ13〉 = −[(α+ + α−)γ − 2]J |Φ13〉, (35)
where
η2 + 4γ2 ± η
, (36)
|Φ14〉 =
1 + (α−)2
(−α−|0001〉+ α−|0010〉 − α−|0100〉+ |0111〉
+α−|1000〉 − |1011〉+ |1101〉 − |1110〉),
|Φ13〉 = 1
1 + (α−)2
(−α−|0001〉 − α−|0010〉 − α−|0100〉+ |0111〉
−α−|1000〉+ |1011〉+ |1101〉+ |1110〉). (37)
Corresponding to these states are |Φ1〉 and |Φ2〉, which can be derived from |Φ14〉 and |Φ13〉
respectively by −α− → α+. They satisfy H4|Φ1,2〉 = [(α+ + α−)γ ± 2]J |Φ1,2〉. The MKB
concurrences for |Φ14〉 are given by
C(4)[|Φ14〉〈Φ14|] = 2α
1 + (α−)2
C4[|Φ14〉〈Φ14|] =
1 + (α−)2
3 + 8(α−)2 + 3(α−)4
. (38)
Hence, |Φ14〉 is also a state with genuine four-partite entanglement [20, 21]. It reduces to a
W state when γ = 0.
Thirdly, we have H4|Φ3,4〉 = ηJ |Φ3,4〉 and H4|Φ11,12〉 = −ηJ |Φ11,12〉, where
|Φ3,4〉 = 1
(|0001〉 ± |0010〉 − |0100〉 ∓ |1000〉),
|Φ11,12〉 = 1
(|0111〉 ± |1011〉 − |1101〉 ∓ |1110〉). (39)
These states belong to the family of W states which are entangled but do not contain genuine
four-partite entanglement [20]. Fourthly, by substituting ω+ in Eq.(30) with ω− we obtain
the corresponding ∆±1 , ∆
2 and ∆
3 in terms of which we express
|Φ10,11〉 = N±∆(∆
1 |0000〉+∆±2 |0011〉+∆±3 |0101〉+∆±2 |0110〉
+∆±2 |1001〉+∆±3 |1010〉+∆±2 |1100〉+ |1111〉). (40)
Here, N±∆ ≡ 1/
1 + (∆±1 )
2 + 4(∆±2 )
2 + 2(∆±3 )
2. They satisfy H4|Φ10,11〉 = ±ω−J |Φ10,11〉.
Lastly, we have the following four degenerate eigenstates with eigenvalue zero:
|Φ6〉 = 1√
(|0011〉 − |1100〉) ,
|Φ7〉 = 1√
(|0101〉 − |1010〉) ,
|Φ8〉 =
(|0110〉 − |1001〉) ,
|Φ9〉 = 1
(|0011〉 − |0110〉 − |1001〉+ |1100〉) . (41)
The eigenstates |Φ6,7,8〉 belong to the family of GHZ states that contain only genuine four-
partite entanglement, while |Φ9〉 is a product state of two Bell states.
As in the case of the two-qubit model, we construct the thermal state of the four-qubit
Heisenberg XY chain as follows:
wi|Φi〉〈Φi| =
e−βH4 . (42)
Here, the partition fuction
Z4 = 4 + 4 cosh βηJ + 2 cosh β[(α
+ + α−)γ + 2]J
+2 cosh β[(α+ + α−)γ − 2]J + 2 cosh βω+J + 2 cosh βω−J. (43)
At non-zero temperatures the state of the system becomes a mixture of the energy eigenstates
with statistical weights
w15 =
, w14 =
eβ[(α
++α−)γ+2]J
, · · · (44)
A. Zero Temperature
In the β → ∞ limit,
[eβ[(α
++α−)γ+2]J |Φ14〉〈Φ14|+ eβω+J |Φ15〉〈Φ15|], (45)
with Z4 = e
β[(α++α−)γ+2]J + eβω
+J . Like in the 2-qubit Heisenberg XY model, there are thus
two possible lowest energy states, namely |Φ14〉 or |Φ15〉, depending on the strength of the
applied magnetic field. As η is increased from zero, there are in general two instances when
(α+ + α−)γ + 2 = ω+. We let η1 and η2 denote the solutions. Their dependence on the
anisotropy of the system are plotted in Fig. 1. As γ is increased from zero, both η1 and η2
become smaller and converge to zero when γ = 1. In fact, for the Ising model (γ = 1), both
|Φ14〉 and |Φ15〉 are the lowest energy states of the system. The system state is an equal
mixture of both states:
[|Φ14〉〈Φ14|+ |Φ15〉〈Φ15|]. (46)
But, once the external magnetic field is turned on, |Φ15〉 becomes the only ground state of
the model regardless of the strength of the field.
