# Role of Steering Inequality In Quantum Key Distribution Protocol

Kaushiki Mukherjee<EMAIL_ADDRESS>Department of Mathematics,
Government Girls’ General Degree College, Ekbalpore, Kolkata-700023, India.
Tapaswini Patro<EMAIL_ADDRESS>Department of
Mathematics, Birla Institute of Technology and Science Pilani,
Hyderabad-500078,Telangana, India  Nirman Ganguly
<EMAIL_ADDRESS>Department of Mathematics, Birla
Institute of Technology and Science Pilani, Hyderabad-500078,Telangana, India

###### Abstract

Violation of Bell’s inequality has been the mainspring for secure key
generation in an entanglement assisted Quantum Key Distribution(QKD) protocol.
Various contributions have relied on the violation of appropriate Bell
inequalities to build an appropriate QKD protocol. Residing between Bell
nonlocality and entanglement, there exist a hybrid trait of correlations,
namely correlations exhibited through the violation of steering
inequalities.However, such correlations have not been put to use in QKD
protocols as much as their stronger counterpart, the Bell violations.In the
present work, we show that the violations of the CJWR(E.G.Cavalcanti,S.J.
Jones,H.M Wiseman and M.D. Reid, Phys.Rev.A 80,032112(2009))steering
inequalities can act as key ingredients in an entanglement assisted QKD
protocol.We work with arbitrary two qubit entangled states, characterize them
in accordance with their utility in such protocols. The characterization is
based on the quantum bit error rate and violation of a CJWR inequality.
Furthermore, we show that subsequent applications of local filtering
operations on initially entangled states exhibiting non violation, lead to
violations necessary for the successful implementation of the protocol. An
additional vindication of our protocol is provided by the use of absolutely
Bell-CHSH local states, states which remain Bell-CHSH local even under global
unitary operations.

##  I Introduction

Quantum Cryptography(QC) promises to bring a paradigmatic change in the domain
of secure information processingQC1 .The state of the art techniques,recently
conceptualized,have led to profound implications in how we deal with secure
messagesQC2 .The archetypal manifestation of quantum mechanics, namely quantum
entanglement lies at the root of QC. Quantum entanglement hor1  makes it
possible to realize protocols unimaginable in classical information theory. It
is indeed fascinating to find QC lie at the intersection of quantum mechanics
and information theory.

An inherent constitution of QC is key distribution, which in the simplest
scenario can be interpreted as the task of generating a private key between
two honest users who can communicate with each other over public channels.As
noted earlier, if one includes quantum resources for secure key generation, it
can outperform protocols availing only classical resources. Such a difference
stems from the fact that quantum protocols basically rely on inherent random
nature of quantum particles contrary to dependence on pseudo randomness and
computational complexity by classical protocolsretsispg10 .

The seminal work by Bennet and Brassardbb84  single handedly pioneered the
study on quantum key distribution protocols. In bb84 , the authors designed a
bipartite key distribution protocol where one of the parties(Alice) prepares a
qubit in one of two conjugate baseswies  and sends it over a public quantum
channel to her counterpart(Bob) who then measures the qubit intercepted. After
repeating the preparation and measurement procedures sufficiently many times,
the parties reconcile their measurement bases publicly and retain results
corresponding to only matching bases(i.e., outputs obtained when they choose
the same basis) for further processing. Since, the communication is made over
a public quantum channel, any third party (Eve) can intercept, measure the
qubits(sent by Alice) and then send some new qubits to Bob thereby posing a
threat to secure key generation. Presence of eavesdroppers can be detected by
Alice and Bob who who then abort the protocol. Detection of eavesdroppers and
successful key generation between Alice and Bob at the end of BB84 protocol
mainly rests on quantum laws such as no-cloning theory, measurement induced
disturbance in quantum systemsnie , etc.It is important to note that the modus
operandi of the BB84 protocol depends on preparation and measurement scheme.
Several key generation protocolscr ; cr1 ; cr2 ; cr3 ; cr4 ; cr5 ; cr6  based
on this scheme have been designed since the BB84 protocol.

A departure from the preparation and measurement scheme is provided by the
entanglement assisted key distribution protocols. The two parties now share an
entangled statehor1  and the presence of the eavesdropper is detected through
the violation of a suitable Bell’s inequalitybell , like the Bell-CHSH cla .
Such a scheme was envisaged by Ekert through his novel contributioneck . Since
its inception,strategies based on violation of Bell’s inequalities have been
used to design many QKD protocolseck1 ; eck2 ; eck3 ; eck4 ; eck5 ; eck6 ;
eck7 .

In entanglement assisted protocols, comparison of Alice and Bob’s information
content with that of an eavesdropper is the most obvious way to verify
securityeck8 . However, violation of a suitable Bell’s inequality provides a
pragmatic alternative eck10 ; eck9 ; eck11 ; eck12 . Although, Bell violation
is a necessary criterion for security,it has been shown that it is not
sufficient eck14 ; eck13 . Consequent to a Bell violation, it becomes
imperative to check the utility of the entangled state in successful key
generation.In main , the authors have characterized two qubit states in this
perspective, in the ambit of Bell-CHSH violation, where the relation between
the violation and quantum error bit rate(QBER)eck8  has been exploited.

Nonlocal correlations being the mainstay of QKD protocols, necessitate
exploitation of other manifestations of it. Quantum steering provides for one
such significant alternative, which remains sandwiched between entanglement
and Bell nonlocal correlations. Schrodinger, pioneered the concept of steering
str1 ; str2 , whose operational significance came much later through the
celebrated work of wis ; jon . A series of work followed, detecting
steerability of correlations derived from a given quantum state red1 ; red2 ;
red3 ; red4 ; zow ; red5 ; san ; ver ; jev ; sjj ; costa .

In the present work, we utilize the steering inequality red3 , commonly known
as the CJWR inequality, to prescribe a QKD protocol.Up to two measurement
settings scenario, notions of Bell-CHSH nonlocality and steering nonlocality
being indistinguishable, CJWR inequality for three measurement settings is
considered. Violation of this symmetric inequality guarantees steerability of
the corresponding bipartite correlations. Such form of steerability is often
referred to as $F_{3}$ steerabilityred3 ; costa . Unlike other criteria, CJWR
inequalityred3  is designed under the assumption that both the parties perform
characterized measurements. Such an assumption makes this inequality a
suitable candidate in our prescription as both the users will be making
quantum measurements.

Our work, which considers an entanglement assisted key distribution protocol,
makes two very significant assumptions:(i)violation of CJWR inequality for
three settings is necessary for security,(ii) minimum secure key rateeck8i  is
dependent on quantum bit error rate only. The protocol designed here is semi-
device independent, as the preparation device(source generating the state) is
untrusted, while the measurement devices of the parties are trusted. A
characterization of arbitrary two qubit state is made to verify the utility of
an $F_{3}$ steerable entangled state in successful implementation of the QKD
protocol. This further entails the detection of useful states for QKD under
the observation of violation of the CJWR inequality.

The above probe raises an imperative query, namely can the useless states be
made useful for QKD with an enhancement in the protocol?. We provide an
affirmative answer, through the application of local filtering operations pop1
; pop2 ; pop3 . Analogous to the procedure which was obtained in main  with
respect to the Bell-CHSH inequality, we modify our protocol to include local
filtering operations to enhance the suitability of the otherwise useless
states. Strikingly, it is also observed that some $F_{3}$ unsteerable two
qubit states can also be used for secure key generation in this modified
protocol.

Usually, in any entanglement assisted QKD protocol, state shared between Alice
and Bob is generated from some unknown source which can be under control of an
eavesdropper. If the source distributes an absolutely Bell-CHSH local
entangled stateverstraeteglobal ; guoch , then secure key generation becomes
impossible if Bell nonlocality is considered necessary for security.In
conjunction with this, we propose a scenario where such states can also be
made useful as steerability is a weaker form of nonlocality than Bell
nonlocality itself.

Remaining part of our work is organized as follows. Motivation underlying the
present discussion is provided in sec.II followed by discussion on some
mathematical pre-requisites in sec.III. Entire characterization of two qubit
states is made in sec.IV. In sec.V, effect of local filtering operations in
our QKD protocol is discussed. In sec.VI, we discuss the case where absolute
Bell-CHSH local states are used in the protocol. We end our discussion with
some concluding remarks in sec.VII.

##  II Motivation

In study of entanglement assisted QKDs, role of Bell nonlocality has undergone
multifaceted analysis. In this perspective, manifestation of nonclassical
quantum correlations beyond Bell-CHSH scenario, needs further exploration.
Connection of quantum steering with security in one sided device independent
QKD has been put forward in bran1  where the authors have used entropic
inequalitiesent1 ; ent2  for their purpose. In temps , concept of temporal
steering is used for the same in preparation and measurement based QKD
schemes. However, to the best of authors’ knowledge, using violation of any
steering inequality has not been considered in any QKD protocol till date.
Violation of any correlator based inequality being experimentally verifiable,
we intended to use a Bell type inequality(as steering detecting criterion). A
closed form of upper bound of CJWR inequalityred3  as a function of state
parameters is presentcosta . Using the closed form, it now becomes possible to
identify useful(in QKD protocol) states for any given amount of violation of
CJWR inequality observed. Besides, some entangled states(absolutely Bell-CHSH
local statesguoch ) turned out to be useful while considering notion of
$F_{3}$ steerability instead of standard Bell nonlocality.

##  III Preliminaries

###  III.1 Bloch Matrix Representationgam

Let $\rho$ denote an arbitrary two qubit state shared between two parties.

$\small{\rho}=\small{\frac{1}{4}(\mathbb{I}_{2}\times\mathbb{I}_{2}+\vec{\mathfrak{a}}.\vec{\sigma}\otimes\mathbb{I}_{2}+\mathbb{I}_{2}\otimes\vec{\mathfrak{b}}.\vec{\sigma}+\sum_{j_{1},j_{2}=1}^{3}w_{j_{1}j_{2}}\sigma_{j_{1}}\otimes\sigma_{j_{2}})},$
(1)

with $\vec{\sigma}$$=$$(\sigma_{1},\sigma_{2},\sigma_{3}),$ $\sigma_{j_{k}}$
denoting Pauli operators along three mutually perpendicular
directions($j_{k}$$=$$1,2,3$). $\vec{\mathfrak{a}}$$=$$(x_{1},x_{2},x_{3})$
and $\vec{\mathfrak{b}}$$=$$(y_{1},y_{2},y_{3})$ denote local bloch
vectors($\vec{\mathfrak{a}},\vec{\mathfrak{b}}$$\in$$\mathbb{R}^{3}$)
corresponding to party $\mathcal{A}$ and $\mathcal{B}$ respectively with
$|\vec{\mathfrak{a}}|,|\vec{\mathfrak{b}}|$$\leq$$1$ and $(w_{i,j})_{3\times
3}$ stands for the correlation tensor matrix $\mathcal{W}$(real matrix).
Components $w_{j_{1}j_{2}}$ of $\mathcal{W}$ are given by
$w_{j_{1}j_{2}}$$=$$\textmd{Tr}[\rho\,\sigma_{j_{1}}\otimes\sigma_{j_{2}}].$
$\mathcal{W}$ can be diagonalized by applying suitable local unitary
operationsgam . Simplified expression of $\rho$ is then given by:

$\small{\rho}^{{}^{\prime}}=\small{\frac{1}{4}(\mathbb{I}_{2}\times\mathbb{I}_{2}+\vec{\mathfrak{m}}.\vec{\sigma}\otimes\mathbb{I}_{2}+\mathbb{I}_{2}\otimes\vec{\mathfrak{n}}.\vec{\sigma}+\sum_{j=1}^{3}\mathfrak{t}_{jj}\sigma_{j}\otimes\sigma_{j})},$
(2)

Correlation tensor in Eq.(2) is given by
$T$$=$$\textmd{diag}(t_{11},t_{22},t_{33})$ where $t_{11},t_{22},t_{33}$ are
the eigen values of $\mathcal{W}.$ Entanglement content of a quantum state
remaining invariant under application of local unitary operations, any notion
of nonlocality of $\rho$ and $\rho^{{}^{\prime}}$ remains the same.

