In this paper, we use a criterion provided by E. Shchukin and W. Vogel [Phys. Rev. Lett. 95, 230502 (2005)] for determining when a bipartite quantum state is entangled. We first show that the entanglement criterion is necessary and sufficient condition for the partial transposition of the bipartite quantum states. Furthermore, previously known entanglement criteria are proved to be special cases of this criterion. We then apply the entanglement criterion to determine entangled property of several non-classical two-mode states. Finally, we provide the conclusion about this criterion

Duc, Truong Minh

Hoai, Nguyen Thi Xuan

Vietnam Academy of Science and Technology: Journals Online

Communications in Physics, Vol. 20, No. 3 (2010), pp. 233-240
ENTANGLEMENT CRITERION FOR BIPARTITE
QUANTUM STATES: APPLICATIONS
TRUONG MINH DUC AND NGUYEN THI XUAN HOAI
Department of Physics, Hue University of Pedagogy
Abstract. In this paper, we use a criterion provided by E. Shchukin and W. Vogel [Phys. Rev.
Lett. 95, 230502 (2005)] for determining when a bipartite quantum state is entangled. We first
show that the entanglement criterion is necessary and sufficient condition for the partial trans-
position of the bipartite quantum states. Furthermore, previously known entanglement criteria
are proved to be special cases of this criterion. We then apply the entanglement criterion to
determine entangled property of several non-classical two-mode states. Finally, we provide the
conclusion about this criterion.
I. INTRODUCTION
For a recent review, it is believed that quantum entanglement plays an essential
role in the rapidly developing field of quantum information processing [1]. An important
matter is to check whether a quantum state is entangled. In order to solve this matter,
Peres proposed an inseparable criterion based on negative partial transposition (NPT)
of the composite density operator of the bipartite system [2]. However, with our best
knowledge, characterizing NPT for such a system is still an completely unsolved problem.
The first set of inseparable criterions based on the NPT that was found is both
necessary and sufficient conditions for the limited dimentional cases such as 2× 2, 2 × 3,
3× 3 and 2× 4 systems [3,4]. After that, a sufficient condition for the NPT was proposed
by Simon [5]. It is based on second-order moments of position and momentum operators
and applied for the continuous variable systems. For the special case of Gaussian states,
it becomes the necessary and sufficient condition. Another inseparable criteria based on
second-order moments of observable quantities was derived without explicitly using the
NPT [6, 7]. These criteria are also complete conditions for two-mode Gaussian states and
sufficient conditions for entanglement in any two-mode states. More recently, sufficient
conditions for entanglement have been found for a wider class of states [8, 9]. Especially,
a criterion was provided by E. Shchukin and W. Vogel [9].
Our aim here is studying and exploring some of the implications of the Shchukin-
Vogel entanglement criterion . We shall first show that this entanglement criterion is
necessary and sufficient condition for the partial transposition of the bipartite quantum
states. We shall then prove previously known entanglement criteria as special cases of this
criterion. Finally, we shall use this criterion to detect entanglement of several non-classical
two-mode states and provide the conclusion about it.
234 ENTANGLEMENT CRITERION FOR BIPARTITE QUANTUM STATES: APPLICATIONS
II. ENTANGLEMENT CRITERION FOR BIPARTITE
QUANTUM STATES
A Hermitian operator Aˆ is called non-negative if and only if for any operator fˆ
whose normally ordered form exists, the following inequality is satisfied〈
fˆ †fˆ
〉
Aˆ
= Tr
(
Aˆfˆ †fˆ
)
≥ 0. (1)
Using for partially transposed density operator ρˆPT , we give an inequality which is equiv-
alent to the positive partial transposition (PPT)
〈f+f〉PT =
〈
fˆ †fˆ
〉
ρˆPT
= Tr
(
ρˆPT fˆ †fˆ
)
≥ 0. (2)
For a form of operator fˆ
fˆ =
+∞∑
n,m,k,l=0
cnmklaˆ
†naˆmbˆ†kbˆl, (3)
where aˆ, bˆ are boson operators and n,m, k, l are the presence of a number of annihilation
or creation operator, the inequality (2) becomes
〈
fˆ †fˆ
〉PT
=
+∞∑
n,m,k,l,p,q,r,s=0
c∗pqrscnmklMpqrs,nmkl ≥ 0, (4)
where
Mpqrs,nmkl =
〈
aˆ†qaˆpaˆ†naˆmbˆ†sbˆrbˆ†kbˆl
〉PT
. (5)
According to the Sylvester criterion, the inequality (4) holds for all cnmkl if and only
if all the subdeterminants of the form (4) are nonnegative. To be convenient, multi-indices
(n,m, k, l) and (p, q, r, s) are respectively replaced by single-indices u and v with following
convention
u < v ⇔
{
|u| < |v| or
|u| = |v| and u <′ v,
(6)
where |u| = n +m + k + l and u <′ v means that the fisrt nonzero difference r − k, s −
l, p− n, q −m is positive. According to this convention, the moments
M0pqrs,nmkl =
〈
aˆ†qaˆpaˆ†naˆmbˆ†sbˆr bˆ†kbˆl
〉
(7)
are written with indices like matrix elements as follows
r s p q k l n m M0ij
0 0 0 0 0 0 0 0 M011 = 1
0 0 0 0 0 0 0 1 M012 = 〈aˆ〉
0 0 0 0 0 0 1 0 M013 = 〈aˆ
†〉
... ... ... ... ... ... ... ... ...
