# Exploring quantum properties of bipartite mixed states under coherent and
incoherent basis

Sovik Roy1, Anushree Bhattacharjee1
1 Department of Mathematics, Techno Main Salt Lake (Engg. Colg.),
Techno India Group, EM 4/1, Sector V, Salt Lake, Kolkata 700091, India

Chandrashekar Radhakrishnan2
2 Centre for Quantum Information, Communication and Computing (CQuICC),
Indian Institute of Technology Madras, Chennai 600036, 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

s.roy2.tmsl@ticollege.orga.bhattacharya.tmsl@ticollege.orgchandrashekar@physics.iitm.ac.inmanirul<EMAIL_ADDRESS>
###### Abstract

Quantum coherence and quantum entanglement are two different manifestations of
the superposition principle. In this article we show that the right choice of
basis to be used to estimate coherence is the separable basis. The quantum
coherence estimated using the Bell basis does not represent the coherence in
the system, since there is a coherence in the sytem due to the choice of the
basis states. We first compute the entanglement and quantum coherence in the
two qubit mixed states prepared using the Bell states and one of the states
from the computational basis. The quantum coherence is estimated using the
$\ell_{1}$-norm of coherence, the entanglement is measured using the
concurrence and the mixedness is measured using the linear entropy. Then we
estimate these quantities in the Bell basis and establish that coherence
should be measured only in separable basis, whereas entanglement and mixedness
can be measured in any basis. We then calculate the teleportation fidelity of
these mixed states and find the regions where the states have a fidelity
greater than the classical teleportation fidelity. We also examine the
violation of the Bell-CHSH inequality to verify the quantum nonlocal
correlations in the system. The estimation of the above mentioned quantum
correlations, teleportation fidelity and the verification of Bell-CHSH
inequality is also done for bipartite states obtained from the tripartite
systems by the tracing out of one of their qubits. We find that for some of
these states teleportation is possible even when the Bell-CHSH inequality is
not violated, signifying that nonlocality is not a necessary condition for
quantum teleportation.

Keywords: Coherence, Concurrence, Linear Entropy, Teleportation Fidelity,
Mixedness, Bell-CHSH inequality

Pacs No.: 03.65.Ud 03.67.Mn 03.67.-a

##  1 Introduction:

In quantum mechanics, superposition gives rise to two interesting phenomena,
one of which is coherence and the other is known as entanglement[1, 2].
Although the entanglement and coherence are two different manifestations of
the superposition principle, they are different in their nature and
properties. The quantum entanglement measures the non-local correlations in
the systems and the quantum coherence is a measure of the total quantumness in
the system. While quantum entanglement, when measured in any reference basis,
retains its trait, quantum coherence is a basis dependent quantity [3, 4].
This has generated an upsurge of interest on coherence and, for the last
several years the exploration of this feature has gained momentum and has been
studied in many different areas, alongside the study of entanglement [5, 6, 7,
8, 9]. A considerable amount of research works on entanglement of bipartite
and as well as of multipartite systems have already been done[10, 11, 12, 13].
The focus of this article will be on bringing coherence and entanglement
together in a bipartite two qubit mixed state scenario and to observe how
these two features together contribute to the capacity of transferring
information via such mixed states in a protocol such as teleportation. Such a
capacity of states is, called teleportation fidelity.

Another interesting fact in the class of mixed states is that there exists a
mixed entangled state, known as Werner state, which does not violate Bell’s
inequality[14]. Moreover Werner state can be used as a quantum teleportation
channel (average optimal teleportation fidelity exceeding $\frac{2}{3}$) even
without violating the Bell-CHSH inequality[13]. We know Bell inequality marks
the boundary between the classical and quantum natures. The relation between
Bell violation and entanglement has been studied already for a class of two
qubit maximally entangled mixed states [10, 11, 12, 13]. Hence it would also
be interesting to examine how coherence and Bell violation are connected to
one another for the class of states considered in this paper. It has been
observed that in most of the cases the computational techniques of measuring
entanglement, coherence, mixedness et. al involve computational basis. In this
article, we shall make a comparative study of the quantification of these
features with respect to both computational basis and Bell basis. Recently in
an article [15] it has been shown that for any general two qubit density
matrix $\rho$, $\ell_{1}-$ norm of coherence, which is a measure of quantum
coherence is always greater than or equal to concurrence of that density
matrix. We will show by considering a class of mixed states that, the relation
between coherence and concurrence is actually dependent on the choice of the
basis under which they are measured and in a product basis where the
multipartite basis is a product of the individual bases, then the $\ell_{1}-$
norm of coherence is greater than or equal to that of the entanglement as
measured by concurrence. When we use an entangled basis like Bell basis this
inequality does not hold. This fact motivates us to explore how teleportation
fidelities of the class of mixed states behave in accordance with coherence,
entanglement and mixedness along with the Bell violation for these class of
mixed states.

The paper is organized as follows. In section $2$ we discuss the quantifiers
of coherence, entanglement and mixedness. It is to be pointed out that,
although relative entropy of coherence is defined in the next section, in most
of the calculations we have considered the $l_{1}-$ norm of coherence. The
relative entropy and $\ell_{1}-$ norm, however, satisfies all the
characteristics of good coherence measure. The mixedness of the class of
states will also be studied and their relationship with both coherence and
concurrence is investigated. The basic objective of studying any quantum
mechanical state (pure or mixed) is to check its utility in information
processing. Also we describe the two different classes of mixed states which
we use in our investigations. In section $3$, we compute the different
measures for the two classes of states introduced in the previous section in
both the computational basis and in the Bell basis. The teleportation fidelity
of these two classes of states is computed in the section $4$ and the Bell-
CHSH violation for these states is investigated in section $5$. A
comprehensive analysis of two qubit reduced density matrices derived from
tripartite states is given in section $6$. For all these states, we find
quantum coherence, concurrence, mixedness, teleportation fidelity and the
Bell-CHSH inequality. It is well known that there is a hierarchy among all
these quantum properties [16] and they are also closely related. In this
scenario,for a complete knowledge of the quantum states it is imperative to
have a holistic view of the qualitative nature of these states. To achieve
this, we study quantum coherence and entanglement separately. The
teleportation fidelity gives us knowledge about the amount of entanglement
that is useful in practical applications like teleportation and all
entanglement cannot be used for practical applications. The Bell-CHSH
inequality verifies the nonlocality of the quantum states. While nonlocality
is related to entanglement, it does not mean that all entangled states are
nonlocal. Thus, the four different quantities we measure give different kind
of information about the quantum states under study and this helps us to have
an overall picture of the states and their possible applications. So, the main
aim of the paper is to characterize the bipartite mixed states in a
comprehensive manner. Finally we present our conclusions in section $7$.

