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Detecting some three-qubit MUB diagonal entangled states via nonlinear optimal entanglement witnesses

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Abstract

The three qubits mutually unbiased bases (MUB) diagonal density matrices with maximally entanglement in Greenberger-Horne-Zeilinger (GHZ) basis are studied. These are a natural generalization of Bell-state diagonal density matrices. The linearity of positive partial transpose (PPT) conditions allows one to specify completely PPT states or feasible region (FR) which form a polygon, where the projection of the feasible region to some two dimensional planes has lead to better visualization. To reveal the PPT entangled regions of these density matrices, we manipulate some appropriate optimal non-decomposable linear entanglement witnesses (EWs) as the envelope of family of linear optimal non-decomposable EWs. These nonlinear EWs are nonlinear functional of MUB diagonal states, so that they are nonnegative valued over all separable, but they are negative valued over some PPT entangled MUB diagonal states. Even though, these nonlinear EWs can not separate completely, the PPT entanglement region from separable one, but however in special cases they lead to necessary and sufficient condition for separability. To support the evidence, we study three categories for special choices of parameters in density matrices, and using the nonlinear EWs we can distinguish the region of PPT entangled states and separable states, completely. At the end some numerical simulations are provided to show the practical applicability of these nonlinear EWs in detecting some PPT entangled MUB diagonal states.
arXiv:0801.3100v1 [quant-ph] 20 Jan 2008
Detecting three-qubit bound MUB diagonal
entangled states via Nonlinear optimal
entanglement witnesses
M. A. Jafarizadeh
a,b,c
, M. Mahdian
a
, A. Heshmati
a
, K. Aghayari
a §
a
Department of Theoretical Physics and Astrophysics, University of Tabriz, Tabriz 51664, Iran.
b
Institute for Studies in Theoretical Physics and Mathematics, Teh ran 19395-1795, Iran.
c
Research Institute for Fundamental Sciences, Tabriz 51664, Iran.
February 2, 2008
E-mail:jafarizadeh@tabrizu.ac.ir
E-mail:Mahdian@tabrizu.ac.ir
E-mail:Heshmati@tabrizu.ac.ir
§
E-mail:Aghayari@tabrizu.ac.ir
1
nonlinear Entanglement Witness using MUB 2
Abstract
One of the important approaches to detect quantum entanglement is using linear en-
tanglement witnesses (EW s). In this paper, by determining the envelope of the boundary
hyper-planes defined by a family of linear EW s, a set of powerful nonlinear optimal EW s
is manipulated. These EW s enable us to detect some three qubits bound MUB (mutu-
ally unbiased bases) diagonal entangled states, i.e., the PPT (positive partial transpose)
entangled states. Also, in some particular cases, the introduced nonlinear optimal EW s
are powerful enough to separate the bound entangled regions from the separable ones.
Finally, we present numerical examples to demonstrate the practical accessibility of this
approach .
Keywords :nonlinear optimal entanglement witnesses, mutually unbiased
bases, MUB diagonal states
nonlinear Entanglement Witness using MUB 3
1 Introduction
In the recent years it became clear that quantum entanglement [1] is one of the most important
resources in the rapidly expanding field of quantum information processing, with remarkable
applications such as quantum parallelism [2], quantum cryptography [3], quantum teleportation
[4, 5], quantum dense coding [6, 7] and reduction of communication complexity [8]. The
above ideas are based on the fact that quantum entanglement, in part icular, the occasionally
occurrence of entangled states produce nonclassical phenomena. Therefore, specifying that a
particular quantum state is entangled or separable is important because if the quantum state
be separable then its statistic properties can be explained entirely by classical statistics.
In this paper, we will deal with three qubit systems with 2
3
-dimensional Hilbert space
H
2
H
2
H
2
, (H
d
denotes the Hilbert space with dimension d). A density matrix ρ on this
Hilbert space, is called fully separable if it can be written as a convex combination of pure
product states as follows
ρ =
X
i
p
i
|α
(1)
i
ihα
(1)
i
| |α
(2)
i
ihα
(2)
i
| |α
(3)
i
ihα
(3)
i
|, (1.1)
where |α
(j)
i
i are arbitrary but normalized vectors lie in H
2
, and p
i
0 satisfy
P
i
p
i
= 1
(hereafter we will refer to fully separable states as separable ones). The first and most widely
used related criterion for distinguishing entangled states from separable ones, is the Positive
Partia l Transpose (PPT ) criterion, introduced by Peres [9]. Furthermore, the necessary and
sufficient condition for separability in H
2
H
2
and H
2
H
3
was shown by Horodecki in
Ref. [10], which was based on a previous work by Woronowicz [11]. Partial transpose means
transposition with respect to one of the subsystems. For a quantum state ρ
AB
with matrix
entries ρ
mn
ij
= hij|ρ
AB
|mni, the partial transposition with respect to the subsystem B, denoted
by ρ
T
B
AB
, is defined by
(ρ
mn
ij
)
T
B
= ρ
mj
in
.
However, as it was shown in Ref. [12], in higher dimensions, there are PPT states that are
nonlinear Entanglement Witness using MUB 4
nonetheless entangled. These states are called PPT entangled states (PPTES ) or bound
entangled states because they possess the peculiar property that no entanglement can be
distilled fro m them by local operations [13]. Another approach to distinguish separable states
from entangled ones involves the so called entanglement witness (EW ) [14]. An EW for a
given entangled state ρ is an observable W whose expectation value over a ll separable states
is nonnegative, but strictly negative on ρ. There is a corresp ondence between EWs and linear
positive (but not completely positive) maps via Jamiolkowski isomorphism [15]. As an example
the partial transposition is a positive map (PM).
In this work, we consider those density matrices which are written a s a linear combina-
tion of maximally commuting observables taken fr om the set of tensor products A
i
B
j
C
k
, where A, B, C {I
2
, σ
x
, σ
y
, σ
z
} and i, j, k {0, 1} (σ
x
, σ
y
and σ
z
are usual Pauli
matrices). We will see later on that common eigenvectros of these observables form mu-
tually unbiased bases (MUB)[16] and so we will refer to a set of such observables as set
of MUB observables, for instance by using the notation σ
i
σ
j
σ
k
σ
i
σ
j
σ
k
, the set
{III, σ
z
σ
z
I, σ
z
Iσ
z
, Iσ
z
σ
z
, σ
x
σ
x
σ
x
, σ
x
σ
y
σ
y
, σ
y
σ
x
σ
y
, σ
y
σ
y
σ
x
} is a set of MUB observables. In
fact, we consider tripartite MUB diagonal density matrices which are written in terms of
MUB observables in a diagonal f orm. Then, we impose the PPT conditions (positivity of
partial tra nsposition with respect to all subsystems) to these density matrices and refer to the
region of those density matrices which satisfy all of the obtained PPT conditions as “feasible
region” (see Fig.1 and Fig.2 for example). In this way, we see that partial transposition plays
an importa nt role because in this type of density matrices, conditions obtained from positivity
of partial transpositions ar e linear and feasible regions are completely contained in polygons;
this allows us to investigate t he separability or entanglement of the density matrices. In order
to distinguish PPT entangled states (PPTES) f rom separable ones we construct some linear
and nonlinear EW s. Namely, we consider density matrices that their common eigenvectors
are maximally entangled states (GHZ -states) and construct a n EW tha t detects such density
nonlinear Entanglement Witness using MUB 5
matrices. Finally, we consider three categories relevant to some special choices of the param-
eters of density matrices, and by using the linear EWs we distinguish the region of PPTES
and separable states completely. In other words, for density matrices contained in one of these
three categories, we show that if the introduced linear EWs can not detect their entanglement,
then they are necessarily separable.
We have also provided some numerical evidence suggesting that PPTES of three qubits
can be detected by using the nonlinear EWs.