0 0 0 0 0 0 0 2 M016 = 〈aˆ
2〉
... ... ... ... ... ... ... ... ...
0 0 0 1 0 0 0 0 M021 = 〈aˆ
†〉
0 0 0 1 0 0 0 1 M022 = 〈aˆ
†aˆ〉
... ... ... ... ... ... ... ... ...
TRUONG MINH DUC AND NGUYEN THI XUAN HOAI 235
Assuming that the partial transposition acts on the density matrix of the second
partite then the partial transposition moments are represented in the form of initial mo-
ment as 〈
aˆ†qaˆpaˆ†naˆmbˆ†sbˆrbˆ†kbˆl
〉PT
=
〈
aˆ†qaˆpaˆ†naˆmb†lbˆkbˆ†rbˆs
〉
. (8)
Now the necessary and sufficient condition for the PPT can be formulated
DN =
∣∣∣∣∣∣∣∣
M11 M12 ... M1N
M21 M22 ... M2N
... ... ... ...
MN1 MN2 ... MNN
∣∣∣∣∣∣∣∣
≥ 0, ∀N (9)
where Mij is defined in Eq (5)and according to Eq (8) it is written
Mij =
〈
aˆ†qaˆpaˆ†naˆmbˆ†lbˆkbˆ†rbˆs
〉
= (M0ij)
PT . (10)
That is the entanglement criterion was proposed by Shchukin and Vogel [9]. It can be
explicitly stated that: The partial transposition of a bipartite quantum state is nonnegative
if and only if all the subdeterminants
DN =
∣∣∣∣∣∣∣∣∣∣∣∣∣
1 〈aˆ〉 〈aˆ†〉 〈bˆ†〉 〈bˆ〉 ...
〈aˆ†〉 〈aˆ†aˆ〉 〈aˆ†2〉 〈aˆ†bˆ†〉 〈aˆ†bˆ〉 ...
〈aˆ〉 〈aˆ2〉 〈aˆaˆ†〉 〈aˆbˆ†〉 〈aˆbˆ〉 ...
〈bˆ〉 〈aˆbˆ〉 〈aˆ†bˆ〉 〈bˆ†bˆ〉 〈bˆ2〉 ...
〈bˆ†〉 〈aˆbˆ†〉 〈aˆ†bˆ†〉 〈bˆ†2〉 〈bˆbˆ†〉 ...
... ... ... ... ... ...
∣∣∣∣∣∣∣∣∣∣∣∣∣
(11)
are nonnegative for allN . The other way around, if there exists a negative subdeterminant
∃N : DN < 0, (12)
the partial transposition matrix defines negative. It means that the criterion (12) is a
sufficient condition for determining entangled property of bipartite quantum systems.
As to be mentioned in Ref. [9], previously known entanglement criteria are proved
to be special cases of this criterion. Now we shall prove this remark in detail. We start
with the separable condition found by Simon [5]. For any separable quantum state, the
quantity S
S = detA1detA2 +
(
1
4
+ detC
)2
− Tr(A1JCJA2JC
T J) −
1
4
(detA1 + detA2) ≥ 0, (13)
where
Ai =
(
〈(∆xˆi)
2〉 〈{∆xˆi,∆pˆi}〉
〈{∆xˆi,∆pˆi}〉 〈(∆pˆi)
2〉
)
, C =
(
〈∆xˆ1∆xˆ2〉 〈∆xˆ1∆pˆ2〉
〈∆pˆ1∆xˆ2〉 〈∆pˆ1∆pˆ2〉
)
(14)
with
〈{∆qˆi,∆pˆi}〉 =
1
2
〈∆qˆi∆pˆi +∆pˆi∆qˆi〉,
and
J =
(
0 1
−1 0
)
.