##  2 Quantumness measures and Quantum States

###  2.1 Measures of coherence, entanglement and mixedness

####  2.1.1 Quantum coherence:

Quantum coherence originates from the principle of superposition and hence it
manifests itself in the off-diagonal elements of the density matrix
corresponding to the state. A widely used quantum coherence quantifier is the
$\ell_{1}$-norm measure and in our work we use the $C_{\ell_{1}}$ to represent
this. Since the quantum coherence is a basis dependent quantity, we fix the
reference basis $|{i}\rangle$ for a given quantum state. The $l_{1}-$ norm of
coherence is defined as

$\displaystyle C_{\ell_{1}}(\rho)=\sum_{i,j;i\neq j}|\rho_{ij}|,$ (1)

where $\rho_{ij}=\langle i|\rho|j\rangle$ is the matrix element corresponding
to the $i^{th}$ row and $j^{th}$ column. Hence if we consider a bipartite two
qubit system in the standard computational basis, the reference basis is fixed
at $\\{|00\rangle,|01\rangle,|10\rangle,|11\rangle\\}$. An alternative method
of quantifying coherence is the relative entropy of coherence, denoted by
$C_{r}$ and which is based on the relative entropy. Thus for a state $\rho$,
the relative entropy of coherence is defined as

$\displaystyle C_{r}(\rho)=\min_{\sigma\in\mathcal{I}}S(\rho\|\sigma),$ (2)

where $S$ is the von-Neumann entropy. The minimum is taken over the set of the
incoherent states $\mathcal{I}$ which are states without quantum coherence. In
Ref.[4] it was proved that the expression for the relative entropy also
reduces to the form

$\displaystyle C_{r}(\rho)=S(\rho_{d})-S(\rho),$ (3)

where $\rho_{d}$ is the dephased state in the reference basis
$\\{|i\rangle\\}$ i.e. the state obtained from $\rho$ by deleting all off-
diagonal entries. Any good measure of coherence must satisfy the properties
for quantum coherence proposed in [4] and both the $l_{1}-$ norm of coherence
and relative entropy of coherence are the most general coherence monotones
satisfying these properties. One can construct a coherence measure using any
distance measure if it satisfies the properties of positivity, monotonicity
and convexity.

An alternative framework for quantifying coherence has been proposed based on
a natural property of coherence, the additivity of coherence for subspace-
independent states. This is described by an operation independent equality
rather than operation-dependent inequalities and therefore applicable to
various physical contexts. This framework is also compatible with all the
known results on coherence measures[17, 18]. Recently a deterministic
coherence distillation scheme has been proposed and the maximum number of
maximally coherent states has been derived[19].

####  2.1.2 Concurrence:

Entanglement is an important quantum resource which arises due to nonlocal
correlations between quantum sytems. There are several measures for
quantifying entanglement, like concurrence, relative entropy of entanglement,
negativity and log negativity. Of these measures concurrence is the widely
accepted measure for two qubit systems and it works equally for pure and mixed
states. The concurrence of a quantum state $\rho$ [3] is

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

where $\lambda_{1}\geq\lambda_{2}\geq\lambda_{3}\geq\lambda_{4}$ are the
eigenvalues of the matrix $\rho\tilde{\rho}$. The spin-flipped density matrix
$\tilde{\rho}$ is

$\displaystyle\tilde{\rho}=(\sigma_{y}\otimes\sigma_{y})\rho^{*}(\sigma_{y}\otimes\sigma_{y}),$
(5)

where $\sigma_{y}$ is the Pauli spin matrix in the $y$-basis and
$\tilde{\rho}$ is in the same basis as $\rho$, and $\rho^{*}$ is the complex
conjugate of the density matrix $\rho$.

####  2.1.3 Mixedness:

A quantum state is more often studied as an isolated system. This is not
exactly true and quantum states are in contact with an external environment
which influences these states externally. The environment causes a degradation
of the quantumness of these states and causes them to decohere and become
classical. The quantumness of a state is maximal when it is pure and once it
interacts with the environment, it usually loses its quantumness and
consequently it appears mixed. A completely classical state is the maximal
mixed state. When the state changes from being pure to becoming mixed, there
is an entropy introduced which is quantified using the linear entropy. For an
arbitrary $d$-dimensional quantum mixed state $\rho$, the mixedness is defined
using the normalized linear entropy $L(\rho)$ and is defined as[20]

$\displaystyle L(\rho)=\frac{d}{d-1}(1-{\rm Tr}(\rho^{2})).$ (6)

Here the quantity ${\rm Tr}(\rho^{2})$ describes the purity of the quantum
system. For a two-qubit system, the value of $L$ ranges from $0$ to $1$. The
entropy $L(\rho)=0$ for any pure state, and the maximum value $L(\rho)=1$ is
attained for the maximally mixed state $\frac{I}{4}$.

###  2.2 Quantum states

The Bell states are a set of maximally entangled two qubit pure states and
they are as follows:

$|\phi^{\pm}\rangle=\frac{1}{\sqrt{2}}\\{|00\rangle_{AB}\pm|11\rangle_{AB}\\};\qquad\qquad\qquad|\varphi^{\pm}\rangle=\frac{1}{\sqrt{2}}\\{|01\rangle_{AB}\pm|10\rangle_{AB}\\}.$
(7)

Here $A$ and $B$ denote the two particles Alice and Bob holding one qubit
each. These four states are orthogonal to each other and form a basis which is
in general referred to as the Bell basis. The elements of the computational
basis corresponding to the two qubit system can be represented in terms of the
Bell basis elements as

$\displaystyle|00\rangle=\frac{1}{\sqrt{2}}\\{|\phi^{+}\rangle_{AB}+|\phi^{-}\rangle_{AB}\\}.$
$\displaystyle|01\rangle=\frac{1}{\sqrt{2}}\\{|\varphi^{+}\rangle_{AB}+|\varphi^{-}\rangle_{AB}\\}.$
$\displaystyle|10\rangle=\frac{1}{\sqrt{2}}\\{|\varphi^{+}\rangle_{AB}-|\varphi^{-}\rangle_{AB}\\}.$
$\displaystyle|11\rangle=\frac{1}{\sqrt{2}}\\{|\phi^{+}\rangle_{AB}-|\phi^{-}\rangle_{AB}\\}.$

While we note that these two different bases can equally describe the Hilbert
space of the two qubit systems, the computational basis is inherently
separable whereas the Bell basis encodes nonlocal correlations in the basis
vectors. Following ref.[21, 22] we can construct a class of mixed states as
given below:

