# Non-maximally entangled mixed states of $X$ and non-$X$ types as
teleportation channels

Anushree Bhattacharjee1, Sovik Roy2
1 2 Department of Mathematics, Techno Main Salt Lake (Engg. Colg.),
Techno India Group, EM 4/1, Sector V, Salt Lake, Kolkata 700091, India

Md. Manirul Ali3
3 Centre for Quantum Science and Technology,
Chennai Institute of Technology, Chennai 600069, India

Biplab Ghosh4
4 Department of Physics,
Vivekananda College for Women, Barisha, Kolkata - 700008, India
a.bhattacharya.tmsl@ticollege.orgs.roy2.tmsl@ticollege.orgmanirul<EMAIL_ADDRESS>
###### Abstract

Mixed spin-$\frac{1}{2}$ states violating Bell-CHSH inequality is useful for
teleportation. There exist states which do not violate Bell-inequality but is
still useful as teleportation channels. Maximally entangled mixed states of
Munro class and Ishizaka-Hiroshima class are such types which although satisfy
Bell-CHSH inequality, yet can perform better as teleportation channels for a
given degree of mixedness[1]. In this work we construct class of mixed states
of non-maximally entangled types whose efficacy as teleportation channels have
been studied. For certain range of state parameters, these non-maximally
entangled mixed states performs better as quantum teleportation channels than
certain maximally entangled mixed states (such as Werner state). These
constructed states, though entangled, satisfy Bell-CHSH inequality implying
further that violation of local inequalities may not be good indicators of
their ability to complete quantum processing tasks such as teleportation.

Keywords: Teleportation Fidelity, Concurrence, Mixedness, $X$ state,
$W\bar{W}$ state, Star state, Werner state.

Pacs No.: 03.65.Ud 03.67.-a

##  1 Introduction:

Recent progresses in the field of quantum information science has marked
significant strides, advancing the comprehension in the quantum environment.
The revolution in quantum information science is propelled by quantum tasks
such as quantum teleportation [2], super dense coding [3], quantum key
distribution [4, 5], quantum computation [6, 7, 8] et. al. Among the above
quantum phenomena, quantum teleportation has garnered widespread attention
among scientific researchers. Being a non-classical technique for transmitting
quantum information across significant distances, quantum teleportation stands
out as a highly pertinent application in the realm of quantum information
processing. The concept of teleportation was first introduced by Bennett et al
[2]. Dik Bouwmeester et. al. successfully executed the experimental
implementation of quantum teleportation [9]. Later Popescu found that the
pairs of mixed states can also be used for teleportation, a feature known as
probabilistic teleportation or imperfect teleportation [10]. Teleportation
requires the separation of a protocol into classical and quantum parts using
which an unknown state is reconstructed with perfect fidelity at another
location. In this process the original copy of the unknown qubit is destroyed.
Not all quantum states are efficient as quantum teleportation channels. This
restriction is captured by the measure known as teleportation fidelity [10,
11, 12]. Classically the maximum possible fidelity achievable by means of
local operations and classical communication (LOCC) is $\frac{2}{3}$. For a
successful achievement of quantum teleportation, the fidelity of the quantum
channel must exceed this classical fidelity. Efficiency of teleportation
reaches its peak only when the sender and the receiver share maximally
entangled pure state. The assessment of teleportation efficiency is quantified
by a parameter known as fidelity, which takes the value $1$ for maximally
entangled states and $0$ for separable states. Almost all teleportation
schemes utilize maximally entangled states which theoretically ensure the
attainment of highest fidelity, however in practical situations, environmental
interactions lead to a gradual degradation of entanglement between particles.
Hence, in ideal scenario, pure maximally entangled states is anticipated to
yield a high fidelity performance in achieving a task whereas, the preparation
of pure states presents a formidable challenge. Instead, pragmatically, states
often dealt with, exhibit mixed characteristics. Quantum states that attain
greatest possible entanglement relative to a given degree of mixedness are
classified as Maximally Entangled Mixed States (MEMS) [13], while states that
do not reach the same pinnacle of entanglement are categorized as Non-
Maximally Entangled Mixed States (NMEMS) [14]. Werner state characterized as a
maximally entangled mixed state proved to be a valuable resource in the
context of teleportation [1]. Munro et. al. investigated a class of states
characterized by the utmost entanglement achievable for a specific level of
purity and formulated an analytical expression to describe that class of MEMS
[15], which can be termed as MJWK type states. Adhikari et. al. demostrated
that the MEMS proposed by MJWK can be used as a reliable teleportation channel
when mixedness of the state falls below the critical threshold value of
$\frac{22}{27}$. However, Werner state outperforms MJWK MEMS class with
respect to its capacity as quantum teleportation channel. They also studied
the bounds of teleportation fidelity of Werner state for which it can be used
as teleportation channel. Adhikari et. al. also proposed a new class of
bipartite non-maximally entangled mixed state by taking convex combination of
GHZ state and W state. This state is more efficient as teleportation channel
than Werner derivative class of states (which is a different type of NMEMS,
obtained by applying a non-local unitary operator on the Werner state)
[1]111Two of the authors S. Roy and B. Ghosh of this work were also the co-
authors of the paper..

In this paper we construct a few special bipartite mixtures and study their
efficiency as quantum teleportation channels. These states are of NMEMS types
and their efficiency as quantum teleportation channel have been ascertained.
For this, we have first considered tripartite pure states such as
$|W\rangle,\>|\bar{W}\rangle,\>|W\bar{W}\rangle$ and $|Star\rangle$ [16]. We
then remove (or trace out) a party from these pure states and take convex
combination of these bipartite mixtures (derived from tripartite states) with
two qubit maximally entangled Bell state. $|W\bar{W}\rangle$ state,
characterized by an equal superposition of a standard $|W\rangle$ state and
its spin-flipped version $|\bar{W}\rangle$ state, is significant because it
has quantum correlations at both bipartite and tripartite levels. Such a state
is a good test bed for studying quantum correlations distributed at different
levels. The presence of different types of correlations is one of the reasons
that $|W\rangle$ states are robust under local decoherence. Similar to
$|W\bar{W}\rangle$ state, the $|Star\rangle$ state also exhibits distribution
of correlations at all conceivable levels, but $|Star\rangle$ state introduces
an intriguing contrast to $|W\bar{W}\rangle$ state by showcasing correlations
in an asymmetric manner. This is because, in $|Star\rangle$ state two types of
qubits are present, viz, central and peripheral qubits. Central qubit is that
qubit which when traced out, the remaining two qubits are left in separable
state. Those two left out qubits, however, are peripheral qubits. Recently Roy
et.al. investigated both $|W\bar{W}\rangle$ and $|Star\rangle$ states and
explored different quantum properties with respect to both coherent and
incoherent basis [17].

The current paper is organised as follows. In sec.2 we have discussed briefly,
the measures used in quantum information science such as to study how
entangled the states are, quantifying mixedness of the state, how to verify
whether states are useful as quantum teleportation channels and to know the
signature status of entanglement of the constructed state by the violation of
Bell-inequality. In sec.3 we build the bipartite mixtures which are non-
maximally entangled mixed states. We find that, among our constructed states
some are $X$ states and others are not of $X$ type, which, in this article,
are termed as non-$X$ states. We, in sec.4, study the teleportation fidelity,
Bell-CHSH inequality violation, nature of entanglement and mixedness of the
$X$ type NMEMS while in sec.5, we study the same for those states that are of
non-$X$ type. Sec.6 gives concluding remarks.

##  2 Quantum measures : A brief overview

### Concurrence, teleportation fidelity and linear entropy:

Entanglement is a fundamental quantum resource emerging from the non-local
correlations among quantum systems. Among the various measures such as
concurrence, relative entropy of entanglement, negativity and log negativity
et. al. that have been developed to quantify entanglement, concurrence stands
out as universally recognized measure for two qubit system, with equal
efficacy in assessing entanglement for both pure and mixed quantum states
[18]. The concurrence for a bipartite state $\rho$ is defined as

$\displaystyle
C(\rho)=max\\{0,\sqrt{\lambda_{1}}-\sqrt{\lambda_{2}}-\sqrt{\lambda_{3}}-\sqrt{\lambda_{4}}\\}$
(1)

where $\lambda_{i},(i=1,2,3,4)$ are the eigenvalues of the matrix
$\rho\tilde{\rho}$ in decreasing order in which $\tilde{\rho}$ is the spin-
flipped density matrix defined as

$\tilde{\rho}=(\sigma_{y}\otimes\sigma_{y}){\rho^{*}}(\sigma_{y}\otimes\sigma_{y})$
(2)

where ${\rho^{*}}$ is the complex conjugate of the density matrix $\rho$ and
$\sigma_{y}$ is the Pauli spin operator.

For an arbitrary density matrix $\rho$, when used as quantum teleportation
channel, it’s optimal teleportation fidelity is given by

$\displaystyle f^{T}(\rho)=\frac{1}{2}\Big{\\{}1+\frac{N(\rho)}{3}\Big{\\}},$
(3)

where $N(\rho)=\sum_{i=1}^{3}\sqrt{u_{i}}$. Here $u_{i}$’s are the eigenvalues
of the matrix $T^{\dagger}T$. The elements of $T$ are denoted by $t_{nm}$ and
these elements are calculated as $t_{nm}=Tr(\rho\sigma_{n}\otimes\sigma_{m})$
where $\sigma_{j}$’s denote the Pauli spin operators. In terms of
teleportation fidelity, a general result holds that any mixed
spin-$\frac{1}{2}$ state $\rho$ is useful for standard teleportation if and
only if $N(\rho)>1$ [19].