It follows that for 0 < γ < 1, depending on η we have the density operator of the system
|Φ15〉〈Φ15| 0 ≤ η < η1
[|Φ15〉〈Φ15|+ |Φ14〉〈Φ14|] η = η1
|Φ14〉〈Φ14| η1 < η < η2
[|Φ14〉〈Φ14|+ |Φ15〉〈Φ15|] η = η2
|Φ15〉〈Φ15| η > η2
FIG. 1: The two transition η’s for the system at zero temperature plotted against the anisotropy
We can therefore calculate the MKB concurrrences of the system at zero temperature (β →
∞) for the different regions of magnetic field strength. Plots of C(4) and C4 against magnetic
field for different values of anisotropy γ are shown in Fig. 2 and 3. For both concurrences,
there are sharp changes at the transition magnetic fields due to quantum phase transition.
FIG. 2: The MKB concurrence C(4), which is an entanglement monotone, plotted against magnetic
field η for different values of anisotropy γ. In general, there are sharp changes in C(4) at the two
transition fields η1 and η2.
B. Non-zero Temperatures
In general, as the temperature of a system is increased, its density operator becomes
closer to the maximally mixed state, 1
I, where n is the dimension of the Hilbert space
FIG. 3: The MKB concurrence C4 plotted against η for different values of γ. Similar to Fig.2, there
are sharp changes of the concurrence at the two transition magnetic fields.
and I is the identity operator. There thus exist critical temperatures Tc beyond which
the MKB concurrences of the system become zero, like in the two-qubit case (see Fig. 4).
The existence of Tc’s is guaranteed by the fact that a state becomes separable when it is
sufficiently close to 1
I [28].
FIG. 4: The points where the MKB concurrence C(4) equals zero are plotted for two different values
of γ. The region above each curve is the region where C(4) = 0.
On the other hand, it can also be seen from Fig. 4 that it is always possible to have
non-zero entanglement in the system by applying a sufficiently large magnetic field to it.
In particular, even when T > Tc, one could reintroduce entanglement into the system by
increasing η. This is always possible for finite temperatures as long as γ 6= 0. Hence, the
thermal entanglement associated with the four-qubit Heisenberg XY model exhibits the
same characteristic robustness as in the two-qubit case. We note that, in contrast to the
two-qubit case (see Fig. 3 in Ref.[15]), each of our graphs in Fig. 4 has two “singular
turning points”. This is due to the fact that there are two transition η’s instead of one in
the two-qubit model.
In order to understand this robustness, we draw inspiration from the two-qubit case. We
observe that for any finite temperature T , the statistical weight w15 when η is large enough,
is given by
w15 =
exp βω+J =
where ξ ≡ 1 + e−2βω+J + 4e−βω+J + 4e−βω+J cosh βηJ + 2e−βω+J cosh β[(α+ + α−)γ + 2]J +
2e−βω
+J cosh β[(α+ + α−)γ − 2]J + 2e−βω+J cosh βω−J . It can be shown that the large η
behavior of w15 is independent of γ. Furthermore, by increasing η appropriately, ξ can be
made to be as close as possible to unity. This results in |Φ15〉 being the only state that
significantly contributes to the thermal state χ4. In the light of Eq.(11), we may conclude
C(4)[χ4] ≈ C(4)[|Φ15〉〈Φ15|]. (49)
That this is indeed the case has been established numerically (see Table I). Since the MKB
concurrence C(4) can be determined exactly, it is calculated for states χ4 and |Φ15〉 under
different combinations of η and T (some results are shown in Table I). The two values for
each combination of η and T are then compared. For a given temperature, the two values,
C(4)[χ4] and C(4)[|Φ15〉〈Φ15|], agree when large enough magnetic field is applied.
