We demonstrate that the Global Entanglement (GE) measure defined by Meyer and
Wallach, J. Math. Phys. 43, 4273 (2002), is maximal at the critical point for
the Ising chain in a transverse magnetic field. Our analysis is based on the
equivalence of GE to the averaged linear entropy, allowing the understanding of
multipartite entanglement (ME) features through a generalization of GE for
bipartite blocks of qubits. Moreover, in contrast to GE, the proposed ME
measure can distinguish three paradigmatic entangled states: $GHZ_{N}$,
$W_{N}$, and $EPR^{\otimes N/2}$. As such the generalized measure can detect
genuine ME and is maximal at the critical point.Comment: 4 pages, 3 figures. Replaced with final published versio

B. d’Espagnat

G. K. Brennen

Gustavo Rigolin

J. I. Latorre

Marcos C. de Oliveira

Thiago R. de Oliveira

English

arXiv

Crossref

Genuine multipartite entanglement in quantum phase transitions

We demonstrate that the global-entanglement (GE) measure defined by Meyer and Wallach [J. Math. Phys. 43, 4273 (2002)] is maximal at the critical point for the Ising chain in a transverse magnetic field. Our analysis is based on the equivalence of GE to the averaged linear entropy, allowing an understanding of multipartite entanglement (ME) features through a generalization of GE for bipartite blocks of qubits. Moreover, in contrast to GE, the proposed ME measure can distinguish three paradigmatic entangled states: GHZ(N) (Greenberger-Horne-Zeilinger), W-N, and EPRcircle times N/2. As such the generalized measure can detect a genuine ME and is maximal at the critical point.73

de Oliveira, TR

Rigolin, G

de Oliveira, MC

Repositorio da Producao Cientifica e Intelectual da Unicamp

Genuine multipartite entanglement in quantum phase transitions
Thiago R. de Oliveira,* Gustavo Rigolin,† and Marcos C. de Oliveira‡
Instituto de Física Gleb Wataghin, Universidade Estadual de Campinas, 13083-970, Campinas, Sao Paulo, Brazil
Received 26 July 2005; published 31 January 2006
We demonstrate that the global-entanglement GE measure defined by Meyer and Wallach J. Math. Phys.
43, 4273 2002 is maximal at the critical point for the Ising chain in a transverse magnetic field. Our analysis
is based on the equivalence of GE to the averaged linear entropy, allowing an understanding of multipartite
entanglement ME features through a generalization of GE for bipartite blocks of qubits. Moreover, in contrast
to GE, the proposed ME measure can distinguish three paradigmatic entangled states: GHZN Greenberger-
Horne-Zeilinger, WN, and EPRN/2. As such the generalized measure can detect a genuine ME and is maximal
at the critical point.
DOI: 10.1103/PhysRevA.73.010305 PACS numbers: 03.67.Mn, 03.65.Ud, 05.30.d, 05.50.q
Entanglement is a correlation of exclusively quantum na-
ture present in principle in any set of postinteracting quan-
tum systems 1. As such, multipartite entanglement ME is
expected to play a key role on quantum-phase transition
QPT phenomena in the same way that classical statistical
correlation does on classical phase transitions 2,3. In ordi-
nary phase transitions, at the critical point, a nonzero order
parameter characterizes a long-range correlation given by
the correlation length divergence. In the same way, in QPTs
the ME is expected to be maximal at the critical point, in the
sense that all the system parties would be entangled with
each other 2. However, this conjecture could not be proved
in general either by measures of pairwise entanglement or by
the proposed ME measures. Even after a considerable effort
a deep understanding of multipartite-entangled states MES
is lacking. Thus it is still a great challenge to capture the
essential features of ME from a conceptual point of view as
well as from a quantitative approach, defining a measure that
among its other properties is able to distinguish MES 4,5.
Indeed, concerning the legitimate quantum correlations in
QPTs it would certainly be important to know exactly what
kind of entanglement we should expect to be maximal at the
critical point. The great majority of efforts trying to answer
this question made use of two kinds of bipartite entangle-
ment measures, both calculated for spin-1 /2 lattice models
such as the Ising model in a transverse magnetic field 6.
The first measure, namely the pairwise entanglement con-
currence between two spins in the chain, was studied in
Refs. 2,3. The second measure, the entropy of entangle-
ment between one part of the chain a block of L spins and
the rest of the chain, was investigated in Refs. 2,7,8. Some
candidates for ME measures were also evaluated in systems
exhibiting QPTs 9–11. Nevertheless, none of the entangle-
ment measures employed in the references cited are maximal
at the critical point except the single site entropy for the Ising
model 2 in the thermodynamical limit and the localizable
entanglement 11 for an Ising chain with 14 spins. Further-
more, in Refs. 2,3 the authors have independently shown
that bipartite entanglement vanishes when the distance be-
tween the two spins is greater than two lattice sites 12. This
is not expected since long-range quantum correlations should
be present at the critical point. It was then suggested that
bipartite entanglement at the critical point should be de-
creased in order to increase ME due to entanglement sharing
2. In other words, ME only appears at the expense of pair-
wise entanglement, and at the critical point we should expect
a genuine MES.
In this paper we demonstrate that the global entanglement
GE introduced in Ref. 13 indeed captures the essential
point to be maximal at the critical point for the Ising model
in a transverse magnetic field in the thermodynamical limit.
We also prove that there exists an interesting relation be-
tween the GE, the von Neumann entropy, linear entropy
LE, and 2-tangle 14–16, showing that they are all equiva-
lent to detect QPTs. Furthermore, this relation helps us to
understand the results obtained in Ref. 2, as outlined in the
previous paragraph, and suggests that they are not unique to
the Ising model but common to all MES with translational
invariance. In addition to this, we generalize the GE and
propose a ME measure that is also maximal at the critical
point for the Ising model and can detect genuine MES, and
in contrast to GE furnishes different values for the entangle-
ment of the GHZN, Greenberger-Horne-Zeilinger, WN, and
EPRN/2 states, thus being able to distinguish between MES.
For a N qubit system spin-1 /2 chain it was noted that
GE is simply related to the N single qubit purities 16–18 by
EG
1 = 2 −
2
N