When interacted with external environment, the quantum states get affected and
there is degradation of quantumness of these states. This phenomenon is known
as decoherence and such a state is called mixed state. Mixed states can be
probabilistically used as quantum teleportation channel and their mixedness
are also studied. Linear entropy is one such measure to quantify the mixedness
of a given mixed state. For an arbitrary $d-$ dimensional quantum mixed state
with density matrix $\rho$, the mixedness, using the normalized linear
entropy, is defined as

$\displaystyle L(\rho)=\frac{d}{d-1}(1-Tr(\rho^{2})),$ (4)

where $Tr(\rho^{2})$ describes the purity of the quantum system [20]. For
bipartite system, the value of $L$ ranges from $0$ to $1$. For any pure state
$\rho$, $L(\rho)=0$ and when $\rho=\frac{I}{4}$, i.e. when $\rho$ is maximally
mixed state, then $L(\rho)$ is $1$.

### Bell-CHSH inequality violation:

Any quantum state described by the density operator $\rho$ violates the Bell-
CHSH inequality if and only if the following condition is satisfied i.e.

$\displaystyle M(\rho)=\max_{i>j}(u_{i}+v_{j})>1.$ (5)

Here $u_{i}$’s are the eigenvalues of the matrix $T^{\dagger}T$.

Moreover, Horodecki et.al [19] had shown that every mixed two
spin-$\frac{1}{2}$ state which violates any generalized Bell-CHSH inequality
is useful for teleportation. If we denote by $P$ the statement that $``$
$\rho$ violates Bell-CHSH inequality” and by $Q$ we mean $``$ the state $\rho$
is useful for teleportation”, then by Horodecki’s paper we have $``$ $P$
$\Rightarrow$ $Q$”. By mathematical logic one can immediately conclude that
$``$ $\neg Q$ $\Rightarrow$ $\neg P$”. This means that the state which is not
useful for teleportation will satisfy Bell-CHSH inequality. But one cannot
immediately conclude that $``$ $Q$ $\Rightarrow$ $P$”, i.e. there may exist
states which satisfy Bell-CHSH inequality, but still can be used as
teleportation channel. In this context we are now going to construct some
classes of non-maximally entangled mixed states to study how these states
behave according to Bell violation and whether they are efficient as quantum
teleportation channels.

##  3 States: Bipartite Mixtures

We know that the Bell states are a set of maximally entangled two qubit pure
states and they are as follows:

$\displaystyle|\phi^{\pm}\rangle=\frac{1}{\sqrt{2}}\\{|00\rangle\pm|11\rangle\\};\qquad\qquad\qquad|\varphi^{\pm}\rangle=\frac{1}{\sqrt{2}}\\{|01\rangle\pm|10\rangle\\}.$
(6)

Here $A$ and $B$ denote the two parties Alice and Bob holding one qubit each.
On the other hand the tripartite $|W\rangle$ state and it’s spin-flipped
version $|\bar{W}\rangle$ are respectively defined as

$\displaystyle|W\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{3}}(|001\rangle+|010\rangle+|100\rangle),$
$\displaystyle|\bar{W}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{3}}(|110\rangle+|101\rangle+|011\rangle),$ (7)

where each party holds qubits individually. $|W\bar{W}\rangle$ state is
characterized by an equal superposition of a standard $|W\rangle$ state and
its spin-flipped version $|\bar{W}\rangle$ state.

$\displaystyle\lvert W\bar{W}\rangle=\frac{1}{\sqrt{2}}(\lvert
W\rangle+\lvert\bar{W}\rangle)$ (8)

Such a state serves as an excellent experimental platform for investigating
quantum correlations distributed across varying levels. The existence of
diverse correlations in $|W\rangle$ state maintains the robustness of the
state under local decoherence. Another tripartite state that has been
considered here is $|Star\rangle$, which is defined as

$\displaystyle|Star\rangle=\frac{1}{2}(|000\rangle+|100\rangle+|101\rangle+|111\rangle).$
(9)

Now from $|W\rangle$ (eq.(3)) state tracing out any of the qubits and then
taking convex combination of the reduced density matrix with that of four Bell
states described in eq.(6), some new class of states are obtained. Similar
construction procedures have been carried by taking into consideration the
$|Star\rangle$ state (eq.(9)) and the Bell states.

### Class $1$ states:

We introduce a distinct class of bipartite states by initially considering a
$3-$ qubit state and subsequently removing the third party. The resulting two
qubit density matrix $\rho^{W}$ is then combined through convex combination
with each of four Bell states (eq.(6)). The resultant class of states are then
written as,

$\displaystyle\rho^{(1)}$ $\displaystyle=$ $\displaystyle
p\rho^{W}+(1-p)\rho^{\phi^{\pm}},$ $\displaystyle\rho^{(2)}$ $\displaystyle=$
$\displaystyle p^{\prime}\rho^{W}+(1-p^{\prime})\rho^{\varphi^{\pm}},\>0\leq
p,p^{\prime}\leq 1.$ (10)

Similarly by taking $|\bar{W}\rangle$ and $|W\bar{W}\rangle$ states, removing
third party as before and taking convex combination with each of the four Bell
states of eq.(6) we obtain the following class of states.

$\displaystyle\rho^{(3)}$ $\displaystyle=$ $\displaystyle
q\rho^{\bar{W}}+(1-q)\rho^{\phi^{\pm}},$ $\displaystyle\rho^{(4)}$
$\displaystyle=$ $\displaystyle
q^{\prime}\rho^{\bar{W}}+(1-q^{\prime})\rho^{\varphi^{\pm}},\>0\leq
q,q^{\prime}\leq 1.$ (11)

and from the symmetric structures of the $|W\rangle$ and $|\bar{W}\rangle$
states, we can also observe that removing qubits respectively held by first
two parties will generate the same set of states as above.

In the limiting case where $p(or\>p^{\prime})=1$ and $q(or\>q^{\prime})=1$ in
eqs.(3) and (3), the states $\rho^{(1)}(or\>\rho^{(2)})$ and
$\rho^{(3)}(or\>\rho^{(4)})$ reduce to the states

$\displaystyle\rho^{W}$ $\displaystyle=$
$\displaystyle\frac{1}{3}|00\rangle\langle
00|+\frac{2}{3}|\psi^{+}\rangle\langle\psi^{+}|,$
$\displaystyle\rho^{\bar{W}}$ $\displaystyle=$
$\displaystyle\frac{1}{3}|11\rangle\langle
11|+\frac{2}{3}|\psi^{+}\rangle\langle\psi^{+}|.$ (12)

These states, as shown in (3), are maximally entangled mixed states (MEMS) as
they fall into the category of Ishizaka-Hiroshima’s proposed class of MEMS
[13]. However, the states represented in eqs.(3) and (3) are non-maximally
entangled mixed states (NMEMS).

### Class $2$ states:

Although the structures of $|W\rangle$ and $|\bar{W}\rangle$ are similar,
$|W\bar{W}\rangle$ is different (which we will shortly emphasize) and removing
third qubit, we construct a mixture by taking convex combination of
$\rho_{AB}^{W\bar{W}}$ and the four possible Bell states (eq.(6)) as follows.

$\displaystyle\rho^{(5)}$ $\displaystyle=$ $\displaystyle
r\rho^{W\bar{W}}+(1-r)\rho^{\phi^{\pm}},$ $\displaystyle\rho^{(6)}$
$\displaystyle=$ $\displaystyle
r^{\prime}\rho^{W\bar{W}}+(1-r^{\prime})\rho^{\varphi^{\pm}},\>0\leq
r,r^{\prime}\leq 1.$ (13)

However, similar to the states $|W\rangle$ and $|\bar{W}\rangle$, the removal
of first two qubits will yield similar states as defined in eq.(3).

Considering the $|Star\rangle$ state defined in eq.(9), we remove first qubit
i.e. the peripheral qubit and take convex combination with each of the Bell
states (eq.(6)) to obtain the following.

$\displaystyle\tau^{(1)}$ $\displaystyle=$ $\displaystyle
s\rho^{star}+(1-s)\rho^{\phi^{\pm}},$ $\displaystyle\tau^{(2)}$
$\displaystyle=$ $\displaystyle
s^{\prime}\rho^{star}+(1-s^{\prime})\rho^{\varphi^{\pm}},\>0\leq
s,s^{\prime}\leq 1.$ (14)

In eqs.(3), (3), (3) and (3), $\rho^{\phi^{\pm}}$ and $\rho^{\varphi^{\pm}}$
are two qubit density matrices corresponding to Bell states. Similar structure
as of eq.(3) can be found if we trace out second qubit (i.e. the second party,
which is also a peripheral qubit). Removing third qubit (central qubit) from
state (9) however makes the $|Star\rangle$ state separable. The class of
states as defined in eqs.(3) and (3) are of NMEMS types. It was also pointed
out that the states of the type $p\rho_{sep}+(1-p)\rho_{ent}$, where
$\rho_{sep}$ is the two qubit $|Star\rangle$ state (after removal of central
qubit) and $\rho_{ent}$ is the two qubit Bell state, are not suitable as
quantum teleportation channels.

Though, to define the above convex mixtures we have used separate symbols for
the state parameters $p,\>\>q,\>\>r$ and $s$, in the following analysis for
the sake of comparative analysis of these states with respect to the study of
fidelity of teleportation and Bell violation we have varies any of these state
parameters in a defined range.