###  III.2 Entanglement Assisted Bipartite QKD Protocol

Consider any entanglement assisted quantum key distribution(QKD) protocoleck ;
main  involving two parties Alice($A$) and Bob($B$) who try to establish
secure key at the end of the protocol. Let a source $\Lambda$(unknown to Alice
and Bob) distribute copies of a bipartite $d\times d$ dimensional entangled
state $\rho$ between the two parties(each party receiving one qubit of
$\rho$). Now Alice and Bob both perform local measurements on their respective
subsystems and record their outcomes. For local measurements, each of them
chooses randomly from a collection of $N$ number of $d$-dimensional mutually
unbiased basis(MUBsmub1 ). Let
$\mathfrak{C}_{A(B)}$$=$$\\{\mathfrak{B}_{A(B)}^{(\beta)}\\}_{\beta=1}^{N}$
denote the collection of $N$ MUBs of Alice(Bob) where $\forall\,\beta,$ MUB
$\mathfrak{B}_{A(B)}^{(\beta)}$ are given by:

$\mathfrak{B}_{A}^{(\beta)}=\\{|\psi^{\beta}_{i}\rangle\\}_{i=1}^{d}\,\textmd{and}\mathfrak{B}_{B}^{(\beta)}=\\{|\phi^{\beta}_{i}\rangle\\}_{i=1}^{d}$
(3)

If $\mathcal{O}_{A(B)}^{(\beta)}$ denote operators corresponding to the basis
$\mathfrak{B}_{A(B)}^{(\beta)}$ then those are given by:

$\mathcal{O}_{A(B)}^{(\beta)}=\\{|\psi(\phi)^{\beta}_{i}\rangle\langle\psi(\phi)^{\beta}_{i}|\\}_{i=1}^{d},\,\forall\beta=1,...,N.$
(4)

Now MUBs of Alice and Bob $\mathfrak{C}_{A}$ and $\mathfrak{C}_{B}$ are
correlated in the following sense. Let for any fixed value of $\beta$ (from
$1,2,...,d$), Alice and Bob performs measurement in MUB
$\mathfrak{B}_{A}^{(\beta)}$ and $\mathfrak{B}_{A}^{(\beta)}$ respectively. In
case $\rho$(shared between them) is a pure entangled state, then perfect
correlations(between Alice and Bob’s outputs) will imply that if Alice gets
outcome $|\psi^{\beta}_{j}\rangle$ then Bob’s outcome must be
$|\phi^{\beta}_{j}\rangle(\forall\,j).$
After performing measurements on $N$ copies of $\rho,$ a fraction of the
measurement outcomes are used to analyze the joint statistics for verifying
whether corresponding correlations are nonlocal or not. Such a verification is
made by testing violation of a Bell inequality. For remaining part of the
measurement outcomes, the parties publicly compare their measurement bases and
keep outcomes only corresponding to the correlated MUBs(discarding the
remaining outcomes). These outcomes(obtained from correlated MUBs) form the
raw keybran1 . The parties can then extract secure key from remaining part of
the raw key by performing information reconciliationeck8i ; in1  and privacy
amplificationeck8i . For our purpose we focus only on the raw key and joint
statistics used for verifying violation of Bell inequality. We next discuss
the quantum bit error rate involved in the protocol.

####  III.2.1 Quantum Bit Error Rate

Ideally, $\rho$ is supposed to be a pure entangled state. Consequently, in
such a scenario the parties generate identical keys. However, in practical
situation, they share copies of mixed entangled state. Such mixedness of
$\rho$ may result from various factors such as imperfections induced during
state preparation at the source itself(source controlled by eavesdropper),
noisy transmission through channels and also disturbance generated by
measurement procedures. In such non idealistic situation the parties end up
sharing non identical raw keys. Quantum bit error is thereby generated. The
parties can measure the quantum bit error rate(QBER) by comparing a small
fraction of their raw key.
Definition: For any given state $\rho,$ QBER($Q$) is defined as the average
mismatch between Alice and Bob’s outcomes obtained when they measure in
correlated bases. With $\mathfrak{C}_{A}$ and $\mathfrak{C}_{B}$ denoting
collection of correlated MUBs(Eq.(3)) of the two parties(as considered above),
QBER can be expressed as:

$Q=\frac{1}{N}\sum_{\beta=1}^{N}\sum_{i\neq
j=1}^{d}\langle\psi^{\beta}_{i}\phi^{\beta}_{j}|\rho|\psi^{\beta}_{i}\phi^{\beta}_{j}\rangle$
(5)

Above expression of $Q$ holds for any $N$$\leq$$d+1$ number of MUBs. For
instance, when source $\Lambda$ generates a two qubit state($d$$=$$2$) and
each party chooses from a collection of two MUBs, i.e.,
$|\mathfrak{C}_{A}|$$=$$|\mathfrak{C}_{B}|$$=$$2,$ QBER is given bymain :

$Q=\frac{1}{4}(2-\vec{u_{1}}.W\vec{v_{1}}-\vec{u_{2}}.W\vec{v_{2}})$ (6)

where $\vec{u}_{i},\vec{v}_{j}\,(i,j$$=$$1,2)$ denote Bloch vectors of the
measurement bases of Alice and Bob respectively and $W$ denotes the
correlation tensor(Eq.(1)). Minimization over all possible measurement
directions $\vec{u}_{1},\vec{u}_{2},\vec{v}_{1},\vec{v}_{2}$ gives:

$Q\geq\frac{1}{4}(2-\textmd{max}_{i,j}(|t_{ii}|+|t_{jj}|)),\,i\neq j$ (7)

where $t_{11},t_{22},t_{33}$ denote the eigen values of correlation tensor $T$
of $\rho^{{}^{\prime}}$(Eq.(2)) and hence eigen values of correlation tensor
$W$ of $\rho$(Eq.(1)).

####  III.2.2 Minimum Secure Key Rate

The minimum secure key rate $\mathfrak{r}_{min}$eck8i  is defined as the
average number of secret bits that can be distilled from each run of the
protocol when Alice and Bob measure in the correlated bases. Secure key can be
extracted from the protocol only if $\mathfrak{r}_{min}$$>$$0.$
$\mathfrak{r}_{min}$ being dependent on various factors involving strategy by
the eavesdropper, no general expression of $\mathfrak{r}_{min}$ as a function
of $Q$ exists. In some special cases of QKDs, such an expression can be
derived. For a particular instance, in case of symmetric attacks on
entanglement assisted bipartite QKD protocol(with two measurement settings per
qubit), secure key rate can be estimated from the following relationeck8 :

$\mathfrak{r}_{min}=1+2Q\,log_{2}Q+2(1-2Q)\,log_{2}(1-Q)$

###  III.3 Steerable Correlations

In a bipartite measurement scenario(involving a bipartite physical state),
steering may be interpreted as the task of one party(Alice,say) to convince
the other party(Bob), that she, by measuring her own subsystem, can either
steer or construct the conditional states on Bob’s side. The bipartite
correlations shared between Alice and Bob are non steerable in case those
satisfy the following relationzow :

$P(a,b|\overrightarrow{x},\overrightarrow{y})=\int
d\lambda{\rho(\lambda)}P(a|\overrightarrow{x},\lambda)\textmd{Tr}[\widehat{\Pi}(b|\overrightarrow{y})\rho_{2}(\lambda)]$
(8)

where $a$, $b$ denote local results, $\overrightarrow{x},\overrightarrow{y}$
denote the corresponding local inputs with $\lambda$ denoting the hidden
variable. $\widehat{\Pi}(b|\overrightarrow{y})$ denotes the projection
operator corresponding to a measurement characterized by Bob’s measurement
setting $\overrightarrow{y}$, associated with the value $b$ and
$\rho_{2}(\lambda)$ denotes some pure state of Bob’s system, characterized by
parameter $\lambda$. When any set of bipartite correlations can be decomposed
as above(Eq.(8)), then those correlations are said to have a local hidden
state(lhs) model. So steerable correlations are inexplicable by lhs modelszow
.

###  III.4 Steering Inequality

In red3  a linear steering inequality was formulated. Under the assumption
that both the parties(Alice and Bob) sharing a bipartite state($\rho$) perform
$n$ dichotomic quantum measurements(on their respective particles),
Cavalcanti, Jones, Wiseman, and Reid(CJWR) derived a series of correlators
based inequalitiesred3  for verifying steerability of $\rho:$

$\mathcal{F}_{n}(\rho,\nu)=\frac{1}{\sqrt{n}}|\sum_{l=1}^{n}\langle
A_{l}\otimes B_{l}\rangle|\leq 1$ (9)

with $A_{l}$$=$$\hat{a}_{l}\cdot\overrightarrow{\sigma}$,
$B_{l}$$=$$\hat{b}_{l}\cdot\overrightarrow{\sigma}$,
$\hat{a}_{l}\in\mathbb{R}^{3},\,\hat{b}_{l}\in\mathbb{R}^{3}$ denote
orthonormal vectors,
$\nu=\\{\hat{a}_{1},\hat{a}_{2},....\hat{a}_{n},\hat{b}_{1},\hat{b}_{2},...,\hat{b}_{n}\\}$
stands for the collection of measurement directions, $\langle A_{l}\otimes
B_{l}\rangle=\textmd{Tr}(\rho(A_{l}\otimes B_{l}))$ and
$\rho\in\mathbb{H_{A}}\otimes\mathbb{H_{B}}$ is any bipartite quantum state.
Violation of Eq.(9) ensures both way steerability of $\rho$ in the sense that
it is steerable from $A$ to $B$ and vice versa. In particular, for $n$$=$$3,$
CJWR inequality(Eq.(9)) for three settings takes the form:

$\mathcal{F}_{3}(\rho,\nu)=\frac{1}{\sqrt{3}}|\sum_{l=1}^{3}\langle
A_{l}\otimes B_{l}\rangle|\leq 1$ (10)

###  III.5 Closed Form of Upper bound of Eq.(10)

In costa , analytical expression for the upper bound of $2$ and $3$ settings
CJWR steering inequality were formulated in terms of correlation tensor
parameters of $\rho.$ As notion of steering nonlocality remains invariant
under local unitary transformations, analytical expressions of the steering
inequality remain unaltered if the simplified form(Eq.(2)) of two qubit state
$\rho$ is considered. Now for $2$ measurement settings scenario, notions of
Bell-CHSH nonlocality and steering nonlocality are indistinguishable. So, for
our purpose we consider $3$ measurement settings scenario. Analytical
expression of the upper bound of corresponding inequality(Eq.(10)) is given
by:

$\displaystyle\textmd{Max}_{\nu}\mathcal{F}_{3}(\rho,\nu)$ $\displaystyle=$
$\displaystyle\sqrt{\mathfrak{t}_{11}^{2}+\mathfrak{t}_{22}^{2}+\mathfrak{t}_{33}^{2}},$
(11) $\displaystyle=$ $\displaystyle\sqrt{\textmd{Tr}(T^{t}T)}$
$\displaystyle=$ $\displaystyle\sqrt{\textmd{Tr}(W^{t}W)}$

where $W$ and $T$ denote the correlation tensor corresponding to bloch matrix
representation of $\rho$ given by Eq.(1) and Eq.(2) respectively. Last
equality in Eq.(11) is due to unitary equivalence of trace operator. So, by
the linear inequality(Eq.10), any two qubit state $\rho$(Eq.(1)), shared
between $A$ and $B$ is both-way $F_{3}$ steerable if:

$\sqrt{Tr[T^{T}T]}=\sqrt{t_{11}^{2}+t_{22}^{2}+t_{33}^{2}}>1.$ (12)

Eq.(12) gives only a sufficient condition detecting steerability. So steerable
states satisfying violating Eq.(12) exist.