j=1
N
Tr j
2 =
1
N

j=1
N
SL j = SL , 1
where GE is hereafter identified as EG
1 , j =Tr j̄	 is the jth
qubit reduced-density matrix obtained by tracing out the
other j̄ qubits, and SL j= d / d−11−Tr j
2 is the stan-
dard definition of LE. This relation shows that EG
1 is just the
mean of LE. It was also noted in Refs. 16,19 that
*Electronic address: tro@ifi.unicamp.br
†Electronic address: rigolin@ifi.unicamp.br
‡Electronic address: marcos@ifi.unicamp.br
PHYSICAL REVIEW A 73, 010305R 2006
RAPID COMMUNICATIONS
1050-2947/2006/731/0103054/$23.00 ©2006 The American Physical Society010305-1
EG
1 =
1
N

j=1
N
 j,rest =  , 2
where  j,rest=C
2 is the 2-tangle 14,16 the square of con-
currence C 20. Both LE and the 2-tangle can thus be used
to quantify the entanglement between any block bipartition
of a system of N qubits. They quantify the entanglement
between one qubit j and the rest of the N−1 qubits of the
chain 16. The proof of Eq. 2 is based on the Schmidt
decomposition 21 that also allows us to use for pure sys-
tems the reduced von Neumann entropy, SV j j̄=−Trj j̄
 j j̄logd j j̄ as a good bipartite-entanglement measure
22. Here d=mindim H j ,dim H j̄	 and dim H j j̄ is the
Hilbert-space dimension of subsystem j j̄. Recalling that SV
is bounded from below by SL and employing Eqs. 1 and 2
we obtain the following important relation:
EG
1 =  = SL  SV , 3
which states that the GE is nothing but the mean LE of single
qubits with the rest of the chain. Furthermore, the GE is also
equal to the mean 2-tangle and a lower bound for the mean
von Neumann entropy. An immediate consequence of this
result shows up when we deal with linear chains with trans-
lational invariance. This implies that SL=SL j and that
SV=SV j. Hence, Eq. 3 becomes EG
1=SL jSV j.
Since SL j and SV j have the same concavity and both
entropies attain their maximal value for a maximally mixed
state this last relation shows that EG
1 is as efficient as the
linear and the von Neumann entropies to detect QPTs. In
Ref. 2 the authors used SV and in Ref. 9 EG
1 was em-
ployed to detect QPTs in the Ising model. Needless to say,
both works arrived at the same results for a given range of
parameters, notwithstanding their use of different entangle-
ment measures, which by that time were thought to be unre-
lated.
Despite its success to detect the Greenberger-Horne-
Zeilinger GHZ state 19,23, EG
1 sometimes fails to distin-
guish different multipartite states. This is best understood if
we study EG
1 for three paradigmatic multipartite states. The
first is 
GHZN= 1/2
0N+ 
1N, where 
0N and