##  4 $X$ type states as teleportation channels and their Bell-CHSH violation:

###  $X$ state:

Mendon _c_ a et al. had shown that there exist two-qubit states which belong
to a wider class of states of the form $X$ in which density matrix contains
elements only along the main diagonal and the complementary diagonal
positions. Such states are called $X$ states [17, 21, 22, 23, 24, 25]. The
density operator structure of such states are given as

$\displaystyle\mathcal{\rho^{(X)}}=\begin{pmatrix}\alpha&0&0&\eta\\\
0&\beta&\xi&0\\\ 0&\xi^{*}&\gamma&0\\\ \eta^{*}&0&0&\delta\\\ \end{pmatrix}.$
(15)

The analytic expression for concurrence of this state is given by

$\displaystyle
C(\mathcal{\rho^{(X)}})=2\>\max(0,\mid\xi\mid-\sqrt{\alpha\>\delta},\mid\eta\mid-\sqrt{\beta\>\gamma}).$
(16)

Using eq.(3), we calculate the teleportation fidelity of this generic $X$
state which is thus obtained as

$\displaystyle
f^{T}(\mathcal{\rho^{(X)}})=\frac{1}{2}+\frac{1}{6}\sqrt{(2\eta+2\xi)^{2}}+\frac{1}{6}\sqrt{(-2\eta+2\xi)^{2}}+\frac{1}{6}\sqrt{(\alpha-\beta-\gamma+\delta)^{2}}.$
(17)

Similarly, for the density matrix of eq.(15), the mixedness of $X$ state
(using eq.(4)) can be calculated as

$\displaystyle
L(\mathcal{\rho^{(X)}})=\frac{4}{3}-\frac{4}{3}(\alpha^{2}+\beta^{2}+\gamma^{2}+\delta^{2})-\frac{8}{3}(\eta^{2}+\xi^{2}).$
(18)

Using eq.(5), we can also immediately identify the eigenvalues of the matrix
$T^{\dagger}T$ corresponding to the $X$ state $\mathcal{\rho^{(X)}}$ of
eq.(15) as

$\displaystyle u_{1}$ $\displaystyle=$ $\displaystyle 8(\xi^{2}+\eta^{2}),$
$\displaystyle u_{2}$ $\displaystyle=$
$\displaystyle(-2\eta+2\xi)^{2}+(\alpha-\beta-\gamma+\delta)^{2},$
$\displaystyle u_{3}$ $\displaystyle=$
$\displaystyle(2\eta+2\xi)^{2}+(\alpha-\beta-\gamma+\delta)^{2}.$ (19)

It is easy to observe that the states of class $1$ defined in eqs.(3) and (3)
take the structure of $X$ state, while the states represented as class $2$,
defined in eqs.(3) and (3) are not of the type of $X$ state. We now analyze
the concurrence, teleportation fidelity and mixedness of the states having $X$
state’s structure.

For example, we can consider well-known Werner state [26] which is a maximally
entangled mixed state of Ishizaka-Hiroshima form[14]. The Werner state has
several forms in $2-$ qubit system, the one which we are interested in, is
given by

$\displaystyle\rho^{(werner)}=\frac{1-m}{3}I_{4}+\frac{4m-1}{3}|\varphi^{-}\rangle\langle\varphi^{-}|,\>\>0\leq
m\leq 1,$ (20)

where $|\varphi^{-}\rangle$ is the singlet state from eq.(6). Consequently,
the density operator form of the state (20) is

$\displaystyle\rho^{(werner)}=\begin{pmatrix}\frac{1-m}{3}&0&0&0\\\
0&\frac{2m+1}{6}&\frac{1-4m}{6}&0\\\ 0&\frac{1-4m}{6}&\frac{2m+1}{6}&0\\\
0&0&0&\frac{1-m}{3}\\\ \end{pmatrix}.$ (21)

We see that the form of the density matrix of the Werner state of eq.(21) is
of $\mathcal{\rho^{(X)}}$ type. Werner state is a type of state that although
is entangled, satisfies Bell-CHSH inequality and that efficacy of Werner state
as teleportation channel, is well established [1]. Using eq.(16), the
concurrence of the state (21) is given as

$\displaystyle
C(\rho^{(werner)})=\begin{array}[]{cc}\Big{\\{}&\begin{array}[]{cc}\frac{2m+1}{6},~{}~{}~{}m>-\frac{1}{2}\\\
\frac{1-2m}{2},~{}~{}~{}m<\frac{1}{2}\\\ \end{array}\end{array}.$ (25)

The state is mixed with mixedness
$L(\rho^{(werner)})=\frac{8}{9}-\frac{16}{9}m^{2}+\frac{8}{9}m$ and from
eq.(17), the teleportation fidelity of the Werner state is found as

$\displaystyle f^{T}(\rho^{(werner)})=\frac{1+2m}{3},$ (26)

where the fidelity exceeds classical limit for $m>\frac{1}{2}$.

We now perform a study of the states defined in sec.3 on the basis of their
utility as teleportation channels. For this, we denote the states defined in
eqs.(3)and (3) as subclass $(A)$ that are of the type of $X$ states and those
defined in eqs.(3) and (3) as subclass $(B)$ which do not fall into the
category of $X$ states. Such states, in this paper, have been termed as
non-$X$ states.

### Teleportation fidelity and Bell-CHSH violation of Subclass $(A)$:

In this section we analyze the structure of the states that are of $X$ type.
These states have been represented in eqs.(3) and (3).

#### Class $\rho^{(1)}$:

For the clarity of the discussion we first of all denote the states
$p\rho^{W}+(1-p)\rho^{\phi^{+}}$ of eq.(3) by $\rho^{(1)_{\phi+}}$ and the
states $p\rho^{W}+(1-p)\rho^{\phi^{-}}$ of eq.(3) by $\rho^{(1)_{\phi-}}$.
When the state is of the form $\rho^{(1)_{\phi+}}$, we see, from eq.(16) and
as concurrence always lies between $0$ and $1$, that when $0.7081<p\leq 1$ the
concurrence $C(\rho^{(1)_{\phi+}})$ is
$\frac{2}{3}p-2\sqrt{\frac{(3-p)(1-p)}{12}}$, while for $0\leq p<0.6$, the
state’s concurrence $C(\rho^{(1)_{\phi+}})$ is $1-p-\frac{2}{3}\sqrt{p^{2}}$.
However for all admissible values of the parameter $p$ (see eq.(3)), the state
is suitable as teleportation channel and the teleportation fidelity exceeds
the classical limit of $\frac{2}{3}$. Using (18), the mixedness
$L(\rho^{(1)_{\phi+}})$ of the state is given as
$\frac{20}{9}p-\frac{44}{27}p^{2}$. It is easy to observe that when $p=0$, the
mixedness is zero which is obvious as the mixture reduces to the Bell state
whose purity is $1$. Again when $p=1$, the mixedness is $\frac{16}{27}$, which
is the mixedness of $\rho_{AB}^{W}$ [1, 17]. The teleportation fidelity
$f^{T}(\rho^{(1)_{\phi+}})$ of the given state, using eq.(17), is found to be
$\frac{1}{2}+\frac{1}{6}\Big{(}\sqrt{(1-\frac{p}{3})^{2}}+\sqrt{(-1+\frac{5p}{3})^{2}}+\sqrt{(1-\frac{4p}{3})^{2}}\Big{)}$.
Also for $p=0$, the teleportation fidelity of the given state is $1$, while
for $p=1$, the teleportation fidelity of the reduced MEMS $\rho^{W}$ is found
to be $\frac{7}{9}$ [1, 17]. Likewise, by taking the state
$\rho^{(1)_{\phi-}}$, we see that the expressions for mixedness
$L(\rho^{(1)_{\phi-}})$ and teleportation fidelity $f^{T}(\rho^{(1)_{\phi-}})$
of the state are exactly same as that of $\rho^{(1)_{\phi+}}$. The concurrence
$C(\rho^{(1)_{\phi-}})$ of the state $\rho^{(1)_{\phi-}}$ is however of the
form $\frac{2}{3}p-2\sqrt{\frac{(3-p)(1-p)}{12}}$ and this is valid for
$0.7081<p\leq 1$.

#### Class $\rho^{(2)}$:

The states $p^{\prime}\rho^{W}+(1-p^{\prime})\varrho^{\varphi^{+}}$ and
$p^{\prime}\rho^{W}+(1-p^{\prime})\rho^{\varphi^{-}}$ of eq.(3) are now
denoted by $\rho^{(2)_{\varphi+}}$ and $\rho^{(2)_{\varphi-}}$ respectively.
Since these states are also of $X$ type, we use eq.(16), (17) and (18) to
calculate concurrence, teleportation fidelity and mixedness. We find that for
$0\leq p^{\prime}\leq 1$, the concurrence $C(\rho^{(2)_{\varphi+}})$ is given
by $1-\frac{p^{\prime}}{3}$. The expression for the mixedness
$L(\rho^{(2)_{\varphi+}})$ of the state is
$p^{\prime}\Big{(}\frac{8}{9}-\frac{8}{27}p^{\prime}\Big{)}$. We also see that
the state $\rho^{(2)_{\varphi+}}$ is useful in teleportation for $0\leq
p^{\prime}\leq 1$ as $N(\rho^{(2)_{\varphi+}})>1$, the teleportation fidelity
$f^{T}(\rho^{(2)_{\varphi+}})$ of the given state is found to be
$1-\frac{2}{9}p^{\prime}$ and it exceeds classical fidelity $\frac{2}{3}$. As
before we also observe that when $p^{\prime}=0$, the state
$\rho^{(2)_{\varphi+}}$ reduces to Bell state with teleportation fidelity $1$
while for $p^{\prime}=1$, the given state reduces to two qubit $\rho^{W}$ with
teleportation fidelity $\frac{7}{9}$ and mixedness $\frac{16}{27}$. Again by
considering the state $\rho^{(2)_{\varphi-}}$, we see from eq.(16) that the
concurrence $C(\rho^{(2)_{\varphi-}})$ is $1-\frac{p^{\prime}}{3}$ when $0\leq
p^{\prime}\leq 1$. Hence we can say the state is entangled for all values of
parameter $p^{\prime}$. Also from eqs.(17) and (18), teleportation fidelity
$f^{T}(\rho^{(2)_{\varphi-}})$ and the mixedness $L(\rho^{(2)_{\varphi-}})$ of
the state is found respectively to be
$\frac{1}{2}+\frac{\sqrt{(\frac{5p^{\prime}}{3}-1)^{2}}}{3}+\frac{\sqrt{(\frac{2p^{\prime}}{3}-1)^{2}}}{6}$
and $8p^{\prime}\Big{(}\frac{1}{3}-\frac{7}{27}p^{\prime}\Big{)}$. It is
observed that although the state $\rho^{(2)_{\varphi-}}$ remains entangled for
all values of parameter $p^{\prime}$, the state is useful for teleportation
with fidelity exceeding classical fidelity of $\frac{2}{3}$ only when $0\leq
p^{\prime}<0.5$ or $0.75\leq p^{\prime}\leq 1$. Here also for the values of
parameter $p^{\prime}=0$ and $p^{\prime}=1$, the state respectively reduce to
Bell state and two qubit $\rho^{W}_{AB}$, in which the teleportation
fidelities are respectively $1$ and $\frac{7}{9}$.