###  III.6 Local Filtering Operations

This type of operations forms a special class of sequential quantum
operationspop1 ; pop2 ; pop3 . By applying suitable local filtering
operations, the entanglement concentration and nonlocal content of any state
$\rho$ can be increased. Let $M_{A(B)}^{(1)},$ $M_{A(B)}^{(2)}$ denote the
local filtering operations applied by Alice(Bob) on their respective
subsystems. Under application of these filtering operation state $\rho$ gets
transformed to a new state $\rho^{{}^{\prime}}$pop2 ; pop3 :

$\rho^{{}^{\prime}}=\frac{M_{A}^{(1)}\otimes
M_{B}^{(1)}\rho(M_{A}^{(1)}\otimes
M_{B}^{(1)})^{\dagger}}{\textmd{Tr}[M_{A}^{(1)}\otimes
M_{B}^{(1)}\rho(M_{A}^{(1)}\otimes M_{B}^{(1)})^{\dagger}]}$ (13)

For simplicity, we consider
$M_{A(B)}^{(2)}$$=$$\sqrt{\mathbb{I}_{2}-M_{A(B)}^{(1)}}$ with the following
specific forms of $M_{A(B)}^{(1)}:$

$\displaystyle M_{A}^{(1)}$ $\displaystyle=$
$\displaystyle\epsilon_{1}|0\rangle\langle 0|+|1\rangle\langle 1|$ (14)
$\displaystyle M_{B}^{(1)}$ $\displaystyle=$
$\displaystyle\epsilon_{2}|0\rangle\langle 0|+|1\rangle\langle
1|,\,\textmd{with}\,\epsilon_{1},\epsilon_{2}\in[0,1]$ (15)

It may be noted here that local filtering operations for two qubit states may
be considered as single copy entanglement distillation operationspop2 ; pop3 .

##  IV Characterizing arbitrary two qubit states based on $Q$

In this section we characterize any given quantum state with respect to its
utility in a QKD protocol. Our analysis will be restricted to only those QKD
protocols where secure key rate is dependent on QBER(Eq.(3)) only. Before
starting our analysis, we first discuss the scenario in details.

###  IV.1 Specifications of QKD Protocol

For our purpose we consider an usual bipartite entanglement assisted QKD
protocol(subsec.III.2) such that $\rho$ shared between Alice and Bob is a two
qubit state($d$$=$$2$). In this protocol each of the parties perform local
measurements on their respective subsystems choosing randomly from a
collection of three MUBs, i.e., Alice(Bob) chooses from
$\mathfrak{C}_{A(B)}$$=$$\\{\mathfrak{B}_{A(B)}^{(\beta)}\\}_{\beta=1}^{3}$
where $\mathfrak{B}_{A(B)}^{(\beta)}$ are given by Eq.(3) for $d$$=$$2.$ As
discussed in subsec.III.2, after making measurements, the parties reconcile
their measurement bases publicly. Now with $x,y$ denoting measurements and
$a,b$ denoting outcomes of Alice and Bob respectively, the correlation
statistics $P(a,b|x,y),$ corresponding to a fraction of raw data(measurement
outcomes) is used for checking $F_{3}$ steerability of $\rho$ via the
violation of CJWR inequality for three settings(Eq.10). As already pointed out
in sec.I, consideration of CJWR inequality for our purpose basically relies on
two reasons. Firstly, Eq.(10) is a Bell type inequality(expressed as a linear
combination of correlators). So its violation can be considered as a necessary
condition for security in QKD protocol involving $\rho$ owing to the fact that
Bell inequality’s violation is necessary for the sameeck4 ; eck5 ; eck13 .
Secondly CJWR inequality is framed as a bipartite steering inequality under
the assumption that both the parties sharing $\rho$ preform
characterized(quantum) measurements. Such an assumption suits our QKD protocol
where each party performs characterized measurements. Precisely speaking, both
Alice and Bob perform projective measurements in arbitrary directions:
$\vec{u}_{i}.\vec{\sigma}$ and $\vec{v}_{i}.\vec{\sigma}\,(i$$=$$1,2,3)$
respectively. $\forall i,\,\vec{u}_{i}$ and $\vec{v}_{i}$ represent bloch
vectors of measurement basis with Alice and Bob respectively. Each party
choosing from a collection of three bases implies that each of them can choose
to perform projective measurement in any one of three arbitrary directions:
$\vec{u}_{1},\vec{u}_{2},\vec{u}_{3}$ for Alice and
$\vec{v}_{1},\vec{v}_{2},\vec{v}_{3}$ for Bob. Having specified the
measurement scenario, we next approach to optimize the QBER($Q$) for our
scenario.

###  IV.2 Optimization of $Q$

In a QKD protocol involving three measurement settings per party, $Q$(Eq.(5))
turns out to be:

$Q=\frac{1}{6}(3-\sum_{i=1}^{3}\vec{u_{i}}.W\vec{v_{i}})$ (16)

where $W$ is the correlation tensor appearing in Bloch matrix representation
of $\rho$(Eq.(1)). Minimization over all possible measurement directions
$\vec{u}_{1},\vec{u}_{2},\vec{u}_{3},\vec{v}_{1},\vec{v}_{2},\vec{v}_{3}$
gives(see appendixVIII):

$\displaystyle Q$ $\displaystyle\geq$ $\displaystyle
Q_{min},\,\textmd{where},\,$ $\displaystyle Q_{min}$ $\displaystyle=$
$\displaystyle\frac{1}{6}(3-(|t_{11}|+|t_{22}|+|t_{33}|)).$ (17)

where $T$$=$$\textmd{diag}(t_{11},t_{22},t_{33})$ denote correlation tensor of
$\rho^{{}^{\prime}}$(Eq.(2)). As discussed in subsec.III,
$t_{11},t_{22},t_{33}$ are the eigen values of correlation tensor $W$ of
$\rho$(Eq.(1)). Now let $\rho$ be an $F_{3}$ unsteerable state, i.e., violate
Eq.(12). For simplicity, let eigen values of its correlation tensor($T$)
satisfies:

$t_{11}^{2}+t_{22}^{2}+t_{33}^{2}=1$ (18)

Now we minimize $Q$ with respect to all such possible $F_{3}$ unsteerable
quantum states. Imposition of such a restriction is required as we are
considering violation of CJWR inequality(Eq.(10)) by $\rho$ necessary for
successful key generation in the protocol(using $\rho$). If $\mathcal{Q}_{0}$
denote the least possible value of $Q$ under such restriction(see appendixIX),
then:

$\mathcal{Q}_{0}=0.211.$ (19)

Consequently, when any $F_{3}$ unsteerable state is used for key distribution,
QBER generated in the protocol cannot be less than $0.211.$ The minimum error
rate($\mathcal{Q}_{0}$) may also be referred to as the critical error rate of
our QKD protocol. Critical error rate($\mathcal{Q}_{0}$) is obtained for
$t_{11}$$=$$t_{22}$$=$$t_{33}$$=$$\pm\frac{1}{\sqrt{3}}$(see appendix IX).
Obtaining the minimal expression of QBER with respect to measurement settings
and thereby estimating its minimum value(with respect to all possible $F_{3}$
unsteerable states) for our protocol, we next characterize two qubit state
space in this context.

###  IV.3 Characterization of Two Qubit State Space

As already discussed before, here we intend to characterize an arbitrary two
qubit state $\rho$ in ambit of analyzing its utility in the QKD scenario
discussed in subsec.III.2. As is evident from our discussion so far in sec.IV,
analyzing the eigen space of correlation tensor $T$ of $\rho$(Eq.(2)) suffices
for our purpose.
$\rho$ being a quantum state, in general, each of
$t_{11},t_{22},t_{33}$$\in$$[-1,1].$ Let $C$ denote a cuboid given by:

$C=\\{t_{11},t_{22},t_{33}:|t_{11}|\leq 1,|t_{22}|\leq
1\,\textmd{and}\,|t_{33}|\leq 1\\}$ (20)

So density matrix corresponding to any point lying outside the cuboid
C(Eq.(20)) does not represent any valid quantum state(see Fig.1). For rest of
our analysis, we denote quantum state corresponding to any point $R$ inside
$C$ as $\rho_{R}.$ Now, inside $C,$ $F_{3}$ unsteerable states reside on and
inside the unit sphere($S$,say) with center at the origin given by Eq.(18). So
any point lying inside $C$ but outside $S$ corresponds to an $F_{3}$ steerable
quantum state(see Fig.1). Now as discussed in subsec.IV.2,when an $F_{3}$
unsteerable state $\rho$ is used then $Q$$\geq$$\mathcal{Q}_{0}.$ This in turn
restricts the eigen values of $T$(corresponding to $\rho$):

$|t_{11}|+|t_{22}|+|t_{33}|\leq\sqrt{3}.$ (21)

For sake of simplicity, in remaining part of our analysis, we are considering
positive octant only(see Fig.1). Analysis made here can however be extended
trivially to the remaining octants. Present analysis then encompasses quantum
states residing inside and on the unit cuboid $C_{+}$(say):

$C_{+}=\\{(t_{11},t_{22},t_{33}):0\leq t_{11},t_{22},t_{33}\leq 1\\}$ (22)

Correspondingly, the positive octant($S_{+}$,say) of the unit sphere(Eq.(18))
is considered as the collection of $F_{3}$ unsteerable states:

$S_{+}=\\{(t_{11},t_{22},t_{33}):t_{11}^{2}+t_{22}^{2}+t_{33}^{2}\leq
1\,\textmd{\small{with}}\,t_{11},t_{22},t_{33}\geq 0\\}.$ (23)

