J
ournal o
f
A
pp
http://dx.doi.or
g
Copyright © 2
0
Coll
e
ABSTRA
C
The numbers
respectively.
W
p
roduct states
Keywords:
G
1. Introdu
c
Graph states
a
associated to
tions in quan
t
tum compute
r
measures ha
v
states, such a
s
[8], the Relat
i
Geometric
M
ment measur
e
known that t
h
Thus these en
state. The ge
o
to its closest
entanglement
b
een calculat
e
[13]. We wil
l
of 9 to 11 q
u
paper. Anoth
e
the Schmidt
m
2. Prelimi
n
The geometr
i
ψ
, is define
where Pro
i
(;)GV
i
s
of edges spe
c
an
nn
sy
m
entries and
0
ab
 oth
e
p
lied Mathemat
i
g
/10.4236/jamp
.
0
13 SciRes.
Enta
n
G
e
ge of Informat
i
C
T
of local com
p
W
e calculate
t
for all these
g
G
raph State; E
n
c
tion
a
re certain pu
r
graphs [1-5]
t
um error corr
e
r
[7]. A vari
e
v
e been propo
s
, the (Global
)
i
ve Entropy o
f
M
easure [11].
e
s are all equa
l
h
e graph state
tanglement m
e
o
metric meas
u
product state
of graph stat
e
e
d and the cl
o
l
consider the
u
bits and their
e
r closely rel
a
m
easure [4,14]
.
n
ar
y
i
c measure o
f
d
as
()
m
g
E
i
s the set o
s
composed o
f
c
ified by the a
m
metric mat
r
1
ab
if ver
t
e
rwise. The gr
a
i
cs and Ph
y
sics
,
.2013.14010 P
u
n
gleme
n
G
raph
Cui
f
i
on and Electro
n
p
limentary in
e
the entangle
m
g
raph states. N
n
tanglemen
t
;
C
r
e multipartite
and have im
p
e
ction [6] and
e
ty of differe
n
sed for multi
p
)
Robustness
o
f
Entanglemen
t
Fortunately,
l
for stabilize
r
is a subset o
f
e
asures are all
u
re is the dist
a
in terms of t
h
e
s of 1 to 8 q
u
o
sest product
s
entanglemen
t
closest prod
u
a
ted entangle
m
.
f
entanglemen
t
2
Pr
log
m
in
o
o
f product s
t
f
a set V of n v
djacency mat
r
r
ix with van
i
t
ices
,ab ar
e
a
ph state rela
t
,
2013, 1, 51-5
5
u
blished Online
O
n
t and
C
States
w
f
eng Wang,
L
n
ic Engineering
,
Email: w
c
Recei
v
e
quivalent gra
p
m
ent, the lowe
r
ew patterns o
f
C
losest Produc
t
quantum stat
e
p
ortant applic
one-way qua
n
nt
entangleme
n
p
artite quantu
m
o
f Entangleme
n
t
[9,10], and t
h
these entangl
r
state [12]. It
f
stabilizer sta
t
equal for gra
p
a
nce of the st
a
h
e fidelity. T
h
u
bit systems h
a
s
tates are fou
n
of graph stat
e
u
ct states in t
h
m
ent measure
t
for pure st
a
2
. (
t
ates. A gra
p
ertices and a s
r
ix , which
i
shing diago
n
e
connected a
n
t
ed to graph
G
5
O
ctobe
r
2013 (
h
C
losest
w
ith 9
t
L
izhen Jian
g
,
Zhejiang, Gon
g
c
f_0815@126.
c
v
e
d
August 201
3
p
h states for
9
r
and upper b
o
f
closest produ
t
State
e
s
a-
n
-
n
t
m
nt
h
e
e-
is
t
e.
p
h
a
te
h
e
a
s
nd
e
s
h
is
is
a
te
1)
p
h
et
is
n
al
n
d
G
is defi
n
where
vector
s
G
c
a
where
p
ure s
t
graph
s
only L
C
concer
n
inequi
v
numbe
r
146, th
and th
e
noted
w
The
ation
a
L
OCC
E
b
er of
fectly
b
b
iparti
t
“matc
h
For
a
have
E
determ
i
h
ttp://www.scir
p
Produ
c
t
o 11 Q
u
g
, Lei Wang
g
shang Univers
i
c
o
m
3
9
, 10 and 11
q
o
unds of the
e
ct states are a
n
n
ed as
G
0 = (0, 0,…,
s
of length n.
a
n be written
a
()EG
(
j
pj
t
ate. All loc
a
s
tates have th
e
C
inequivale
n
n
ing with the
v
alent graph
s
r
of LC inequ
i
e entangleme
n
e
closest prod
u
w
ith
.1No
t
o
entanglement
a
nd classica
l
2
lognN
,
w
graph basis s
t
b
y LOCC [1
3
t
e entangleme
n
h
ing” bound
E
E
a
large numbe
r
bi LOGG
E
E, t
h
i
ned. While
fo
p
.org/journal/
ja
m
c
t State
u
bits
i
ty, Hangzhou,
3
q
ubit systems
e
ntanglement
n
alyzed.
1
1
2
0
1(1)
2n
0) and 1 = (
The entangle
m
a
s
()
m
g
EG
01
pj
pj
e