#### Classes $\rho^{(3)}$ and $\rho^{(4)}$:

It is being further observed that for the states $\rho^{(3)}$ and $\rho^{(4)}$
defined in eq.(3), the concurrence, teleportation fidelity and mixedness are
same as the states $\rho^{(1)}$ and $\rho^{(2)}$ (defined in eq.(3))
respectively.

#### Comparison of teleportation fidelities of mixtures of Bell states with
Subclass $(A)$ and two qubit state derived from $|GHZ\rangle$ state:

The $|GHZ\rangle$ state is a three qubit entangled state [27], but the state
is different from $|W\rangle$ state as when a qubit is lost from
$|GHZ\rangle$, the reduced two-qubit state is separable whereas under similar
conditions $|W\rangle$ is inseparable. Hence $|GHZ\rangle$ and $|W\rangle$ are
two different classifications under SLOCC operations in the tripartite
scenario. The $|GHZ\rangle$ state is defined as [28, 29, 30]

$\displaystyle|GHZ\rangle=\frac{1}{\sqrt{2}}(|000\rangle+|111\rangle).$ (27)

Now as per our construction of states as defined in the previous sections we
eliminate any one qubit from state (27) to get state $\rho^{ghz}$ and then
taking convex combination with any of the four possible Bell states (say
$|\phi^{+}\rangle$) as

$\displaystyle\rho^{g}$ $\displaystyle=$ $\displaystyle
t\rho^{ghz}+(1-t)\rho^{\phi^{+}}.$ (28)

These states are type of $X$ state. It can easily be shown that the above
states are useful as quantum teleportation channels and violate Bell-CHSH
inequality for $0\leq t\leq 1$. Using eq.(17) the teleportation fidelity of
the state is thus given by

$\displaystyle
f^{T}(\mathcal{\rho}^{g})=\frac{2}{3}+\frac{1}{3}\Big{(}1-t\Big{)}.$ (29)

We can also construct states like $\rho^{g}$ of eq.(28) by taking convex
combinations of the state $\rho^{ghz}$ with other Bell states from eq.(6) and
conduct similar study.

### Bell-CHSH violation of Subclass $(A)$:

#### Class $\rho^{(1)}$:

Using eqs.(5) and (4), we see that for the states of the type
$\rho^{(1)_{\phi+}}$, $M(\rho^{(1)_{\phi+}})=4-10p+\frac{67}{9}p^{2}$. Now
$M(\rho^{(1)_{\phi+}})>1$ when $0\leq p\leq 0.45$ and when $0.89\leq p\leq 1$.
Hence in these two ranges, the state $\rho^{(1)_{\phi+}}$ violates Bell-
inequality. But when $0.45<p<0.89$, the state $\rho^{(1)_{\phi+}}$ satisfies
Bell’s inequality, although the state is entangled there. Again for the state
$\rho^{(1)_{\phi-}}$ ,
$M(\rho^{(1)_{\phi-}})=4-\frac{22}{3}p+\frac{43}{9}p^{2}$. It is being
observed that for the state $\rho^{(1)_{\phi-}}$, $M(\rho^{(1)_{\phi-}})>1$
for all values of the state parameter $p$. Therefore, the state
$\rho^{(1)_{\phi-}}$, violates Bell’s inequality $\forall$ values of parameter
$p$. Since the concurrence of the state $\rho^{(1)_{\phi+}}$ is positive in
the range $[0,0.6]$ and in $(0.7,1]$ and as per the nature of Bell violation
by the state $\rho^{(1)_{\phi+}}$, as discussed, it is thereby observed that
the states of the form $\rho^{(1)_{\phi+}}$ satisfies Bell’s inequality
although being entangled in the ranges $0.45\leq p<0.6$ and for $0.70<p\leq
0.89$. The state $\rho^{(1)_{\phi-}}$ is having positive concurrence only when
$0.70<p\leq 1$ and in this range it violates Bell’s inequality too.

Thus we see that the class of states of the form $\rho^{(1)}$ is useful for
teleportation although for certain ranges of the state parameter $p$,
$\rho^{(1)}$ satisfies Bell’s inequality although being entangled in the given
range.

#### Class $\rho^{(2)}$:

As before using eqs.(5) and (4), we see that for the states of the type
$\rho^{(2)_{\varphi+}}$,
$M(\rho^{(2)_{\varphi+}})=\frac{7}{9}(p^{\prime})^{2}-\frac{10}{3}p^{\prime}+4$.
Now $M(\rho^{(2)_{\varphi+}})>1$, $\forall$ $p^{\prime}$. On the other hand,
for the states $\rho^{(2)_{\varphi-}}$,
$M(\rho^{(2)_{\varphi-}})=\frac{79}{9}(p^{\prime})^{2}-\frac{34}{2}p^{\prime}+4$
which is greater than unity when $0\leq p^{\prime}<0.37$ and when
$0.91<p^{\prime}\leq 1$. For $p^{\prime}\in[0.37,\>0.91]$, the state
$\rho^{(2)_{\varphi-}}$ satisfies Bell’s inequality. Now both the types of the
states $\rho^{(2)_{\varphi+}}$ and $\rho^{(2)_{\varphi-}}$ are entangled for
$0\leq p^{\prime}\leq 1$, but for certain ranges of the parameter
$p^{\prime}$, the states of the type $\rho^{(2)}$ satisfy Bell’s inequality.

Thus it is being observed that, in the two regions of the parameter
$p^{\prime}$ such as $[0,\>0.5)$ and $[0.75,\>1]$, the states $\rho^{(2)}$ is
entangled, satisfies Bell’s inequality and moreover are useful as quantum
teleportation channels.

We now plot the teleportation fidelities of the states (3), (3) and (28)
(although for explicit representation of the states we have used different
symbols for state parameters, these state parameters all vary between $0$ and
$1$ and hence to compare we have varied one such state parameter $p$).

Figure 1: The figure shows the plots of teleportation fidelities of the states
$\rho^{(1)}$ (or $\rho^{(3)}$) (blue line), $\rho^{(2)}$ (or $\rho^{(4)}$)
(red line) and $\rho^{g}$ (yellow line) defined respectively in eqs.(3),(3)
and (28) against the parameter $p$ (the other symbols $q$ and $t$ used to
defined the respective mixtures are similar to $p$ and ranges from $0$ to
$1$).

The figure $1$ shows the plots of the teleportation fidelity
$\rho^{(1)_{\phi+}}$ (and $\rho^{(1)_{\phi-}}$), $\rho^{(1)_{\psi-}}$,
$\rho^{(1)_{\psi+}}$ and $\rho^{g}$. The state parameters represented in
eqs.(3), (3) and (27) are $p,\>p^{\prime},\>t$ but as all these state
parameters vary from $0$ to $1$. For the sake of discussion we have plotted
the teleportation fidelities of these states with respect to a common
framework and hence we have arbitrarily taken parameter $p$ along horizontal
axis varying from $0$ to $1$. We see from fig.$1$ that the teleportation
fidelity of $\rho^{(1)_{\phi+}}$ (and $\rho^{(1)_{\phi-}}$) (the expression
for both of these states are similar) as well as that of $\rho^{(1)_{\psi-}}$
are exceeding the teleportation fidelity of $\rho^{g}$ when $0.85\leq p\leq 1$
while the teleportation fidelity of $\rho^{(1)_{\psi+}}$ is always exceeding
that of $\rho^{g}$ for $0\leq p\leq 1$. This implies that the $X$ type states
$\rho^{(1)}$ and $\rho^{(2)}$ described in eqs.(3) and (3) give better
performance as quantum teleportation channels than the state $\rho^{g}$
described in eq.(27) for certain ranges of states’ parameter whereas for some
other range the state $\rho^{g}$ described in eq.(27) is better performer as
teleportation channel. The analysis of fig.$1$ has been summarized in the
following table.

Table 1: Summary of teleportation fidelities of the states $\rho^{(1)}$, $\rho^{(2)}$ and $\rho^{g}$ $p$ | Telep. Fid. of $\rho^{(1)_{\phi+}}$ (or $\rho^{(1)_{\phi-}}$) | Telep. Fid. of $\rho^{(2)_{\psi+}}$ | Telep. Fid. of $\rho^{(2)_{\psi-}}$ | Telep. Fid. of $\rho^{g}$
---|---|---|---|---
$0.0$ | $1$ | $1$ | $1$ | $1$
$0.1$ | $0.94$ | $0.98$ | $0.93$ | $0.97$
$0.2$ | $0.89$ | $0.96$ | $0.87$ | $0.93$
$0.3$ | $0.83$ | $0.93$ | $0.80$ | $0.90$
$0.4$ | $0.78$ | $0.91$ | $0.73$ | $0.87$
$0.5$ | $0.72$ | $0.89$ | $0.67$ | $0.83$
$0.6$ | $0.67$ | $0.87$ | $0.60$ | $0.80$
$0.7$ | $0.67$ | $0.84$ | $0.64$ | $0.77$
$0.8$ | $0.69$ | $0.82$ | $0.69$ | $0.73$
$0.9$ | $0.73$ | $0.80$ | $0.73$ | $0.70$
$1$ | $0.78$ | $0.78$ | $0.78$ | $0.67$

##  5 Non-$X$ type states as teleportation channels and their Bell-CHSH
violation:

The states whose structures are not similar to that of $X$ state as defined in
eq.(15), are termed in this paper as non-$X$ states.

### Teleportation fidelity and Bell-CHSH violation of Subclass $(B)$:

To begin with the analysis of the states of subclass $(B)$, we observe that
the convex combination of the two qubit state derived from $|W\bar{W}\rangle$
with $\phi^{\pm}$ of class of states $\rho^{(5)}$ defined in eq.(3) can be
represented by the following general density matrix form.