1N represent N tensor products of 
0 and 
1 respectively.
The second is a tensor product of N /2 Bell states 18,

EPRN= 
+N/2, where 
+= 1/2
00+ 
11. This
state is obviously not a MES. Only the pairs of qubits
2j–1,2j, where j=1,2 ,… ,N, are entangled. Nevertheless,
for both states EG
1=1. The last is the W state 4, 
WN
= 1/N j=1N 
00¯1 j¯00. The state 
00¯1 j¯00 repre-
sents N qubits in which the jth is 
1 and the others are 
0.
As shown in Ref. 13, EG
1WN=4N−1 /N2.
We now present a generalization of GE. There are three
main features of this approach. First, it becomes clear that we
have different classes of ME measures, where EG
1 is the first
one. Second, the first nontrivial class, EG
2, furnishes different
values for the three states considered above. Third, it gives
new insights into the study of QPT and ME.
In order to define EG
2 we need the following function:
G2,l 
4
3
1 − 1
N − lj=1
N−l
Tr j,j+l
2  , 4
where  j,j+l is the density matrix of qubits j and j+ l obtained
by tracing out the other N−2 qubits. The index 0 lN is
the distance in the chain of two qubits and 4/3 is a normal-
ization constant assuring that G2, l1. Of interest here are
two quantities that can be considered ME measures in the
same sense that EG
1 is:
G2,1 
4
3
1 − 1
N − 1 j=1
N−1
Tr j,j+1
2  , 5
and
EG
2 =
1
N − 1 l=1
N−1
G2,l . 6
We can interpret G2,1 as the mean LE of all two-qubit
nearest neighbors with the rest of the chain. Similar interpre-
tations are valid for the others G2, l. EG
2 is the mean of all
G2, l and it gives the mean LE of all two qubits, indepen-
dent of their distance, with the rest of the chain 24. To
define EG
3 we need the function G3, l1 , l2 with one more
parameter, since now we can have different distances be-
tween the three qubits of the reduced state. A complete
analysis of this ME measure and its usefulness to detect
MES is discussed elsewhere 25.
Table I shows the quantities given by Eqs. 5 and 6 for
GHZN ,EPRN, and WN. We note that due to translational sym-
metry, G2,1 and EG
2 are identical for GHZN and WN. It is
worthy of mention that depending on the value of N the
states are classified differently by G2,1. A similar behavior
is observed for EG
2 24. In this case, however, EPRN is the
most entangled state for long chains. The reason for that lies
in the definition of EG
2. For EPRN , G2, l=1 for any l	2.
Thus, since EG
2 is the average of all G2, l, for long chains
G2,1 does not contribute significantly and EG
2→1.
It is worth noting that even at the thermodynamical limit,
N→
 , EG
2, and G2,1 still distinguish the three states.
However, the ordering of the states is different. As already
explained, this is due to the contribution of G2, l , l	2, in
the calculation of EG
2EPRN.
TABLE I. Comparison among the three paradigmatic states.
EG
1 G2,1 EG
2
GHZN 1 2/3 2/3
EPRN 1
N−2
2N−1
2N−1N−2
2N−12
WN
4N−1
N2
16N−2
3N2
16N−2
3N2
DE OLIVEIRA, RIGOLIN, AND DE OLIVEIRA PHYSICAL REVIEW A 73, 010305R 2006
RAPID COMMUNICATIONS
010305-2
Now we specify to the one-dimensional Ising model in a
transverse magnetic field, which is given by the following
Hamiltonian:
H = 
i=1
N
i
xi+1
x + 
i=1
N
i
z, 7
where i represents the ith qubit,  is a free parameter related
to the inverse strength of the magnetic field, and we work in
the thermodynamical limit. We assume periodic boundary
conditions: N+1=1. As we have shown, for a system with
translational symmetry GE is nothing but the LE of one spin
with the rest of the chain. We only need, then, the LE to
obtain the GE. For that purpose we must calculate the single-
qubit or single-site reduced-density matrix that is obtained
from the two-qubits two-sites reduced-density matrix. It is
a 44 matrix and can be written as
ij = Tr ij =
1
4