$\displaystyle\rho^{1}=p_{1}|\phi^{\pm}\rangle_{AB}\langle\phi^{\pm}|+(1-p_{1})|00\rangle_{AB}\langle
00|$
$\displaystyle\rho^{2}=p_{1}|\phi^{\pm}\rangle_{AB}\langle\phi^{\pm}|+(1-p_{1})|11\rangle_{AB}\langle
11|$
$\displaystyle\rho^{3}=p_{1}|\varphi^{\pm}\rangle_{AB}\langle\varphi^{\pm}|+(1-p_{1})|01\rangle_{AB}\langle
01|$
$\displaystyle\rho^{4}=p_{1}|\varphi^{\pm}\rangle_{AB}\langle\varphi^{\pm}|+(1-p_{1})|10\rangle_{AB}\langle
10|,$ (9)

and

$\displaystyle\varrho^{1}=p_{2}|\phi^{\pm}\rangle_{AB}\langle\phi^{\pm}|+(1-p_{2})|01\rangle_{AB}\langle
01|$
$\displaystyle\varrho^{2}=p_{2}|\phi^{\pm}\rangle_{AB}\langle\phi^{\pm}|+(1-p_{2})|10\rangle_{AB}\langle
10|$
$\displaystyle\varrho^{3}=p_{2}|\varphi^{\pm}\rangle_{AB}\langle\varphi^{\pm}|+(1-p_{2})|00\rangle_{AB}\langle
00|$
$\displaystyle\varrho^{4}=p_{2}|\varphi^{\pm}\rangle_{AB}\langle\varphi^{\pm}|+(1-p_{2})|11\rangle_{AB}\langle
11|,$ (10)

where $0\leq p_{1}\leq 1$ and $0\leq p_{2}\leq 1$ are the respective mixing
parameters. In future discussions in this manuscript, we refer to the states
described in Eq.(2.2) as the class-$1$ mixed states. Similarly, the states
given in Eq.(2.2) are referred to as the class-$2$ mixed states. Each one of
these equations, i.e. Eq.(2.2) and Eq.(2.2) contain $16$ different mixed
states and hence in total we have thirty two different mixed states. A quantum
state in $\mathcal{C}^{2}\otimes\mathcal{C}^{2}$ or
$\mathcal{C}^{2}\otimes\mathcal{C}^{3}$ is separable if and only if it obeys
the PPT criteria[23]. Any violation of this criteria in the
$\mathcal{C}^{2}\otimes\mathcal{C}^{2}$ or
$\mathcal{C}^{2}\otimes\mathcal{C}^{3}$ implies that the state is entangled.
Using this criteria we can show that the classes shown in Eq.(2.2)and (2.2)
are mixed entangled states under the conditions $0<p_{1}<1$ and $0<p_{2}<1$.
When $p_{1}$ and $p_{2}$ are zero, they are separable pure states and when
$p_{1}$ and $p_{2}$ are both one, they are pure entangled states. In the
following sections we investigate the coherence, concurrence and mixedness of
these two classes of states. For the sake of simplicity we denote the
computational basis with the notation
$B_{1}=\\{|00\rangle,|01\rangle,|10\rangle,|11\rangle\\}$ and the Bell basis
as
$B_{2}=\\{|\phi^{+}\rangle,|\varphi^{+}\rangle,|\varphi^{-}\rangle,|\phi^{-}\rangle\\}$.

Figure 1: The quantum coherence measured using the $\ell_{1}$ norm of
coherence $C_{\ell_{1}}$, the entanglement measured using the concurrence and
the mixedness estimated using the linear entropy $L$ are given for (a)
class-$1$ type of states as a function of the probability $p_{1}$ and (b)
class-$2$ type of states as a function of the probability $p_{2}$. In this
plot we use the computational basis for estimating the above mentioned
quantifiers.

##  3 Choice of measurement basis

###  3.1 Measurement in the computational basis

The quantum coherence, concurrence and the mixedness of the two classes of
states viz class-$1$ states given through Eq.(2.2) and the class-$2$ states
given through Eq.(2.2) are computed in the computational basis. For the
class-$1$ states the results obtained are

$\displaystyle C_{\ell_{1}}(\rho^{i})$ $\displaystyle=$ $\displaystyle p_{1},$
$\displaystyle C(\rho^{i})$ $\displaystyle=$ $\displaystyle p_{1},$
$\displaystyle L(\rho^{i})$ $\displaystyle=$
$\displaystyle\frac{4}{3}p_{1}(1-p_{1}),\forall i\in\\{1,2,3,4\\},$ (11)

The results corresponding to the class-$2$ states are

$\displaystyle C_{\ell_{1}}(\varrho^{i})$ $\displaystyle=$ $\displaystyle
p_{2},$ $\displaystyle C(\varrho^{i})$ $\displaystyle=$ $\displaystyle p_{2},$
$\displaystyle L(\varrho^{i})$ $\displaystyle=$
$\displaystyle\frac{8}{3}p_{2}(1-p_{2}),\forall i\in\\{1,2,3,4\\}.$ (12)

From the analytic expressions we find that the coherence and concurrence are
equal to each other. This happens due to the following reason: These two
classes of states belong to the wider classification of states called the
$X$-states in which the density matrix contains elements only along the main
diagonal and the anti-diagonal. A generic $X$-state has the following form as
given below:

$\displaystyle\rho^{m}=\left(\begin{array}[]{cccc}a&0&0&g\\\ 0&b&f&0\\\
0&f^{*}&c&0\\\ g^{*}&0&0&d\\\ \end{array}\right).$ (17)

For this state one can find the analytic expression of concurrence to be:

$C(\rho^{m})=2~{}{\rm max}\left\\{0,|f|-\sqrt{ad},|g|-\sqrt{bc}\right\\}.$
(18)

For the class-$1$ set of states in Eq.(2.2) either the set $\\{a,g,g^{*},d\\}$
or the set $\\{b,c,f,f^{*}\\}$ which will give us the result to be either
$C(\rho)=2g$ or $C(\rho)=2f$. The class-$2$ set of states in Eq.(2.2) has five
elements which again gives us the result of either $C(\rho)=2g$ or
$C(\rho)=2f$ depending on which element is present. The variation of quantum
coherence, concurrence and mixedness is shown in Fig. (1) as a function of the
mixing parameters. For the class-$1$ set of states the results are shown in
Fig. 1(a). In Fig. 1(b), the results corresponding to class-$2$ set of states
are given. From both these plots we find that the coherence and the
concurrence increase linearly with $p_{1}$ and $p_{2}$ as expected from the
analytic result. The mixedness is maximum at $p_{1}=0.5$ and $p_{2}=0.5$ when
the states are equally present. In the extreme cases when $p_{1}=p_{2}=0$ and
$p_{1}=p_{2}=1$ the states are pure and consequently the mixedness vanishes.