$\displaystyle\mathcal{\rho^{C}}^{1}=\begin{pmatrix}\alpha+\delta&\alpha&\alpha&\delta\\\
\alpha&2\alpha&2\alpha&\alpha\\\ \alpha&2\alpha&2\alpha&\alpha\\\
\delta&\alpha&\alpha&\alpha+\delta\\\ \end{pmatrix}$ (30)

Using eqs. (1), (3) and (4) we calculate the concurrence, teleportation
fidelity and mixedness of the state. The concurrence of the state is thus
denoted by $C(\mathcal{\rho^{C}}^{1})$ and is given by

$\displaystyle
C(\mathcal{\rho^{C}}^{1})=\sqrt{d_{1}+\frac{\sqrt{d_{2}}}{2}}-\sqrt{d_{1}-\frac{\sqrt{d_{2}}}{2}}-\sqrt{d_{3}},$
(31)

where
$d_{1}=\frac{9}{2}\alpha^{2}+2\alpha\delta+2\delta^{2}$,$d_{2}=81\alpha^{4}+72\alpha^{3}\delta-168\alpha^{2}\delta^{2}+32\alpha\delta^{3}+16\delta^{4}$
and $d_{3}=\alpha^{2}$.

Now the eigenvalues of of the matrix $T^{\dagger}T$ are

$\displaystyle v_{1}$ $\displaystyle=$ $\displaystyle(2\alpha-2\delta)^{2},$
$\displaystyle v_{2}$ $\displaystyle=$ $\displaystyle(4\alpha+2\delta)^{2},$
$\displaystyle v_{3}$ $\displaystyle=$ $\displaystyle(4\alpha-2\delta)^{2},$
(32)

and consequently the teleportation fidelity of the given state (30) is denoted
by $f^{T}(\mathcal{\rho^{C}}^{1})$ and is found as

$\displaystyle
f^{T}(\mathcal{\rho^{C}}^{1})=\frac{1}{2}+\frac{1}{6}\sqrt{(2\alpha-2\delta)^{2}}+\frac{1}{6}\sqrt{(2\delta+4\alpha)^{2}}+\frac{1}{6}\sqrt{(-2\delta+4\alpha)^{2}},$
(33)

while the mixedness of state (30) is denoted by $L(\mathcal{\rho^{C}}^{1})$
and is obtained as

$\displaystyle
L(\mathcal{\rho^{C}}^{1})=\frac{4}{3}-\frac{8}{3}(\alpha+\delta)^{2}-32\alpha^{2}-\frac{8}{3}\delta^{2}.$
(34)

Again the states $\rho^{(6)}$ defined in eq.(3) can be categorized in two
different forms, which we respectively denote by
$\rho^{(6a)}=r^{\prime}\rho^{W\bar{W}}+(1-r^{\prime})|\varphi^{+}\rangle\langle\varphi^{+}|$
and
$\rho^{(6b)}=r^{\prime}\rho^{W\bar{W}}+(1-r^{\prime})|\varphi^{-}\rangle\langle\varphi^{-}|$.
The density matrices of $\rho^{(6a)}$ and $\rho^{(6b)}$ are respectively
denoted by $\mathcal{\rho^{C}}^{2a}$ and $\mathcal{\rho^{C}}^{2b}$ and is
found of the following forms.

$\displaystyle\mathcal{\rho^{C}}^{2a}=\begin{pmatrix}\alpha&\alpha&\alpha&0\\\
\alpha&2\alpha+\beta&2\alpha+\beta&\alpha\\\
\alpha&2\alpha+\beta&2\alpha+\beta&\alpha\\\ 0&\alpha&\alpha&\alpha\\\
\end{pmatrix},\>\>\mathcal{\rho^{C}}^{2b}=\begin{pmatrix}\alpha&\alpha&\alpha&0\\\
\alpha&2\alpha+\beta&2\alpha-\beta&\alpha\\\
\alpha&2\alpha-\beta&2\alpha+\beta&\alpha\\\ 0&\alpha&\alpha&\alpha\\\
\end{pmatrix}.$ (35)

As before we use eqs. (1), (3) and (4) to calculate the concurrence,
teleportation fidelity and mixedness of the states defined in eq.(35).

The concurrence of $\mathcal{\rho^{C}}^{2a}$ is given by

$\displaystyle
C(\mathcal{\rho^{C}}^{2a})=\sqrt{f_{1}+\frac{\sqrt{f_{2}}}{2}}-\sqrt{f_{1}-\frac{\sqrt{f_{2}}}{2}}-\sqrt{f_{3}},$
(36)

where
$f_{1}=\frac{9}{2}\alpha^{2}+8\alpha\beta+2\beta^{2}$,$f_{2}=81\alpha^{4}+288\alpha^{3}\beta+312\alpha^{2}\beta^{2}+128\alpha\beta^{3}+16\beta^{4}$
and $f_{3}=\alpha^{2}$.

The eigenvalues of the matrix $T^{\dagger}T$ are

$\displaystyle w_{1}$ $\displaystyle=$ $\displaystyle(4\alpha+2\beta)^{2},$
$\displaystyle w_{2}$ $\displaystyle=$ $\displaystyle(4\alpha+2\beta)^{2},$
$\displaystyle w_{3}$ $\displaystyle=$ $\displaystyle(-2\alpha-2\beta)^{2}.$
(37)

so that the teleportation fidelity of the state $\mathcal{\rho^{C}}^{2a}$ is
found to be

$\displaystyle
f^{T}(\mathcal{\rho^{C}}^{2a})=\frac{1}{2}+\frac{1}{3}\sqrt{(4\alpha+2\beta)^{2}}+\frac{1}{6}\sqrt{(-2\alpha-2\beta)^{2}},$
(38)

while using eq.(4), the mixedness of the state $\mathcal{\rho^{C}}^{2a}$ is

$\displaystyle
L(\mathcal{\rho^{C}}^{2a})=\frac{4}{3}-\frac{40}{3}\alpha^{2}-\frac{16}{3}(2\alpha+\beta)^{2}.$
(39)

Again using eqs.(1),(3) and (4) we calculate the concurrence, teleportation
fidelity and mixedness of the state $\mathcal{\rho^{C}}^{2b}$. The concurrence
of $\mathcal{\rho^{C}}^{2b}$ is given by

$\displaystyle
C(\mathcal{\rho^{C}}^{2b})=2\sqrt{\alpha^{2}}-2\sqrt{\beta^{2}}.$ (40)

The eigenvalues of the matrix $T^{\dagger}T$ are

$\displaystyle z_{1}$ $\displaystyle=$ $\displaystyle(4\alpha-2\beta)^{2},$
$\displaystyle z_{2}$ $\displaystyle=$ $\displaystyle(4\alpha-2\beta)^{2},$
$\displaystyle z_{3}$ $\displaystyle=$ $\displaystyle(-2\alpha-2\beta)^{2}.$
(41)

The teleportation fidelity and the mixedness of the state
$\mathcal{\rho^{C}}^{2b}$ are however obtained as

$\displaystyle f^{T}(\mathcal{\rho^{C}}^{2b})$ $\displaystyle=$
$\displaystyle\frac{1}{2}+\frac{\sqrt{(4\alpha-2\beta)^{2}}}{3}+\frac{\sqrt{(-2\alpha-2\beta)^{2}}}{6},$
$\displaystyle L(\mathcal{\rho^{C}}^{2b})$ $\displaystyle=$
$\displaystyle\frac{4}{3}-\frac{104\alpha^{2}}{3}-\frac{16\beta^{2}}{3}.$ (42)

#### Teleportation fidelity of class $\rho^{(5)}$:

First of all, the convex combination of $\rho^{W\bar{W}}$ and $\phi^{+}$ is
denoted by $\mathcal{\rho^{C}}^{1}_{\phi^{+}}$ and the the convex combination
of $\rho^{W\bar{W}}$ and $\phi^{-}$ is denoted by
$\mathcal{\rho^{C}}^{1}_{\phi^{-}}$. Using eqs.(31), we observe that the
concurrence of state $\mathcal{\rho^{C}}^{1}_{\phi^{+}}$ are respectively
given as
$C(\mathcal{\rho^{C}}^{1}_{\phi^{+}})=\frac{\sqrt{u+6\sqrt{v}}}{12}-\frac{\sqrt{u-6\sqrt{v}}}{12}-\frac{\sqrt{r^{2}}}{6}$,
where $u=18r^{2}+48r\frac{1-r}{2}+288\frac{(1-r)^{2}}{4}$ and
$v=-135r^{4}+72r^{3}+408r^{2}-480r+144$. The concurrence is positive when
$0\leq r<0.6$ and when $0.75<r\leq 1$. From eqs.(33) and (34) we find the
teleportation fidelity and mixedness of the state
$\mathcal{\rho^{C}}^{1}_{\phi^{+}}$ respectively as
$f^{T}(\mathcal{\rho^{C}}^{1}_{\phi^{+}})=\frac{1}{2}+\frac{1}{6}[\sqrt{(\frac{4r}{3}-1)^{2}}+\sqrt{(1-\frac{r}{3})^{2}}+\sqrt{(-1+\frac{5r}{3})^{2}}]$
and $L(\mathcal{\rho^{C}}^{1}_{\phi^{+}})=-\frac{50}{27}r^{2}+\frac{20}{9}r$
while the state is mixed for all $r$. For parameter $0\leq r\leq 1$, the state
$\mathcal{\rho^{C}}^{1}_{\phi^{+}}$ of eq.(3)
$N(\mathcal{\rho^{C}}^{1}_{\phi^{+}})>1$ and teleportation fidelity of the
state exceeding the classical fidelity of $\frac{2}{3}$. But the state
$\mathcal{\rho^{C}}^{1}_{\phi^{+}}$ is entangled for $r\in[0,\>0.6)$ and
$(0.75,\>1]$. So, in this two ranges the state can bes suitably used as
quantum teleportation channel successfully. It is also to be noted that at the
extreme points i.e. when $r=0$ or $r=1$, the state’s concurrence,
teleportation fidelity and mixedness are aligned with that of Bell state and
two qubit $\rho^{W\bar{W}}$ from eq.(3). Similarly, for the state
$\mathcal{\rho^{C}}^{1}_{\phi^{-}}$, using eq.(31), it can be easily shown
that the state $\mathcal{\rho^{C}}^{1}_{\phi^{-}}$ is entangled when $0\leq
r\leq 0.39$ and when $0.87<r\leq 1$. Also,
$N(\mathcal{\rho^{C}}^{1}_{\phi^{-}})>1$ and
$f^{T}(\mathcal{\rho^{C}}^{1}_{\phi^{-}})>\frac{2}{3}$ for all $r$. Hence the
state $\mathcal{\rho^{C}}^{1}_{\phi^{-}}$ can suitably used as quantum
teleportation channel for $0\leq r\leq 0.39$ and when $0.87<r\leq 1$.