Hence, for γ 6= 0, the revival and robustness of entanglement in the Heiseberg four-qubit
model can be understood, as in the two-qubit case, in terms of the large η behaviors of both
the MKB concurrence C(4)[|Φ15〉〈Φ15|] (Eq.(15)) and the statistical weight w15 (Eq.(48)). We
may conclude, when w15 ≈ 1, that the entanglement associated with χ4 is of the genuine four-
partite kind [20, 21]. The total entanglement as measured by C4 thus also undergoes revival
since genuine four-partite entanglement is only one kind of the entanglement measured by
C4. In contrast, for γ = 0 and large η, the ground state is the product state |1111〉. This is an
important distinction between the four-qubit XX model and the XY model. Consequently,
T η =0 η =100 η =1000
χ4 |Φ15〉〈Φ15| χ4 |Φ15〉〈Φ15| χ4 |Φ15〉〈Φ15|
1 0 1 0.0000180069 0.0000180069 1.79177×10−7 1.79177×10−7
5 0 1 0.0000180068 0.0000180069 1.79177×10−7 1.79177×10−7
10 0 1 0.0000174316 0.0000180069 1.79177×10−7 1.79177×10−7
50 0 1 0 0.0000180069 1.79175×10−7 1.79177×10−7
100 0 1 0 0.0000180069 1.07513×10−7 1.79177×10−7
TABLE I: A comparison between C(4)[χ4] and C(4)[|Φ15〉〈Φ15|] for some combinations of η and T .
no revival of entanglement is observed for the case of four-qubit isotropic Heisenberg XX
chain.
We may apply the above analysis employing C(n) to study Heisenberg XY chains with
even number n of qubits. Firstly, we determine if the ground state |Φg〉 remains genuinely
multipartite entangled at large η. Secondly, we determine if the ground state statistical
weight wg can be made very close to 1 at large η. Through Eq.(11), these two properties
together imply that for any finite temperature, nonzero genuine multipartite entanglement
can always be obtained by applying a large enough magnetic field. We show that the six-
qubit Heisenberg XY chain has both properties in the next section.
V. SIX-QUBIT HEISENBERG XY MODEL AND BEYOND
In this section, we show that the six-qubit Heisenberg XY chain (with γ 6= 0) does
indeed possess the characteristic properties, which enable the model to have non-zero thermal
entanglement at any given finite temperature when subject to an external magnetic field of
appropriate strength. To this end, we determine the eigenstate |Φg〉 of H6 whose eigenvalue
Eg = −λJ is the minimum when η is large. Here,
3(2 + 2γ2 + η2) + 2κ+ 2
(4γ2 + η2)((3 + γ2 + η2) + κ) (50)
with κ ≡
γ4 + (−3 + η2)2 + 2γ2(3 + η2). And,
|Φg〉 = N(|000000〉Θ1 + |000011〉Θ2 + |000101〉Θ3 + |000110〉Θ4 + |001001〉Θ4 + |001010〉Θ3
+|001100〉Θ2 + |001111〉Θ5 + |010001〉Θ3 + |010010〉Θ4 + |010100〉Θ3 + |010111〉Θ6
+|011000〉Θ2 + |011011〉Θ7 + |011101〉Θ6 + |011110〉Θ5 + |100001〉Θ2 + |100010〉Θ3
+|100100〉Θ4 + |100111〉Θ5 + |101000〉Θ3 + |101011〉Θ6 + |101101〉Θ7 + |101110〉Θ6
+|110000〉Θ2 + |110011〉Θ5 + |110101〉Θ6 + |110110〉Θ7 + |111001〉Θ5 + |111010〉Θ6
+|111100〉Θ5 + |111111〉Θ8) (51)
{24J2 − (λ− 3Jη)(λ− Jη)}Θ8 + 2J2(λ− Jη)ζ + 8J2τ
2Jγ(λ+ 3Jη)
Θ2 = −
(λ + 3Jη)Θ1
Θ3 = −
(λ + 3Jη)Θ8 + (λ− Jη)τ + 2J2ζ
3(2− γ2)Θ8 + (λ− Jη)ζ + 2τ
Θ5 = −
(λ− 3Jη)Θ8
3Θ8 + τ
Θ7 = −
(λ− 3Jη)Θ8 − 2J2ζ
Θ8 = λ
4 − 2J2λ2(2 + 2γ2 + η2) + J4{η2(η2 − 12) + 4γ2(η2 + 4)},
Θ21 +Θ
8 + 3(2Θ
2 + 2Θ
4 + 2Θ
5 + 2Θ
, (52)
τ = −8Jλη{λ2 − J2(6− 2γ2 + η2)}, (53)
ζ = 4λ{λ2(γ2 − 1)− J2(4γ4 − 9η2 − 4γ2 + γ2η2)}. (54)
The MKB concurrence C(6) is calculated for this state, giving
C(6)[|Φg〉〈Φg|] = 2N2|Θ1Θ8 + 6Θ2Θ5 + 6Θ3Θ6 + 3Θ4Θ7|. (55)
In the limit of large η,
C(6)[|Φg〉〈Φg|] ≈ 2γ3η−3. (56)
It can again be shown that |Φg〉 is a genuine six-partite entangled state [20, 21]. In addition,
the statistical weight wg of the ground state |Φg〉 can be shown numerically to be close
to unity when an appropriately large magnetic field is applied. The ground state |Φg〉
therefore possesses the two desired properties for the system to have non-zero genuine six-