The paper is organized as follows: In section 2, we introduce the MUB- ( zzz)
G
diagonal
density matr ices and consider the corresponding PPT conditions and feasible region. Section
3 is devoted to definition of an EW and construction of optimal linear EW s. In section 4,
we obtain an envelope of family of linear EWs and construct some nonlinear EW s. Section
5 is devoted to classification and detection of PPTES for MUB diagonal density matrices in
three categories. In section 6, we discuss some numerical analysis for evaluating the feasible
region and the region of PPTES. The paper is ended with a brief conclusion together with two
appendices.
2 MUB diago nal density matrices
In this section we introduce the so called MUB diagonal density matrices. The basic notions
and definitions of MUB states relevant to our study are given in the Appendix I.
2.1 MUB -(zzz)
G
diagonal density matrices
In this subsection we introduce the MUB-(zzz)
G
diagonal density matrices which are consid-
ered through the paper. A MUB-(zzz)
G
diagonal density matrix for three qubits is defined as
ρ =
8
X
i=1
p
i
|ψ
i
ihψ
i
| , 0 p
i
1 ,
8
X
i=1
p
i
= 1, (2.2)
nonlinear Entanglement Witness using MUB 6
where three qubit GHZ states |ψ
i
i for i = 1, 2, ..., 8 are given by
|ψ
1
i =
1
2
[|000i + |111i], |ψ
2
i =
1
2
[|000i |111i],
|ψ
3
i =
1
2
[|001i + |110i], |ψ
4
i =
1
2
[|001i |110i],
|ψ
5
i =
1
2
[|010i + |101i], |ψ
6
i =
1
2
[|010i |101i],
|ψ
7
i =
1
2
[|011i + |100i], |ψ
8
i =
1
2
[|011i |100i]. (2.3)
Then, by using the MUB states in line 6 of Table I given in Appendix A, the density matrix
ρ can be rewritten as follows
ρ =
1
8
[III + r
1
σ
z
σ
z
I + r
2
σ
z
Iσ
z
+ r
3
Iσ
z
σ
z
+ r
4
σ
x
σ
x
σ
x
+ r
5
σ
x
σ
y
σ
y
+ r
6
σ
y
σ
x
σ
y
+ r
7
σ
y
σ
y
σ
x
], (2.4)
where
r
1
= +p
1
+ p
2
+ p
3
+ p
4
p
5
p
6
p
7
p
8
,
r
2
= +p
1
+ p
2
p
3
p
4
+ p
5
+ p
6
p
7
p
8
,
r
3
= +p
1
+ p
2
p
3
p
4
p
5
p
6
+ p
7
+ p
8
,
r
4
= +p
1
p
2
+ p
3
p
4
+ p
5
p
6
+ p
7
p
8
,
r
5
= p
1
+ p
2
+ p
3
p
4
+ p
5
p
6
p
7
+ p
8
,
r
6
= p
1
+ p
2
+ p
3
p
4
p
5
+ p
6
+ p
7
p
8
,
r
7
= p
1
+ p
2
p
3
+ p
4
+ p
5
p
6
+ p
7
p
8
. (2.5)
It should be noticed that the MUB states o f any line of Table I except for the states in the
first three lines which a re associated with separable states, can define a MUB diagonal density
matrix, similarly. In the next subsection we impose the PPT conditions to the density matrix
(2.4) in order to obtain the corresponding feasible region (region of those MUB diagonal density
matrices which satisfy all of the PPT conditions).
nonlinear Entanglement Witness using MUB 7
2.2 Feasible regions
Here we are concerned with MUB diagonal density matrices of type (2.4) and by imposing
the conditions obtained from positivity of partial transpositions with resp ect to each qubit,
we obtain the so called feasible region. For these particular density matrices, the positivity of
partial t ranspo sitions gives linear constraints on the parameters p
i
for i = 1, 2, ..., 8. In order to
obtain the feasible region for density matrix (2.4), first we group the conditions obtained from
positivity of partia l transpositions with respect to each subsystem (each qubit) in six partitions
(p
3
, p
4
, p
5
, p
6
), (p
1
, p
2
, p
7
, p
8
), (p
1
, p
2
, p
5
, p
6
), (p
3
, p
4
, p
7
, p
8
), (p
1
, p
2
, p
3
, p
4
) and (p
5
, p
6
, p
7
, p
8
) as
follows:
The positivity of partial transposition with respect to the first qubit gives the following
constraints:
(p
3
, p
4
, p
5
, p
6
)
p
3
+ p
4
+ p
5
p
6
0
p
3
+ p
4
p
5
+ p
6
0
p
3
p
4
+ p
5
+ p
6
0
p
3
+ p
4
+ p
5
+ p
6
0
(2.6)
(p
1
, p
2
, p
7
, p
8
)
p
1
+ p
2
+ p
7
p
8
0
p
1
+ p
2
p
7
+ p
8
0
p
1
p
2
+ p
7
+ p
8
0
p
1
+ p
2
+ p
7
+ p
8
0
(2.7)
The positivity of partial transp osition with respect to the second qubit g ives:
(p
1
, p
2
, p
5
, p
6
)
p
1
+ p
2
+ p
5
p
6
0
p
1
+ p
2
p
5
+ p
6
0
p
1
p
2
+ p
5
+ p
6
0
p
1
+ p
2
+ p
5
+ p
6
0
(2.8)
nonlinear Entanglement Witness using MUB 8
(p
3
, p
4
, p
7
, p
8
)
p
3
+ p
4
+ p
7
p
8
0
p
3
+ p
4
p
7
+ p
8
0
p
3
p
4
+ p
7
+ p
8
0
p
3
+ p
4
+ p
7
+ p
8
0
(2.9)
The positivity of partial transp osition with respect to the third qubit g ives:
(p
1
, p
2
, p
3
, p
4
)
p
1
+ p
2
+ p
3
p
4
0
p
1
+ p
2
p
3
+ p
4
0
p
1
p
2
+ p
3
+ p
4
0
p
1
+ p
2
+ p
3
+ p
4
0
(2.10)
(p
5
, p
6
, p
7
, p
8
)
p
5
+ p
6
+ p
7
p
8
0
p
5
+ p
6
p
7
+ p
8
0
p
5
p
6
+ p
7
+ p
8
0
p
5
+ p
6
+ p
7
+ p
8
0
(2.11)
The region of tho se density matrices of type (2.4) which satisfy the above 24 constraints, is the
feasible region. In order to specify the new perspective from this feasible region, we consider
the parameters p
i
in four pairs (p
1
, p
2
), (p
3
, p
4
), (p
5
, p
6
) and (p
7
, p
8
).
Now if we choose one of the pairs, say (p
1
, p
2
), then we can sp ecify the projection of the
feasible region to (p
1
, p
2
) plane with the following three inequalities ( the last inequalities of
(2.7), (2.8) a nd (2.10), respectivelyPPT conditions)
p
1
p
2
+ p
7
+ p
8
p
1
p
2
+ p
5
+ p
6
p
1
p
2
+ p
3
+ p
4
By adding right hand side and left hand side of the above inequalities and using the equality
P
8
i=1
p
i
= 1 , we get the following inequality
4p
1
2p
2
1. (2.12)
nonlinear Entanglement Witness using MUB 9
Similarly, one can obtain the inequality
4p
2
2p
1
1, (2.13)
by exchanging 1 and 2 in the above steps. For an illustration see Fig. 1. It should be noticed
that if we chose any other pair from (p
3
, p
4
), (p
5
, p
6
) and (p
7
, p
8
) instead of the pair (p
1
, p
2
),
we would obtain the similar inequalities as in (2.12) and (2.13) for each pair.
According to Fig. 1, since the vertex points (
1
2
,
1
2
), (
1
4
, 0), (0 ,
1
4
) and (0,0) satisfy the criterion
(2.12), all of the points inside the shape will fulfill the PPT conditions (this is due to the f act
that the feasible region is a convex region).