#### Teleportation fidelity of class $\rho^{(6)}$:

The concurrence of state $\mathcal{\rho^{C}}^{2a}$ of (35) has concurrence of
the form $C(\mathcal{\rho^{C}}^{2a})$ which is
$\frac{x+6\sqrt{y}}{12}-\frac{x-6\sqrt{y}}{12}-\frac{\sqrt{r^{\prime^{2}}}}{6}$,
where
$x=18r^{\prime^{2}}+192r^{\prime}\frac{1-r^{\prime}}{2}+288\frac{(1-r^{\prime})^{2}}{2}$
and $y=-15r^{\prime^{4}}+48r^{\prime^{3}}+24r^{\prime^{2}}-192r+144$. The
mixedness of the state is
$L(\mathcal{\rho^{C}}^{2a})=\frac{8}{9}r^{\prime}-\frac{14}{27}r^{\prime^{2}}$
and the teleportation fidelity $f^{T}(\mathcal{\rho^{C}}^{2a})$ is
$\frac{1}{2}+\frac{1}{3}\sqrt{(1-\frac{r^{\prime}}{3})^{2}}+\frac{\sqrt{(\frac{2r^{\prime}}{3}-1)^{2}}}{6}$.
It is being observed that the state $\mathcal{\rho^{C}}^{2a}$ is entangled
with non-negative concurrence $0\leq r^{\prime}\leq 1$. Also
$N(\mathcal{\rho^{C}}^{2a})>1$ and
$f^{T}(\mathcal{\rho^{C}}^{2a})>\frac{2}{3}$ for all $r^{\prime}$.

Again for the state $\mathcal{\rho^{C}}^{2b}$ of eq.(3), the teleportation
fidelity, concurrence and mixedness as obtained by eqs.(40)and (5), are
$C(\mathcal{\rho^{C}}^{2b})=\frac{\sqrt{r^{\prime^{2}}}}{3}-2\sqrt{(\frac{1-r^{\prime}}{2})^{2}}$,
$f^{T}(\mathcal{\rho^{C}}^{2b})=\frac{1}{2}+\frac{\sqrt{(\frac{2r^{\prime}}{3}-1)^{2}}}{6}+\frac{\sqrt{(\frac{5r^{\prime}}{3}-1)^{2}}}{3}$,
and
$L(\mathcal{\rho^{C}}^{2b})=\frac{8}{3}r^{\prime}-\frac{62}{27}r^{\prime^{2}}$.
It is seen that the state $\mathcal{\rho^{C}}^{2b}$ is entangled with non-
negative concurrence when $0.75\leq r^{\prime}\leq 1$. The state is useful as
quantum teleportation channel when $0\leq r^{\prime}\leq 0.5$ and $0.75\leq
r^{\prime}\leq 1$ as in these ranges $N(\mathcal{\rho^{C}}^{2b})>1$ and
teleportation fidelity exceeds classical limit of $\frac{2}{3}$.

#### Bell-CHSH violation of class $\rho^{(5)}$:

For the class of states $\mathcal{\rho^{C}}^{1}_{\phi^{+}}$ of eq.(3), we
observed that the concurrence is greater than zero $\forall$ $r$. Also using
eqs.(5) and (5) we calculate $M(\mathcal{\rho^{C}}^{1}_{\phi^{+}})$ which is
$(\frac{4r}{3}-1)^{2}+(1-\frac{r}{3})^{2}$. Now
$M(\mathcal{\rho^{C}}^{1}_{\phi^{+}})>1$ when $0\leq r\leq 0.38$ whereas for
$0.38<r\leq 1$ satisfies Bell’s inequality. Therefore the given state is
entangled violating Bell’s inequality for $0\leq r\leq 0.38$ while for
$0.38<r<0.6$, the state is entangled satisfying Bell’s inequality. The given
state is also entangled satisfying Bell’s inequality when $0.75<r\leq 1$.
Again,
$M(\mathcal{\rho^{C}}^{1}_{\phi^{-}})=(1-\frac{r}{3})^{2}+(-1+\frac{5r}{3})^{2}$
and it is greater than unity when $0\leq r\leq 0.32$ whereas
$M(\mathcal{\rho^{C}}^{1}_{\phi^{-}})<1$ when $0.32<r\leq 1$. Combining the
facts, we can immediately conclude that the state
$\mathcal{\rho^{C}}^{1}_{\phi^{-}}$ is entangled violating Bell’s inequality
for $r\in[0,\>0.32]$ and is entangled satisfying Bell’s inequality for
$r\in(0.32,0.39)$ and also for $r\in(0.87,\>1]$.

Thus we see that the state $\mathcal{\rho^{C}}^{1}_{\phi^{+}}$, although
satisfies Bell’s inequality in the ranges $(0.38,\>0.6)$ and $(0.75,\>1]$, is
useful as quantum teleportation channel. Also the state
$\mathcal{\rho^{C}}^{1}_{\phi^{-}}$ is useful as quantum teleportation channel
although the state satisfies Bell’s inequality in the ranges $0.32<r<0.38$ and
$0.87<r\leq 1$. Hence is is seen that the class of states $\rho^{(5)}$ is
another example of states which can be suitably used as teleportation channel
while satisfying Bell’s inequality for some specified ranges of the state
parameter $r$.

#### Bell-CHSH violation of class $\rho^{(6)}$:

With respect to the states defined by $\rho^{(6)}$ of eq.(3), next we observe
that the state $\mathcal{\rho^{C}}^{2a}$ of eq.(35) is entangled and useful
for the purpose of teleportation in the entire admissible range of parameter
$r^{\prime}$ in spite of satisfying Bell-inequality in the range $0.878\leq
r^{\prime}\leq 1$. Also we found that the state $\mathcal{\rho^{C}}^{2b}$
represented in eq.(35) which also corresponds to the state $\rho^{(6)}$ of
eq.(3), satisfies Bell-CHSH inequality when $0.87<r\leq 1$.

This again leads to the conclusion that the states defined by $\rho^{(6)}$ can
be useful as teleportation channel although such states satisfy Bell-CHSH
inequality at certain regions of the states’ parameter.

#### Comparison of teleportation fidelities of NMEMS with MEMS:

The teleportation fidelities of the states $\rho^{W}$ and $\rho^{\bar{W}}$ of
eq.(3) are $\frac{7}{9}$ and these states are Ishizaka-Hiroshima class of
maximally entangled mixed states. As compared to these MEMS, the states
$\rho^{(1)}$ (or $\rho^{(2)}$), $\rho^{(3)}$ (or $\rho^{(4)}$) ($X$ type NMEMS
of eqs.(3) and (3)) and $\rho^{(5)}$ (or $\rho^{(6)}$) (non-$X$ type NEMS of
eq.(3)) give better performance as quantum teleportation channels for certain
range of state parameters. The following figure shows this comparative
assessment of teleportation fidelities of these states.

Figure 2: The figure shows the plots of teleportation fidelity of the MEMS
i.e. $f^{T}(\rho^{W})$ (or of $f^{T}(\rho^{\bar{W}})$) with value
$\frac{7}{9}$ for $0\leq p\leq 1$ (line parallel to horizontal axis) and the
teleportation fidelities of the other NMEMSs $\rho^{(5)}$ and $\rho^{(6)}$.

It is to be noted that $\rho^{(5)}$ states of eq.(3) whose form is of non-$X$
type has the teleportation fidelity similar to that of $X$ type states
$\rho^{(1)}$ and $\rho^{(3)}$ of eqs.(3) and (3). $\rho^{(6)}$ has been
described by two forms viz. $\rho^{C^{2a}}$ and $\rho^{C^{2b}}$ of eq.(35).
They are non-maximally entangled mixed states of non-$X$ type. The
teleportation fidelity of $\rho^{C^{2a}}$ is much higher than $\rho^{C^{2b}}$.
The horizontal line in the fig.$2$ represents the teleportation fidelity of
$\rho^{W}$ (and $\rho^{\bar{W}}$) of eq.(3) which is $\frac{7}{9}$ for both of
these MEMS. It is being observed that $\rho^{C^{1}}$ (which are basically of
$\rho^{(5)}$ types) performs better as quantum teleportation channel than
$\rho^{C^{2b}}$ (which are of $\rho^{(6)}$ type). Also for $0\leq p\leq 0.35$
the teleportation fidelity of $\rho^{C^{2b}}$ is higher than that of
$\rho^{W}$ while for $0\leq p\leq 0.4$, the teleportation fidelity of
$\rho^{C^{1}}$ is greater than that of $\rho^{W}$. Also it is interesting to
note down that the states $\rho^{C^{2a}}$ acts as better candidate as quantum
teleportation channel than the maximally entangled mixed state for $0\leq
p\leq 1$. (It is to be remembered that in this comparative study too, we have
plotted all the teleportation fidelities against a common state parameter, say
$p$).

###  5.1 Teleportation fidelity and Bell-CHSH violation of mixture of two
qubit state derived from $|Star\rangle$ and Bell-states:

We here consider the state $\tau^{(1)}$ defined in eq.(3) and study the
teleportation fidelitiy of the states along with its behaviour with respect to
Bell-CHSH violation. We also observe that the states $\tau^{(1)}$ fall into
the class of non $X$ type. For the clarity of the inspection we define the
mixture of $\rho^{star}$ and Bell-state $|\phi^{+}\rangle$ from state
$\tau^{(1)}$ as $\tau^{\phi+}$ and the mixture of $\rho^{star}$ and Bell-state
$|\phi^{-}\rangle$ from state $\tau^{(1)}$ as $\tau^{\phi-}$. The density
matrices of the non-$X$ type classes to which they belong are given as
follows.

