Decrease of entanglement by local operations in the D\"ur-Cirac method
Decrease of entanglement by local operations in the Dür–Cirac method
Yukihiro Ota,1, 2, ∗ Motoyuki Yoshida,1, † and Ichiro Ohba1, 2, 3, ‡
Department of Physics, Waseda University, Tokyo 169–8555, Japan
Advanced Research Institute for Science and Technology, Waseda University, Tokyo 169–8555, Japan
Kagami Memorial Laboratory for Material Science and Technology, Waseda University, Tokyo 169–0051, Japan
(Dated: October 27, 2018)
One cannot always obtain information about entanglement by the Dür–Cirac (DC) method. The
impracticality is attributed to the decrease of entanglement by local operations in the DC method.
We show that, even in 2–qubit systems, there exist states whose entangled property the DC method
never evaluates. The class of such states in 2–qubit systems is completely characterized by the value
of the fully entangled fraction. Actually, a state whose fully entangled fraction is less than or equal
is always transformed into a separable state by local operations in the DC method, even if it
has negative partial transposition.
PACS numbers: 03.67.–a, 03.67.Mn, 03.65.Ca
I. INTRODUCTION
Quantum mechanics has a quite different mathemati-
cal and conceptual structure from that of classical me-
chanics. Quantum entanglement vividly illustrates this
point [1]. Investigation into the character of entangle-
ment is necessary for not only the deep understanding of
quantum theory but also its application. Indeed, entan-
glement is regarded as a key concept of quantum infor-
mation processing [2].
The classification and quantification of bipartite entan-
glement (i.e., entanglement between two subsystems in a
total quantum system) are well established [3, 4, 5, 6, 7,
8, 9, 10]. In particular, the positive partial transposition
criterion (PPT) [3, 4, 5, 6] is very useful, because one
can readily obtain a sufficient condition for an entangled
state, a necessary condition for a separable state (i.e., a
state with no quantum correlation) [11], or a necessary
condition for a distillable state [10], by linear algebra.
The situation becomes more complicated as the num-
ber of subsystems in a total system increases. In 3–qubit
systems, for example, there are two inequivalent classes
of entanglement. By stochastic local operations and clas-
sical communication [12], a Greenberger–Horn–Zeilinger
(GHZ) state cannot be transformed into a W state, and
vice versa [13]. However, multiparticle entanglement can
play an important role in quantum protocol (e.g., quan-
tum telecloning [14]) and quantum computing. More-
over, its classification will be useful for deeply under-
standing quantum phase transitions in condensed mat-
ter physics [15]. Thus, research into multiparticle entan-
glement is a crucial and popular issue in both quantum
physics and quantum information theory.
Various attempts to classify and quantify multiparti-
cle entanglement have been made [16, 17, 18, 19, 20, 21,
∗Electronic address: oota@hep.phys.waseda.ac.jp
†Electronic address: motoyuki@hep.phys.waseda.ac.jp
‡Electronic address: ohba@waseda.jp
22, 23]. Among them, Dür and Cirac [17] proposed a
systematic way of classifying multiparticle entanglement
in N–qubit systems. Hereafter, we call it the Dür–Cirac
(DC) method. The main idea is that, using a sequence of
local operations, one can transform an arbitrary density
matrix of an N–qubit system into a state whose entan-
gled property is easily examined. It should be noted that
entanglement cannot increase through local operations.
Accordingly, if the density matrix transformed by local
operations is entangled, then the original density matrix
represents an entangled state.
However, one cannot always obtain an entangled prop-
erty by the DC method. In our previous paper [24], we
suggested that there exists an impracticality in the DC
method through an example.
In this letter, we reveal the possibility that one can-
not obtain the desired information on entanglement by
the DC method, though it is a very simple and effec-
tive method for examining multiqubit entanglement. We
show that there is such a possibility even in 2–qubit
systems. The most important quantity in our discus-
sion is a fully entangled fraction [7, 10]. Our main re-
sult is that, in 2–qubit systems, one can never determine
whether a quantum state is entangled or not through the
DC method if the fully entangled fraction is less than or
equal to 1
. Then, we completely characterize the class
of the state in 2–qubit systems whose entangled property
is never obtained by the DC method. The impracticality
of the DC method is due to the decrease of entanglement
by local operations in the DC method. Additionally, we
investigate what parts of the local operations reduce the
entanglement in 2–qubit system.