We can find another new projection of the feasible region in the (p
1
, p
3
) plane, concerning
the following inequalities
p
1
p
2
+ p
5
+ p
6
p
3
p
4
+ p
7
+ p
8
p
1
+ p
3
1
2
. (2.14)
This region is illustrated in Fig. 2; therefore we have presented a projection of the spatial
shape in a two-dimensional space.
2.3 A special case of feasible region
Here, we discuss a special case of feasible region which will be appeared in subsection 5.3 as
a region of bound entangled MUB diagonal density matrices (PPT entangled states). To this
aim, we consider the line p
1
+ p
3
=
1
2
of the feasible region (2.14) in the (p
1
, p
3
) plane (see Fig.
2).
First we take the feasible region for the (p
3
, p
4
) plane (see Fig.3). If, we consider the
following parametric line equation
p
3
= αp
4
+
1
4
, (2.15)
then (according to equations similar to (2.12) and (2.13) for the pair (p
3
, p
4
)), we obtain
4p
4
2p
3
= 1 p
4
=
p
3
2
+
1
4
(2.16)
nonlinear Entanglement Witness using MUB 10
By substituting (2.16) in (2.15) and using the fact that 0 p
3
1/2 (see Eq. (2.14) ), one can
obtain
p
3
=
αp
3
2
+
α + 1
4
0 p
3
=
α + 1
4 2α
1
2
,
so we get 1 α
1
2
. According to the boundary condition p
1
+ p
3
=
1
2
and (2.15) we obtain:
p
1
= αp
4
+
1
4
. (2.17)
Now by considering the PPT conditions
p
3
+ p
4
+ p
5
+ p
6
0
p
3
+ p
4
+ p
7
+ p
8
0
, (2.18)
from equations (2.6) and (2.9), and adding the sides of the them, we obta in
(1 2α)p
4
p
2
0. (2.19)
Also from the PPT conditions
p
1
+ p
2
+ p
5
+ p
6
0
p
1
+ p
2
+ p
7
+ p
8
0
, (2.20)
given in (2.7) and (2.8), we get
(1 2α)p
4
+ p
2
0. (2.21)
Then, from (2.19) and (2.21) we conclude that p
2
= (1 2α)p
4
. Then, by using (2.15) and
(2.17) one can easily conclude the following equations
p
3
p
4
= (α 1)p
4
+
1
4
p
1
p
2
= αp
4
+
1
4
(1 2α)p
4
= (α 1)p
4
+
1
4
,
which indicate that p
3
p
4
= p
1
p
2
. On the other hand, from the inequalities (2.19) and
(2.21), one can deduce that the left hand sides of the inequalities (2.18) and (2.20) must be
equal to zero. Therefore, for PPT density matrices with positive p
i
’s, we obtain
p
5
+ p
6
= p
7
+ p
8
= p
3
p
4
= (α 1)p
4
+
1
4
0 p
4
1
4(1 α)
nonlinear Entanglement Witness using MUB 11
furthermore we obtain p
3
p
4
. Clearly, the PPT conditions (2.10) and (2.11) a r e satisfied
by using the relations p
1
+ p
4
= p
2
+ p
3
and p
5
+ p
6
= p
7
+ p
8
. The bo und entanglement or
separability of density matrices belonging to this special case of feasible region will be discussed
in subsection 5.3 (as third category). For an illustration see Fig.3.
2.4 MUB-(zzz)
G
diagonal density matrices for which PPT condi-
tions are necessary and sufficient for separability
In this section we consider the family of MUB-(zzz)
G
diagonal density matrices where the
PPT criterions are necessary and sufficient for their separability.
2.4.1 Case (1)
In this case, we will put one of the pairs (p
1
, p
2
), (p
3
, p
4
), (p
5
, p
6
) and (p
7
, p
8
) equal to (0, 0),
then we will see that the PPT conditions given in (2.6)-(2.11) are necessary and sufficient for
separability of MUB-(zzz)
G
diagonal density matrices. For example, if we choose
p
1
= p
2
= 0,
then, by using the PPT conditions, we obtain
p
3
= p
4
, p
5
= p
6
, p
7
= p
8
.
Then, the density matrices satisfying these conditions can be written as
ρ =
1
8
[III + r
1
σ
z
σ
z
I + r
2
σ
z
Iσ
z
+ r
3
Iσ
z
σ
z
]. (2.22)
The density matrix ρ in (2.22) is separable, since by using (2.5) we can rewrite ρ as
ρ =
1
4
{p
3
(III + σ
z
σ
z
I σ
z
Iσ
z
Iσ
z
σ
z
)
+p
5
(III + σ
z
Iσ
z
σ
z
σ
z
I Iσ
z
σ
z
) + p
7
(III + Iσ
z
σ
z
I σ
z
Iσ
z
σ
z
σ
z
I)} =
p
3
(|ψ
3
ihψ
3
| + |ψ
4
ihψ
4
|) + p
5
(|ψ
5
ihψ
5
| + |ψ
6
ihψ
6
|) + p
7
(|ψ
7
ihψ
7
| + |ψ
8
ihψ
8
|),
which is clearly a separable state, since it is a convex combination of projection operators.
nonlinear Entanglement Witness using MUB 12
2.4.2 Case (2)
In this case, we choose p
i
’s in each pair except for one of them to be equal, then we show that
the PPT conditions (2.6)-(2.11) are necessary and sufficient for separability of MUB-(zzz)
G
diagonal density matrices. For example, we consider
p
1
6= p
2
, p
3
= p
4
, p
5
= p
6
, p
7
= p
8
.
Then, we can write the density matrix (2.2) as follows
ρ = (
p
1
+ p
2
2
)(|ψ
1
ihψ
1
| + |ψ
2
ihψ
2
|) + (
p
1
p
2
2
)(|ψ
1
ihψ
1
| |ψ
2
ihψ
2
|) + p
3
(|ψ
3
ihψ
3
| + |ψ
4
ihψ
4
|)+
p
5
(|ψ
5
ihψ
5
| + |ψ
6
ihψ
6
|) + p
7
(|ψ
7
ihψ
7
| + |ψ
8
ihψ
8
|). (2.23)
We assume that p
3
< p
5
< p
7
and p
2
< p
1
(the other cases give the same results as those which
is obtained by this assumption in the following).
By substituting p
3
= p
4
in PPT conditions ( 2.10), we obtain p
1
p
2
+ 2 p
3
, so that we can
write p
1
= p
2
+ 2p
3
2ǫ
1
(p
1
> p
2
0 ǫ
1
p
3
). Also, from the assumptions p
3
< p
5
and
p
3
< p
7
, one can write p
5
= p
3
+ ǫ
5
and p
7
= p
3
+ ǫ
7
, resp ectively. By substituting t hese values
of p
i
’s in the density matrix (2.23) and using the resolution of identity
P
8
i=1
|ψ
i
ihψ
i
| = III,
one can write
ρ = ǫ
1
III + (p
2
ǫ
1
)(|ψ
1
ihψ
1
| + |ψ
2
ihψ
2
|) + (p
3
ǫ
1
)(III + |ψ
1
ihψ
1
| |ψ
2
ihψ
2
|)
+ ǫ
5
(|ψ
5
ihψ
5
| + |ψ
6
ihψ
6
|) + ǫ
7
(|ψ
7
ihψ
7
| + |ψ
8
ihψ
8
|). (2.24)
Then, f r om the fact that (III ±|ψ
1
ihψ
1
|| ψ
2
ihψ
2
|) are separable states, one can see that for
ǫ
1
< p
2
in (2.24), the density matrix ρ is separable (since it is written as a convex combination
of product states). For ǫ
1
> p
2
, we can write ρ as follows
ρ = p
2
III + (ǫ
1
p
2
)(III |ψ
1
ihψ
1
| |ψ
2
ihψ
2
|) + (p
3
ǫ
1
)(III + |ψ
1
ihψ
1
| |ψ
2
ihψ
2
|)
+ǫ
5
(|ψ
5
ihψ
5
| + |ψ
6
ihψ
6
|) + ǫ
7
(|ψ
7
ihψ
7
| + |ψ
8
ihψ
8
|),
which is again a separable state. So in t he second case, the PPT conditions are necessary and
sufficient for separability, too.