$\displaystyle\mathcal{\tau^{\phi+}}=\begin{pmatrix}2\alpha&\alpha-\beta&0&\alpha+\beta\\\
\alpha-\beta&\alpha-\beta&0&\alpha-\beta\\\ 0&0&0&0\\\
\alpha+\beta&\alpha-\beta&0&\alpha+\beta\\\
\end{pmatrix},\>\>\mathcal{\tau^{\phi-}}=\begin{pmatrix}2\alpha&\alpha-\beta&0&\alpha-3\beta\\\
\alpha-\beta&\alpha-\beta&0&\alpha-\beta\\\ 0&0&0&0\\\
\alpha-3\beta&\alpha-\beta&0&\alpha+\beta\\\ \end{pmatrix}.$ (43)

Using eqs.(1),(3) and (4) we determine the concurrence, teleportation fidelity
and mixedness of the state (43) which are summarized below. Thus the
concurrence, teleportation fidelity and mixedness of the states are denoted
respectively by $C(\tau^{\phi\pm})$, $f^{T}(\tau^{\phi\pm})$ and
$L(\tau^{\phi\pm})$ and are given by,

$\displaystyle C(\tau^{\phi+})$ $\displaystyle=$
$\displaystyle\sqrt{\alpha+\beta}\Big{(}\sqrt{3\alpha+\beta+2\sqrt{2\alpha(\alpha+\beta)}}-\sqrt{3\alpha+\beta-2\sqrt{2\alpha(\alpha+\beta)}}\Big{)},$
$\displaystyle C(\tau^{\phi-})$ $\displaystyle=$
$\displaystyle\sqrt{u+2\sqrt{v}}-\sqrt{u-2\sqrt{v}},$ (44)

where $u=3\alpha^{2}-4\alpha\beta+9\beta^{2}$ and
$v=2\alpha^{4}-10\alpha^{3}\beta+6(\alpha\beta)^{2}+18\alpha\beta^{3}$.

$\displaystyle f^{T}(\tau^{\phi+})$ $\displaystyle=$
$\displaystyle\frac{1}{2}+\frac{1}{3}\sqrt{(\alpha+\beta)^{2}}+\frac{2}{3}\sqrt{2(\alpha^{2}+\beta^{2})},$
$\displaystyle f^{T}(\tau^{\phi-})$ $\displaystyle=$
$\displaystyle\frac{1}{2}+\frac{\sqrt{(\alpha+\beta)^{2}}}{3}+\frac{1}{3}\Big{(}\sqrt{u^{\prime}+v^{\prime}}+\sqrt{u^{\prime}-v^{\prime}}\Big{)},$
(45)

where $u^{\prime}=2\alpha^{2}-4\alpha\beta+6\beta^{2}$ and
$v^{\prime}=4\sqrt{2}(\beta^{2}-\alpha\beta)$.

and

$\displaystyle L(\tau^{\phi+})$ $\displaystyle=$
$\displaystyle\frac{4}{3}-16\alpha^{2}+\frac{16}{3}(\alpha\beta-2\beta^{2}),$
$\displaystyle L(\tau^{\phi-})$ $\displaystyle=$
$\displaystyle\frac{4}{3}-16\alpha^{2}+\frac{80}{3}\alpha\beta-32\beta^{2}.$
(46)

#### Teleportation fidelities of $\mathcal{\tau^{\phi\pm}}$ and
$\mathcal{\tau^{\psi\pm}}$:

Using eqs.(5.1),(5.1) and (5.1), it is being observed that for the state
$\tau^{\phi+}$, the concurrence, teleportation fidelity and mixedness are
given to be
$C(\tau^{\phi+})=\sqrt{\frac{1+s}{4}}\sqrt{\frac{3}{4}+\frac{s}{4}+\frac{1}{2}\sqrt{2+2s}}-\sqrt{\frac{3}{4}+\frac{s}{4}-\frac{1}{2}\sqrt{2+2s}}$,
$f^{T}(\tau^{\phi+})=\frac{1}{2}+\frac{1}{3}\sqrt{\frac{1}{4}(1+s)^{2}}+\frac{1}{6}\sqrt{2s^{2}+2}$
and $L(\tau^{\phi+})=\frac{1}{3}(1+s-2s^{2})$. It is found that the state
$\tau^{\phi+}$ is entangled for $0\leq s\leq 1$ and can be used as quantum
teleportation channel as $N(\tau^{\phi+})>1$ in this range, teleportation
fidelity being greater than $\frac{2}{3}$ also. Moreover, for the state
$\tau^{\phi-}$, the the concurrence, teleportation fidelity and mixedness are
given to be
$C(\tau^{\phi-})=\frac{1}{4}\Big{(}\sqrt{3-4s+9s^{2}+2\sqrt{18s^{3}+6s^{2}-10s+2}}-\sqrt{3-4s+9s^{2}-2\sqrt{18s^{3}+6s^{2}-10s+2}}\Big{)}$,
$f^{T}(\tau^{\phi-})=\frac{1}{2}+\frac{1}{3}\sqrt{(\frac{3s+1}{4})^{2}}+\frac{1}{12}\Big{(}\sqrt{4\sqrt{2}(s^{2}-s)+2-4s+6s^{2}}+\sqrt{4\sqrt{2}(s-s^{2})+2-4s+6s^{2}}\Big{)}$
and $L(\tau^{\phi-})=\frac{1}{3}(1+5s)-2s^{2}$. Just like the state
$\tau^{\phi+}$, the state $\tau^{\phi-}$ is also entangled for $0\leq s\leq 1$
and the state is mixed in this region. The state can be used as teleportation
channel as $N(\tau^{\phi-})>1$ when $0\leq s\leq 1$ with teleportation
fidelity $f^{T}(\tau^{\phi-})$ exceeding $\frac{2}{3}$ in two different
regions, one when $0\leq s\leq 0.314$ and another when $0.43\leq s\leq 1$. It
is also to be noted that when $s=0$ state $\tau^{(1)}$ is maximally entangled
pure state ($|\phi^{+}\rangle$ or $|\phi^{-}\rangle$) which is a pure state of
course with unit teleportation fidelity whereas when $s=1$, the state is
$\rho^{star}$ which is a two qubit state derive from $3-$ qubit $|Star\rangle$
state with concurrence $\frac{1}{2}$ and teleportation fidelity as $0.81$.

#### Bell-CHSH violation of $\mathcal{\tau^{\phi\pm}}$ and
$\mathcal{\tau^{\psi\pm}}$:

Using eq.(5), we can see that the state $\tau^{\phi+}$ violates Bell-
inequality as $M(\tau^{\phi+})=\frac{1+s^{2}}{2}>1$ implying that the state is
entangled $0\leq s\leq 1$. Again from eq.(5), we see that the state
$\tau^{\phi-}$ violates Bell-inequality as
$M(\tau^{\phi-})=\sqrt{2}s(1-s)+\frac{1}{2}+s\Big{(}\frac{3}{2}s-1\Big{)}>1$
implying that the state is entangled $0\leq s\leq 1$.

To conclude the analysis of mixture of two qubit state derived from
$|Star\rangle$ and Bell states, we next consider the states $\tau^{(2)}$
defined in eq.(3). The density matrices of mixture of $\rho^{star}_{BC}$ with
Bell states $|\psi^{+}_{BC}\rangle$ and $|\psi^{-}_{BC}\rangle$ are given
respectively as follows:

$\displaystyle\mathcal{\tau^{\psi+}}=\begin{pmatrix}2(\alpha-\beta)&\alpha-\beta&0&\alpha-\beta\\\
\alpha-\beta&\alpha+\beta&2\beta&\alpha-\beta\\\ 0&2\beta&2\beta&0\\\
\alpha-\beta&\alpha-\beta&0&\alpha-\beta\\\
\end{pmatrix},\>\>\mathcal{\tau^{\psi-}}=\begin{pmatrix}2\beta&\beta&0&\beta\\\
\beta&2\alpha-\beta&-2(\alpha-\beta)&\beta\\\
0&-2(\alpha-\beta)&2(\alpha-\beta)&0\\\ \beta&\beta&0&\beta\\\ \end{pmatrix}.$
(47)

Next we proceed for similar analysis of teleportation fidelity of states
$\tau^{(2)}$ as done for states $\tau^{(1)}$. With minor moderations in the
terms of the density matrices defined in eq.(47) it can easily be shown that
$C(\tau^{\phi^{\pm}})=C(\tau^{\psi^{\pm}})$. Likewise it is observed that
$L(\tau^{\phi^{\pm}})=L(\tau^{\psi^{\pm}})$ and
$f^{T}(\tau^{\phi^{\pm}})=f^{T}(\tau^{\psi^{\pm}})$ as well.

We now plot below the teleportation fidelities of the bipartite mixture of
states that were defined in sec.3 such as $\rho^{(1)}(and\>\rho^{(2)})$ from
eq.(3), $\rho^{(3)}(and\>\rho^{(4)})$ from eq.(3),
$\rho^{(5)}(and\>\rho^{(6)})$ from eq.(3) and $\tau^{(1)}(and\>\tau^{(2)})$
from eq.(3) by taking a common framework for comparison of teleportation
fidelities of the given states, for which we have varied a common state
parameter (say $p$) along the horizontal axis from $0$ to $1$. The states of
eqs.(3) and (3) are of $X$ type and those of eqs.(3) and (3) are of non-$X$
types. It is also to be remembered that as $|\bar{W}\rangle$ is the spin
flipped version of $|W\rangle$, and as $|W\bar{W}\rangle$ is the state
constructed as linear superposition of $|W\rangle$ and $|\bar{W}\rangle$ and
although $|W\bar{W}\rangle$ is of non-$X$ type state while both $|W\rangle$
and $|\bar{W}\rangle$ are of $X$ type, yet the nature of the teleportation
fidelities of these states are similar. In the following figure we plot
teleporatation fidelities of $\rho^{(1)}$, $\rho^{(6)}$, $\tau^{\phi+}$ and
$\tau^{\phi-}$ to compare how these states behave as quantum teleportation
channels.