a
l complime
n
e
same amoun
n
t graph states
entanglemen
t
tates increase
i
valent graph
s
n
t is calculate
d
u
ct states are
f
o
.146No
.
is upper bou
n
l
communica
w
here
N
is
t
t
ates that can
3
,15], and lo
w
n
t deduced fro
m
bi
E
[16]. Thus
()
bi
E
EG
r
of graph stat
e
h
e entangle
m
fo
r graph stat
e
m
p
)
s of
3
10018, China
are 440, 313
2
and obtain th
e
1
2
T
z

1, 1,…, 1) a
r
m
ent of a gr
a
2
log
m
in G
1)
j
i
e
is the
n
tary (LC) e
q
ts of entangle
m
should be c
o
t
. The numb
e
s with n rapi
d
s
tates up to 8
d
with iterativ
e
f
ound [13] an
d
n
ded by the lo
c
tion (LOCC
)
t
he the maxi
m
be discrimin
a
w
er bounded
b
m
the state, t
h
LOGG
E
e
s considered
m
ent can be
e
e
s with unequ
JAMP
2
, 40457,
e
closest
(2)
r
e binary
a
ph state
2
(3)
product
q
uivalent
m
ent, so
o
nsidered
e
r of LC
d
ly. The
qubits is
e
method
d
are de-
c
al oper-
boun
m
al nu
m
-
a
ted per-
b
y some
h
at is, the
(4)
later, we
e
asily be
u
al lower
C. F. WANG ET AL.
Copyright © 2013 SciRes. JAMP
52
and upper bounds, we will use iterative method [13] to
calculate the entanglement and obtain the closest product
states.
3. Classifications of the Graph States of Nine,
Ten and Eleven Qubits
All the graphs of 9, 10 and 11 vertices that are LC ine-
quivalent for their graph states can be found in Ref. [17].
The total number of inequivalent classes of 9-qubit graph
states is 440. We denote them from No.147 to No.586.
The entanglement of the graph states is classified and
listed in Table 1. The entanglement of graph states with
equal lower and upper bounds can be calculated with the
methods in Ref [16], and we use computer program to
determine the bounds. The entanglement of graph states
with unequal bounds can be calculated with the iterative
algorithm [13].
These two graph states whose entanglement values is
5.5124 and 5.8381 are new in the sense that they have
different values of entanglement after the decimal point
with respect to the former (1 to 8 qubit) graph states.
The total number of inequivalent classes of 10-qubit
graph states is 3132, and we denote the graphs from
No.587 to No.3718. The entanglement is classified and
listed in Table 2.
The total number of inequivalent classes of 11-qubit
graph states is 40457, and we denote them from No.3719
to No.44175. The entanglement is classified and listed in
Table 3.
4. Structures of the Closest Product States of
Nine, Ten and Eleven Qubits
4.1. The Nine Qubit System
Denote 01 1,(1,,4),
j
i
jppej
 
with
11
(1) 0.4597,
23
p12 3
3
,,,
444

 

4
3;
4
 11
(0 1);(01),
22
Oi   
1
'(01).
2
Oi
The closest product state of ring 5
graph state is 5
5.81ring No