###  3.2 Measurement in the Bell basis

In this part we calculate the concurrence, quantum coherence and mixedness of
the two classes of states namely the class-$1$ and the class-$2$ in the Bell
basis. It is well known that the Bell states are the set of maximally
entangled states in the two qubit space. They are linearly independent, hence
forms a basis and hence any two-qubit state whether entangled or not can be
expressed in the Bell basis. In this subsection we look at the implications of
computing the quantifiers like concurrence, coherence and mixedness in the
Bell basis. We begin by expressing the density matrix in the Bell basis
$\rho^{(b)}$. To compute the concurrence we find $\tilde{\rho}^{(b)}$ as well
in the same Bell basis using Eq.(5). From the eigenvalues of the density
matrix we can find the concurrence of the system. For computing the quantum
coherence, we use the density matrix in the Bell basis $\rho^{(b)}$ and find
the distance to the closest diagonal state $\rho^{(b)}_{d}$ using the
$\ell_{1}$-norm coherence measure. The mixedness of the state $\rho^{(b)}$ is
found using the expression in Eq.(6). Below we summarize the results of our
calculations. For the class-$1$ type mixed states we have

$\displaystyle C_{\ell_{1}}(\rho^{i})$ $\displaystyle=$ $\displaystyle
1-p_{1},$ $\displaystyle C(\rho^{i})$ $\displaystyle=$ $\displaystyle p_{1},$
$\displaystyle L(\rho^{i})$ $\displaystyle=$
$\displaystyle\frac{4}{3}p_{1}(1-p_{1}),\forall i\in\\{1,2,3,4\\},$ (19)

and for the class-$2$ type mixed states we have

$\displaystyle C_{\ell_{1}}(\varrho^{i})$ $\displaystyle=$ $\displaystyle
1-p_{2},$ $\displaystyle C(\varrho^{i})$ $\displaystyle=$ $\displaystyle
p_{2},$ $\displaystyle L(\varrho^{i})$ $\displaystyle=$
$\displaystyle\frac{8}{3}p_{2}(1-p_{2}),\forall i\in\\{1,2,3,4\\}.$ (20)

Figure 2: The quantum coherence measured using the $\ell_{1}$ norm of
coherence $C_{\ell_{1}}$, the entanglement measured using the concurrence and
the mixedness estimated using the linear entropy $L$ are given for (a)
class-$1$ type of states as a function of the probability $p_{1}$ and (b)
class-$2$ type of states as a function of the probability $p_{2}$. In this
plot we use the Bell basis for estimating the above mentioned quantifiers.

The variation of the quantum coherence, concurrence and mixedness with the
mixing probabilities for the class-$1$ and class-$2$ types of states are shown
through the plots in Fig. 2(a) and Fig. 2(b) respectively. From the plots we
can see that the coherence and the concurrence are inversely proportional to
each other. When $p_{1}(p_{2})$ is zero, the coherence $C_{\ell_{1}}(\rho)$
($C_{\ell_{1}}(\varrho)$) is maximum and goes to zero when $p_{1}(p_{2})$ is
unity. In the case of concurrence we find that $C(\rho)(C(\varrho))$ is
minimum when the probabilities are zero and maximum when the probabilities are
set to one. This behavior is in contrast to the results obtained when these
quantifiers are measured in the computational basis where we observed that the
coherence and concurrence varied in the same manner. Now we explain the reason
behind the contrasting results when we use these two different measurement
bases. The computational basis is a separable basis in the sense that the two
qubit basis states can be written as a tensor product of the single qubit
basis states. Hence these basis states do not contain any type of quantum
correlations or coherence within them. Meanwhile if we look at the Bell basis
the two qubit basis states are the entangled states and they cannot be written
as tensor product of the individual basis states. Hence these basis states
inherently contain entanglement and coherence within them. When we use the
concurrence measure Eq.(4) to calculate the entanglement, we find the
eigenvalues of the product of the density matrix and its spin-flipped version,
with both these matrices expressed in the same Bell basis. This computation
gives us a basis independent result and hence the concurrence is the same when
we use either the computational basis or the Bell basis. In fact the
concurrence measure is an invariant quantity with respect to the choice of the
basis. From the results on quantum coherence we find that the two different
calculations arising from the computational basis and the Bell basis give
opposite results. This is because when we use the Bell basis for calculating
quantum coherence, the density matrix is transformed to the Bell basis and
then we use the same procedure as described through Eq.(1). It is expected
that this coherence need not be quantitatively same as the coherence measured
using the computational basis, since quantum coherence as measured using
Eq.(1) is a basis dependent quantity. But what we are observing is a
qualitative difference in behaviour where the relation between entanglement
and coherence changes from being directly proportional to inversely
proportional. This difference is because in the Bell basis there is coherence
hidden within the basis which is not accounted for. Hence in the calculation
of coherence we should always choose a basis in which the multiqubit basis
states are tensor product of the single qubit basis states to ensure no
coherence is left unaccounted for.

The mixedness of these quantum states can be computed from the linear entropy
of the system which is defined in Eq.(6). In general the mixedness can take
values from $0$ to $1$. From the plots of the two class of states we find that
the maximal value of mixedness of the class-$1$ states is lower than that of
the class-$2$ states. This is because the class-$1$ states have only two
diagonal elements whereas the class-$2$ states have three diagonal elements
and mixedness is the distance to the maximally mixed state $I/4$ where $I$ is
the Identity matrix which has four diagonal elements. When the mixing
parameter is $0.5$, the class-$1$ states and the class-$2$ states attain their
maximal mixedness which are $L(\rho)=1/3$ and $L(\varrho)=2/3$ respectively.
The qualitative and quantitative nature of mixedness is the same when it is
computed either in the computational basis or in the Bell basis.

##  4 Teleportation fidelity of the class of mixed states:

Teleportation provides us a different notion of quantum inseparability. In
this method the information in a given quantum state is separated into the
classical information and the quantum information and they are shared via a
classical and quantum channel respectively. Later these quantum and classical
parts can be combined to reconstruct the state with perfect fidelity. For this
process, we use an entangled pair of states as a quantum channel. In general a
singlet state $|\psi^{-}\rangle=\frac{1}{\sqrt{2}}(|01\rangle-|10\rangle)$ is
used and the state is shared between Alice (sender) and Bob (receiver). Now
Alice decides to send an unknown reference state of the form
$\alpha|0\rangle+\beta|1\rangle$ to Bob. To achieve this Alice performs Bell
measurements jointly on her qubit and the reference qubit. Her measurement
results are then conveyed to Bob by classical communication. Finally Bob
applies some unitary operators on his qubit depending upon Alice’s measurement
results. In this process the unknown quantum state is destroyed at Alice’s end
and is recreated at Bob’s end.

The teleportation protocol as introduced by Bennett et. al [24] was based on
using pure states as quantum channel. Later Popescu in Ref.[25] found that the
pairs of mixed states can also be used for teleportation, a feature known as
probabilistic teleportation [26] or imperfect teleportation. However there is
a limitation to kind of quantum state which can be used in teleportation. This
limitation is captured by the measure called teleportation fidelity. A quantum
channel is useful for teleportation only if its fidelity exceeds $2/3$ which
is the maximum possible fidelity achievable by means of Local Operations and
Classical Communication (LOCC) and is known as the classical fidelity [25,
27]. A well known two qubit mixed state is the Werner state and has the form

$\rho_{w}=\frac{1-F_{w}}{3}I_{4}+\frac{4F_{w}-1}{3}|\psi^{-}\rangle\langle\psi^{-}|$
(21)

where $|\psi^{-}\rangle=(|01\rangle-|10\rangle)/\sqrt{2}$ is the maximally
entangled singlet and $F_{w}$ is the maximal singlet fraction in the Werner
state. When the maximal singlet fraction $F_{w}$ lies in the range
$\frac{1}{2}\leq F_{w}\leq\frac{3+\sqrt{2}}{4\sqrt{2}}$ the Werner state
satisfies the Bell-CHSH inequality, but the state is still entangled in this
region. This is because a quantum state which violates Bell-CHSH inequality is
said to exhibit nonlocality a form of quantum correlation which is much
stronger than entanglement and in the hierarchy of quantum correlations is
rarer than entanglement [16]. Hence Werner states can be used as quantum
teleportation channel with average optimal fidelity exceeding $2/3$ even
though it does not violate the Bell-CHSH inequality in the given region [13].