Restricted in positive octant, Eq.(21) now represents region lying below a
tangent plane to $S_{+}$(Eq.(23)), hence to the unit sphere $S$(Eq.(18)) at
the point $P(\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}})$:

$t_{11}+t_{22}+t_{33}\leq\sqrt{3}.$ (24)

Clearly when $\rho$ corresponding to any point lying below or on the tangent
plane(Eq.(24)) is used in the protocol, estimated QBER is greater than or at
most equal to the critical error rate($\mathcal{Q}_{0}$). On the contrary,
when $\rho$ used in the protocol corresponds to any point($R$,say) lying above
the same plane, QBER is less than $\mathcal{Q}_{0}.$ Clearly, in such a case,
the point $R$ lies outside $S_{+}$(see Fig.1). Consequently, $\rho_{R}$ is
$F_{3}$ steerable. Under our assumption of CJWR inequality’s violation
necessary for ensuring secure key generation in the protocol, $\rho_{R}$ thus
turns out to be useful. In this context, we consider any state $\rho$ as
useful in our QKD protocol if QBER obtained in the protocol(using $\rho$) is
less than $\mathcal{Q}_{0}.$ Hence two qubit state $\rho$ corresponding to any
point lying in $C_{+}$ is useful if and only if:

$t_{11}+t_{22}+t_{33}>\sqrt{3}.$ (25)

So, in general when one considers the entire $C$(Eq.(20)), i.e., for an
arbitrary two qubit state, above restriction(Eq.(25)) becomes:

$|t_{11}|+|t_{22}|+|t_{33}|>\sqrt{3}.$ (26)

Now let us focus on the region lying outside $S_{+}$ and inside $C_{+}.$ Let
$L$ be any point lying in that region. So $\rho_{L}$ is $F_{3}$ steerable. $L$
may lie below or above the tangent plane(Eq.(24)). Utility of $\rho_{L}$ thus
depends on position of $L.$ To be precise, if $L$ lies below the tangent plane
then $\rho_{L}$ is useful in our protocol whereas $\rho_{L}$ turns out to be
useless in the other case($L$ lying above the plane). This in turn points out
the insufficiency of the $F_{3}$ steerability criterion to ensure secure key
generation. Three settings CJWR inequality(Eq.(10)) being a Bell type
inequality, our observation simply points out the following:
Violation of a Bell type inequality by any two qubit state $\rho$ is necessary
but not sufficient to guarantee secure key generation in an entanglement
assisted protocol involving $\rho.$

---

(a) Sub figure-I (b) Sub figure-II

Figure 1: _Spectrum of correlation tensor of an arbitrary two qubit state
considered here. In sub figure-I, cuboid indicates all possible two qubit
states whereas any point from the sphere gives $F_{3}$ unsteerable state. In
sub figure -II, only positive octant is considered. Any point inside the
portion($S_{+}$) of the sphere corresponds to an unsteerable two qubit state.
Region lying outside $S_{+}$ and below the tangent plane at
$P(\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}})$ indicate
steerable but useless states whereas that lying above the tangent plane give
useful states._

In practical situation owing to unavailability of pure entangled state for key
distribution, observation of maximal violation of CJWR inequality becomes
impossible. Hence, based on amount of violation, identifying two qubit
entangled states useful in entanglement assisted QKD protocol is important
from practical view point.
Identifying Useful States: Let in a QKD protocol, CJWR inequality(Eq.(10)) be
violated by some fixed amount $\mathcal{V}$(say). Let $\varrho$ be some
unknown two qubit state used in the corresponding protocol. Let
$\lambda_{11},\lambda_{22},\lambda_{33}$ denote the eigen values of the
correlation tensor of $\varrho.$ Violation being observed, restriction is
imposed on these three unknown(as $\varrho$ is unknown) quantities:

$\displaystyle\lambda_{11}^{2}+\lambda_{22}^{2}+\lambda_{33}^{2}$
$\displaystyle=$ $\displaystyle\mathcal{V}^{2}$
$\displaystyle\textmd{Alternatively,}\,\lambda_{11}$ $\displaystyle=$
$\displaystyle\pm\sqrt{\mathcal{V}^{2}-\lambda_{22}^{2}-\lambda_{33}^{2}}$
(27)

Now $\varrho$ is useful for secure key generation in the protocol if it
satisfies Eq.(26). Hence, $\varrho$ is useful if:

$\sqrt{3}-\lambda_{11}-\lambda_{22}<\sqrt{\mathcal{V}^{2}-\lambda_{22}^{2}-\lambda_{33}^{2}}\leq
1$ (28)

Eq.(28) thus specifies the criterion required to be satisfied by an unknown
state provided $\mathcal{V}$ amount of violation of CJWR inequality is
observed in the protocol(see Fig.3).

Figure 2: _Shaded regions give two qubit states useful for QKD for some
specific violation amounts of CJWR inequality observed in the protocol._

###  IV.4 Illustrations

Let us now analyze above characterization for a few well known class of two
qubit states.
Bell Diagonal states: The class of Bell diagonal statesnie  is represented as
follows:

$\displaystyle\varrho_{Bell}=w_{1}|\psi^{-}\rangle\langle\psi^{-}|+w_{2}|\phi^{+}\rangle\langle\phi^{+}|+w_{3}|\phi^{-}\rangle\langle\phi^{-}|$
$\displaystyle+w_{4}|\psi^{+}\rangle\langle\psi^{+}|$ , (29)

with $w_{i}$$\small{\in}$$[0,1]\,\forall i$$=$$1,2,3,4,$
$\sum_{i=1}^{4}w_{i}$$=$$1$ and
$|\phi^{\pm}\rangle$$=$$\frac{|00\rangle\pm|11\rangle}{\sqrt{2}},$
$|\psi^{\pm}\rangle$$=$$\frac{|01\rangle\pm|10\rangle}{\sqrt{2}}$ denote the
Bell states. Eq.(IV.4) is often referred to as the class of states having
maximally mixed marginals.
Correlation matrix is given by:
$\textmd{diag}(1-2(w_{1}+w_{3}),1-2(w_{2}+w_{3}),1-2(w_{1}+w_{2}))$. Bell
diagonal states are $F_{3}$ steerable provided the following relation holds:

$\sqrt{8(\sum_{i,j=1}^{3}w_{i}*w_{j}+w_{4})-5}>1$ (30)

On the other hand, Bell diagonal states useful for QKD protocol(satisfying
Eq.(26)) are characterized by:

$\sum_{i,j=1}^{3}|1-2(w_{i}+w_{j})|>\sqrt{3},\,\textmd{where}\,i\neq j$ (31)

Combination of Eqs.(30,31) points out that not all $F_{3}$ steerable states
from this family are useful for our QKD protocol(see Fig.3).

Figure 3: _Shaded region forms a part of parameter space of the Bell diagonal
family(Eq.( IV.4)). Bell diagonal state corresponding to any point in the
shaded region is useful in QKD protocol._

Now let us consider the Werner class of states from the family of Bell
diagonal states:

$\varrho_{W}=\omega|\psi^{-}\rangle\langle\psi^{-}|+\frac{(1-\omega)}{4}\mathbb{I}_{2}\times\mathbb{I}_{2},\,\omega\in[0,1].$
(32)

Eigen values for the correlation tensor are all given by $-v$. Clearly for
$v$$\small{\in}$$(0.5772,1]$ corresponding member from Werner class(Eq.(32))
is $F_{3}$ steerable. Again Eq.(26) is satisfied for the same range of values
of $v.$ Consequently for this subclass of Bell diagonal states(Eq.(IV.4)), any
$F_{3}$ steerable state is always useful in QKD protocol.
Family of States Not Diagonal in Bell Basis: Consider the following classgrud
; woj :

$\gamma=q|\varphi\rangle\langle\varphi|+(1-q)|00\rangle\langle 00|$ (33)

where $|\varphi\rangle=\cos\alpha|10\rangle+\sin\alpha|01\rangle,$ with
$\alpha$$\in$$[0,\frac{\pi}{4}]$ and $0$$\leq$$q$$\leq$$1$. This class of
states was used in grud  for increasing maximally entangled fraction in an
entanglement swapping network. Correlation tensor is given by
$\textmd{diag}(q\sin 2\alpha,q\sin 2\alpha,1-2q).$ A member from this family
is $F_{3}$ steerable if:

$2q^{2}\sin^{2}2\alpha+(1-2q)^{2}>1$ (34)

Again any state from this class is useful in QKD protocol in case it satisfies
the following relation:

$2q\sin 2\alpha+|1-2q|>\sqrt{3}.$ (35)

Clearly, QKD protocol will run successfully if states satisfying both the
above relations(Eqs.(34,35) are used in the protocol(see Fig.4).

Figure 4: _Shaded region gives the states(Eq.( 33)) which can be used for
secure key generation._

##  V Incorporating Local Filtering Operations in QKD

Let us now consider an entanglement assisted protocol where both the parties
can perform local filtering operations before measuring their subsystems in
correlated MUBs. We next discuss the protocol in details.

###  V.1 Modified QKD Protocol

Steps of the protocol are specified below:

  * •

Source $\Lambda$ distributes many copies of a two qubit state($\rho$) between
the two parties Alice and Bob.

  * •

On receiving the qubits the parties perform full state tomography.

  * •

Being ensured that the shared state is an entangled state they perform local
filtering operations(Eq.(14)): Alice using operators
$\\{M_{A}^{(1)},M_{A}^{(2)}\\}$ and Bob performing
$\\{M_{B}^{(1)},M_{B}^{(2)}\\}$ where the operators are specified in
subsec.III.6.

  * •

After completion of measurements, they announce outcome of measurements in
$\\{M_{A}^{(i)}\\}$ and $\\{M_{B}^{(j)}\\}.$ They select those qubit pairs for
which $M_{A}^{(1)}$ and $M_{B}^{(1)}$ clicked. Only these qubit pairs are
considered further in the protocol discarding the remaining ones. Let each
pair of two qubit state selected be denoted by $\rho^{{}^{\prime}}.$

  * •

Rest of the protocol runs as usual(subsec.IV.1).

In the protocol, QBER ($Q$) for transformed states($\rho^{{}^{\prime}}$,say)
is given by:

$Q^{{}^{\prime}}_{min}=P_{succ}*Q_{min}$ (36)

where $Q_{min}$ is given by Eq.(IV.2) and $P_{succ}$(possibility of
measurement $M_{A}^{(1)}$ and $M_{B}^{(1)}$ clicking) is given by:

$P_{succ}=\textmd{Tr}[M_{A}^{(1)}\otimes M_{B}^{(1)}\rho(M_{A}^{(1)}\otimes
M_{B}^{(1)})^{\dagger}],$ (37)

where $M_{A}^{(1)},M_{B}^{(1)}$ are given by Eq.(14).
For further discussion in this section we will refer to this QKD protocol(with
filtering operations) as Modified QKD protocol while that without any
filtering operations as QKD protocol only.

