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Journal of Quantum Information Science, 2012, 2, 78-81
http://dx.doi.org/10.4236/jqis.2012.23013 Published Online September 2012 (http://www.SciRP.org/journal/jqis)
An Entanglement Criterion for States in Infinite
Dimensional Bipartite Quantum Systems
Yinzhu Wang1,2, Danxia Wang2
1Department of Mathematics, Taiyuan University of Science and Technology, Taiyuan, China
2School of Mathematics, Taiyuan University of Technology, Taiyuan, China
Email: 2006wang.[email protected]
Received July 10, 2012; revised August 10, 2012; accepted September 10, 2012
ABSTRACT
In this paper, an entanglement criterion for states in infinite dimensional bipartite quantum systems is presented. We
generalize some of separability criterion that was recently introduced by Wu and Anandan in (Phys. Lett. A, 2002, 297,
4-8) to infinite dimensional bipartite quantum systems. In addition, we give an example aimed to illustrate the applica-
tion of the theorem.
Keywords: Entanglement Criterion; Infinite Dimensional Quantum Systems; Bochner Integral Representation
1. Introduction
Quantum entanglement plays a crucial role in the rapidly
developing theory of quantum information and quantum
computation [1]. In the cases of finite dimensional quan-
tum systems, there are many methods to quantify the
entanglement of bipartite and multipartite quantum sys-
tems [2-8]. However, most of them have not explicit
formula, or it is hard to calculate. For the cases of infinite
dimensional systems, the method of entanglement detec-
tion is a very difficult problem. But the case of infinite
dimensional quantum systems can’t be neglected since
they do exist in quantum world [9]. Recently, Shengjun
Wu and Jeeva Anandan [10] proposed a necessary cri-
terion based on Pauli matrices re-presentation. Their re-
sult is as follows:
Let ,
AB
H
H
2
Adi
be separable complex Hilbert spaces,
,
dim HmB
H
n
n
 . By
we denote the set of all states in A
AB
SH H
B
H
H. A given
state
in
AB
H
H can be written as:
11 12
21 22



(1)
where eachkl
1, 2;1, 2kl
is an n by n matrix,
can also be written as:
0
0
z
xy
xy z
MM MiM
MiM MM




(2)
or
0
xy yz z
I
MMMM

    (3)
where the four matrices


01122 12
122111 22
11
,
22
1
,
22
x
yz
MM
i
MM
21


 
 
(4)
are n-dimensional Hermitian matrices. Let R be a
3-dimensional real matrix, and
R
be a transformation
on the density matrix
with the following form:

0
0
RRR
zxy
RRR R
xy z
MM MiM
MiM MM







(5)
where
R
x
x
R
y
y
R
z
z
MM
MRM
MM






(6)
Shengjun Wu and Jeeva Anandan [10] give the fol-
lowing results:
1) If
is separable, then

R
must be positively
defined for any 3-dimensional real matrix R which satis-
fies 0R
T
IR
.
2) If
is separable, then
a) 0
M
rM
(, ,)
is positively defined for any vector
x
yz
r with 1
r,(, ,)
x
yz
M
MMM;
b)
2222
00
xyz
TrMMMM
 ;
where ,,
x
yz
M
MMare defined in Equation (4).
Then, a nature problem is arisen: whether or not there
is counterpart result for the infinite-dimensional bipartite
quantum systems? We find that the answer is “yes”. The
aim of the present paper is to establish this criterion for
the infinite dimensional bipartite quantum systems.
C
opyright © 2012 SciRes. JQIS
Y. Z. WANG, D. X. WANG 79
The paper is organized as follows: In Section 2, we
give the main results and the proof of the main results. In
Section 3, we give an example to illustrate the application
of the theorem.
2. Some Notations and Main Results
In this section, we mainly generalize the finite dimen-
sional results, which be proposed by Wu and Anandan
[10], to infinite-dimensional bipartite quantum sys-
tems AB
H
H, where dim 2,dim.
AB
HH
,

Let’s fix some notations. Let AB
H
H be separable
complex Hilbert spaces, by B
, we denote
the set of all states in AB
A
H
SH
H
H. The set of all separable
pure states in AB
H
H is denoted by

s
pA B
.
Throughout the paper we use the Dirac’s symbols. The
bra-ket notation stands for the inner product in the
given Hilbert spaces. Recall that a quantum state
AB
SHH

| 

AB
HSH
HHdim
 , which is
positive and has trace one, is said to be separable if
can
be written as:
1
n
ii i
i
p


(7)
where i
and i
are pure states in the subsystems
A
H
and
B
H
respectively, Otherwise, 0, 1.
ii
i
pp
i
is
by Werner[8], a state
called an entangled state. If

dim AB
HH
,
acting on AB
H
His called sepa-
can be approximated in the trace norm by the
states of the form
rable if it
i
1
n
nii
i
p


(8)
Furthermore, it is shown in [11] that any separable
state
admits a representation of the Bochner interal

d
AB AB
Sp
HH HH
S

 
(9)
where
is a Borel probability measure on

s
pA B
SHH
, and :Sp Sp
SS

is a measurable function. It is known that, from the defi-
nition of the Bochner integral, there exists a sequences of
step function

n
, such that

lim
AB AB
HH HH
n
n
  

 
(10)
with respect to the trace norm. Where

1
n
A
BABA
i
k
HH HHHH
nEi
i
B
i
 
 
(11)

i
E is the characteristic function of i, and 1
{}
En
k
ii
E
is a partition of
lim
A
B
i
HH
ii i
EEi
E
 