The utility of the class of maximally and non-maximally entangled states in
quantum teleportation has been studied in Ref.[13]. Below we examine the
teleportation fidelity of the class of states defined in the previous section.
For an arbitrary density matrix $\rho$, the optimal teleportation fidelity is

$f_{T}(\rho)=\frac{1}{2}\Bigg{[}1+\frac{N(\rho)}{3}\Bigg{]},$ (22)

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 $T$, denoted by
$t_{st}$ are

$t_{st}=Tr(\rho\sigma_{s}\otimes\sigma_{t}),$ (23)

and $\sigma_{j}$’s denote the Pauli spin matrices. In terms of the quantity
$N(\rho)$, one can say that any mixed state $\rho$ is useful for standard
teleportation if and only if [28] $N(\rho)>1$. For the class-$1$ states
defined in Eq. (2.2), $\forall i$, $N(\rho^{i})>1$ with $0<p_{1}\leq 1$ and
the optimal teleportation fidelity defined in Eq.(22) for these states is

$f_{T}(\rho^{i})=\frac{2+p_{1}}{3},$ (24)

Now for the class-$2$ states defined in Eq.(2.2), we have $\forall j$,
$N(\varrho^{j})>1$ when $0.5\leq p_{2}\leq 1$. The optimal teleportation
fidelity of these states is

$f_{T}(\varrho^{j})=\frac{1+2p_{2}}{3}.$ (25)

These results hold both in the computational basis and Bell basis. It is easy
to observe analytically that the teleportation fidelity of the class-$1$
states always exceeds the classical teleportation value of $2/3$. Hence these
states can be used as teleportation channels for all values of mixing
parameter. As for class-$2$ states it is observed that in the range $0\leq
p_{2}<1/2$ the teleportation fidelity of the states is less than the classical
teleportation fidelity and is not suitable for quantum teleportation. For the
range $1/2\leq p_{2}\leq 1$ however, the teleportation fidelity of the states
is greater than the classical teleportation fidelity of $2/3$ and hence it is
suitable for quantum teleportation. Hence we find the class-$1$ states are
better candidates for quantum teleportation in comparison to class-$2$.

##  5 Bell-CHSH violation for the class of mixed states

The paradox which arose due to the Einstein-Podolsky-Rosen thought experiment
also known as EPR paradox, where they concluded that quantum mechanical
description of nature is incomplete since it goes against our common sense
perceptions of locality and reality[29]. To verify the existence of non-local
correlations, Bell proposed an inequality and a simpler form of this
inequality is the CHSH inequality[30, 31]. The verification of the Bell-CHSH
inequality for the class-$1$ and class-$2$ mixed states is defined in Eq.(2.2)
and in Eq.(2.2) is carried out in this section.

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

$M(\rho)=\max_{i>j}(u_{i}+u_{j})>1.$ (26)

Here $u_{i}$’s are eigenvalues of the matrix $T^{\dagger}T$ [32]. The elements
of the matrix $T$ are defined in the Eq.(23) and $\sigma_{i}$’s are the Pauli
matrices.

Bell violation of the class-$1$ states in the computational basis: For the
class-$1$ states defined in Eq.(2.2), the eigenvalues of the matrix
$T^{\dagger}T$ are $u_{1}=p_{1}^{2}+1$, $u_{2}=p_{1}^{2}+1$ and
$u_{3}=2p_{1}^{2}$. The state $M(\rho^{i})>1$ for $0<p_{1}\leq 1$, $\forall
i$. Hence we can conclude that the class-$1$ type of mixed states violates the
Bell-CHSH inequality for all values of the mixing parameter.

Bell violation of the class-$2$ states in the computational basis: The
eigenvalues of the $T^{\dagger}T$ matrix are $v_{1}=p_{2}^{2}$,
$v_{2}=p_{2}^{2}$ and $v_{3}=(2p_{2}-1)^{2}$. Using this we compute the three
possible different summations namely $e_{1}=v_{2}+v_{1}=2p_{2}^{2}$,
$e_{2}=v_{3}+v_{1}=5p_{2}^{2}-4p_{2}+1$ and
$e_{3}=v_{3}+v_{2}=5p_{2}^{2}-4p_{2}+1$. In the range $\frac{1}{3}<p_{2}<1$,
$M(\varrho^{j})=2p_{2}^{2}$ and for the region $0<p_{2}<\frac{1}{3}$,
$M(\varrho^{j})=5p_{2}^{2}-4p_{2}+1$ based on the maximum value. We find that
the quantity $M(\varrho^{j})=5p_{2}^{2}-4p_{2}+1>1$ for $0.8<p_{2}<1$.
Similarly, the quantity $M(\varrho^{j})=2p_{2}^{2}>1$ when $0.7071<p_{2}<1$.
Hence we see that the class $2$ mixed states $\varrho^{j}$ violate Bell-CHSH
inequality $0.7071<p_{2}<1$, $\forall j$. While these quantum states are
correlated for any value of $p_{2}$, the states have non-locality only in the
region $0.7071<p_{2}<1$.

From these observations we find that the class-$1$ type mixed states violate
the Bell-CHSH inequality for all values of the mixing parameter $p_{1}$ . In
the case of the class-$2$ type mixed states, the state violates Bell
inequality when $p_{2}>0.7071$, while for the region $0.5<p_{2}\leq 0.7071$
the states satisfy the Bell-CHSH inequality and in both these cases these type
of mixed states have a teleportation fidelity which is higher than the
classical teleportation fidelity of $2/3$. If we use the Bell basis the
results obtained are consistent with those of the computational basis which is
shown in this section.

##  6 Mixed bipartite states derived from tripartite states

The bipartite mixed states in Eq.(2.2) and Eq.(2.2) are probabilistic mixtures
of two qubit pure states. To achieve this experimentally one might have to
generate pure states and it is well known that perfect generation of pure
states is an experimental challenge. An alternative is to generate tripartite
quantum states and obtain the two qubit mixed states by tracing out one of the
qubits. We use the bipartite mixed states obtained from the reduced states of
some well known pure as well as three qubit mixed states. In the first
subsection we look at the reduced two qubit density matrices obtained from
pure three qubit systems. The second subsection presents results where the two
qubit mixed states are obtained from three qubit mixed states.