236 ENTANGLEMENT CRITERION FOR BIPARTITE QUANTUM STATES: APPLICATIONS
Expanding S in form the annihilation and creation operators, then comparing with a
following subdeterminant of DN
D5 =
∣∣∣∣∣∣∣∣∣∣∣
1 〈aˆ〉 〈aˆ†〉 〈bˆ†〉 〈bˆ〉
〈aˆ†〉 〈aˆ†aˆ〉 〈aˆ†2〉 〈aˆ†bˆ†〉 〈aˆ†bˆ〉
〈aˆ〉 〈aˆ2〉 〈aˆaˆ†〉 〈aˆbˆ†〉 〈aˆbˆ〉
〈bˆ〉 〈aˆbˆ〉 〈aˆ†bˆ〉 〈bˆ†bˆ〉 〈bˆ2〉
〈bˆ†〉 〈aˆbˆ†〉 〈aˆ†bˆ†〉 〈bˆ†2〉 〈bˆbˆ†〉
∣∣∣∣∣∣∣∣∣∣∣
, (15)
we find that S coincides withD5. Hence, Simon criterion is only a special case of Shchukin-
Vogel criterion.
Another separable condition was formulated in Ref. [6]. For any separable state,
the inequality 〈
(∆uˆ)2
〉
ρ
+
〈
(∆vˆ)2
〉
ρ
≥ r2 +
1
r2
(16)
holds true for all nonzero real parameter r, where uˆ = |r|xˆ1 +
1
r xˆ2 and vˆ = |r|pˆ1 −
1
r pˆ2.
By representing operators uˆ and vˆ in form the creation and annihilation operators of the
fisrt mode aˆ†, aˆ and the second mode bˆ†, bˆ we find〈
(∆uˆ)2
〉
+
〈
(∆vˆ)2
〉
= 2r2〈∆aˆ†∆aˆ〉+
2
r2
〈∆bˆ†∆bˆ〉+ 2
r
|r|
[
〈∆aˆ∆bˆ〉+ 〈∆aˆ†∆bˆ†〉
]
+ r2 +
1
r2
. (17)
Inserting (17) in (16) and noticing 〈∆aˆ∆bˆ〉+ 〈∆aˆ†∆bˆ†〉 = 2Re〈∆aˆ∆bˆ〉 we have
r4〈∆aˆ†∆aˆ〉+ 2
r3
|r|
Re〈∆aˆ∆bˆ〉+ 〈∆bˆ†∆bˆ〉 ≥ 0. (18)
Let the inequality (18) be true for all nonzero real parameter r then
Re2〈∆aˆ∆bˆ〉 ≤ 〈∆aˆ†∆aˆ〉〈∆bˆ†∆bˆ〉. (19)
Considering a subdeterminant of DN as follows
d =
∣∣∣∣∣∣
1 〈aˆ〉 〈bˆ†〉
〈aˆ†〉 〈aˆ†aˆ〉 〈aˆ†bˆ†〉
〈bˆ〉 〈aˆbˆ〉 〈bˆ†bˆ〉
∣∣∣∣∣∣ . (20)
We have
d = 〈aˆ†aˆ〉〈bˆ†bˆ〉+ 〈aˆ†〉〈bˆ†〉〈aˆbˆ〉+ 〈aˆ†bˆ†〉〈aˆ〉〈bˆ〉
−〈aˆ†aˆ〉〈bˆ†〉〈bˆ〉 − 〈aˆ†〉〈aˆ〉〈bˆ†bˆ〉 − 〈aˆ†bˆ†〉〈aˆbˆ〉
= 〈∆aˆ†∆aˆ〉〈∆bˆ†∆bˆ〉 − |〈∆aˆ∆bˆ〉|2. (21)
According to Eq. (21) we find d ≥ 0 when
〈∆aˆ†∆aˆ〉〈∆bˆ†∆bˆ〉 ≥ |〈∆aˆ∆bˆ〉|2. (22)
It becomes clear that the condition (19) is weaker than the condition (22).