(12)
with respect to the trace norm, as well as with respect to the
Hilbert Schmidt norm. Where there exists an ensemble
,A
H
ii
p
(or
,B
H
jj
p
) of A
H
(or
B
H
) such that
A
A
HHH
iii

A
(or BB
HHH
iii

B
) (13)
Next, we give the main results as follows:
Theorem 2.1 Let HA and HB be separable complex
Hilbert spaces, dim 2
A
H
, . If dim B
H
AB
SH H
 is a separable state, then
1)
R
must be positively defined for any 3 by 3
real matrix R which satisfies , where 0
T
IRR
R
M
is defined in Equations (5) and (6).
2) 0
rM
(, ,)
is positively defined for any vector
x
yz
rwith 1
r,

,,
x
yz
M
MMMand
2222
00
xyz
TrMMMM
 , 0,,,
x
yz
M
MMM
are defined in Equation (4).
Proof. 1) Since
is separable, according by Equa-
tions (9)-(12), we have

lim
A
AB
i
HH HH
iiii i
EEi
E


B
(14)
where A
HH
ii

A
are pure states of ,
A
H
B
B
HH
ii

are pure states of
B
H
, respectively. Furthermore, ac-
cording by Bloch representation [1], we have





1
lim 2
1
lim
2
B
B
i
BB
i
BB
BB
BB
HH
iii
EEi
HH
ii i
EE i
HH
xiiii
i
HH
yiiii
i
HH
ziiii
i
EI
IE
Ex
Ey
Ez
 









r
i
(15)
with respect to the trace norm, where ,,
x
yz

are
Pauli matrixes and
,,
x
yz

,

,,
iiii
x
yzr are
real vectors on the Bloch sphere and satisfies
222
1
iii
xyz
. Comparing Equations (3) and (15), we
have


0
1
1
1lim
2
1lim
2
BB
i
BB
i
HH
ii i
EE i
HH
xiii
EE i
ME
MEx


i
s
pA B
SHH
. By E we denote the set
of all partitions of
s
pA
SH
B
H, Thus we have
Copyright © 2012 SciRes. JQIS
Y. Z. WANG, D. X. WANG
80


1
1
1lim
2
1lim
2
BB
i
BB
i
HH
yiii
EE i
HH
ziii
EE i
MEy
MEz










i
i




1
() lim[||
2
BB
i
BB
BB
BB
HH
Ri
EE i
HH
xiiii
i
HH
yiiii
i
HH
ziiii
i
IE
Ex
Ey
Ez
 



 



ii
i
(16)
where
ii
i
ii
x
x
y
Ry
zz
 
 
 
 
 
(17)
Since for any 3-dimensional real matrix R which satis-
fies , means that ,
on the other hand, according by Equation (16), we have
0
T
IRR
  
222
1
iii
xyz




1lim
2
BB
i
HH
Ri
EE i
AE
 



ii
,
where
1'' '
''1'
iii
ii i
zxiy
Axiy z




(18)
It is obvious that we have, in fact since
, so
0A

  
222
1
iii
xyz

 R
is still a density ma-
trix, i.e., .

0
R






0
0
1
1
2)
1
lim
2
1
lim
2
1
lim
2
1
lim
2
1
lim
2
BB
i
BB
i
BB
i
BB
i
i
xyz
HH
ii i
EE i
HH
iii i
EE i
HH
iiii
EE i
HH
iiii
EE i
i
EE i
M
MxMyMzM
E
xEx
yEy
zEz
Ez





 
















rM
1
BB
HH
iiiii i
x
xyyzz








Since  aba b, so we have
and
10
iii
xxyy zz
0
On the other hand, we have
0.MrM



2222
0
2
Tr
1lim
4
10
i
BB
xyz
ij
EE ij
HH
ijij ijii
MMMM
EE
xx yy zz



 

This completes the proof.
3. Example
Next, we give an example to illustrate the application of
the Theorem 2.1. We consider a bipartite infinite dimen-
sional state with the following forms:

1
00 0001100110
2
x
x


(19)
with ()
AB
SH H
dim 2,dim,
AB
HH
 01
x
,
where
0,1 is the orthogonal real basis of A
H
,
0,1,

 is the orthogonal real basis of
B
H
.
For simplicity, we assume . In
this case,we can obtain that

dim 2
B
Hn n
0
100
4
1
00
4
00
1
00
4
100
4
00
1
00
2
100
2
2
00
31 00
4
1
00
4
00
x
y
z
x
x
M
x
x
M
x
x
i
M
x
x
M












































Copyright © 2012 SciRes. JQIS
Y. Z. WANG, D. X. WANG
Copyright © 2012 SciRes. JQIS
81
According by Theorem 2.1, by a straightforward cal-
culation, we obtain the following result, if
1
05
x,
then

2222
00
xyz
TrMMMM  (20)
so
is entangled.
Remark: It is obvious that this criterion of Theorem
2.1 is weaker than PPT criterion [4], in fact if x ≠ 1,
is also entangled, but this criterion give us a method to
detect the entanglement for states in infinite bipartite
quantum systems.
4. Acknowledgements
This work was partially supported by the National Natu-
ral Science Foundation of China (11171249) and the
Natural Science Foundation of Shanxi Province
(2011021002-2). The authors also wish to give their
thanks to the referees for their comments to improve the
presentation of this paper.
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