###  6.1 Bipartite mixed states derived from pure tripartite states

In this part we consider three qubit pure entangled states and consider the
reduced density matrices arising out of them by tracing out any one of the
qubits. The three qubit pure states can be entangled in two different ways
namely the three way entangled $GHZ$ state[33, 34] and the two way entangled
$W$ state[35]. In the $GHZ$ state all the three qubits are entangled in such a
way that loss of any one of the qubits destroys all the entanglement in the
system. When a qubit is traced out in a three qubit $W$ state the remaining
bipartite system still has residual entanglement in the system. These two
states are classified into two distinct SLOCC (Stochastic Local Operations and
Classical Communication) classes since a state in a given class cannot be
transformed to a state in another class using only SLOCC operations[36, 37,
38, 39].

SLOCC class of states: The GHZ state is tripartite maximally entangled state
in which the loss of even a single qubit makes the quantum state disentangled.
The GHZ state has the form $|GHZ\rangle=(|000\rangle+|111\rangle)/\sqrt{2}$
and for this state the two qubit reduced density matrix is

$\rho^{g}_{AB}=\rho^{g}_{BC}=\rho^{g}_{AC}=\left(\begin{array}[]{cccc}\frac{1}{2}&0&0&0\\\
0&0&0&0\\\ 0&0&0&0\\\ 0&0&0&\frac{1}{2}\\\ \end{array}\right),$ (27)

where $\rho^{g}_{AB}={\rm Tr}_{C}\;\rho^{g}_{ABC}$ ($\rho^{g}_{BC}={\rm
Tr}_{A}\;\rho^{g}_{ABC}$, $\rho^{g}_{AC}={\rm Tr}_{B}\;\rho^{g}_{ABC}$) is the
reduced two qubit density matrix and $\rho^{g}_{ABC}=|GHZ\rangle\langle GHZ|$.
Computing the various quantum correlations for this state (27) we find the
concurrence $C=0$, coherence $C_{\ell_{1}}=0$ and mixedness $L=2/3$. The
teleportation fidelity of the quantum state
$f_{T}(\rho_{AB}^{g})=f_{T}(\rho_{BC}^{g})=f_{T}(\rho_{CA}^{g})=0$. Since
there is no entanglement in the two qubit reduced density matrix of a $GHZ$
state, the state (27) is not useful for any kind of teleportation.

A $W$ state is a three qubit entangled state, in which entanglement is shared
only between the pairs of qubits and there is no genuine entanglement as in a
$GHZ$ state. The three qubit $W$ state reads:
$|W\rangle=(|001\rangle+|010\rangle+|100\rangle)/\sqrt{3}$. The reduced
density of the $W$ state is

$\rho^{W}_{AB}=\rho^{W}_{BC}=\rho^{W}_{AC}=\left(\begin{array}[]{cccc}\frac{1}{3}&0&0&0\\\
0&\frac{1}{3}&\frac{1}{3}&0\\\ 0&\frac{1}{3}&\frac{1}{3}&0\\\ 0&0&0&0\\\
\end{array}\right),$ (28)

where $\rho^{W}_{AB}={\rm Tr}_{C}\;\rho^{W}_{ABC}$ ($\rho^{W}_{BC}={\rm
Tr}_{A}\;\rho^{W}_{ABC}$, $\rho^{W}_{AC}={\rm Tr}_{B}\;\rho^{W}_{ABC}$) is the
reduced two qubit density matrix and $\rho^{W}_{ABC}=|W\rangle\langle W|$
Representing the two qubit reduced density matrix
$\rho^{w}=\rho^{W}_{AB}=\rho^{W}_{BC}=\rho^{W}_{AC}$, we give the coherence,
concurrence, mixedness and teleportation fidelity below:

$\displaystyle C_{\ell_{1}}(\rho^{w})$ $\displaystyle=$
$\displaystyle\frac{2}{3},$ $\displaystyle C(\rho^{w})$ $\displaystyle=$
$\displaystyle\frac{2}{3},$ (29) $\displaystyle L(\rho^{w})$ $\displaystyle=$
$\displaystyle\frac{16}{27},$ $\displaystyle f_{T}(\rho^{w})$ $\displaystyle=$
$\displaystyle 7/9.$ (30)

The bipartite reduced density matrices obtained from the $W$-states have a
teleportation fidelity higher than $2/3$ and so they can be used in quantum
teleportation.

$W\tilde{W}$ state: The three qubit $W\tilde{W}$ state is a pure state which
is a linear superposition of the $W$ state and the $\tilde{W}$ state (which in
fact is a spin flipped version of the $W$ state)[40]. The expression for this
tripartite state reads:

$\displaystyle|W\tilde{W}\rangle=\frac{1}{\sqrt{2}}(|W\rangle+|\tilde{W}\rangle).$
(31)

where

$\displaystyle|W\rangle=\frac{1}{\sqrt{3}}(|001\rangle+|010\rangle+|100\rangle),$
$\displaystyle|\tilde{W}\rangle=\frac{1}{\sqrt{3}}(|110\rangle+|101\rangle+|011\rangle).$
(32)

The $W\tilde{W}$ state has its coherence distributed in both three way and two
way manner, apart from this single qubit coherences are also present in the
system. For this system, we find the tangle $\tau=1/3$ and hence we know that
the three qubit entanglement is also present in the system. The two qubit
reduced density matrix also has entanglement suggesting that the bipartite
entanglement is also present in the system. In the computational basis the two
qubit reduced density matrix of this state is

$\displaystyle\rho^{W\tilde{W}}_{AB}=\rho^{W\tilde{W}}_{BC}=\rho^{W\tilde{W}}_{AC}=\left(\begin{array}[]{cccc}\frac{1}{6}&\frac{1}{6}&\frac{1}{6}&0\\\
\frac{1}{6}&\frac{1}{3}&\frac{1}{3}&\frac{1}{6}\\\
\frac{1}{6}&\frac{1}{3}&\frac{1}{3}&\frac{1}{6}\\\
0&\frac{1}{6}&\frac{1}{6}&\frac{1}{6}\\\ \end{array}\right),$ (37)

This two qubit mixed state has coherence value of $C_{\ell_{1}}=2$,
concurrence $C=1/3$, mixedness $L=10/27$ and teleportation fidelity
$f_{T}=7/9$. To examine the Bell-CHSH inequality violation we compute
$M(\rho^{W\bar{W}})$ (where
$\rho^{W\tilde{W}}=\rho^{W\tilde{W}}_{AB}=\rho^{W\tilde{W}}_{BC}=\rho^{W\tilde{W}}_{AC}$)
which comes out to be $8/9$. As $M(\rho^{W\tilde{W}})<1$, the state satisfies
Bell-CHSH inequality.