,
pi

  j
, 8
where  is the broken-symmetry ground state in the thermo-
dynamical limit and p=Tri
 j
ij= i
 j
. Tr ij is the
partial trace over all degrees of freedom except the spins at
sites i and j ,i
 is the Pauli matrix acting on the site i , ,
=0,x ,y ,z where 0 is the identity matrix, and p is real.
Therefore, all we need are the ground state two-point corre-
lation functions CFs. Using symmetry arguments for the
ground state 2 the only nonzero CFs are
p00, pxx , pyy , pzz , p0x= px0 , p0z= pz0, and pxz= pzx. Due to
normalization p00=1 and a direct calculation gives pxz= pzx
=0 for 1. On the other hand, the Schwartz inequality
necessarily gives 0 
pxz
 
i
xi
z
, thus allowing that the
lower and upper bounds for entanglement be calculated for
1. We plot the upper bound for entanglement by taking
pxz=0. By continuity the true value for entanglement must
show a similar behavior.
Those CFs have already been calculated 6, and we just
highlight the main results. The two-point CFs and the mean
values of x and z are
1
xl
x = 
g− 1 g− 2 ¯ g− l
g0 g− 1 ¯ g− l + 1
] ]  ]
gl − 2 gl − 3 ¯ g− 1
 , 9
1
yl
y = 
g1 g0 ¯ g− l + 2
g2 g1 ¯ g− l + 3
] ]  ]
gl gl − 1 ¯ g1
 , 10
where 1
zl
z= 1
z2−glg−l , 1
z=g0, and 1
x=0 for
1 or 1
x= 1−−21/8 for 1. Here gl=Ll+Ll
+1 ,Ll= 1/0dkcoskl / 1+2+2 cosk, and l	1
is the lattice-site distance between two qubits. By tracing out
one of the qubits we obtain the single-qubit density matrix,
which allows us to obtain EG
1 as a function of . This is
shown in Fig. 1. As a matter of fact EG
1 is maximal with a
singular derivative at the critical point =1. For compari-
son, in Fig. 1 we plot SV j, which was already shown to be
also maximal at the critical point for the broken-symmetry
state 2. We emphasize that these measures quantify en-
tanglement in the global system by measuring how mixed the
subsystems are. The physical meaning behind studying “mix-
edness” lies in the fact that the more entangled two sub-
systems are the more mixed their reduced-density matrix
should be 9,18. However, in a many-body system there are
many ways in which one could divide the global system into
subsystems. The first nontrivial generalization is to study the
LE of two sites with the rest of the chain. Using ij we can
calculate G2, l for the Ising model Fig. 2. It has a similar
behavior to EG
1, since it is also maximal with a singular
derivative at the critical point. This feature demonstrates
that both a pair of nearest-neighbors sites and the sites them-
selves are maximally entangled with the rest of the chain at
the critical point. But this is not unique to nearest neighbors
as shown in Fig. 3, where G2,1 , G2,15, and EG
2
= 1/15i=1
15 G2, i is plotted. G2,15 is also maximal at the
critical point indicating that in a QPT entanglement sharing
at the critical point is favored by an increase of all types of
MEs. Moreover, Fig. 3 shows that G2,15 is only slightly
different from EG
2=
1
15
i=1
15 G2, i. This is due to the rapid
convergence of G2, l as l is increased. At the critical point
liml→
 G2, l is 0.675, and thus higher than the values for
FIG. 1. Color online von Neumann entropy dashed curve and
GE/LE solid curve as a function of .
FIG. 2. Color online EG
1 solid curve and G2,1 dashed
curve as a function of . Both quantities are maximal at the critical
point =1.
GENUINE MULTIPARTITE ENTANGLEMENT IN… PHYSICAL REVIEW A 73, 010305R 2006
RAPID COMMUNICATIONS
010305-3
GHZN ,EPRN, and WN, we obtained in the thermodynamical
limit, indicating thus a genuine MES. We also note that in
addition to EG
1 ,G2, l, and EG
2 all being maximal at the
critical point, EG
1EG
2 for every value of . However, an
interesting change of ordering for EG
1 and G2,1 occurs
around the critical point. For 1,EG
1G2,1, but for 
1,EG
1G2,1. Thus one type of ME is favored in detri-
ment to the others, depending on the system phase. Also, the