TRUONG MINH DUC AND NGUYEN THI XUAN HOAI 237
We now consider the condition derived recently in Ref. [8]: For any separable state,
we have the inequality 〈
aˆ†maˆmbˆ†nbˆn
〉
≥ |〈aˆmbˆ†n〉|2. (23)
In Eq. (3), if we only retain two terms and identify other terms with zero, we have
fˆ = C1aˆ
†naˆmbˆ†kbˆl + C2aˆ
†paˆq bˆ†rbˆs. (24)
Then Shchukin-Vogel criterion forms
〈
fˆ †fˆ
〉PT
=
〈
aˆ†maˆnaˆ†naˆmbˆ†lbˆkbˆ†kbˆl
〉〈
aˆ†qaˆpaˆ†paˆq bˆ†sbˆr bˆ†rbˆs
〉
−
〈
aˆ†maˆnaˆ†paˆq bˆ†sbˆrbˆ†kbˆl
〉〈
aˆ†qaˆpaˆ†naˆmbˆ†lbˆkbˆ†r bˆs
〉
≥ 0.
(25)
For n = k = p = q = r = s = 0, the inequality (25) becomes
〈aˆ†maˆmbˆ†lbˆl〉 ≥ |〈aˆmbˆ†l〉|2. (26)
By comparing (23) and (26) we find that for fˆ is defined by Eq. (24), Shchukin-Vogel
criterion coincides with Hillery-Zubairy criterion.
III. APPLICATION
In order to gain a better understanding of the applicability of this entanglement
criterion, we shall study several examples. The fisrt example is that of a pair coherent
state. As defined by Agarwal [10], a pair coherent state of a two mode radiation field is
simultaneous eigenstate of the pair annihilation operator aˆbˆ and the difference in photon
number operator aˆ†aˆ− bˆ†bˆ:
aˆbˆ|ξ, q〉 = ξ|ξ, q〉, (27)
(aˆ†aˆ− bˆ†bˆ)|ξ, q〉 = q|ξ, q〉,
where aˆ and bˆ are two annihilation operators associated with two modes. The pair coherent
state is specified by two parameters, ξ is a complex number and q, which is called the
degeneracy parameter, takes on integer values and can be assumed to be positive without
any loss of generality. This is the non-Gaussian entangled state and is written using the
Fock basic as
|ξ, q〉 = Nq
∞∑
n=0
ξn√
n!(n+ q)!
|n+ q, n〉, (28)
where
Nq =
[
∞∑
n=0
|ξ|2n
n!(n+ q)!
]−1/2
=
[
|ξ|−qIq(2|ξ|)
]−1/2
(29)
is the normalization coefficient, Iq(2|ξ|) being the modified Bessel function.
238 ENTANGLEMENT CRITERION FOR BIPARTITE QUANTUM STATES: APPLICATIONS
According to Shchukin-Vogel criterion, we have a subdeterminant origined from
operator fˆ = c1 + c2aˆ
2 + c3bˆ
2 as following:
D =
∣∣∣∣∣∣
1 〈aˆ2〉 〈bˆ†2〉
〈aˆ†2〉 〈aˆ†2aˆ2〉 〈aˆ†2bˆ†2〉
〈bˆ2〉 〈aˆ2bˆ2〉 〈bˆ†2bˆ2〉
∣∣∣∣∣∣ . (30)
Applying this subdeterminant for the pair coherent states we accquire
D = −x2|ξ|4
[
x2(q2 − 1) + 2
]
− q|ξ|2
[
x2q2 − q + 1− x2
]
. (31)
For q = 0, we find
D0 = −
N 20
N 21
|ξ|4
(
2−
N 20
N 21
)
. (32)
It becomes clear that D0 is negative for all values of ξ. In addition, for other parameters
q, D < 0 is still true. It means that the pair coherent states satisfy the inequality (12)
and it is entangled. Whereas the criteria in Refs. [5–8] do not either provide sufficient
conditions to determine the existence of entanglement or confirm that the pair coherent
states are entangled [11].