###  V.2 Illustrations

Now, consider the family of states given by Eq.(33). $F_{3}$ steerable members
from this class which are useless in the QKD violate Eq.(35). Let any such
$F_{3}$ steerable state be used in the modified protocol. For some suitable
local filtering operations(specific values of $\epsilon_{1},\epsilon_{2}$ in
Eq.(14)), the protocol runs successfully(see fig.5). For a particular
instance, consider $\alpha$$=$$0.25,$ $q$$=$$0.9$(Eq.(33)) Before filtering
$Q_{min}$$=$$0.22284$$>$$\mathcal{Q}_{0}.$ On using this state in the modified
QKD protocol with $\epsilon_{1}$$=$$0.02119$ and $\epsilon_{2}$$=$$0.02563,$
$P_{succ}$$\approx$$0.001,$ and by Eq.(36),
$Q_{min}^{{}^{\prime}}$$=$$0.19862.$ As
$Q_{min}^{{}^{\prime}}$$<$$\mathcal{Q}_{0},$ so successful key generation
takes place in the protocol. For some fixed values of $\alpha,$ range of noise
level parameter $q$ for useful states(Eq.(34)) are given in Table.(1). Now,
for obvious reasons, not all local filtering operations(Eq.(14)) turn out to
be useful in the modified protocol. Depending on the state to be used,
($\epsilon_{1},\epsilon_{2}$), parameterizing these operations(Eq.(14)) are to
be selected. For the above class of states(Eq.(33) considered, suitable range
of $(\epsilon_{1},\epsilon_{2})$ is shown in Fig.(7).

Figure 5: _Shaded region in subfig.I gives the unsteerable states(Eq.( 33))
useful in the modified QKD protocol for $\epsilon_{1}$$=$$.02119$ and
$\epsilon_{2}$$=$$0.02563.$ Similarly, shaded region in subfig.II gives
steerable states that are useful in the same modified protocol(i.e., for
$\epsilon_{1}$$=$$.02119$ and $\epsilon_{2}$$=$$0.02563$). None of these
states(both in subfig.I, subfig.II) is useful in the QKD protocol(without
filtering operations)._
Figure 6: _In both sub figures parameter space $(\epsilon_{1},\epsilon_{2})$
characterizing local filtering operations is considered. Shaded regions in
each characterize suitable local filtering operations(used in the modified QKD
protocol) for some specific states from the class given by Eq.(33). Consider
two $F_{3}$ unsteerable states specified by $(q,\alpha)$$=$$(0.6,0.7)$ and
$(q,\alpha)$$=$$(0.5,0.4)$ In sub figure-I, any point in the region $R_{1}\cup
R_{2}$ gives suitable local filters for the first state whereas that for the
second state is given by any point from the region $R_{2}\cup R_{3}.$ Next
consider two steerable states:$(q,\alpha)$$=$$(0.9,0.26)$ and
$(q,\alpha)$$=$$(0.68,0.7)$ In sub figure-II, suitable local filters for these
two steerable states are given by any point from region $R_{1}\cup R_{2}$ and
$R_{2}\cup R_{3}$ respectively. _

Next, let $F_{3}$ unsteerable states(violating Eq.(34)) from this family be
used in the modified protocol. Again for some suitable filtering operations
made by the users, secure key generation becomes possible for some of these
$F_{3}$ unsteerable states(see Fig.(5)). Some specific instances are given in
Table.(1).

Table 1: Modified QKD protocol with $\epsilon_{1}$$=$$0.15,$ $\epsilon_{2}$$=$$0.02563$ is considered. For some specific values of state parameter $\alpha,$ range of other parameter $q$ is specified or which corresponding state is useful for secure key generation in this modified QKD protocol. Second column in the table indicates whether the state used in the protocol violates CJWR inequality or not. Last two instances point out the fact that initially $F_{3}$ unsteerable states can also be used for the modified QKD protocol. State | $F_{3}$ | Range of
---|---|---
parameter | Steerability | $q$
$\alpha$$=$$0.24$ | Steerable | $[0.904,1]$
$\alpha$$=$$0.7$ | Steerable | $[0.674,1]$
$\alpha$$=$$0.2$ | Unsteerable | $[0.5,1]$
$\alpha$$=$$0.6$ | Unsteerable | $[0.52,1]$

After discussing how our QKD protocol can be modified by allowing the users to
apply local filtering operations, we next consider another significant aspect
of our protocol.

##  VI Absolute Bell-CHSH Local States In Secure Key Generation

Our QKD protocol being semi device independent in nature, entangled state is
distributed from some unknown source $\Lambda.$ Let an untrusted third party
Eve has access to $\Lambda.$ So if $\rho$ be the state generated from
$\Lambda,$ then Eve has access to both the qubits of $\rho.$ Let, Eve measure
$\rho$ in some suitable global basis such that $\rho$ transforms into
$\rho^{{}^{\prime}}$ where $\rho^{{}^{\prime}}$ remains entangled but becomes
Bell-CHSH local in the new basis. Under control of Eve source $\Lambda$ thus
distributes an absolutely Bell-CHSH local state $\rho^{{}^{\prime}}$ between
Alice and Bob in the protocol. Unlike that of any standard QKD protocol
relying on Bell-CHSH violation, our protocol can securely generate key for
some of these states. Owing to existence of absolutely Bell-CHSH local $F_{3}$
steerable two qubit states, our QKD protocol gives advantage over the standard
ones. We next discuss a few examples in this respect.
Consider the family of Bell diagonal states(Eq.(IV.4)). Parameters of
absolutely Bell-CHSH local states from this family satisfyguoch :

$\displaystyle\textmd{Max}_{(i,j,k)}[1-4(w_{i}-w_{i}^{2}-w_{i}*w_{j}-w_{i}*w_{k})$
$\displaystyle-2(w_{j}+w_{k}-w_{j}^{2}-w_{k}^{2})]$ $\displaystyle\leq$
$\displaystyle\frac{1}{2},$ (38)

where $(i,j,k)$ denote all possible cyclic permutations(of length three) over
$1,2,3:\,(1,2,3),\,(2,3,1)\,(3,1,2).$ Let any such state gets distributed
between the two users of the protocol. Some of these states satisfy Eq.(31).
Consequently the protocol runs successfully for them(see Fig.7).

Figure 7: _Corresponding to any point from the shaded regions of the parameter
space $(w_{1},w_{2},w_{3})$, absolutely local Bell diagonal state(Eq.(IV.4))
is useful in our QKD protocol._

##  VII Discussions

Notion of Bell nonlocality has been rigorously analyzed with respect to the
study of QKD in entanglement assisted protocols. However, whether Bell
violation provides a sufficient criterion is a matter of great debate. Present
discussion points out the insufficiency of a Bell type inequality in this
perspective. CJWR inequality(Eq.(10)), a Bell type inequality for detecting
steerability is used here, to prescribe a QKD protocol. Using notion of
$F_{3}$ steerability, characterization of arbitrary two qubit state is
obtained in the context of its usefulness in QKD protocol. Interestingly, any
$F_{3}$ steerable Werner state is useful in QKD protocol. For any amount of
violation $\mathcal{V}$ of Eq.(10), one can identify whether the state(giving
$\mathcal{V}$ amount of violation) is useful in the protocol or not. Such an
identification is completely based on the eigen values of the correlation
tensor of the corresponding state.

Furthermore, in case local filtering operations are allowed(before the users
perform local bases measurements) in the protocol, some $F_{3}$ unsteerable
states become useful in the modified QKD protocol. The utility of absolutely
local Bell-CHSH states in our modified protocol further buttresses our work.

Critical quantum bit error rate $\mathcal{Q}_{0}$(Eq.(19)) in our protocol is
greater than that obtained for protocol depending on Bell-CHSH
nonlocality($\mathcal{Q}_{0}$$=$$0.14$ main ). Increased number of measurement
settings per party($2$ in main  and $3$ here) is one of the potent causes for
such a contrast.

As already specified before, the entire analysis in our work is applicable for
those QKD protocols where secure key rate is a function of QBER($Q$) onlyeck8
. However owing to complexity of practical situations, $\mathfrak{r}_{min}$
may depend on many other factors. So, characterizing arbitrary two qubit
states for more general QKDs is a potential source of future research. A
generalization of the scheme to include other Bell-like inequalities also
warrants attention.

## Acknowledgement

Tapaswini Patro would like to acknowledge the support from DST-Inspire
fellowship No. DST/INSPIRE Fellowship/2019/IF190357.

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##  VIII Appendix.I

Proof of Eq(16):
Let $\vec{\mathfrak{m}}$$=$$(m_{1},m_{2},m_{3})$ denote an arbitrary
direction. Eigen basis operators corresponding to projective measurement along
$\vec{\mathfrak{m}}$ are given by:
$\\{\frac{\mathbb{I}_{2}+\vec{\mathfrak{m}}.\vec{\sigma}}{2},\frac{\mathbb{I}_{2}-\vec{\mathfrak{m}}.\vec{\sigma}}{2}\\}.$
Now, as discussed in the main text, in our QKD protocol each of Alice and Bob
perform projective measurements along any one of three arbitrary directions:
$\vec{u}_{1},\vec{u}_{2},\vec{u}_{3}$ for Alice and
$\vec{v}_{1},\vec{v}_{2},\vec{v}_{3}$ for Bob. So for Alice the collection of
measurement basis operators(Eq.(4)) is given by:

$\mathcal{O}_{A}^{(j)}=\\{\frac{\mathbb{I}_{2}+\vec{u_{j}}.\vec{\sigma}}{2},\frac{\mathbb{I}_{2}-\vec{u_{j}}.\vec{\sigma}}{2}\\},\,j=1,2,3$
(39)

Similarly, collection of measurement basis operators of Bob is given by:

$\mathcal{O}_{B}^{(j)}=\\{\frac{\mathbb{I}_{2}+\vec{v_{j}}.\vec{\sigma}}{2},\frac{\mathbb{I}_{2}-\vec{v_{j}}.\vec{\sigma}}{2}\\},\,j=1,2,3$
(40)

As discussed in the main text, corresponding to correlated bases of Alice and
Bob, the operator bases are given by
$\mathcal{O}_{A}^{(j)},\mathcal{O}_{B}^{(j)}(j$$=$$1,2,3).$
An arbitrary two qubit state $\rho$(Eq.(1)) is shared between Alice and Bob.
In case Alice and Bob measure
$\vec{u_{i}}.\vec{\sigma},\vec{v_{i}}.\vec{\sigma},$ probability of them
obtaining mismatching outputs while measuring in correlated bases is given by:

$P_{j}=\frac{1}{4}\sum_{i=0,1}\textmd{Tr}[(\mathbb{I}_{2}+(-1)^{i}\vec{u_{j}}.\vec{\sigma})\otimes(\mathbb{I}_{2}+(-1)^{i+1}\vec{v_{j}}.\vec{\sigma}).\rho],$
(41)

$\forall\,j$$=$$1,2,3.$ Using Eq.(41) the expression for quantum bit error
rate $Q$(Eq.(5)) becomes:

$\displaystyle Q$ $\displaystyle=$ $\displaystyle\sum_{i=1}^{3}P_{i}$ (42)
$\displaystyle=$
$\displaystyle\frac{1}{6}(3-\sum_{i=1}^{3}\sum_{j=1}^{3}u_{ij}\sum_{k=1}^{3}w_{jk}v_{ik})$
$\displaystyle=$
$\displaystyle\frac{1}{6}(3-\sum_{i=1}^{3}\vec{u_{i}}.W\vec{v_{i}}),$

where $\vec{u_{i}}$$=$$(u_{i1},u_{i2},u_{i3})^{t}$ and
$\vec{v_{i}}$$=$$(v_{i1},v_{i2},v_{i3}),\forall i$$=$$1,2,3.$ Eq.(42) gives
Eq.(16).