fact that at the critical point both EG
1 and EG
2 are maximal
indicates entanglement sharing, such that all the sites of the
chain are strongly quantum correlated. Of course this state-
ment is true only if EG
m is also shown to be maximal for any
2mN−1 all possible partitions. Furthermore, the fact
that G2, l always increases as l→
 at the critical point
suggests a kind of diverging entanglement length. However,
its precise definition demands a careful calculation of the
scaling of entanglement such as in Refs. 7,9. These points
are left for further investigation 25.
In conclusion we have demonstrated that for an infinite
Ising chain both EG
1 and its generalization, EG
2, are maximal
at the critical point. Furthermore, EG
2 as defined here is able
to detect a genuine ME. We note that the behavior of the ME
measures presented here for an infinite chain is in agreement
with the localizable entanglement calculated for a finite N
=14 Ising chain for the broken-symmetry state 11. Yet our
results were obtained in a relatively simpler fashion and
could be used to infer a genuine ME for systems where the
localizable entanglement has failed to detect the QPT 26.
Finally, our results reinforced the conjecture of Osborne and
Nielsen 5 that at the critical point ME should be high due
to entanglement sharing, to the detriment of bipartite en-
tanglement.
We thank A. O. Caldeira, E. Miranda and J. A. Hoyos for
helpful discussions about ME and QPT. We acknowledge
support from FAEPEX-UNICAMP, CNPq, and FAPESP.
1 B. d’Espagnat, Conceptual Foundations of Quantum Physics
Princeton University Press, Princeton, 1973.
2 T. J. Osborne and M. A. Nielsen, Phys. Rev. A 66, 032110
2002.
3 A. Osterloh et al., Nature London 416, 608 2002.
4 W. Dür, G. Vidal, and J. I. Cirac, Phys. Rev. A 62, 062314
2000.
5 G. Rigolin, Phys. Rev. A 71, 032303 2005.
6 P. Pfeuty, Ann. Phys. N.Y. 57, 79 1970.
7 J. I. Latorre, E. Rico, and G. Vidal, Quantum Inf. Comput. 4,
48 2004 and references therein.
8 G. Vidal et al., Phys. Rev. Lett. 90, 227902 2003.
9 R. Somma et al., Phys. Rev. A 70, 042311 2004.
10 T. -C. Wei et al., Phys. Rev. A 71, 060305 2005.
11 F. Verstraete, M. Popp, and J. I. Cirac, Phys. Rev. Lett. 92,
027901 2004; M. Popp et al., Phys. Rev. A 71, 042306
2005.
12 The authors in Ref. 6 also realized that the derivative of the
concurrence as a function of the magnetic field diverges at the
critical point.
13 D. A. Meyer and N. R. Wallach, J. Math. Phys. 43, 4273
2002.
14 V. Coffman, J. Kundu, and W. K. Wootters, Phys. Rev. A 61,
052306 2000.
15 A. Wong and N. Christensen, Phys. Rev. A 63, 044301 2001.
16 A. Lakshminarayan and V. Subrahmanyam, Phys. Rev. A 71,
062334 2005.
17 G. K. Brennen, Quantum Inf. Comput. 3, 619 2003.
18 H. Barnum et al., Phys. Rev. Lett. 92, 107902 2004.
19 J. Endrejat and H. Büttner, Phys. Rev. A 71, 012305 2005.
20 W. K. Wootters, Phys. Rev. Lett. 80, 2245 1998.
21 S. L. Braunstein and P. van Loock, Rev. Mod. Phys. 77, 513
2005.
22 C. H. Bennett et al., Phys. Rev. A 54, 3824 1996.
23 D. M. Greenberger et al., Am. J. Phys. 58, 1131 1990.
24 A. J. Scott, Phys. Rev. A 69, 052330 2004.
25 G. Rigolin, T. R. de Oliveira, and M. C. de Oliveira unpub-
lished.
26 L. C. Venuti and M. Roncaglia, Phys. Rev. Lett. 94, 207207
2005.
FIG. 3. Color online G2,1 dashed/black, G2,15 solid/
red, and EG
2 dotted-dashed/blue as a function of . We see that
EG
2 is slightly different from G2,15 showing that G2, l saturates
as l→
.
DE OLIVEIRA, RIGOLIN, AND DE OLIVEIRA PHYSICAL REVIEW A 73, 010305R 2006
RAPID COMMUNICATIONS
010305-4


Physical Review A

Genuine Multipartite Entanglement in Quantum Phase Transitions

Genuine Multipartite Entanglement in Quantum Phase Transitions

Abstract

Similar works

Full text

Available Versions

Crossref

Repositorio da Producao Cientifica e Intelectual da Unicamp