As a second example, let us consider a symmetric superposition of two two-mode
coherent states
|ψ〉 =
(
2 + 2x2
)−1/2
(|α, β〉+ |β, α〉) , (33)
with x = |〈α|β〉|. When considering a subdeterminant derived from Schukin-Vogel by
using operator fˆ = c1 + c2aˆ+ c3bˆ
2:
D =
∣∣∣∣∣∣
1 〈aˆ〉 〈bˆ†2〉
〈aˆ†〉 〈aˆ†aˆ〉 〈aˆ†bˆ†2〉
〈bˆ2〉 〈aˆbˆ2〉 〈bˆ†2bˆ2〉
∣∣∣∣∣∣ , (34)
we obtain
D = −N2x2
[
|α|2 + |β|2 − (α∗β + αβ∗)
]2
|α+ β|2, (35)
with
N = (2 + 2x2)−1. (36)
According to (35) the symmetric superposition of two two-mode coherent states will be
entangled for all α 6= β. We will not gain this result with the criterion in Ref. [5, 7, 8]
again.
Finally, we consider a two mode photon-added coherent state
|ψ〉ab = (2 + |α+ β|
2)−1/2(a† + b†)|α, β〉. (37)
We may apply a subdeterminant ofDN to the one resulting from operator fˆ = c1+c2aˆ+c3bˆ:
D =
∣∣∣∣∣∣
1 〈aˆ〉 〈bˆ†〉
〈aˆ†〉 〈aˆ†aˆ〉 〈aˆ†bˆ†〉
〈bˆ〉 〈aˆbˆ〉 〈bˆ†bˆ〉
∣∣∣∣∣∣ . (38)
TRUONG MINH DUC AND NGUYEN THI XUAN HOAI 239
For this state, it is convenient to use
〈α|aka†l|α〉 =
{
k!Ll−kk (−|α|
2)α∗
(l−k)
if l ≥ k
l!Lk−ll (−|α|
2)αk−l if k ≥ l,
(39)
where Lmn (x) is the Laguerre polynomial
Lmn (x) =
n∑
k=0
(m+ n)!(−x)k
(m+ k)!(n− k)!k!
. (40)
By using (39) we get the expressions of the elements in the subdeterminant (38) and final
result
D = N3αβ × (2− |α+ β|
2), (41)
where
Nαβ = (2 + |α+ β|
2)−1. (42)
It can be realized that according to this criterion, the two-mode photon-added coherent
state will be entangled for all α and β satisfying |α + β|2 > 2. Comparing with result
achieved in Ref. [12], it is clear the entanglement condition obtained in this paper is more
easy to gain.
IV. CONCLUSION
In conclusion, we have proved that several previously known entanglement criteria
are special cases of the criterion in Ref. [9] in detail. Specially, we have used this criterion
for finding entangled property of two mode nonclassical states. Our calculations yield the
pair coherent state will be entangled for all ξ for some value of q parameters. This result
would not gain if we used previously known entanglement criteria. Furthermore, for the
symmetric superposition of two two-mode coherent states and the two mode photon added
coherent state, the entanglement conditions calculated by this criterion also are stronger
than one calculated by some known entanglement criteria. It means that the criterion
studied in this paper is not only general but also stronger than previously known entan-
glement criteria. Hence, we have great expectations of the applicability of this criterion
for determining when a bipartite quantum state is entangled. Finally, it is necessary to
notice that we can use this criterion for higher mode bipartite quantum states.
ACKNOWLEDGMENT
The authors would like to thank Prof. Nguyen Ba An for his valuable comments.
This work was supported by the NAFOSTED.
REFERENCES
[1] Samuel L.Braustein, Peter van Loock , Review of Modern Physics 77 (2005) 513.
[2] Asher Peres, Physical Review Letters 77 (1996) 1413.
[3] Michal Horodecki, Pawel Horodecki, Ryszard Horodecki , Phys. Lett. A223 (1996) 1.
[4] Pawel Horodecki, Physics Letter A, 232 (1997) 333.
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[6] L. M. Duan, G. Giedke, J. I. Cirac, P. Zoller, Physical Review Letters, 84 (2000) 2722.
[7] M. G. Raymer, A. C. Funk, B. C. Sander, H. Guise, Physics Review A, 67 (2003) 052104.
[8] Mark Hillery, M. Suhail Zubairy, Physical Review Letters, 96 (2006) 050503.
[9] E. Shchukin, W. Vogel, Physical Review Letters, 95 (2005) 230502.
[10] G. S. Agarwal, Journal of the Optical Society of America B, 5 (1988) 1940.
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Received 30 September 2009.


Entanglement Criterion for Bipartite Quantum States: Applications

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Entanglement Criterion for Bipartite Quantum States: Applications

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