, the entanglement
is .822
1log3log(33)2.9275
No
E  [13]. The
graph set with non-integer entanglement (k + 0.9275)
graph states is closely related with ring 5 graph. The
closest product state of graph state whose entanglement
equals 3.9275 can be
323
41
0


 (5)
The closest product state of graph state whose entan-
Table 1. Classification of the graph state of 9 qubits.
E
bi
E
locc
E
NUM
1
2
3
3.9275
4
4.5850
4.9275
5
5.5124
5.8381
1
2
3
3
4
4
4
4
4
4
1
2
3
4
4
5
5
5
6
6
1
10
61
3
207
2
42
112
1
1
Table 2. Classification of the graph state of 10 qubits.
E
bi
E
locc
E
NUM
1
2
3
3.9275
4
4
4.5850
4.9275
5
5
5.5850
5.9275
6
5
5.5850
5.8549
5.9275
6
6.1669
1
2
3
3
4
4
4
4
5
4
4
4
4
5
5
5
5
5
5
1
2
3
4
4
5
5
5
5
5
6
6
6
6
6
6
6
6
7
1
11
103
3
631
5
2
76
1536
100
1
13
13
40
28
12
217
339
1
glement equals 4.9275 can be
323
41
0


 (6)
The graph set with non-integer entanglement (k +
0.5850) graph states is closely related with No.19 graph.
The closest product state of graph state whose entangle-
ment equals 4.5850 can be
5
2
31
0

 (7)
No.510 has a new structure in 9-qubit graph states. Its
entanglement is
.5102 2
22log (3)log (33)5.5124
No
E (8)
its closest product state can be
C. F. WANG ET AL.
Copyright © 2013 SciRes. JAMP
53
Table 3. Classification of the graph state of 11 qubits.
E
bi
E
locc
E
NUM
1
2
3
3.9275
4
4
4.5850
4.9275
5
5
5
5.5850
5.5850
5.8549
5.9275
5.9275
5.9275
6
6
6.5124
6.5850
6.7824
6.8381
6.8549
6.9275
7
1
2
3
3
4
4
4
4
5
5
4
4
5
5
5
5
4
5
5
5
5
5
5
5
5
5
1
2
3
4
4
5
5
5
5
6
5
6
6
6
6
7
6
6
7
7
7
7
7
7
7
7
1
13
163
3
1561
14
2
121
9951
286
125
1
48
12
1936
1
2
22573
351
4
67
1
1
35
1145
2040
9
.510 2No

. (9)
The entanglement of No.582 is 5.8381, and its closest
product state has a totally new structure. The known
closest product state include components
(1,,4),0,1,,'
jjOO
, however, there are 5
new components in the closest product state of No.582.
Denote 01 1,(5,,9)
j
i
jj j
ppej
 .
The new components are listed as follows,
5:0.9357,7.3 ,p


6:0.9255,21.4 ,p


7:0.4365,43.0 ,p


8:0.4208,44.7,p


9:0.8807, 43.8p


.
In fact, we can find 15 kinds of new component qubits
in the closest product state of graph state No.582. How-
ever, we can transform these 15 new component qubits
into the above five kinds of qubits by using unitary trans-
formation 1
11
1
11
2
U


or 2
1
1
1
2
i
Ui



. Using
iterative algorithm, we find the closest product states of
graph state No.582, and denote them as
1EDDBCAABC ,
2DEAACBDCB ,
3DAECBDBAC ,
4BACEACDBD,
5CCBAEABDD,
6ABDCAECDB ,
7ADBDBCECA ,
8BCABDDCEA ,
9CBCDDBAAE . (10)
where A, B, C, D, E represent 59
,,
, respectively.
If we ignore the order of qubits, the closest product state
of graph state No.582 can be written as
222 2
.582
No EA B CD
 
(11)
4.2. The Ten Qubit System
In Table 2, the graph set with non-integer entanglement
(k+0.9275) graph states is characterized by ring 5 graph.
The closest product state can be
45
20


 (12)
for ten qubit graph state with entanglement 3.9275, it can
be
3
322
31
0


 (13)
for graph state with entanglement 4.9275, it can be
3
22
323
00


 (14)
when with entanglement 5.9275. The graph set with non-
integer entanglement (k+0.8549) graph states is characte-
rized by graph state No.133. The entanglement of graph
state No.133 is
.1332 2
12 log32 log(33)4.8549
No
E , its closest
product state can be 222
13341 4 1 4
No
 