Figure 3: The figure represents the teleportation fidelities of the states of
subclass (A) and subclass (B) The states $\tau^{\phi+}$, $\tau^{\phi-}$,
$\rho^{(1)}$ and $\rho^{(6)}$ have been plotted against common state parameter
$p$ ranging from $0$ to $0.45$.

It is being observed from fig.$3$ that the states of subclass (A) and subclass
(B) are both useful as quantum teleportation channels. The states $\rho^{(1)}$
of subclass (A) is a non-maximally entangled mixed state of $X$ type whereas
the states $\tau^{\phi+}$, $\tau^{\phi-}$ and $\rho^{(6)}$ are non-maximally
entangled mixed state of non-$X$ type. The state $\rho^{(6)}$ which has been
constructed by taking convex mixture of two qubit state derived from
$|W\bar{W}\rangle$ and two qubit singlet state performs better as quantum
teleportation channels than the state $\tau^{\phi+}$ (which is a convex
mixture of two qubit state derived from $|Star\rangle$ and Bell state) for
$0\leq p\leq 0.25$ and again for $0.25<p\leq 0.45$. The state $\tau^{\phi+}$
proves to be better candidate as the teleportation channel than $\rho^{(6)}$.
However the state $\rho^{(6)}$ is always a better performer than the state
$\rho^{(1)}$ (of subclass (A)) as teleportation channel. On the other hand,
the state $\tau^{\phi-}$ of eq.(3) also acts as a better candidate as quantum
teleportation channel than the states $\rho^{(1)}$ and $\rho^{(6)}$ when
$p>0.5$.

Finally we also observe that all the non-maximally entangled mixed states
defined in this paper outperforms MEMS $\rho^{W}$ and $\rho^{\bar{W}}$ as
quantum teleportation channels (note that
$f^{T}(\rho^{W})=f^{T}(\rho^{\bar{W}})=\frac{7}{9}$). We also plot in the
following the teleportation fidelity of the states $\tau^{\phi+}$,
$\tau^{\phi-}$ and $\rho^{(werner)}$ against the common state parameter
$\frac{1}{2}\leq p\leq 1$, since in this range the teleportation fidelity of
Werner state exceeds classical limit of $\frac{2}{3}$.

Figure 4: The figure shows the plot of teleportation fidelities of the states
$\tau^{\phi+}$ , $\tau^{\phi-}$ and $\rho^{(werner)}$, set against the
parameter $p$.

We see from the fig.$3$ that the teleportation fidelities of $\tau^{\phi+}$
and $\tau^{\phi-}$ surpasses the teleportation fidelity of Werner state for
$p>0.5$. This implies that the teleportation fidelities of non-maximally
entangled mixed states of eq.(3) outperform that of maximally entangled mixed
state such as Werner state. We know that Werner state and $\rho^{W}$ of eq.(3)
are both Ishizaka-Hiroshima class of maximally entangled mixed states. Werner
state outperforms $\rho^{W}$ when $p>0.6$ which is also clear from fig.$4$.
The class of NMEMS of non-$X$ type defined in eq.(3) outperforms both of these
MEMS of Ishizaka-Hiroshima class.

##  6 Conclusion:

To summarize, in this paper we have studied the efficiency of a few non-
maximally entangled mixed states (NMEMS) as resources for quantum
teleportation. Horodecki et.al showed that there exist inseparable states
which are not useful for teleportation within the standard scheme. It was also
proposed that the states which violate generalized Bell-CHSH inequality are
useful for teleportation[19]. Later Adhikari et.al had shown that not all
maximally entangled mixed states (MEMS) are useful for teleportation. Munro
class of states is one such example. These states are not useful as quantum
teleportation channel when their mixedness exceeds a certain bound while
Werner state is another type of maximally entangled mixed states of Ishizaka-
Hiroshima type, which are although less entangled for a given degree of
mixedness than Munro class, could be more useful as a quantum teleportation
channel[1]. Moreover Werner states are such states which do not violate Bell-
CHSH inequality but are entangled. Adhikari et.al also proposed a class of
NMEMS states and had shown that they could perform as quantum teleportation
channels in spite of satisfying Bell-CHSH inequality. It has been found that,
magnitude of entanglement of the states and violation of Bell-CHSH inequality
by the states are both not good indicators of their capacity to perform as
quantum teleportation channels[1]. In this light, we have proposed here a few
class of non-maximally entangled mixed states and explored their ability to
act as quantum teleportation channels and to know how they behave with respect
to Bell-violation.

For this we have constructed mixtures of class of Bell states with two qubit
mixed states derived from three qubit pure states like
$|W\rangle,\>|\bar{W}\rangle$, $|W\bar{W}\rangle$ and $|Star\rangle$ by
removing parties. It is found that some mixtures are of $X$ type and others
are not of $X$ type. Our motivation in this work is to explore the
characteristics of these two different types of states from the perspective of
their usefulness as quantum teleportation channels and how they behave with
respect to Bell-CHSH inequality violation. The states $\rho^{(1)}$ (or
$\rho^{(2)}$) and $\rho^{(3)}$ (or $\rho^{(4)}$) are of similar nature as both
of these class of states have been obtained as convex combination of two qubit
state derived from $|W\rangle$ and $|\bar{W}\rangle$ respectively and Bell
states where $|\bar{W}\rangle$ is the spin flipped version of $|W\rangle$.
Moreover the states $\rho^{(1)}$ (or $\rho^{(2)}$) and $\rho^{(3)}$ (or
$\rho^{(4)}$) belong to class of $X$ type states. On the other hand the state
$\rho^{(5)}$ (or $\rho^{(6)}$), which has been constructed as mixtures of two-
qubit states derived from $3-$ qubit $|W\bar{W}\rangle$ and Bell states are
found not only to be useful as teleportation channels but also one of such
non-$X$ type states viz. $\rho^{C^{2a}}$ of eq.(35) have higher teleportation
fidelity for specific range of state parameter than that of $\rho^{(1)}$ (or
$\rho^{(2)}$) and $\rho^{(3)}$ (or $\rho^{(4)}$). We know that tripartite
$|GHZ\rangle$ and $|W\rangle$ are two different SLOCC class of states. We have
also constructed mixtures of two qubit state derived from $|GHZ\rangle$ and
Bell states, which we have designated in the paper by the notation $\rho^{g}$
and when compared with states like $\rho^{(1)}$ (and $\rho^{(2)}$),
$\rho^{(3)}$ (and $\rho^{(4)}$) and $\rho^{(5)}$ (and $\rho^{(6)}$), it is
observed that the teleportation fidelity of the state $\rho^{g}$ is less than
that of the rest for certain range of state parameter. It is to be noted that
$\rho^{g}$ is $X$ type states. It is interesting to note down that the
teleportation fidelities of these bipartite non-maximally entangled mixed
states $\rho^{(1)}$ (and $\rho^{(2)}$), $\rho^{(3)}$ (and $\rho^{(4)}$) (of
$X$ type) and $\rho^{(5)}$ (and $\rho^{(6)}$) (of non-$X$ type) exceed the
teleportation fidelity of MEMS i.e. $f^{T}(\rho^{W})$ (or
$f^{T}(\rho^{\bar{W}})$) of eq.(3). Bipartite NMEMS obtained from mixtures of
two qubit states derived from tripartite $|Star\rangle$ and Bell states
(denoted as $\tau^{\phi+}$ (and $\tau^{\phi-}$)) are good candidates for
quantum teleportation task. First of all, these states are of non-$X$ type.
Secondly, unlike the other states which are similar with respect to whichever
qubit is removed from the tripartite counterpart, to build states like
$\tau^{\phi+}$ (and $\tau^{\phi-}$)), only the peripheral qubits can be
removed from the tripartite $|Star\rangle$ and not the central qubit (as
removal of central qubit makes the states separable). It is found however that
the states $\tau^{\phi+}$ (and $\tau^{\phi-}$)) give better performance as
quantum teleportation channels than those of $\rho^{(1)}$ (and $\rho^{(2)}$),
$\rho^{(3)}$ (and $\rho^{(4)}$) and $\rho^{(5)}$ (and $\rho^{(6)}$) and
$\rho^{g}$. The states $\tau^{\phi+}$ (and $\tau^{\phi-}$)) are NMEMS and we
also observe that the teleportation fidelities of such NMEMS exceed the
teleportation fidelity of Werner state $\rho^{(werner)}$ (which is a MEMS).

The study of Bell-CHSH inequality violation by the states is another aspect
which have been looked upon in this work. Althoughit is known that a state
indicating Bell-violation is suitable as teleportation channel[19], yet there
are states which are entangled but satisfy Bell-inequality and are also useful
for teleportation [1]. The states $\rho^{(1)}$ (and $\rho^{(2)}$),
$\rho^{(3)}$ (and $\rho^{(4)}$) of eqs.(3) and (3) of $X$ type satisfy Bell-
inequality and yet they are entangled, shown to be useful as teleportation
channels for specific ranges of state parameter. In case of states of non-$X$
type, that we have constructed, behave differently. The states $\rho^{(5)}$
(and $\rho^{(6)}$) of eq.(3) are examples of states which satisfy Bell-
inequality for specified ranges of state parameter, yet they are entangled and
useful for teleportation whereas another non-$X$ types of states
$\tau^{\phi+}$ (and $\tau^{\phi-}$)) are found to violate Bell-inequality.
Consequently these states are entangled can be useful as quantum teleportation
channels.

In future, we can analyze the decoherence effects on these NMEMS states. The
experimental realization of these non-maximally entangled mixed states can
also be studied.

## Data Availability Statement

The authors confirm that the data supporting the findings of this study are
available within the article.

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