nonlinear Entanglement Witness using MUB 13
3 Entanglement witnes ses
An entanglement witness acting on the Hilbert space H = H
2
H
2
H
2
is a Hermitian
operator W = W
, that satisfies T r(Wρ
s
) 0 for any separable state ρ
s
in B (H) (Hilbert
space of bounded operators), and has at least one negative eigenvalue. If a density matrix ρ
satisfies T r(Wρ) < 0, then ρ is an entangled state and we say that W detects entanglement
of the density matrix ρ. Note that in the aforementioned definition of EW s, we are not worry
about the kind of entanglement of the quantum state and we are rather looking for EW s which
possess nonnegative expectation values over all separable states despite of the fact that they
possess some negative eigenvalues. The existence of an EW for any entangled state is a direct
consequence of Hahn-Banach theorem [20 ] and the f act that the subspace of separable density
operators is convex and closed [21]. Geometrically, EW s can be viewed as hyper planes which
separate some entangled states from the set of separable states and hyper plane indicated as
a line corresponds to the state with T r[W ρ] = 0.
Based on the notion of partial transpose, the EW s are classified into two classes: decomp os-
able EW s (d-EW ) and non-decomposable EW s (nd-EW ). An EW W is called decomposable
if there exist positive operators P, Q
K
so that
W = P + Q
T
A
1
+ Q
T
B
2
+ Q
T
c
3
, (3.25)
where T
K
, K = A, B, C denotes the partial transposition with respect to subsystems A, B
and C, respectively. W is called non-decomposable if it can not be written in this form [2 2].
Clearly a d-EW can not detect bound entangled states (entangled states with positive partial
transpose (PPT) with respect to all subsystems) whereas there are some bound entangled
states which can be detected by a nd-EW.
Usually one is interested in finding optimal EWs W which detect entangled states in an
optimal way. An EW W is said to be optimal, if f or all positive operators P and ε > 0, the
nonlinear Entanglement Witness using MUB 14
new Hermitian operator
W
= (1 + ε)W εP (3.26)
is not anymore an EW [23]. Suppo se that there is a positive operator P and ǫ 0 such that
W
= (1 + ε)W εP is yet an EW (T r(W
ρ
s
) 0 for all separable states ρ
s
). This means
that if T r(W ρ
s
) = 0, then T r(P ρ
s
) = 0, for all separable states ρ
s
which indicates that, the
operator P is necessarily orthogonal to the kernel of W denoted by Ker(W ). By using the
fact that every separable state is convex combination of pure product states, one can take ρ
s
as a pure product state |ψihψ|. Also, one can assume that the positive operato r P is a pure
projection operator, since an arbitrary positive operator can be written as convex combination
of pure projection operators with positive coefficients.
3.1 EWs detecting bound M UB diagonal density matrices
By employing tensor products of pauli operators relevant to MUB-(zzz)
G
state of Table I of
the Appendix A, we introduce the following linear t hree qubit EW [24]
W = A
0
III + A
1
Iσ
z
σ
z
+ A
2
(σ
x
σ
x
σ
x
+ σ
x
σ
y
σ
y
) + A
3
(σ
y
σ
x
σ
y
+ σ
y
σ
y
σ
x
), (3.27)
where A
0
, A
2
, A
3
0 and A
1
can be negative or positive. Now evaluating the trace of EW
(3.27) over a pure product state,
ρ
s
= |αihα| |βih β| |γihγ|,
we get
T r[W ρ
s
] = A
0
+ A
1
b
3
c
3
+ A
2
(a
1
b
1
c
1
+ a
1
b
2
c
2
) + A
3
(a
2
b
1
c
2
+ a
2
b
2
c
1
),
where a
i
b
j
c
k
:= T r[σ
i
σ
j
σ
k
ρ
s
] for i, j, k = 0, 1, 2, 3 with a
0
= b
0
= c
0
= 1. We parameterize
points on the unit sphere S
2
using traditional spherical co ordinates, so that θ and ϕ stand for
the angles of colatitude and longitude, respectively (θ [0, π], ϕ [0, 2π]). Thus, the points
nonlinear Entanglement Witness using MUB 15
a = (a
1
, a
2
, a
3
), b = (b
1
, b
2
, b
3
) and c = (c
1
, c
2
, c
3
), can be uniquely represented a s the unit
vectors with the following coordinates
a
1
= sin θ
1
cos ϕ
1
, a
2
= sin θ
1
sin ϕ
1
, a
3
= cos θ
1
b
1
= sin θ
2
cos ϕ
2
, b
2
= sin θ
2
sin ϕ
2
, b
3
= cos θ
2
c
1
= sin θ
3
cos ϕ
3
, c
2
= sin θ
3
sin ϕ
3
, c
3
= cos θ
3
,
so that, we obtain
T r[W ρ
s
] = A
0
+A
1
cos θ
3
cos θ
2
+sin θ
1
sin θ
2
sin θ
3
(A
2
cos ϕ
1
cos (ϕ
2
ϕ
3
)+A
3
sin ϕ
1
sin (ϕ
2
+ ϕ
3
)).
By appropriate choice of the angles, one can minimize t he above expression. In fact, in the
Appendix B, it has been proved that by taking A
0
=
p
A
2
2
+ A
2
3
and A
1
=
p
A
2
2
+ A
2
3
, the
minimum value of T r[W ρ
s
] is attained to zero, i.e., we obtain
min(T r[W ρ
s
]) = 0.
Consequently, EW (3.27) takes the following form
W
(ψ)
=
q
A
2
2
+ A
2
3
(III Iσ
z
σ
z
+ cos ψ(σ
x
σ
x
σ
x
+ σ
x
σ
y
σ
y
) + sin ψ(σ
y
σ
x
σ
y
+ σ
y
σ
y
σ
x
)), (3.28)
where
cos ψ =
A
2
p
A
2
2
+ A
2
3
, sin ψ =
A
3
p
A
2
2
+ A
2
3
.
In the following, we discuss the optimality of the obta ined linear EW W
ψ
.