The letter is organized as follows. We briefly review
the DC method in section II. Then, we illustrate the
impracticality of the DC method through an example
in a 2–qubit system, and show the relation to the local
operations in section III. After that, we show our main
results in section IV. Our results are shown only in 2–
qubit systems, but they clearly reveal the limitation of
the DC method. Section V is devoted to a summary.
http://arxiv.org/abs/0704.1375v1
mailto:oota@hep.phys.waseda.ac.jp
mailto:motoyuki@hep.phys.waseda.ac.jp
mailto:ohba@waseda.jp
II. REVIEW OF THE DC METHOD
We briefly review the DC method [16, 17, 18, 19]. Its
main idea is that, using a sequence of local operations,
one can transform an arbitrary density matrix of an N–
qubit system into a state whose property of entanglement
is easily examined.
First, we explain how to specify a bipartition of the
system concerned. We divide an N–qubit system into
two subsystems, system A and system B, as follows. Let
us consider a set of binary numbers, {ki}Ni=1 (ki = 0, 1).
When ki is equal to 0 (1), the ith qubit is in system
A (B). We always set k1 = 0; the first qubit is al-
ways in system A. Representing the number by a binary,
i=2 ki2
i−2, a partition is specified in the N–qubit
system if an integer k(∈ [1, 2N−1 − 1]) is chosen; we call
such a partition the bipartition k.
The authors in Ref. [16, 17, 18, 19] introduced a special
family of density matrices as follows:
ρN = λ
0 〉〈Ψ
0 |+ λ
0 〉〈Ψ
2N−1−1
|Ψ+j 〉〈Ψ
j |+ |Ψ
j 〉〈Ψ
, (1)
where the coefficients λ±0 and λj are real and positive,
and λ+0 +λ
∑2N−1−1
j=1 λj = 1 because trρN = 1. These
coefficients are related to the information on an arbitrary
density matrix of an N–qubit system, as shown below.
The generalized GHZ state [16, 17, 18, 19] in an N–qubit
system |Ψ±j 〉 is defined as follows:
|Ψ±j 〉 =
(|0j〉 ± |1̄〉) (0 ≤ j ≤ 2N−1 − 1), (2)
where j ≡
i=2 ji2
i−2 for the binary number ji (= 0, 1),
|0j〉 ≡ |0〉1 ⊗
i=2 |ji〉i and |1̄〉 ≡ |1〉1 ⊗
i=2 |1− ji〉i.
The symbol ̄ means a bit–flip of j: ̄ = 2N−1−1− j. We
write the computational basis for the ith qubit as |0〉i and
|1〉i ( i〈0|0〉i = 1, i〈1|1〉i = 1, and i〈0|1〉i = 0). The sub-
scription i(= 1, 2, . . .N) is the label of the qubit. We can
easily find the generalized GHZ states are the elements
of an orthonormal basis of the Hilbert space correspond-
ing to the N–qubit system. Note that the convention of
generalized GHZ states (2) is slightly different from the
corresponding one in Ref. [16, 17, 18, 19], but such a dif-
ference doesn’t matter in our discussion.
We summarize the several useful properties of ρN . The
compact consequences for partial transposition with re-
spect to any bipartition are known [16, 17, 18, 19]. First,
ρN has positive partial transposition (PPT) with re-
spect to a bipartition k if and only if ∆ ≤ 2λk, where
∆ = |λ+0 − λ
0 |. On the other hand, ρN has negative
partial transposition (NPT) with respect to a biparti-
tion k if and only if ∆ > 2λk. Furthermore, the authors
in Ref. [17, 18] proved the theorems about multiparticle
entanglement. Among them, we explain an important
one [17]. We concentrate on two qubits, for example the
ith and jth qubits, in an N–qubit system. Let us con-
sider all possible bipartitions, Pij under which the ith
and jth qubits belong to different parties. The theorem
is that ρN has NPT with respect to
∀k ∈ Pij if and only
if the maximal entangled states between the ith and jth
qubits can be distilled.
The most important result in Ref. [16, 17] is that an
arbitrary density matrix, ρ of an N–qubit system, can
be transformed into ρN by local operations, and local
operations cannot increase entanglement. Accordingly, if
ρN is an entangled state with respect to a bipartition,
ρ is also such a state. Moreover, according to the theo-
rem explained at the end of the above paragraph, if ρN
has NPT with respect to ∀k ∈ Pij , the maximal entan-
gled state between the ith and jth qubits can be distilled
from ρN . Then, one should be able to distill the max-
imal entangled state between such qubits from ρ. This
result implies that one can know the sufficient condition
for the distillability of ρ for an arbitrary N . Note that,
through the PPT criterion, one can only obtain the neces-
sary condition for the distillability in an N -qubit system
when N > 2 [6].