partite entanglement at any finite temperature, provided an appropriate magnetic field is
applied. We conjecture that the robustness of genuine multipartite entanglement is a general
property of Heisenberg XY chain with even number of particles. This conjecture can be
experimentally tested, as will be discussed in the next section.
VI. CONCLUSIONS
In this study, we have investigated in detail the origin of the robustness of genuine mul-
tipartite entanglement in two-, four-, and six-qubit Heisenberg XY models. Two important
properties possessed by the ground states |Φg〉 of these models, which enable them to ex-
hibit the characteristic robustness, were identified. The first property, namely the statistical
weight wg associated with |Φg〉 can be made very close to unity by applying a large enough
magnetic field, allows us to use Eq.(11) and conclude that the MKB concurrence of the
thermal state χ equals that of the ground state. It follows that we could obtain fairly ac-
curately the MKB concurrence of χ (a mixed state) by calculating the MKB concurrence
of |Φg〉 (a pure state) in this case. The second property being that |Φg〉 remains genuinely
multipartite entangled under such a magentic field, then guarantees that there is non-zero
genuine multipartite entanglement for any finite temperature as long as sufficiently large
magnetic field is applied. These properties allow us to extend our study to XY chains with
any even number of qubits. Heisenberg XY chains with odd number of qubits were not
considered in this study because C(n) = 0 for odd n. However, one could similarly study the
robustness of other kinds of entanglement employing the other MKB concurrences.
In fact, the applicability of our analysis is not only restricted to the Heisenberg XY
chains. As long as the ground state of a system can be made to dominate the thermal state
at any finite temperature by adjusting some parameters of the system Hamiltonian, Eq.(11)
tells us that the MKB concurrence of the thermal state is given by that of the ground state.
So, if in addition, the ground state remains multipartite entangled under these conditions,
similar robustness of multipartite entanglement is expected to be observed. Therefore, our
analysis can be used to identify possible candidates for realization of quantum computation
at finite temperatures.
The MKB concurrences C(n) and Cn can be directly measured for pure states, if two
copies of the states are available [25, 26]. This is obviously due to the fact that the MKB
concurrences are defined in terms of expectation values of Hermitean operators. Indeed,
this has lead to the direct measurement of Wootters concurrence, previously thought to be
not directly measurable, for pure states in laboratory [25]. Since we are interested in the
region where the ground state |Φg〉 is the only state that contributes significantly to the
thermal state χ, the state χ is “almost pure” and therefore it is possible to measure the
MKB concurrences of χ with a high degree of success. The results obtained here can thus
be experimentally verified.
In conclusion, we have established a rather general method of identifying systems that
exhibit robustness of multipartite entanglement at finite temperatures. Our analysis rest on
the mathematical property of the MKB concurrences, namely Eq.(11). The MKB concur-
rences can be directly measured and therefore experiments can be carried out to verify the
results of any analysis.
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http://arxiv.org/abs/quant-ph/0506073
	Introduction
	The MKB Concurrences
	Two-qubit Heisenberg XY Model
	Four-qubit Heisenberg XY model
	Zero Temperature
	Non-zero Temperatures
	Six-qubit Heisenberg XY model and beyond
	Conclusions
	References