3.2 Optimality of the linear EW W
(ψ)
According to the arguments about optimal EWs given in section 3, in order to prove the
optimality of the EW W
(ψ)
given in (3.28), it suffices to show that there exists no positive
operator P such that W
:= (1 + ε)W
(ψ)
εP be an EW, namely it must be proved that for
any pure product state |νi so that T r(W
(ψ)
|νihν|) = 0, there exists no positive operator P
nonlinear Entanglement Witness using MUB 16
with the constraint T r(P |νih ν|) = 0. By considering a general three qubit pure product state
as
|νi =
1
2
cos (
θ
1
2
)
e
1
sin (
θ
1
2
)
cos (
θ
2
2
)
e
2
sin (
θ
2
2
)
cos (
θ
3
2
)
e
3
sin (
θ
3
2
)
, (3.29)
one can evaluate
T r[W
(ψ)
ρ
s
] = 1cos θ
2
cos θ
3
sin θ
1
sin θ
2
sin θ
3
(cos ψ cos ϕ
1
cos (ϕ
2
ϕ
3
)+sin ψ sin ϕ
1
sin (ϕ
2
+ ϕ
3
)),
where ρ
s
= | νihν|. Now, it is easily seen that by choosing the angles θ and ϕ as follows
(1) :
cos (ϕ
2
ϕ
3
) = 1
sin (ϕ
2
+ ϕ
3
) = 1
ϕ
2
= ϕ
3
=
π
4
, ϕ
1
= ψ, θ
1
=
π
2
, θ
2
= θ
3
,
(2) :
cos (ϕ
2
ϕ
3
) = 1
sin (ϕ
2
+ ϕ
3
) = 1
ϕ
2
= ϕ
3
=
π
4
, ϕ
1
= ψ, θ
1
=
π
2
, θ
2
= θ
3
,
(3) :
cos (ϕ
2
ϕ
3
) = 1
sin (ϕ
2
+ ϕ
3
) = 1
ϕ
2
=
π
4
, ϕ
3
=
3π
4
, ϕ
1
= ψ π, θ
1
=
π
2
, θ
2
= θ
3
,
(4) :
cos(ϕ
2
ϕ
3
) = 1
sin (ϕ
2
+ ϕ
3
) = 1
ϕ
2
=
3π
4
ϕ
3
=
π
4
, ϕ
1
= π ψ, θ
1
=
π
2
, θ
2
= θ
3
, (3.30)
we obtain T r[W
(ψ)
ρ
s
] = 0. Now, in order to prove that W
(ψ)
is an optimal EW, we proceed
as follows: Let P be a pure projection operator that one can subtract from W
(ψ)
, so that
(1 + ǫ)W
(ψ)
ǫP is an EW for some ǫ > 0. From Eq. (3.28), one can easily see that, any
pure state of the form |Ψi = |αi|z
+
z
+
i + |βi|z
z
i, (where, |αi and |βi are arbitrary states)
belongs to the Ker(W
(ψ)
), i.e., we have T r[W
(ψ)
|ΨihΨ|] = 0. Then, due to the fact that, the
pure projection operator P must be orthogonal to Ker(W
(ψ)
) (and so orthogonal to |ΨihΨ|),
we have P = | ΦihΦ| with
|Φi = |αi|z
+
z
i + |βi|z
z
+
i (3.31)
nonlinear Entanglement Witness using MUB 17
Now, the pure projection operator defined as above, must be orthogonal to pure product states
|ν
(i)
i, i = 1, 2, 3, 4 obtained by substituting the angles given by (3.30) in (3.29), since these
states belong to Ker(W
(ψ)
). But, this is possible only if α
1
, α
2
, β
1
and β
2
satisfy the f ollowing
equations:
hν
1
|Φi = (α
1
+ α
2
e
) + (β
1
+ β
2
e
) = 0,
hν
2
|Φi = (α
1
+ α
2
e
+
) + (β
1
+ β
2
e
+
) = 0,
hν
3
|Φi = (α
1
α
2
e
) + (β
1
β
2
e
) = 0,
hν
4
|Φi = (α
1
α
2
e
+
) + (β
1
β
2
e
+
) = 0.
Above equations imply that for
ψ 6= 0, π
we have
α
1
= α
2
= β
1
= β
2
= 0.
Therefore, there is no positive operator P to subtract from W
(ψ)
.
In general, linear optimal EWs can be written as
W
(ψ)
±i,±(j,k),(l,m)
= III ± O
i
+ cos ψ(O
j
± O
k
) + sin ψ(O
l
± O
m
), (3.32)
where i = 1, 2, 3 while the indices j 6= k 6= l 6= m t ake values between 4, 5, 6, 7.
The observables O
i
for i = 1, 2, 3 and O
j
for j = 4, 5, 6, 7 are defined as
O
1
= Iσ
z
σ
z
, O
2
= σ
z
Iσ
z
, O
3
= σ
z
σ
z
I,
O
4
= σ
x
σ
x
σ
x
, O
5
= σ
x
σ
y
σ
y
, O
6
= σ
y
σ
x
σ
y
, O
7
= σ
y
σ
y
σ
x
.
4 Non-linear optimal EWs
Actually with a given entangled density matrix, one can associate a non-linear EW, simply
by defining a non-linear functional, so that it is nonnegative valued over all separable density
nonlinear Entanglement Witness using MUB 18
matrices, but it is negative valued over the density matrix. In ot her words, we optimize
T r[W
(ψ)
±i,±(j,k),(l,m)
ρ] where, W
(ψ)
±i,±(j,k),(l,m)
are the linear optimal EWs given by (3.32) and ρ is
the MUB-(zzz)
G
diagonal density matrix given by (2.4). Then, one can easily get
T r[W
(ψ)
±i,±(j,k),(l,m)
ρ] = (1 ± r
i
) + (r
j
+ r
k
) cos ψ + (r
l
+ r
m
) sin ψ,
which indicates that, by appropriate choice of the parameter ψ as a functional o f ρ, one can
obtain a non-linear function of the parameters o f ρ which is nonnegative over all separable
states. To this aim, we define
cos θ =
r
j
+ r
k
p
(r
j
+ r
k
)
2
+ (r
l
+ r
m
)
2
,
then T r[W
(ψ)
±i,±(j,k),(l,m)
ρ] can be written as
T r[W
(ψ)
±i,±(j,k),(l,m)
ρ] = 1 ± r
i
+
q
(r
j
+ r
k
)
2
+ (r
l
+ r
m
)
2
cos(ψ θ).
Now, by choosing (ψ θ) = π, we obtain
T r[W
±i,±(j,k),(l,m)
ρ] = 1 ± r
i
q
(r
j
+ r
k
)
2
+ (r
l
+ r
m
)
2
. (4.33)
The above expression is the required non-linear function in terms of the parameters of ρ and it
is definitely nonnegative valued function of separable states, hence it is the non-linear optimal
EW associated with ρ (since it is obtained from optimal linear EWs).
4.1 Non-linear EWs as an envelop of family of linear EWs
As the parameter ψ of linear EW’s W
(ψ)
±i,±(j,k),(l,m)
varies, the envelope of hyper planes defined
by
T r[W
(ψ)
±i,±(j,k),(l,m)
ρ
s
] = 0, (4.34)
namely their intersections, define the boundary of PPT bound ent angled states that can be
detected by the linear EWs. Obviously, the envelope of these curves can be obtained simply
nonlinear Entanglement Witness using MUB 19
by eliminating the parameter ψ from the Eq.(4.34). To this aim, we need to determine cos ψ
and sin ψ by solving above equation together with the equation that can be obta ined by taking
its derivative with respect to ψ equal to zero, i.e., we consider
T r[W
(ψ)
±i,+(4,5),(6,7)
ρ] = (1 ± r
i
) + (r
4
+ r
5
) cos ψ + (r
6
+ r
7
) sin ψ = 0,
d
(T r[W
(ψ)
±i,+(4,5),(6,7)
ρ]) = sin ψ(r
4
+ r
5
) + cos ψ(r
6
+ r
7
) = 0.
By solving the above equations one can obtain
cos ψ =
(r
j
+ r
k
)(1 ± r
i
)
(r
j
+ r
k
)
2
+ (r
l
+ r
m
)
2
,
sin ψ =
(r
l
+ r
m
)(1 ± r
i
)
(r
j
+ r
k
)
2
+ (r
l
+ r
m
)
2
.
Now, using the identity cos
2
ψ + sin
2
ψ = 1 we obtain the required envelope of curves defined
by the following equations
(1 ± r
i
)
2
= (r
j
+ r
k
)
2
+ (r
l
+ r
m
)
2
.
5 Bound entangled MUB diagonal density matrices
In this section, we consider three main categories of bound entangled states according to
equations (B-i) and (B-ii) given in the Appendix B. In these categories, the relations |(r
j
±
r
k
)| (1 ± r
i
), are a lways satisfied since if we consider for example the inequality
|(r
4
r
7
)| > (1 + r
1
),
then we conclude the inequality p
2
+ p
4
< 0 which is clearly impossible.