Under the local operations, the coefficients λ±0 and λj
of ρN are given by the following relations:
0 = 〈Ψ
0 |ρ|Ψ
0 〉, 2λj = 〈Ψ
|ρ|Ψ+
〉+〈Ψ−
|ρ|Ψ−
〉. (3)
Consequently, one can systematically treat the evalua-
tion of multiparticle entanglement as a task for bipar-
tite entanglement, because it is only necessary to cal-
culate some specific matrix elements of ρ. In addition,
this point will be useful for investigating entanglement in
experiments [19].
III. LOCAL OPERATIONS IN THE DC
METHOD
As shown in the previous section, one can readily evalu-
ate the information of multiparticle entanglement by the
DC method. However, the desired information about en-
tanglement isn’t always obtained. Let us illustrate such
an impractical case by an example in a 2–qubit system.
One can easily find that, by the PPT criterion, the fol-
lowing density matrix has NPT (i.e., entangled):
|Ψ+0 〉〈Ψ
|Ψ+1 〉〈Ψ
|Ψ−1 〉〈Ψ
|Ψ+1 〉〈Ψ
1 |+ |Ψ
1 〉〈Ψ
. (4)
One needs only to calculate ∆ and 2λ1 to apply the DC
method to a 2–qubit system. According to Eq. (3), one
can readily obtain the following results for ρf : ∆ =
2λ1 =
. Then, it is not possible to determine whether
ρf is entangled or not, because ∆ = 2λ1.
We will show that the above problem should be at-
tributed to the decrease of entanglement by local opera-
tions in the DC method. Let us explain Dür and Cirac’s
explicit expressions to clarify this point. The local oper-
ations in the DC method are sequence of the following
three steps. First, we perform the following probabilistic
unitary operator on an arbitrary density operator of an
N–qubit system:
L1ρ =
1 , (5)
where W1 =
i=1 σ
x and σ
x = |0〉i〈1|+ |1〉i〈0|. Note
j, j′=0
|Ψ+j 〉〈Ψ
|+ µ+−
|Ψ+j 〉〈Ψ
|Ψ−j 〉〈Ψ
|+ µ−−
|Ψ−j 〉〈Ψ
where µσσ
jj′ s are the matrix elements of ρ for the gener-
alized GHZ states (σ, σ′ = ±). As a result of this opera-
tion, the terms corresponding to |Ψ+j 〉〈Ψ
| and |Ψ−j 〉〈Ψ
are vanishing because W1|Ψ±j 〉 = ±|Ψ
The following probabilistic unitary operators are nec-
essary for the second step:
Llρ =
(l = 2, 3, . . . , N), (7)
where Wl = σ
z ⊗σ(l)z and σ(i)z = |0〉i〈0|− |1〉i〈1|. Equa-
tion (7) is a local operation with respect to the first and
lth qubit. Note that we abbreviate the identity opera-
tors for the other qubits in Wl. In the second step, we
perform
l=2 Ll on the result of the first step. By this
operation, the terms corresponding to |Ψ±j 〉〈Ψ
| (j 6= j′)
are vanishing because Wl|Ψ±j 〉 = (−1)jl |Ψ
j 〉. In this
stage, the resultant state is a diagonal form with respect
to the generalized GHZ states.
Finally, we perform the local random phase–shift, Lr
on the result of the second step:
Lrρ =
2π δ(Φ− 2π)Rφ ρR†φ, (8)
where Rφ =
i=1 R
(i)(φi), R
(i)(φi)|0〉i = eiφi |0〉i,
R(i)(φi)|1〉i = |1〉i, and Φ =
i=1 φi. Note
that Lr |Ψ±0 〉〈Ψ
0 | = |Ψ
0 〉〈Ψ
0 | and Lr |Ψ
j 〉〈Ψ
j | =
(|0j〉〈0j| + |1̄〉〈1̄|) (j 6= 0). After the final step, we
can find that the resultant state is equivalent to Eq. (1).