Star state: The three qubit quantum states considered above viz $GHZ$, $W$ and
the $W\tilde{W}$ states are all symmetric states in that the reduced density
matrices are all identical irrespective of which qubit is traced over. As an
example of asymmetric states we consider the three qubit star states[40]. In
these states there is a central qubit which is entangled to two peripheral
qubits. These peripheral qubits are not entangled between themselves and the
form of a three qubit star state is:

$\displaystyle|S\rangle_{ABC}=\frac{1}{2}[|000\rangle+|100\rangle+|101\rangle+|111\rangle].$
(38)

When we trace out the central qubit, the reduced density matrix of the two
qubit system is

$\displaystyle\rho_{AB}^{s}=\left(\begin{array}[]{cccc}\frac{1}{4}&0&\frac{1}{4}&0\\\
0&0&0&0\\\ \frac{1}{4}&0&\frac{1}{2}&\frac{1}{4}\\\
0&0&\frac{1}{4}&\frac{1}{4}\\\ \end{array}\right).$ (43)

Although the state has zero concurrence, it has finite amount of coherence
$C_{\ell_{1}}=1$. This bipartite partition is separable and hence is not of
use in computing quantum teleportation fidelity. If we trace out any one of
the peripheral qubit we get the following reduced density matrix:

$\displaystyle\rho_{BC}^{s}=\rho_{AC}^{s}=\left(\begin{array}[]{cccc}\frac{1}{2}&\frac{1}{4}&0&\frac{1}{4}\\\
\frac{1}{4}&\frac{1}{4}&0&\frac{1}{4}\\\ 0&0&0&0\\\
\frac{1}{4}&\frac{1}{4}&0&\frac{1}{4}\\\ \end{array}\right).$ (48)

The state has a concurrence $C=1/2$, coherence $C_{\ell_{1}}=3/2$ and
mixedness $L=1/3$. The teleportation fidelity of this state is
$(7+2\sqrt{2})/2$ which is higher than the classical teleportation fidelity of
$2/3$ and hence can be used for quantum teleportation. These reduced density
matrices satisfy the Bell-CHSH inequality.

###  6.2 Bipartite mixed states derived from mixed tripartite states

In this subsection, we consider bipartite mixed states obtained by tracing out
one qubit in a three qubit mixed state which is a convex combination of three
qubit pure states. We investigate various quantum features like quantum
coherence, entanglement, mixedness and teleportation fidelity for these
states.

Mixture of $GHZ$ and $W$ states: We consider the genuinely entangled three
qubit $GHZ$ state and the bipartite entangled $W$ state and consider a
tripartite state which is a convex mixture of both these states. The form of
this tripartite mixed state is

$\rho^{gw}_{ABC}=p|GHZ\rangle_{ABC}\langle GHZ|+(1-p)|W\rangle_{ABC}\langle
W|,~{}~{}~{}~{}0\leq p\leq 1,$ (49)

where $p$ is the mixing parameter. Now if we trace out any one of the qubits
($A$, $B$ or $C$) from the tripartite state Eq. (49), the reduced density
matrix $\rho^{gw}$ in the computational basis is

$\displaystyle\rho^{gw}=\left(\begin{array}[]{cccc}\frac{p+2}{6}&0&0&0\\\
0&\frac{1-p}{3}&\frac{1-p}{3}&0\\\ 0&\frac{1-p}{3}&\frac{1-p}{3}&0\\\
0&0&0&\frac{p}{2}\\\ \end{array}\right).$ (54)

The coherence, concurrence and mixedness along with teleportation fidelity of
the state (54) in computational basis are

$\displaystyle C_{\ell_{1}}(\rho^{gw})$ $\displaystyle=$
$\displaystyle\frac{2(1-p)}{3},$ $\displaystyle C(\rho^{gw})$ $\displaystyle=$
$\displaystyle\frac{2(1-p)}{3}-2\sqrt{\frac{p(p+2)}{6}},~{}~{}0\leq p<0.292$
$\displaystyle L(\rho^{gw})$ $\displaystyle=$
$\displaystyle\frac{2}{27}\left(8-13p^{2}+14p\right),$ $\displaystyle
f_{T}(\rho^{gw})$ $\displaystyle=$ $\displaystyle\frac{7-4p}{9},0\leq
p<\frac{1}{4}.$ (55)

The application of this state as a teleportation channel has been investigated
in [13] where it has been shown that the state is useful for quantum
teleportation when $p$ lies between $0$ and $0.25$. It is also to be noted
that for $0.25<p<0.292$ though the state is entangled yet it cannot be used as
a teleportation channel. The results of these investigations are shown in the
plots in Fig. (3).

Figure 3: The quantum coherence measured using the $\ell_{1}$ norm of
coherence $C_{\ell_{1}}$, the entanglement measured using the concurrence and
the teleportation fidelity $f_{T}$ are given for (a) mixture of $GHZ$ and $W$
states and (b) a mixture of $W$ state and $\tilde{W}$ states both as a
function of the probability $p$.

Mixture of $W$ and $\tilde{W}$ states: Next we consider a convex mixture of
$|W\rangle$ and $|\tilde{W}\rangle$ states and investigate the teleportation
capability of two qubit reduced states obtained from these states. The
expression for the three qubit mixed states read:

$\displaystyle\varrho^{W\tilde{W}}_{ABC}$ $\displaystyle=$ $\displaystyle
p\rho_{W}+(1-p)\rho_{\tilde{W}}$ (56) $\displaystyle=$ $\displaystyle
p|W\rangle_{ABC}\langle
W|+(1-p)|\tilde{W}\rangle_{ABC}\langle\tilde{W}|,~{}~{}~{}~{}0\leq p\leq 1.$

Tracing out any one of the three qubits i.e., $A$ or $B$ or $C$, the three
qubit state $\varrho^{W\tilde{W}}_{ABC}$ reduces to a two qubit density matrix
$\varrho^{w\tilde{w}}$ which is

$\displaystyle\varrho^{w\tilde{w}}=\left(\begin{array}[]{cccc}\frac{p}{3}&0&0&0\\\
0&\frac{1}{3}&\frac{1}{3}&0\\\ 0&\frac{1}{3}&\frac{1}{3}&0\\\
0&0&0&\frac{1-p}{3}\\\ \end{array}\right),$ (61)

where
$\varrho^{w\tilde{w}}=\varrho^{W\tilde{W}}_{AB}=\varrho^{W\tilde{W}}_{BC}=\varrho^{W\tilde{W}}_{AC}$.
For this state, the coherence, concurrence, mixedness and teleportation
fidelity are

$\displaystyle C_{\ell_{1}}(\varrho^{w\tilde{w}})$ $\displaystyle=$
$\displaystyle\frac{2}{3},$ $\displaystyle C(\varrho^{w\tilde{w}})$
$\displaystyle=$ $\displaystyle\frac{2}{3}-\frac{2}{3}\sqrt{p(1-p)},$
$\displaystyle L(\varrho^{w\tilde{w}})$ $\displaystyle=$
$\displaystyle\frac{8}{27}\left(2-p^{2}+p\right),$ $\displaystyle
f_{T}(\varrho^{w\tilde{w}})$ $\displaystyle=$ $\displaystyle\frac{7}{9}.$ (62)

Thus we find the teleportation fidelity to be greater than the classical
teleportation fidelity of $2/3$ and hence the states can be useful in quantum
teleportation.