5.1 First category
The first interesting family of three qubit bound entangled states is introduced for the choices
of the parameters r
i
so that:
1 ± r
1
= r
5
± r
6
, 1 ± r
1
= r
4
r
7
, 1 ± r
2
= r
5
± r
7
,
nonlinear Entanglement Witness using MUB 20
1 ± r
2
= r
4
r
6
, 1 ± r
3
= r
6
± r
7
, 1 ± r
3
= r
4
r
5
, (5.35)
for example, if we consider 1 + r
1
= r
4
r
7
, then we obtain p
2
= p
4
= 0. Then, the PPT
conditions (2.6)-(2.11) lead t o p
1
= p
3
, and triangle inequalities for the cases (p
1
, p
5
, p
6
) and
(p
1
, p
7
, p
8
) are established. This state can be detected by non-linear EW W
+1,(4,7),(5,6)
, since
by using the result (4.33), we have
T r[W
+1,(4,7),(5,6)
ρ] = ( 1 + r
1
)
p
(1 + r
1
)
2
+ (r
5
r
6
)
2
, (5.36)
which indicates that for r
5
6= r
6
, T r[W
+1,(4,7),(5,6)
ρ] < 0.
On the other hand, by imposing the condition (r
5
r
6
) = p
5
p
6
p
7
+ p
8
= 0, the state
ρ will be separable, since we have
p
5
p
6
p
7
+ p
8
= 0 p
5
+ p
8
= p
6
+ p
7
.
So, by using the relations (2.5), one can see that if r
2
> 0 (r
2
= 2(p
6
p
8
) since we have
p
1
= p
3
and p
2
= p
4
= 0 ), then we get p
6
> p
8
and t herefore we have
ρ = (1 r
2
)III + (r
1
+ r
2
)Iσ
z
σ
z
+ r
2
[(III σ
z
σ
z
I)(III + σ
z
Iσ
z
)] + r
4
σ
x
σ
x
σ
x
+ r
5
σ
x
σ
y
σ
y
with
r
4
r
7
= 4p
1
r
1
+ r
2
= 2(p
1
p
7
p
8
)
r
4
r
7
(r
1
+ r
2
) = 4p
1
2p
1
+ 2p
7
+ 2p
8
= 2(p
1
+ p
7
+ p
8
)
1 r
2
= 1 2p
6
+ 2p
8
= 2 (p
1
+ p
7
+ p
8
)
1 r
2
(r
1
+ r
2
) + r
4
r
7
= 4 (p
1
+ p
7
+ p
8
) 1,
which is separable state.
If r
2
< 0 then we will have p
8
> p
6
and
ρ = (1 + r
2
)III + (r
1
r
2
)Iσ
z
σ
z
r
2
[(III σ
z
σ
z
I)(III + σ
z
Iσ
z
)] + r
4
σ
x
σ
x
σ
x
+ r
5
σ
x
σ
y
σ
y
nonlinear Entanglement Witness using MUB 21
with
r
1
r
2
= 2(p
1
p
5
p
6
) 0
r
4
r
7
(r
1
r
2
) = 2(p
1
+ p
5
+ p
6
) 0
1 + r
2
= 2 (p
1
+ p
5
+ p
6
) 0
(1 + r
2
) + r
4
r
7
(r
1
r
2
) = 4(p
1
+ p
5
+ p
6
) 1.
For the above cases we had p
2
= p
4
= 0, so, this category consists two vanishing non-paired
(p
2
and p
4
belong to different pairs) parameters. The other cases in (5.35) can be discussed
similarly.
A special case
As a special case, if we consider ρ in (2.4) with the following parameters
p
4
= p
8
= p
6
= 0,
p
3
= p
5
= p
7
= p,
we get r
1
= r
2
= r
3
and r
5
= r
6
= r
7
. Then, ρ can be written as
ρ =
1
8
[III + r
1
(Iσ
z
σ
z
+ σ
z
Iσ
z
+ σ
z
σ
z
I) + r
4
σ
x
σ
x
σ
x
+ r
5
(σ
x
σ
y
σ
y
+ σ
y
σ
x
σ
y
+ σ
y
σ
y
σ
x
)]. (5 .3 7)
Concerning the following PPT and normalization conditions
p
1
+ p
2
p
3
0
p
1
p
2
+ p
3
0
p
1
+ p
2
+ p
3
0
8
X
i=1
p
i
= 1 p
1
+ p
2
+ 3p
3
= 1
we construct the convex hull of the following boundary planes
p
1
+ p
2
p
3
= 0
nonlinear Entanglement Witness using MUB 22
p
1
+ p
2
+ 3p
3
= 1 4p
3
= 1 p
3
=
1
4
p
1
p
2
+ p
3
= 0
p
1
+ p
2
+ 3p
3
= 1 2p
1
+ 4p
3
= 1
p
1
+ p
2
+ p
3
= 0
p
1
+ p
2
+ 3p
3
= 1 2p
2
+ 4p
3
= 1,
which define a tria ngular bound entangled region in (p
1
, p
2
, p
3
) space (as it is shown in
Fig.4). With the boundary defined by the following lines where the boundaries of triangular
region corresponding to the lines passing through the points (
1
4
, 0,
1
4
), (0,
1
4
,
1
4
) and (
1
2
,
1
2
, 0),
where the states corresponding to the sides defined by the thick line passing through the
points (
1
4
, 0,
1
4
) and (
1
2
,
1
2
, 0) are separable.
5.2 Second category
Another interesting family for bound entangled states is obtained by considering the following
cases:
1 ± r
1
= r
4
+ r
5
, 1 ± r
1
= r
4
r
5
, 1 ± r
1
= r
6
+ r
7
, 1 ± r
1
= r
6
r
7
,
1 ± r
1
= r
4
+ r
6
, 1 ± r
1
= r
4
r
6
, 1 ± r
1
= r
5
+ r
7
, 1 ± r
1
= r
5
r
7
,
1 ± r
2
= r
4
+ r
5
, 1 ± r
2
= r
4
r
5
, 1 ± r
2
= r
6
+ r
7
, 1 ± r
2
= r
6
r
7
,
1 ± r
2
= r
4
+ r
7
, 1 ± r
2
= r
4
r
7
, 1 ± r
2
= r
5
+ r
6
, 1 ± r
2
= r
5
r
6
,
1 ± r
3
= r
4
+ r
6
, 1 ± r
3
= r
4
r
6
, 1 ± r
3
= r
5
+ r
7
, 1 ± r
3
= r
5
r
7
,
1 ± r
3
= r
4
+ r
7
, 1 ± r
3
= r
4
r
7
, 1 ± r
3
= r
5
+ r
6
, 1 ± r
3
= r
5
r
6
. (5.38)
If we choose one case such as 1 + r
1
= r
4
+ r
6
, we obtain p
7
= p
1
+ p
2
+ 2p
4
+ p
8
. Then, by
using PPT conditions we obtain
p
1
+ p
2
p
7
+ p
8
0 p
4
= 0, (5.39)
p
3
p
7
+ p
8
0 p
3
= p
1
+ p
2
, (5.40)
nonlinear Entanglement Witness using MUB 23
and satisfy t he t r ia ngle inequality for (p
3
, p
5
, p
6
) case. So according to (5.39) and (5.40) we get
p
7
= p
3
+ p
8
. (5.41)
Applying the EW W
+1,+(4,6),(5,7)
to the state (5.37), we obtain
T r[W ρ] = (1 + r
1
)
p
(1 + r
1
)
2
+ (r
5
+ r
7
)
2
< 0
the condition (r
5
+ r
7
) = p
1
+ p
2
+ p
5
p
6
= 0 corresponds to separable state.