Now, let us go back to Eq. (4). We only need to per-
form L1 on ρf to transform it into the form of Eq. (1):
L1ρf = 12 |Ψ
0 〉〈Ψ
0 | + 14 |Ψ
1 〉〈Ψ
1 | + 14 |Ψ
1 〉〈Ψ
1 |. Obvi-
ously, the resultant state is separable. It implies that
the entanglement decreases by the local operation L1. In
the subsequent section, we will characterize the class of
the quantum states in a 2–qubit system whose entangled
property is not obtained by the DC method due to its
decrease by the local operations.
IV. LIMITATION OF THE DC METHOD IN
2–QUBIT SYSTEMS
We attempt to reveal the class of the quantum states
whose entangled property is not obtained by the DC
method. In this section, we focus on the case N = 2
because its entanglement structure is well known.
Let us first introduce an important quantity for our
consideration:
F(ρ) = max
〈Ψ+0 |(U ⊗ V ) ρ (U ⊗ V )
†|Ψ+0 〉, (9)
where U and V are unitary operators on the Hilbert
spaces for the first and second qubits, respectively. Equa-
tion (9) is called a fully entangled fraction [7, 10].
We show that the value of a fully entangled fraction
plays an important role in determining whether the DC
method works or not. According to the DC method, the
sufficient condition for an entangled state in a 2–qubit
system is ∆ > 2λ1. Using trρ = 1 and Eq. (3), we readily
obtain the following relation:
∆ > 2λ1 ⇐⇒ 〈Ψ+0 |ρ|Ψ
0 〉 >
or 〈Ψ−0 |ρ|Ψ
0 〉 >
.(10)
The right–hand side of Eq. (10) implies F(ρ) > 1
. Note
that |Ψ−0 〉 = (I(1) ⊗ σ
z )|Ψ+0 〉, where I(1) = |0〉1〈0| +
|1〉1〈1|. Summarizing the above argument, we obtain the
following statements:
∆ > 2λ1 =⇒ F(ρ) >
, (11)
F(ρ) ≤ 1
=⇒ ∆ ≤ 2λ1. (12)
Accordingly, we obtain the following conclusion. Let us
consider the density matrix in a 2–qubit system which
has NPT; it is an entangled state. However, if its fully
entangled fraction is less than or equal to 1
, then it is not
possible to determine whether such a state is entangled
or not by the DC method. Actually, Eq. (4) is just such
an example.
Next, we investigate the density matrix ρ whose fully
entangled fraction is greater than 1
. In general, the con-
dition F(ρ) > 1
does not imply 〈Ψ±0 |ρ|Ψ
0 〉 > 12 . How-
ever, the following statement is always true:
∃Ũ ⊗ Ṽ s.t. |Ψ+0 〉 = Ũ ⊗ Ṽ |ψ̃〉, (13)
where Ũ and Ṽ are unitary operators on the Hilbert
spaces for the first and second qubits, respectively,
and |ψ̃〉 is the maximally entangled state that satisfies
〈ψ̃|ρ|ψ̃〉 = F(ρ). Consequently, using the above local
unitary operator, we obtain
〈Ψ+0 |ρ̃|Ψ
0 〉 >
, (14)
ρ̃ = (Ũ ⊗ Ṽ ) ρ (Ũ ⊗ Ṽ )†. (15)
According to Eqs. (11) and (14), we obtain the following
statement:
F(ρ) > 1
=⇒ ∃Ũ ⊗ Ṽ s.t. ∆̃ ≡ |λ̃+0 − λ̃
0 | > 2λ̃1,
where λ̃±0 = 〈Ψ
0 |ρ̃|Ψ
0 〉 and 2λ̃1 = 〈Ψ
1 |ρ̃|Ψ
1 〉 +
〈Ψ−1 |ρ̃|Ψ
1 〉. The local unitary transformed state ρ̃ is en-
tangled if ∆̃ > 2λ̃1; one can obtain the entangled prop-
erty of ρ̃ by the DC method. On the other hand, the
original density matrix ρ is related to ρ̃ through the local
unitary operator Ũ⊗ Ṽ from Eq. (15); ρ̃ is equivalent to ρ
with respect to entanglement. Therefore, one can obtain
the entangled property of a density matrix whose fully
entangled fraction is greater than 1
by the DC method
with a suitable local unitary operator. Let us show an
example for such a case. We consider a Bell–diagonal
state. Such a state is defined by as follows:
ρBD =
|+ µ−
, (17)
where µ±j ≥ 0 and
j=0(µ
j + µ
j ) = 1. Note that our
example in Ref. [24] was a special case of Eq. (17). We
can show that ρBD has NPT if and only if
|µ+0 − µ
0 | > µ
1 + µ
1 or |µ
1 − µ
1 | > µ
0 + µ
0 . (18)
According to tr ρBD = 1 and Eq. (18), if one of µ
j s (σ =
±) is at least greater than 1
, then ρBD has NPT, and
vice versa. In addition, we easily obtain the following
relation:
F(ρBD) = max
σ=±, j=0, 1
µσj . (19)
Then, if ρBD is entangled, F(ρBD) is greater than 12 . In
this case, we can obtain the information of the entangle-
ment for the Bell–diagonal state in a 2–qubit system by
the DC method with a suitable local operator. Note that
one only needs to use Lr to transform ρBD into ρN .