[13]. The closest product state of graph state whose en-
tanglement equals 5.8549 can be
22
3 21323
0


 (15)
The only ten qubit graph state with new pattern of
closest product state is graph state No.3599 whose en-
C. F. WANG ET AL.
Copyright © 2013 SciRes. JAMP
54
tanglement is
.3599 2
32log 36.1699
No
E (16)
the closest product state can be
5
22
.35993 431
No

(17)
4.3. The Eleven Qubit System
For eleven qubit graph states show in Table 3, a detail
comparison of computed closest product states of
No.3724, No.3765 shows that all these closest states have
a substructure of the closest product state of ring 5 graph.
Ring 5 graph is essential to all these graph states with
entanglement k + 0.9275 (integer k). The closest product
states can be
4
5
.37242 30
No

 (18)
422
.37652 42
10
No


 (19)
The graph set (No.3764, No.3936) with non-integer
entanglement (k + 0.5850) graph states is characterized
by graph No.19. The entanglement of graph state No.19
is .19 2
2log 33.5850
No
E  Typically, the closest
product state is 33
.193 4
No

[13]. The closest
state for No.3764 and No.3936 graph state can be
2
4
.3764232 31
0
No
 
 (20)
32
.39363 2 442
00
No
 

 (21)
respectively. The graph set (No.4113, No.30597) with
non-integer entanglement (k + 0.8549) graph states is
characterized by graph state No.133. The closest state for
No.4113 and No.30597 graph state can be
23
.41131 3 1 4 1 40
No


,
222
.305971111 4
00
No


 .
The entanglement of No.30505 is a new type in
11-qubit graph states. Its entanglement is
.30505 2
23log(33)6.7824
No
E  (22)
its closest product state can be
42
.305051414141
No
 