By using the nor malization condition
8
X
i=1
p
i
= 1 3p
3
+ p
5
+ p
6
+ 2p
8
= 1
and
p
1
+ p
2
+ p
5
p
6
= 0 p
1
+ p
6
= p
2
+ p
5
,
we obtain fo r r
2
> 0
ρ = (1 r
2
)III + (r
1
+ r
2
)Iσ
z
σ
z
+ r
2
[(III + σ
z
Iσ
z
)(III σ
z
σ
z
I)] + r
4
σ
x
σ
x
σ
x
+ r
6
σ
y
σ
x
σ
y
r
4
+ r
6
= 4(p
1
+ p
3
) = 4p
3
r
1
+ r
2
= 4p
8
r
4
+ r
6
(r
1
+ r
2
) = 4p
7
1 r
2
= 4p
7
(1 r
2
) + (r
4
+ r
6
) (r
1
+ r
2
) = 8p
7
while for r
2
< 0, we get
ρ = (1 + r
2
)III + (r
1
r
2
)Iσ
z
σ
z
r
2
[(III + σ
z
σ
z
I)(III σ
z
Iσ
z
)] + r
4
σ
x
σ
x
σ
x
+ r
6
σ
y
σ
x
σ
y
r
1
r
2
= 2(p
3
p
5
p
6
) 0
r
4
+ r
6
(r
1
r
2
) = 2(p
3
+ p
5
+ p
6
) 0
nonlinear Entanglement Witness using MUB 24
1 + r
2
= 2(1 2p
7
) = 2(p
3
+ p
5
+ p
6
) 0
(1 + r
2
) + (r
4
+ r
6
) (r
1
r
2
) = 4(p
3
+ p
5
+ p
6
) 1.
So, in this case according to (B-i), if we choose parameters as p
i
= p
j
+ p
k
, (p
i
and p
j
are
in the same pairs) and p
k
belong to another pairs, then we o bta in second category of bound
entangled states.
The other family ( 5.2) can be considered similarly.
5.3 Third category
The last category is given by the following cases:
1 ± r
1
= r
4
± r
7
, 1 ± r
1
= r
5
r
6
, 1 ± r
2
= r
4
± r
6
,
1 ± r
2
= r
5
r
7
, 1 ± r
3
= r
4
± r
5
, 1 ± r
3
= r
6
r
7
. (5.42)
If 1r
1
= r
4
r
7
then p
1
+p
3
= p
2
+p
4
+p
5
+p
6
+p
7
+p
8
=
1
2
and r
5
r
6
= 2(p
5
p
6
p
7
+p
8
).
The PPT conditions for this case have been previously considered (section 2.4.2). Therefore, if
r
5
r
6
6= 0 this state will be bound entangled and can be detected by W
1,(4,7),(5,6)
; otherwise
it is separable since we have
r
5
r
6
= 0 p
6
+ p
7
= p
5
+ p
8
and so fro m the PPT conditions we get
p
5
= p
7
, p
6
= p
8
after calculation of r
i
’s, we obtain
ρ = 2p
2
(III + σ
z
σ
z
I)(III + Iσ
z
σ
z
) + 2p
4
(III + σ
z
σ
z
I)(III σ
z
Iσ
z
)
+(1 2(p
2
+ p
4
))III + 2p
5
σ
x
σ
x
σ
x
4p
6
σ
x
σ
y
σ
y
.
nonlinear Entanglement Witness using MUB 25
We know that the first and second cases in the density matrix are separable and for other cases
|(1 2(p
2
+ p
4
))| + |2p
5
| + | 4p
6
| 1.
So, if we choose two parameters and add them as p
i
+ p
j
=
1
2
,( p
i
, p
j
are in different pairs)
then, the category consists the bound entangled states.
The other cases (5.2) can be discussed similarly.
6 Numerical analysis of three-qu bit boun d MUB diag-
onal entangled states
This section is devoted to some numerical studies of three-qubit bound MUB diagonal entan-
gled states as follows: The feasible regions in (p
1
, p
2
) and (p
1
, p
3
) planes, defined by equations
(2.12) a nd (2.14), are supported numerically. Using the nonlinear EWs, about 2.7% of bound
MUB-(zzz)
G
diagonal density matrices are detected numerically. The numerical results are
plotted in (p
1
, p
3
, p
5
) , (p
2
, p
4
, p
8
) , (p
6
, p
7
) and (p
1
, p
3
) , (p
2
, p
4
) , (p
5
, p
6
) , (p
7
, p
8
) phase spaces
and the bound density matrix (p
1
= 0.043425, p
2
= 0.15308, p
3
= 0.016132, p
4
= 0.19387, p
5
=
0.059793, p
6
= 0.24806, p
7
= 0.18207, p
8
= 0.10357) is shown in Fig.5 and Fig.6, as a proto type
of a bound MUB diagonal density matrix.
7 Conclus i on
The feasible region of PPT MUB-(zzz)
G
diagonal density matrices is determined, where it is
a convex polyatope due to linearity of PPT conditions. In order to detect three-qubit bound
MUB diagonal entangled states, some nonlinear optimal EWs are manipulated, such that they
form the envelope of the boundary hyper-planes defined by a family o f optimal linear EWs.
By using these nonlinear EWs, the region o f bound entangled states a nd separable ones are
nonlinear Entanglement Witness using MUB 26
determined ana lytically in some particular cases three categories, where the numerical analysis
suppo r t them. The results thus obtained in this paper indicate that, the proposed methods
in this work, can be used in studying entanglement of more g eneral systems with linear PPT
conditions such as multiqubit MUB diagonal systems. .
Appendix A
I. Mutually unbiased basis
Let V be a d-dimensional Hilbert space with two orthonormal basis
B
1
= {|e
1
i, |e
2
i, ..., |e
d
i} and
B
2
= {|f
1
i, |f
2
i, ..., |f
d
i},
where |e
i
i and |f
i
i for i = 1, 2, ..., d belong to C
d
(the standard Hilbert space o f dimension d
endowed with usual inner product denoted by h | i).
The basis B
1
and B
2
are called mutually unbiased if and only if
|he
i
|f
j
i| =
1
d
. (A-i)
As an example, for a two-level system there is such a set of bases that can be represented in
terms of eigenvectors of the usual Pauli matr ices σ
x
, σ
y
, σ
z
as f ollows
B
x
= {
1
2
(|0i + |1i),
1
2
(|0i |1i)},
B
y
= {
1
2
(|0i + i|1i),
1
2
(|0i i|1i)} and
B
z
= {|0i, |1i} .
When d is a prime or power of a prime, the maximum number o f such MUB’s is equal to d + 1,
otherwise there is no clear number of sets.
nonlinear Entanglement Witness using MUB 27
According to Refs. [18, 19] and Table I, in the case of three qubits, we have nine sets of
mutually unbiased bases and corresponding maximally commuting sets of observables, where
we will refer to them as generalized Pauli matrices; each of these sets consist of seven commut-
ing observables. In the Table I, the first three rows contains product common eigenvectors,
(xyz)
π
, (yzx)
π
and (zxy)
π
(subscript π means product state). For example, eight states for basis
(xyz)
π
could be written as |n
x
n
y
n
z
i where n
i
= 1 and n
i
= 0 correspo nd to spin down and
spin up along the ith axis for i = x, y, z, respectively. These product states are separable so
we will not use them for construction of EWs. Other bases consist of six maximally entangled
states (xxx)
Gi
, (yyy)
G
, (zzz)
G
, (xzy)
G
, (yxz)
G
and (zyx)
G
, here subscript G denotes a family of
Greenberger-Ho r ne-Zeilinger (GHZ ) states. For example, eight states f or basis (zzz)
G
and
(xxx)
Gi
can be written as
(zzz)
G
= | n
z
n
z
n
z
, ±i = (| n
z
n
z
n
z
i ± | ¯n
z
¯n
z
¯n
z
i), n
z
= 0, 1 (A-ii)
(xxx)
Gi
= |n
x
n
x
n
x
, ±i = (| n
x
n
x
n
x
i ± i| ¯n
x
¯n
x
¯n
x
i), n
x
= 0, 1 (A-iii)
where labels bar show that if n
z
= 0 or 1, then ¯n
z
= 1 or 0, r espectively. In section 2, we have
considered only the state (zzz)
G
; since the other cases can be obtained from this state by local
unitary operations.