Proof of Eq(IV.2):Minimization of $Q$ over all possible MUBs of the two
parties.

Clearly $Q$ is summation of the probability terms appearing in Eq.(41). Let us
introduce a few notations $G_{i}^{z_{1}},\,H_{i}^{z_{2}},\forall i$$=$$1,2,3$
for ease of use in further calculations:

$\displaystyle G_{i}^{z_{1}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}(\mathbb{I}_{2}+(-1)^{z_{1}}\vec{u_{i}}.\vec{\sigma}),\,z_{1}=0,1$
(43) $\displaystyle H_{i}^{z_{2}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}(\mathbb{I}_{2}+(-1)^{z_{2}}\vec{v_{i}}.\vec{\sigma}),\,z_{2}=0,1$
(44)

Now, let $L_{a}$ and $L_{b}$ denote the local unitary operations such that:

$\rho=(L_{a}\otimes L_{b})\rho^{{}^{\prime}}(L_{a}\otimes L_{b})^{\dagger}$
(45)

Let the unitary operations be specified as follows:

$L_{a(b)}=\left(\begin{array}[]{cc}l_{a(b)}^{(11)}&l_{a(b)}^{(12)}\\\
l_{a(b)}^{(21)}&l_{a(b)}^{(22)}\\\ \end{array}\right)$ (46)

Using Eq.(45), $\forall j$$=$$1,2,3,$ from Eq.(41), we get:

$\displaystyle P_{j}$ $\displaystyle=$
$\displaystyle\sum_{i=0,1}\textmd{Tr}[G_{j}^{i}\otimes
H_{j}^{i+1}((L_{a}\otimes L_{b})\rho^{{}^{\prime}}(L_{a}\otimes
L_{b})^{\dagger})]$ (47) $\displaystyle=$
$\displaystyle\sum_{i=0,1}\textmd{Tr}[(L_{a}\otimes
L_{b})^{\dagger}G_{j}^{i}\otimes H_{j}^{i+1}(L_{a}\otimes
L_{b})\rho^{{}^{\prime}}]$ $\displaystyle=$
$\displaystyle\sum_{i=0,1}\textmd{Tr}[(L_{a}^{\dagger}G_{j}^{i}L_{a})\otimes(L_{b}^{\dagger}H_{j}^{i+1}L_{b})\,\rho^{{}^{\prime}}]$

From Eq.(47), consider the term $L_{a}^{\dagger}G_{j}^{i}L_{a}$ for
$i$$=$$0$(say). Let us now further analyze this term. Using Eqs.(43,46), we
get:

$\displaystyle L_{a}^{\dagger}G_{j}^{0}L_{a}$ $\displaystyle=$ $\displaystyle
L_{a}^{\dagger}\frac{1}{2}(\mathbb{I}_{2}+\vec{u_{j}}.\vec{\sigma})L_{a}$ (48)
$\displaystyle=$
$\displaystyle\frac{1}{2}(\mathbb{I}_{2}+L_{a}^{\dagger}\vec{u_{j}}.\vec{\sigma}L_{a})$

Now using Eq.(46), $L_{a},L_{a}^{\dagger}$ can be expressed in terms of Pauli
matrices($\sigma_{1},\sigma_{2},\sigma_{3}$) as follows:

$\displaystyle L_{a}$ $\displaystyle=$
$\displaystyle\frac{1}{2}((l_{a}^{(11)}+l_{a}^{(22)})\mathbb{I}_{2}+(l_{a}^{(11)}-l_{a}^{(22)}))\sigma_{3}$
$\displaystyle+(l_{a}^{(12)}$ $\displaystyle+$ $\displaystyle
l_{a}^{(21)})\sigma_{1}+\imath(l_{a}^{(12)}-l_{a}^{(21)})\sigma_{2}$
$\displaystyle L_{a}^{\dagger}$ $\displaystyle=$
$\displaystyle\frac{1}{2}((\overline{l_{a}^{(11)}}+\overline{l_{a}^{(22)}})\mathbb{I}_{2}+(\overline{l_{a}^{(11)}}-\overline{l_{a}^{(22)}}))\sigma_{3}$
$\displaystyle+(\overline{l_{a}^{(12)}}$ $\displaystyle+$
$\displaystyle\overline{l_{a}^{(21)}})\sigma_{1}+\imath(\overline{l_{a}^{(21)}}-\overline{l_{a}^{(12)}})\sigma_{2}$
(50)

with $\overline{l_{a}^{(ij)}}$ denoting complex conjugate of $l_{a}^{(ij)}.$
Using Eq.(VIII), from Eq.(48), we get:

$\displaystyle L_{a}^{\dagger}\vec{u_{j}}.\vec{\sigma}L_{a}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}A_{11}&A_{12}\\\ A_{21}&A_{22}\\\
\end{array}\right),\,\textmd{where,}$ (53) $\displaystyle A_{11}$
$\displaystyle=$
$\displaystyle(l_{a(b)}^{(21)}(u_{j1}-\imath\,u_{j2})+l_{a(b)}^{(11)}u_{j3})\overline{l_{a(b)}^{(11)}}$
$\displaystyle+$
$\displaystyle(l_{a(b)}^{(11)}(u_{j1}+\imath\,u_{j2})-l_{a(b)}^{(21)}u_{j3})\overline{l_{a(b)}^{(21)}}$
$\displaystyle A_{12}$ $\displaystyle=$
$\displaystyle(l_{a(b)}^{(21)}(u_{j1}-\imath\,u_{j2})+l_{a(b)}^{(11)}u_{j3})\overline{l_{a(b)}^{(12)}}$
$\displaystyle+$
$\displaystyle(l_{a(b)}^{(11)}(u_{j1}+\imath\,u_{j2})-l_{a(b)}^{(21)}u_{j3})\overline{l_{a(b)}^{(22)}}$
$\displaystyle A_{21}$ $\displaystyle=$
$\displaystyle(l_{a(b)}^{(22)}(u_{j1}-\imath\,u_{j2})+l_{a(b)}^{(12)}u_{j3})\overline{l_{a(b)}^{(11)}}$
$\displaystyle+$
$\displaystyle(l_{a(b)}^{(12)}(u_{j1}+\imath\,u_{j2})-l_{a(b)}^{(22)}u_{j3})\overline{l_{a(b)}^{(21)}}$
$\displaystyle A_{22}$ $\displaystyle=$
$\displaystyle(l_{a(b)}^{(22)}(u_{j1}-\imath\,u_{j2})+l_{a(b)}^{(12)}u_{j3})\overline{l_{a(b)}^{(12)}}$
(54) $\displaystyle+$
$\displaystyle(l_{a(b)}^{(12)}(u_{j1}+\imath\,u_{j2})-l_{a(b)}^{(22)}u_{j3})\overline{l_{a(b)}^{(22)}}$

$L_{a}$ being an unitary matrix,
$L_{a}.L_{a}^{\dagger}$$=$$L_{a}^{\dagger}.L_{a}$$=$$\mathbb{I}_{2}.$ Using
that we get:

$\displaystyle|\small{l_{a}^{(11)}}|^{2}+|\small{l_{a}^{(12)}}|^{2}$
$\displaystyle=$ $\displaystyle 1$
$\displaystyle|\small{l_{a}^{(21)}}|^{2}+|\small{l_{a}^{(22)}}|^{2}$
$\displaystyle=$ $\displaystyle 1$
$\displaystyle\small{l_{a}^{(22)}}\overline{\small{l_{a}^{(12)}}}$
$\displaystyle=$
$\displaystyle-\small{l_{a}^{(21)}}\overline{\small{l_{a}^{(11)}}}$
$\displaystyle\small{l_{a}^{(11)}}\overline{\small{l_{a}^{(21)}}}$
$\displaystyle=$
$\displaystyle-\small{l_{a}^{(12)}}\overline{\small{l_{a}^{(22)}}}$ (55)

Using above set of relations(Eq.(VIII)),
$L_{a}^{\dagger}\vec{u_{j}}.\vec{\sigma}L_{a}$(Eq.(53)) can be expressed in
terms of Pauli matrices as follows:

$L_{a}^{\dagger}\vec{u_{j}}.\vec{\sigma}L_{a}=\vec{u_{j}}^{{}^{\prime}}.\vec{\sigma}$
(56)

where
$\vec{u_{j}}^{{}^{\prime}}$$=$$(u_{j1}^{{}^{\prime}},u_{j2}^{{}^{\prime}},u_{j3}^{{}^{\prime}}).$
It components are specified as follows:

$\displaystyle
u_{j1}^{{}^{\prime}}=\frac{1}{2}(\imath\small{u_{j2}}(-\small{l_{a}^{(22)}}\overline{\small{\small{l_{a}^{(11)}}}}$
$\displaystyle-$
$\displaystyle\small{l_{a}^{(21)}}\overline{\small{l_{a}^{(12)}}}+\small{l_{a}^{(12)}}\overline{\small{l_{a}^{(21)}}}+\small{\small{l_{a}^{(11)}}}\overline{\small{l_{a}^{(22)}}})$
$\displaystyle+\small{u_{j1}}(\small{l_{a}^{(22)}}\overline{\small{\small{l_{a}^{(11)}}}}$
$\displaystyle+$
$\displaystyle\small{l_{a}^{(21)}}\overline{\small{l_{a}^{(12)}}}+\small{l_{a}^{(12)}}\overline{\small{l_{a}^{(21)}}}+$
$\displaystyle\small{\small{l_{a}^{(11)}}}\overline{\small{l_{a}^{(22)}}})$
$\displaystyle+$
$\displaystyle\small{u_{j3}}(-\small{l_{a}^{(22)}}\overline{\small{l_{a}^{(21)}}}-\small{l_{a}^{(21)}}\overline{\small{l_{a}^{(22)}}}))$
$\displaystyle
u_{j2}^{{}^{\prime}}=\frac{\imath}{2}(\small{u_{j1}}(\small{l_{a}^{(22)}}\overline{\small{\small{l_{a}^{(11)}}}}$
$\displaystyle-$
$\displaystyle\small{l_{a}^{(21)}}\overline{\small{l_{a}^{(12)}}}+\small{l_{a}^{(12)}}\overline{\small{l_{a}^{(21)}}}-\small{\small{l_{a}^{(11)}}}\overline{\small{l_{a}^{(22)}}})$
$\displaystyle+\small{u_{j2}}(\small{l_{a}^{(22)}}\overline{\small{\small{l_{a}^{(11)}}}}$
$\displaystyle-$
$\displaystyle\small{l_{a}^{(21)}}\overline{\small{l_{a}^{(12)}}}-\small{l_{a}^{(12)}}\overline{\small{l_{a}^{(21)}}}+$
$\displaystyle\small{\small{l_{a}^{(11)}}}\overline{\small{l_{a}^{(22)}}})$
$\displaystyle+$
$\displaystyle\imath\small{u_{j3}}(-\small{l_{a}^{(22)}}\overline{\small{l_{a}^{(21)}}}+\small{l_{a}^{(21)}}\overline{\small{l_{a}^{(22)}}}))$
$\displaystyle
u_{j3}^{{}^{\prime}}=\small{u_{j1}}(-\small{l_{a}^{(22)}}\overline{\small{l_{a}^{(12)}}}$
$\displaystyle-$
$\displaystyle\small{l_{a}^{(12)}}\overline{\small{l_{a}^{(22)}}})+\imath\small{u_{j2}}(\small{l_{a}^{(22)}}\overline{\small{l_{a}^{(12)}}}$
$\displaystyle-\small{l_{a}^{(12)}}\overline{\small{l_{a}^{(22)}}})$
$\displaystyle+$
$\displaystyle\small{u_{j3}}(-1+2\small{l_{a}^{(22)}}\overline{\small{l_{a}^{(22)}}})$
(57)