Finally, we consider whether, through the DC method
with appropriate local unitary operators, we can obtain
the entangled property of the quantum state whose fully
entangled fraction is less than or equal to 1
. It should
be noted that the converse statement of Eq. (16) can be
easily shown. Therefore, we conclude that one never ob-
tains the entangled property for a density matrix whose
fully entangled fraction is less than or equal to 1
by the
DC method, even if one uses local unitary operators.
In summary, we have completely classified the states
in 2–qubit systems whose entangled property is not ob-
tained by the DC method, or by the DC method with lo-
cal unitary operators. The limitation of the method is de-
termined by the value of the fully entangled fraction. If it
is greater than 1
, we can always obtain the desired infor-
mation on entanglement by the DC method with suitable
local unitary operators. Otherwise, we never obtain it.
The impracticality of the DC method is attributed to the
decrease of entanglement by the local operations. Note
that the Bell–diagonal state is entangled if F(ρBD) > 12 .
Moreover, we can easily find L1 ρBD = L2 ρBD = ρBD
and LrρBD = ρN . Accordingly, in 2–qubit systems, the
crucial decrease of entanglement occurs in L1 and L2.
V. SUMMARY
We have shown that one cannot always obtain an en-
tangled property by the DC method, even in 2–qubit
systems. The most important quantity in our discussion
is a fully entangled fraction. One can never determine
whether a quantum state is entangled or not through the
DC method, if the fully entangled fraction is less than
or equal to 1
. On the other hand, one can make such a
determination by the DC method with suitable local uni-
tary operators, if the fully entangled fraction is greater
than 1
The impracticality of the DC method is attributed to
the decrease of entanglement by the local operations. Ac-
tually, from Eqs. (4) and (5), we have easily shown that
L1ρf is separable, even if ρf is entangled. The Bell–
diagonal state (17) is invariant under L1 and L2; we only
need to use Lr for transforming it into the form of Eq. (1).
In addition, the Bell–diagonal state which is entangled
has a fully entangled fraction greater than 1
. Therefore,
the crucial decrease of entanglement for examining it by
the DC method occurs in L1 and L2 in 2–qubit systems.
Finally, we would like to comment on the case of mul-
tiqubit systems. The DC method has been proposed as
a systematic estimation of multiparticle entanglement.
Therefore, it is necessary to study the limitation of the
method in N–qubit systems when N > 2. However, the
situation will be more complicated in this case. Nev-
ertheless, the results in this letter can hint at a solu-
tion. Namely, we will consider the following question: (i)
Is it possible to obtain the entangled property of Bell–
diagonal states in N–qubit systems,
ρBD =
2N−1−1
j 〉〈Ψ
j |+ µ
j 〉〈Ψ
, (20)
by the DC method with suitable local unitary operators?
(ii) How are fragile quantum states with respect to en-
tanglement, for example ρf , under the local operations
characterized in N–qubit systems? We think the above
questions are related to the decrease of quantum entan-
glement under local operations and decoherence. In ad-
dition, our examination of the above questions will lead
to the understanding of the structure of quantum states
in N–qubit systems.
Acknowledgments
The authors acknowledge H. Nakazato for valu-
able discussions. This research is partially supported
by a Grant–in–Aid for Priority Area B (No. 763),
MEXT, by the 21st Century COE Program (Physics of
Self-Organization Systems) at Waseda University from
MEXT, and by a Waseda University Grant for Special
Research Projects (Nos. 2004B–872 and 2007A–044).
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