II. MUB sets for three qubit systems
We know MUB can be constructed using a number o f methods that depend on the dimension-
ality of the space. These methods using for different case such as dimension space is prime, a
product of primes, or a power of a prime, and if it is odd or even. We confine our study to the
case of three qubits, that is, to an eight-dimensional Hilbert space, in t his space there exist
four MUB formation, where denotes sets of MUBs where the basis vectors are either separable
, biseparable or entangled states. The fo ur MUB formations are (2, 3, 4) (here 2 means two
separable states, 3 means there is three biseparable state, 4 means four maximally entangled
states), (0, 9, 0), (3, 0, 6) and (1, 6, 2). There are other MUB sets for three qubits but one can
nonlinear Entanglement Witness using MUB 28
get them by local transformations from the previous ones. In this paper we work with table
(3, 0, 6), line (6) in order to introduce our EWs. This table is given by
1 (xyz)
π
σ
x
II Iσ
y
I IIσ
z
σ
x
σ
y
σ
z
σ
x
σ
y
I σ
x
Iσ
z
Iσ
y
σ
z
2 (yzx)
π
σ
y
II Iσ
z
I IIσ
x
σ
y
σ
z
σ
x
σ
y
σ
z
I σ
y
Iσ
x
Iσ
z
σ
x
3 (zxy)
π
σ
z
II Iσ
x
I IIσ
y
σ
z
σ
x
σ
y
σ
z
σ
x
I σ
z
Iσ
y
Iσ
x
σ
y
4 (xxx)
Gi
σ
y
σ
z
σ
z
σ
z
σ
y
σ
z
σ
z
σ
z
σ
y
σ
y
σ
y
σ
y
σ
x
σ
x
I σ
x
Iσ
x
Iσ
x
σ
x
5 (yyy)
G
σ
z
σ
x
σ
x
σ
x
σ
z
σ
x
σ
x
σ
x
σ
z
σ
z
σ
z
σ
z
σ
y
σ
y
I σ
y
Iσ
y
Iσ
y
σ
y
6 (zzz)
G
σ
x
σ
y
σ
y
σ
y
σ
x
σ
y
σ
y
σ
y
σ
x
σ
x
σ
x
σ
x
σ
z
σ
z
I σ
z
Iσ
z
Iσ
z
σ
z
7 (xzy)
G
σ
z
σ
x
σ
z
σ
y
σ
x
σ
x
σ
y
σ
y
σ
z
σ
z
σ
y
σ
x
σ
x
σ
z
I σ
x
Iσ
y
Iσ
z
σ
y
8 (yxz)
G
σ
x
σ
y
σ
x
σ
z
σ
y
σ
y
σ
z
σ
z
σ
x
σ
x
σ
z
σ
y
σ
y
σ
x
I σ
y
Iσ
z
Iσ
x
σ
z
9 (zyx)
G
σ
y
σ
z
σ
y
σ
x
σ
z
σ
z
σ
x
σ
x
σ
y
σ
y
σ
x
σ
z
σ
z
σ
y
I σ
z
Iσ
x
Iσ
y
σ
x
Ta ble 1: Nine sets of operators defining a (3,0,6) MUB .
In this table three states
(xxx)
Gi
, (yyy)
G
, (zzz)
G
(A-iv)
can be reversibly converted into each other by local unitary operations (permutation), (i.e.
σ
y
σ
z
σ
x
), and e.g., if we construct EW using (zzz)
G
then this EW can be converted by
another EW for (xxx)
G
state by applying the local unitary transformation σ
z
σ
x
.
Another states
(xzy)
G
, (yxz)
G
, (zyx)
G
(A-v)
can be transformed into each other by local unitary operations, e.g., for state (xzy)
G
we have
(xzy)
G
= |n
1
x
n
2
z
n
3
y
, ±i = (|n
1
x
n
2
z
n
3
y
i ± |
¯
n
1
x
¯
n
2
z
¯
n
3
y
i), (A-vi)
we can convert the states (A-v) into other three maximally entangled states (A-iv) by the
nonlinear Entanglement Witness using MUB 29
following local operations
U
xz
=
1
2
1 1
1 1
, U
yx
=
e
4
0
0 e
4
, U
yz
=
1
2
1 i
i 1
.
where, we have applied the permutation x z to the first qubit, y x to the middle qubit
and y z to the rightmost qubit.
The above discussion was used for table (3, 0, 6), but if we choose table (2, 3, 4), we will
obtain maximally entangled states ((xzy)
G
, (yyz)
G
, (yxy)
G
, (zyx)
Gi
) so that, according to the
following local operators and permutations we can convert the states (A-v) into ot her three
maximally entangled states.
(yyz)
G
(yxy)
G
with local operators U(1)
zx
, U(2)
yx
, U(3)
zy
.(the notation U( i)
means that U acting on t he (i)-th qubit)
(xzy)
G
(zyx)
Gi
with permutatio n (σ
x
σ
z
σ
y
) only for first and second qubits and
local operators U(3)
yx
(xzy)
G
(yyz)
G
with local operators U(1 )
xy
, U(2)
zy
and with permutation (σ
y
σ
z
σ
x
) for the rightmost qubit.
Appendix B
The following cases are used for construction of EWs:
r
4
+ r
5
= 2(p
3
p
4
+ p
5
p
6
), r
6
+ r
7
= 2(p
1
+ p
2
+ p
7
p
8
) (B-i)
r
4
r
5
= 2(p
1
p
2
+ p
7
p
8
), r
6
r
7
= 2 (p
3
p
4
p
5
+ p
6
)
r
4
+ r
6
= 2(p
3
p
4
+ p
7
p
8
), r
5
+ r
7
= 2(p
1
+ p
2
+ p
5
p
6
)
r
4
r
6
= 2(p
1
p
2
+ p
5
p
6
), r
5
r
7
= 2 (p
3
p
4
p
7
+ p
8
)
r
4
+ r
7
= 2(p
5
p
6
+ p
7
p
8
), r
5
+ r
6
= 2(p
1
+ p
2
+ p
3
p
4
)
nonlinear Entanglement Witness using MUB 30
r
4
r
7
= 2 (p
1
p
2
+ p
3
p
4
), r
5
r
6
= 2(p
5
p
6
p
7
+ p
8
).
We choose one the following cases for our EW:
1 + r
1
= 2(p
1
+ p
2
+ p
3
+ p
4
), 1 r
1
= 2(p
5
+ p
6
+ p
7
+ p
8
) (B-ii)
1 + r
2
= 2(p
1
+ p
2
+ p
5
+ p
6
), 1 r
2
= 2(p
3
+ p
4
+ p
7
+ p
8
)
1 + r
3
= 2 (p
1
+ p
2
+ p
7
+ p
8
), 1 r
3
= 2 (p
3
+ p
4
+ p
5
+ p
6
).
nonlinear Entanglement Witness using MUB 31
Proof for EWs detecting bound MUB-(zzz)
G
diagonal density matrices
For more detail, we have
T r[W ρ
s
] = A
0
+A
1
cos θ
3
cos θ
2
+sin θ
1
sin θ
2
sin θ
3
(A
2
cos ϕ
1
cos (ϕ
2
ϕ
3
)+A
3
sin ϕ
1
sin (ϕ
2
+ ϕ
3
)).
Ta king ϕ
2
= ϕ
3
=
π
4
, θ
1
=
π
2
and according to relation (
a
2
+ b
2
a sin θ + b cos θ
a
2
+ b
2
), we have
T r[W ρ
s
] = A
0
+ A
1
cos θ
3
cos θ
2
± sin θ
2
sin θ
3
q
A
2
2
+ A
3
3
,
according to cos θ
3
, we obtain
T r[W ρ
s
] = A
0
±
q
A
2
1
cos θ
2
2
+ sin θ
2
2
(A
2
2
+ A
2
3
),
and from t he condition (A
2
2
+ A
2
3
A
2
1
, θ
2
=
π
2
), one can get