On simplification of the above equations, each component of
$\vec{u_{j}}^{{}^{\prime}}$ turns out to be real quantity:

$\displaystyle u_{j1}^{{}^{\prime}}$ $\displaystyle=$
$\displaystyle\small{u_{j2}}(\textmd{\small{Im}}[\small{l_{a}^{(22)}}\overline{\small{\small{l_{a}^{(11)}}}}]+\textmd{\small{Im}}[\small{l_{a}^{(21)}}\overline{\small{l_{a}^{(12)}}}])+$
$\displaystyle\small{u_{j1}}(\textmd{\small{Re}}[\small{l_{a}^{(22)}}\overline{\small{\small{l_{a}^{(11)}}}}]$
$\displaystyle+$
$\displaystyle\textmd{\small{Re}}[\small{l_{a}^{(21)}}\overline{\small{l_{a}^{(12)}}}])-2\small{u_{j3}}\textmd{\small{Re}}[\small{l_{a}^{(22)}}\overline{\small{l_{a}^{(21)}}}]$
$\displaystyle u_{j2}^{{}^{\prime}}$ $\displaystyle=$
$\displaystyle\small{u_{j1}}(-\textmd{\small{Im}}[\small{l_{a}^{(22)}}\overline{\small{\small{l_{a}^{(11)}}}}]+\textmd{\small{Im}}[\small{l_{a}^{(21)}}\overline{\small{l_{a}^{(12)}}}])+$
$\displaystyle
2\small{u_{j3}}\textmd{\small{Im}}[\small{l_{a}^{(22)}}\overline{\small{l_{a}^{(21)}}}]$
$\displaystyle+$
$\displaystyle\small{u_{j2}}(\textmd{\small{Re}}[\small{l_{a}^{(22)}}\overline{\small{\small{l_{a}^{(11)}}}}]-\textmd{\small{Re}}[\small{l_{a}^{(21)}}\overline{\small{l_{a}^{(12)}}}])$
$\displaystyle u_{j3}^{{}^{\prime}}=-1+2|\small{l_{a}^{(22)}}|^{2}$
$\displaystyle-$ $\displaystyle
2\small{u_{j2}}\textmd{\small{Im}}[\small{l_{a}^{(22)}}\overline{\small{l_{a}^{(12)}}}]-2\small{u_{j1}}\textmd{\small{Re}}[\small{l_{a}^{(22)}}\overline{\small{l_{a}^{(12)}}}]$
(58)

Using above relations, length of $\vec{u_{j}}^{{}^{\prime}}$ turns out to be
$1.$
So, in totality, Eq.(48) can be expressed as:

$L_{a}^{\dagger}G_{j}^{0}L_{a}=\frac{1}{2}(\mathbb{I}_{2}+\vec{u_{j}}^{{}^{\prime}}.\vec{\sigma})$
(59)

where $\vec{u_{j}}^{{}^{\prime}}$ is a unit length real vector for each of
$j$$=$$1,2,3.$
Analogous argument can be put for each of
$L_{a}^{\dagger}G_{j}^{1}L_{a},\,L_{a}^{\dagger}H_{j}^{0}L_{a}$ and
$L_{a}^{\dagger}H_{j}^{1}L_{a}.$ So $P_{j}$(Eq.(47)) now becomes:

$\small{P_{j}}=\frac{1}{4}\sum_{i=0,1}\textmd{\small{Tr}}\small{[(\mathbb{I}_{2}+(-1)^{i}\vec{u_{j}}^{{}^{\prime}}.\vec{\sigma})\otimes(\mathbb{I}_{2}+(-1)^{i+1}\vec{v_{j}}^{{}^{\prime}}.\vec{\sigma}).\rho^{{}^{\prime}}]},$
(60)

where
$\vec{v_{j}}^{{}^{\prime}}.\vec{\sigma}$$=$$L_{b}^{\dagger}\vec{v_{j}}.\vec{\sigma}L_{b}.$
Using Eq.(60), $Q$ now becomes:

$\displaystyle Q$ $\displaystyle=$
$\displaystyle\frac{1}{6}(3-\sum_{i=1}^{3}\sum_{j=1}^{3}u_{ij}^{{}^{\prime}}\sum_{k=1}^{3}T_{jk}v_{ik}^{{}^{\prime}})$
(61) $\displaystyle=$
$\displaystyle\frac{1}{6}(3-\sum_{i=1}^{3}\vec{u_{i}}^{{}^{\prime}}.T\vec{v_{i}}^{{}^{\prime}}),$

where $T$$=\textmd{diag}(t_{11},t_{22},t_{33})$ is the correlation tensor of
$\rho^{{}^{\prime}}$(Eq.(2)) To prove Eq.(IV.2), we now need to minimize
$Q$(Eq.61) over all possible measurement directions
$\vec{u_{j}}^{{}^{\prime}},\vec{v_{j}}^{{}^{\prime}}(j$$=$$1,2,3).$
As said in the main text, each of Alice and Bob performs measurement in
mutually unbiased bases(MUBs). Now for local dimension $d$$=$$2,$ up to global
phase factor, there exist three possible MUBsmub1
:$\\{|0\rangle,|1\rangle\\},\,\\{\frac{|0\rangle\pm|1\rangle}{2}\\}$ and
$\\{\frac{|0\rangle\pm\imath\,|1\rangle}{2}\\}.$ Collection of possible
operator bases for each of Alice and Bob are enlisted in Table.2. Minimization
of $Q$ is now performed over all these measurement operators.

Table 2:  All possible mutually unbiased operator bases for local dimension $2$ are specified here. Corresponding direction $\vec{m}$ of projective measurement $\vec{m}.\vec{\sigma}$ is given. Clearly, up to global phase, the three possible MUB operator bases are given corresponding to directions $\vec{m_{1}},\vec{m_{3}}$ and $\vec{m_{5}}.$ Each of three measurement directions of Alice and Bob are thus given by third column of the table, i.e., $\forall j$$=$$1,2,3,\,\vec{u_{j}}^{{}^{\prime}},\vec{v_{j}}^{{}^{\prime}}$$\in$$\\{\vec{m_{1}},\vec{m_{2}},\vec{m_{3}},\vec{m_{4}},\vec{m_{5}},\vec{m_{6}}\\}.$ As Alice performs three measurement in MUBs, so if $\vec{u_{1}}^{{}^{\prime}}$ is one of $\vec{m_{1}}$ or $\vec{m_{2}}$(say), then $\vec{u_{2}}^{{}^{\prime}}$$\neq$$\vec{m_{1}},\vec{m_{2}}.$ It can be any one of from ($\vec{m_{3}}$,$\vec{m_{4}}$) or from ($\vec{m_{5}}$,$\vec{m_{6}}$). If say $\vec{u_{2}}^{{}^{\prime}}$$=\vec{m_{3}},$ then $\vec{u_{3}}^{{}^{\prime}}$ is any one of $\vec{m_{5}}$ or $\vec{m_{6}}.$ Same is the case for the other party. $i$ | $\mathcal{O}_{i}$ | $\vec{m_{i}}$
---|---|---
$1$ | $\\{|0\rangle\langle 0|,|1\rangle\langle 1|\\}$ |  $\vec{m_{1}}$$=$$(0,0,1)$
$2$ | $\\{-|0\rangle\langle 0|,-|1\rangle\langle 1|\\}$ |  $\vec{m_{2}}$$=$$(0,0,-1)$
$3$ | $\\{\frac{1}{2}(|0\rangle\langle 0|+|0\rangle\langle 1|+|1\rangle\langle 0|+|1\rangle\langle 1|),$ |  $\vec{m_{3}}$$=$$(1,0,0)$
| $\frac{1}{2}(|0\rangle\langle 0|-|0\rangle\langle 1|-|1\rangle\langle 0|+|1\rangle\langle 1|)\\}$ |
$4$ | $\\{-\frac{1}{2}(|0\rangle\langle 0|+|0\rangle\langle 1|+|1\rangle\langle 0|+|1\rangle\langle 1|),$ |  $\vec{m_{4}}$$=$$(-1,0,0)$
| $-\frac{1}{2}(|0\rangle\langle 0|-|0\rangle\langle 1|-|1\rangle\langle 0|+|1\rangle\langle 1|)\\}$ |
$5$ | $\\{\frac{1}{2}(|0\rangle\langle 0|+\imath|0\rangle\langle 1|+\imath|1\rangle\langle 0|-|1\rangle\langle 1|),$ |  $\vec{m_{5}}$$=$$(0,1,0)$
| $\frac{1}{2}(|0\rangle\langle 0|-\imath|0\rangle\langle 1|-\imath|1\rangle\langle 0|-|1\rangle\langle 1|)\\}$ |
$6$ | $\\{-\frac{1}{2}(|0\rangle\langle 0|+\imath|0\rangle\langle 1|+\imath|1\rangle\langle 0|-|1\rangle\langle 1|),$ |  $\vec{m_{6}}$$=$$(0,-1,0)$
| $-\frac{1}{2}(|0\rangle\langle 0|-\imath|0\rangle\langle 1|-\imath|1\rangle\langle 0|-|1\rangle\langle 1|)\\}$ |

Depending on the signature of $t_{11},t_{22},t_{33},$ we divide the process of
minimization in different cases.
Case.i: All $t_{ii}$$\geq$$0$
In this case, minimum value of R.H.S. of Eq.(61) is obtained for
$\vec{u^{{}^{\prime}}_{1}}$$=$$\vec{v^{{}^{\prime}}_{1}}$$=$$\vec{m_{1}},$
$\vec{u^{{}^{\prime}}_{2}}$$=$$\vec{v^{{}^{\prime}}_{2}}$$=$$\vec{m_{3}}$ and
$\vec{u^{{}^{\prime}}_{3}}$$=$$\vec{v^{{}^{\prime}}_{3}}$$=$$\vec{m_{5}}$(see
Table2):

$Q=\frac{1}{6}(3-t_{11}-t_{22}-t_{33}).$ (62)