Table 1: Summary of the quantum correlations, teleportation fidelity and Bell-CHSH inequality Quantum | Quantum | Concurrence | Mixedness | Teleportation
---|---|---|---|---
states | Coherence $C_{\ell_{1}}$ | $C$ | $L$ | Fidelity $f_{T}$
class-$1$ states with | $p_{1}$ | $p_{1}$ | $4(p_{1}(1-p_{1}))/3$ | $(2+p_{1})/3$
mixing parameter $p_{1}$ |  |  |  |
class-$2$ states with | $p_{2}$ | $p_{2}$ | $8(p_{2}(1-p_{2}))/3$ | $(1+2p_{2})/3$
mixing parameter $p_{2}$ |  |  |  |
$GHZ$ state | 0 | 0 | 2/3 | 0
2-qubit reduced form |  |  |  |
$W$ state | 2/3 | 2/3 | 16/27 | 7/9
2-qubit reduced form |  |  |  |
$|W\bar{W}\rangle$ | $2$ | $1/3$ | $10/27$ | $7/9$
2-qubit reduced form |  |  |  |
Star state | $2$ | $3/2$ | $1/3$ | $(7+2\sqrt{2})/2$
2-qubit reduced form |  |  |  |
$p\rho_{GHZ}+(1-p)\rho_{W}$ | $\frac{2}{3}(1-p)$ | $2\left(\frac{1-p}{3}-\sqrt{\frac{p(p+2)}{6}}\right)$ | $\frac{2}{27}(8-13p^{2}+14p)$ | $(7-4p)/9$
2-qubit reduced form |  | $0\leq p\leq 0.292$ |  | $0\leq p\leq 0.25$
$p\rho_{W}+(1-p)\rho_{\tilde{W}}$ | $2/3$ | $2/3-2(\sqrt{p(1-p)})/3$ | $\frac{8}{27}(2-p^{2}+p)$ | $7/9$
2-qubit reduced form |  |  |  |

##  7 Conclusion:

In this work, we study the quantum coherence and correlations of several
bipartite mixed states. Particularly, we find the quantum coherence,
entanglement and mixedness of these quantum systems. To estimate quantum
coherence we use the well-known $\ell_{1}$ norm of coherence, in which we sum
over the off-diagonal elements of the two qubit density matrix. The
entanglement is measured using the concurrence measure which is suitable for
both pure and mixed states. Using the linear entropy we find the amount of
mixedness in the states. For the quantum states we consider a wider classes of
states considering all possibilities. Initially we consider the two qubit
states created using a mixture of the maximally entangled Bell state with any
one of the states from the computational basis. Here we find that we can
create two classes of states viz class-$1$ type and the class-$2$ type of
states. In the class-$1$ type of states the basis vector being mixed with is
also a constituent of the Bell state, whereas in the class-$2$ type of states
it is not a constitutent of the Bell state with which it is being mixed. Apart
from this we also consider bipartite mixed states derived from either a
tripartite pure or mixed state by tracing out one of their qubits. A complete
list of the states considered in this study are given in Table 1, where we
have summarized the properties of each individual state.

The quantum coherence, entanglement and mixedness of both the class-$1$ and
class-$2$ type of states are computed both in the computational as well as in
the Bell basis. In the computational basis, the quantum coherence measured
using the $\ell_{1}$-norm of coherence is equal to the entanglement estimated
using the concurrence. This is because we are using a specific class of
$X$-states in which only two diagonal elements are present. The mixedness of
the class-$1$ (class-$2$) states varies such that they are zero when either
$p_{1}$ ($p_{2}$) is zero or when $p_{1}$ ($p_{2}$) is unity and the maximal
value is attained midway when the probabilities are half. When we estimate the
coherence and concurrence in the Bell basis we find them to have opposite
behavior where the coherence is maximal when the concurrence is zero and the
concurrence is maximal for vanishing coherence. This is because in the Bell
basis there is a coherence within the basis elements which is not accounted
for when we compute the $\ell_{1}$-norm of coherence. But concurrence
measurements are basis independent and will give the same result irrespective
of the basis chosen for the computation. From this observation we learn that
one should always use a incoherent basis (separable basis) for the computation
of quantum coherence in the system. The use of either a coherent basis or an
entangled basis might always lead to wrong results. The mixedness in the bell
basis is identical to the one obtained in the computational basis because the
linear entropy is a basis independent evaluation. We would like to highlight
that quantum coherence is a basis dependent quantity and the amount of
coherence measured depends on the choice of the basis. Quantum coherence can
occur either due to the inter-qubit correlations which is the global coherence
or due to the superposition of the levels which is the local coherence. Any
measure of coherence should estimate the total coherence which is a sum of
both the global and local coherence. To measure the total coherence, we need
to use a separable basis. If we use an entangled basis, we are not measuring
the total coherence, since the basis choice does not allow us to estimate the
global coherence arising due to inter-qubit correlations. This give rise to
spurious results where we can have an incoherent state which on being
expressed in Bell basis will show some coherence. Now this origin of coherence
is not from the state, but rather from the basis in which the state is
expressed. Here we have to make a distinction that the choice of the basis is
an operation carried out to estimate the coherence and this operation should
not increase the amount of coherence in the state. But when we use the
entangled basis the coherence content is increased. So, a right choice of
basis is the one where the choice of basis does not introduce any coherence
into the system. Since the issue of the right kind of basis is settled, we
computed the teleportation fidelity of both the class-$1$ and class-$2$ type
of states and examined their suitability for quantum teleportation. The
teleportation fidelity of the class-$1$ type of states is greater than the
classical teleportation fidelity of $2/3$ for all values of the mixing
parameter. Hence we conclude that these type of states can always be used for
quantum teleportation. In the case of the class-$2$ type of states, the
teleportation fidelity is higher than the classical value only when $0.5\leq
p\leq 1$ and so these states can be used in quantum teleportation only in this
range. Next we verify the Bell-CHSH inequality for both these class of states.
We find that the class-$1$ type of states violates the Bell-CHSH inequality
for all values of the mixing parameter. For the class-$2$ type of states we
observe that the Bell-CHSH inequality is violated for $p>0.7071$. In the
region $0.5\leq p\leq 0.7071$, the states satisfy the Bell-CHSH inequality but
they can still be used for quantum teleportation. Finally we also investigate
reduced two qubit states obtained from the three qubit pure and mixed states.
Here we consider a comprehensive set of pure states viz $GHZ$ state, $W$
state, $W\tilde{W}$ state, star state. For the mixed states we consider the
mixture of (i) $GHZ$ and $W$ state and (ii) $W$ and $\tilde{W}$ state. The
computed correlations as well as their teleportation fidelity is listed in
Table 1. These results give us an idea of the suitability of various states
for quantum teleportation. An interesting future work will be the examination
of the teleportation fidelity of these different states under classical noise
like telegraphic noise as well as dissipative and dephasing noise. We are
currently working on these problems and the results will form the discussions
of our future work.

## Acknowledgements

Chandrashekar Radhakrishnan was supported in part by a seed grant from IIT
Madras to the Centre for Quantum Information, Communication and Computing. Md.
Manirul Ali was supported by the Centre for Quantum Science and Technology,
Chennai Institute of Technology, India, vide funding number
CIT/CQST/2021/RD-007.

## Data Availability Statement

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

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