(23)
The entanglement of No.23813 is 6.8381, the closest
product state of No.23813 can be
.23813 1
No BEDB DCAAC
 (24)
It contains the closest product of graph state No.582
.582No
as its substructure.
Note that, we have just found one of the many closest
product states for each LC inequivalent graph state. Ac-
tually, there are many local equivalent closest product
states for the same graph state.
5. Conclusions
We calculate the entanglement of all local complimenta-
ry inequivalent graph states with nine, ten and eleven
qubits. The total number of the graph states we treated is
44029, and we list the numbers of them in Tables 1-3
according to their entanglement. Further calculation for
twelve qubits is quite difficult since there are more than
1.27 million of local complimentary inequivalent graph
states. Four new types of non-integer entanglement val-
ues appear in 9, 10 and 11 qubit graph states. The detail
results and some special characters of the closest product
states are as follows: (1) The graph sets with non-integer
entanglement (k + 0.9275, k + 0.5850 and k + 0.8549)
graph states are specified by ring 5 graph, No.19 graph
and No.133 graph, respectively. Their closest product
states contain five, six and eight (1,,4)
jj
; (2)
The closest product states of graph states with non-in-
teger entanglement (5.5124 for nine qubits, 6.1669 for
ten qubits, 6.7824 for eleven qubits) contains 9, 10 and
11 (1,,4)
jj
, respectively; (3) The graph states
with non-integer entanglement (k + 0.8381) have a new
structure in their closest product states. We find 5 new
components (5,,9)
jj
in the closest product
states. (4). The closest product state for graph state with
integer entanglement does not contain (5,,9)
jj
.
6. Acknowledgements
Funding by the National Natural Science Foundation of
China (Grant No. 60972071), Natural Science Founda-
tion of Zhejiang Province (Grant No. Y6100421), Zhe-
jiang Province Science and Technology Project (Grant
No. 2009C31060) are gratefully acknowledged.
REFERENCES
[1] R. Raussendorf, D. E. Browne and H. J. Briegel, “Mea-
surement-Based Quantum Computation on Cluster States,”
Physical Review A, Vol. 68, No. 2, 2003, Article ID:
022312. http://dx.doi.org/10.1103/PhysRevA.68.022312
[2] D.-M. Schlingemann, Quant. Inf. Comp., Vol. 2, 2002, p.
307.
[3] D.-M. Schlingemann, Quant. Inf. Comp., Vol. 4, 2002, p.
287.
[4] M. Hein, J. Eisert and H. J. Briegel, “Multiparty Entan-
glement in Graph States,” Physical Review A, Vol. 69, No.
6, 2004, Article ID: 062311.
http://dx.doi.org/10.1103/PhysRevA.69.062311
[5] M. Hein, W. Dur, J. Eisert, R. Raussendorf, M. Van den
Nest and H. J. Briegel, In G. Casati, D. L. Shepelyansky,
P. Zoller and G. Benenti, Eds., Quantum Computers, Al-
gorithms and Chaos, IOS Press, Amsterdam, 2006.
[6] M. Grassl, A. Klappenecker and M. Rotteler,Graphs,
C. F. WANG ET AL.
Copyright © 2013 SciRes. JAMP
55
Quadratic Forms, and Quantum Codes,” Proceedings of
2002 IEEE International Symposium on Information
Theory, Lausanne, Switzerland, p. 45.
http://dx.doi.org/10.1109/ISIT.2002.1023317
[7] R. Raussendorf and H. J. Briegel, “A One-Way Quantum
Computer,” Physical Review Letters, Vol. 86, No. 22,
2001, pp. 5188-5191.
http://dx.doi.org/10.1103/PhysRevLett.86.5188
[8] G. Vidal and R. Tarrach, “Robustness of Entanglement,”
Physical Review A, Vol. 59, No. 1, 1999, pp. 141-150.
http://dx.doi.org/10.1103/PhysRevA.59.141
[9] V. Vedral, M. B. Plenio, M. A. Rippin and P. L. Knight,
“Quantifying Entanglement,” Physical Review Letters,
Vol. 78, No. 12, 1997, pp. 2275-2279.
http://dx.doi.org/10.1103/PhysRevLett.78.2275
[10] V. Vedral and M. B. Plenio, “Entanglement Measures and
Purification Procedures,” Physical Review A, Vol. 57, No.
3, 1998, pp. 1619-1633.
http://dx.doi.org/10.1103/PhysRevA.57.1619
[11] T.-C. Wei and P. M. Goldbart, “Geometric Measure of
Entanglement and Applications to Bipartite and Multipar-
tite Quantum States,” Physical Review A, Vol. 68, No. 4,
2003, Article ID: 042307.
http://dx.doi.org/10.1103/PhysRevA.68.042307
[12] M. Hayashi, D. Markham, M. Murao, M. Owari and S.
Virmani, “Entanglement of Multiparty-Stabilizer, Sym-
metric, and Antisymmetric States,” Physical Review A,
Vol. 77, No. 1, 2008, Article ID: 012104.
http://dx.doi.org/10.1103/PhysRevA.77.012104
[13] X. Y. Chen, “Entanglement of Graph States up to Eight
Qubits,” Journal of Physics B, Vol. 43, No. 8, 2010, Ar-
ticle ID: 085507.
http://dx.doi.org/10.1088/0953-4075/43/8/085507
[14] A. Cabello, A. J. Lopez-Tarrida, P. Moreno and J. R.
Portillo, “Entanglement in Eight-Qubit Graph States,”
Physics Letters A, Vol. 373, No. 26, 2009, pp. 2219-2225.
http://dx.doi.org/10.1016/j.physleta.2009.04.055
[15] M. Hayashi, D. Markham, M. Murao, M. Owari and S.
Virmani, “Bounds on Multipartite Entangled Orthogonal
State Discrimination Using Local Operations and Clas-
sical Communication,” Physical Review Letters, Vol. 96,
No. 4, 2006, Article ID: 040501.
http://dx.doi.org/10.1103/PhysRevLett.96.040501
[16] D. Markham, A. Miyake and S. Virmani, “Entanglement
and Local Information Access for Graph States,” New
Journal of Physics, Vol. 9, 2007, pp. 194.
http://dx.doi.org/10.1088/1367-2630/9/6/194
[17] L. E. Danielsen, “Database of Self-Dual Quantum Codes.”
http://www.ii.uib.no/larsed/vncorbits/