Case.ii: One of $t_{ii}$$<$$0.$ Without loss of any generality, let
$t_{11}$$<$$0.$
In this case, minimum value of R.H.S. of Eq.(61) is obtained for
$\vec{u^{{}^{\prime}}_{1}}$$=$$\vec{v^{{}^{\prime}}_{1}}$$=$$\vec{m_{2}},$
$\vec{u^{{}^{\prime}}_{2}}$$=$$\vec{v^{{}^{\prime}}_{2}}$$=$$\vec{m_{3}}$ and
$\vec{u^{{}^{\prime}}_{3}}$$=$$\vec{v^{{}^{\prime}}_{3}}$$=$$\vec{m_{5}}$(see
Table2). Minimal expression of $Q$ is given by:

$Q=\frac{1}{6}(3+t_{11}-t_{22}-t_{33}).$ (63)

Case.iii: Only one of $t_{ii}$$\geq$$0.$ Without loss of any generality, let
$t_{11},t_{22}$$<$$0.$
Minimum value in this case is obtained for
$\vec{u^{{}^{\prime}}_{1}}$$=$$\vec{v^{{}^{\prime}}_{1}}$$=$$\vec{m_{2}},$
$\vec{u^{{}^{\prime}}_{2}}$$=$$\vec{v^{{}^{\prime}}_{2}}$$=$$\vec{m_{4}}$ and
$\vec{u^{{}^{\prime}}_{3}}$$=$$\vec{v^{{}^{\prime}}_{3}}$$=$$\vec{m_{5}}$(see
Table2):

$Q=\frac{1}{6}(3+t_{11}+t_{22}-t_{33}).$ (64)

Case.i: All $t_{ii}$$<$$0$
Minimization is now obtained for
$\vec{u^{{}^{\prime}}_{1}}$$=$$\vec{v^{{}^{\prime}}_{1}}$$=$$\vec{m_{2}},$
$\vec{u^{{}^{\prime}}_{2}}$$=$$\vec{v^{{}^{\prime}}_{2}}$$=$$\vec{m_{4}}$ and
$\vec{u^{{}^{\prime}}_{3}}$$=$$\vec{v^{{}^{\prime}}_{3}}$$=$$\vec{m_{6}}$:

$Q=\frac{1}{6}(3+t_{11}+t_{22}+t_{33}).$ (65)

Combining all the possibilities(see Table3), minimal expression of $Q$ turns
out to be:

$Q=\frac{1}{6}(3-|t_{11}|-|t_{22}|-|t_{33}|).$ (66)

Expression for $Q_{min}$(Eq.(IV.2)) is thus obtained.

Table 3:  Minimal expression of $Q$(with respect to measurement settings) for different possible cases are enlisted here. Suitable measurement directions $\vec{u_{j}}^{{}^{\prime}},\vec{v_{j}}^{{}^{\prime}}$ for obtaining minimum are specified for each case in $3^{rd}$ column. It may be noted here that alternative minima exists. Signature of | Expression | Minimizing measurement
---|---|---
$t_{11},t_{22},t_{33}$ | of $Q$ | Directions
$t_{11},t_{22},t_{33}$$\geq$$0$ | $\frac{1}{6}(3-t_{11}-t_{22}-t_{33})$ |  $\vec{u^{{}^{\prime}}_{1}}$$=$$\vec{v^{{}^{\prime}}_{1}}$$=$$\vec{m_{1}},$ $\vec{u^{{}^{\prime}}_{2}}$$=$$\vec{v^{{}^{\prime}}_{2}}$$=$$\vec{m_{3}}$
|  | and $\vec{u^{{}^{\prime}}_{3}}$$=$$\vec{v^{{}^{\prime}}_{3}}$$=$$\vec{m_{5}}$
$t_{11}$$<$$0,$ | $\frac{1}{6}(3+t_{11}-t_{22}-t_{33})$ |  $\vec{u^{{}^{\prime}}_{1}}$$=$$\vec{v^{{}^{\prime}}_{1}}$$=$$\vec{m_{2}},$ $\vec{u^{{}^{\prime}}_{2}}$$=$$\vec{v^{{}^{\prime}}_{2}}$$=$$\vec{m_{3}}$
$t_{22},t_{33}$$\geq$$0$ |  | and $\vec{u^{{}^{\prime}}_{3}}$$=$$\vec{v^{{}^{\prime}}_{3}}$$=$$\vec{m_{5}}$
$t_{22}$$<$$0,$ | $\frac{1}{6}(3-t_{11}+t_{22}-t_{33})$ |  $\vec{u^{{}^{\prime}}_{1}}$$=$$\vec{v^{{}^{\prime}}_{1}}$$=$$\vec{m_{1}},$ $\vec{u^{{}^{\prime}}_{2}}$$=$$\vec{v^{{}^{\prime}}_{2}}$$=$$\vec{m_{4}}$
$t_{11},t_{33}$$\geq$$0$ |  | and $\vec{u^{{}^{\prime}}_{3}}$$=$$\vec{v^{{}^{\prime}}_{3}}$$=$$\vec{m_{5}}$
$t_{33}$$<$$0,$ | $\frac{1}{6}(3-t_{11}-t_{22}+t_{33})$ |  $\vec{u^{{}^{\prime}}_{1}}$$=$$\vec{v^{{}^{\prime}}_{1}}$$=$$\vec{m_{1}},$ $\vec{u^{{}^{\prime}}_{2}}$$=$$\vec{v^{{}^{\prime}}_{2}}$$=$$\vec{m_{3}}$
$t_{22},t_{11}$$\geq$$0$ |  | and $\vec{u^{{}^{\prime}}_{3}}$$=$$\vec{v^{{}^{\prime}}_{3}}$$=$$\vec{m_{6}}$
$t_{11}$$\geq$$0,$ | $\frac{1}{6}(3-t_{11}+t_{22}+t_{33})$ |  $\vec{u^{{}^{\prime}}_{1}}$$=$$\vec{v^{{}^{\prime}}_{1}}$$=$$\vec{m_{1}},$ $\vec{u^{{}^{\prime}}_{2}}$$=$$\vec{v^{{}^{\prime}}_{2}}$$=$$\vec{m_{4}}$
$t_{22},t_{33}$$<$$0$ |  | and $\vec{u^{{}^{\prime}}_{3}}$$=$$\vec{v^{{}^{\prime}}_{3}}$$=$$\vec{m_{6}}$
$t_{22}$$\geq$$0,$ | $\frac{1}{6}(3+t_{11}-t_{22}+t_{33})$ |  $\vec{u^{{}^{\prime}}_{1}}$$=$$\vec{v^{{}^{\prime}}_{1}}$$=$$\vec{m_{2}},$ $\vec{u^{{}^{\prime}}_{2}}$$=$$\vec{v^{{}^{\prime}}_{2}}$$=$$\vec{m_{3}}$
$t_{11},t_{33}$$<$$0$ |  | and $\vec{u^{{}^{\prime}}_{3}}$$=$$\vec{v^{{}^{\prime}}_{3}}$$=$$\vec{m_{6}}$
$t_{33}$$\geq$$0,$ | $\frac{1}{6}(3+t_{11}+t_{22}-t_{33})$ |  $\vec{u^{{}^{\prime}}_{1}}$$=$$\vec{v^{{}^{\prime}}_{1}}$$=$$\vec{m_{2}},$ $\vec{u^{{}^{\prime}}_{2}}$$=$$\vec{v^{{}^{\prime}}_{2}}$$=$$\vec{m_{4}}$
$t_{22},t_{11}$$<$$0$ |  | and $\vec{u^{{}^{\prime}}_{3}}$$=$$\vec{v^{{}^{\prime}}_{3}}$$=$$\vec{m_{5}}$
$t_{11},t_{22},t_{33}$$<$$0$ | $\frac{1}{6}(3+t_{11}+t_{22}+t_{33})$ |  $\vec{u^{{}^{\prime}}_{1}}$$=$$\vec{v^{{}^{\prime}}_{1}}$$=$$\vec{m_{2}},$ $\vec{u^{{}^{\prime}}_{2}}$$=$$\vec{v^{{}^{\prime}}_{2}}$$=$$\vec{m_{4}}$
|  | and $\vec{u^{{}^{\prime}}_{3}}$$=$$\vec{v^{{}^{\prime}}_{3}}$$=$$\vec{m_{6}}$

##  IX Appendix.II

Proof of Eq.(19):
It may be noted that minimizing $Q_{min}$(Eq.(IV.2)), is equivalent to
maximizing the following expression:

$f(t_{11},t_{22},t_{33})=|t_{11}|+|t_{22}|+|t_{33}|$ (67)

where $f(t_{11},t_{22},t_{33})$ is a symmetric function of eigen values of the
correlation tensor of $\rho$(Eq.1). Here we use Lagrange multiplier’s method
to maximize $f(t_{11},t_{22},t_{33})$ subjected to the constraint provided by
Eq.(18). Depending on the position of $(t_{11},t_{22},t_{33})$ in $8$ possible
octants of $C$(Eq.(20)), we perform the minimization considering separate
cases.
Case.i: $t_{11},t_{22},t_{33}$ all positive. Then

$f(t_{11},t_{22},t_{33})=t_{11}+t_{22}+t_{33}.$ (68)

Let $\gamma_{1}$ be the lagrange multiplier. Consider the following function:

$F_{1}(t_{11},t_{22},t_{33},\gamma_{1})=t_{11}+t_{22}+t_{33}+\gamma_{1}(t_{11}^{2}+t_{22}^{2}+t_{33}^{2}-1).$
(69)

Partial differentiation of $F_{1}(t_{11},t_{22},t_{33},\gamma_{1})$ with
respect to each of the variables $t_{11},t_{22},t_{33}$ gives

$\frac{\partial F_{1}}{\partial t_{ii}}=1+2\gamma_{1}t_{ii},\,i=1,2,3.$ (70)

Critical point is then given by $\frac{\partial F_{1}}{\partial t_{ii}}$$=$$0$
which in turn gives:

$t_{ii}=-\frac{1}{2\gamma_{1}},\,i=1,2,3.$ (71)

Using Eq.(71), in Eq.(18), we get:

$\gamma_{1}=\pm\frac{\sqrt{3}}{2}$ (72)

Now, for this case, as all $t_{ii}$$\geq$$0,$ so by Eqs.(70,71),
$\gamma_{1}$$=$$-\frac{\sqrt{3}}{2}.$ Critical point($K_{1}$,say) is thus
given by

$K_{1}=(\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}})$ (73)

Now calculating second order differential of
$F_{1}(t_{11},t_{22},t_{33},\gamma_{1}),$ we get:

$\displaystyle d^{2}F_{1}(t_{11},t_{22},t_{33},\gamma_{1})$ $\displaystyle=$
$\displaystyle\sum_{i,j=1}^{3}\frac{\partial^{2}F_{1}}{\partial t_{ii}\partial
t_{jj}}(dt_{ii}dt_{jj})$ (74) $\displaystyle=$ $\displaystyle
2\gamma_{1}(dt_{ii})^{2}$ $\displaystyle=$ $\displaystyle-\frac{1}{\sqrt{3}}$
$\displaystyle<$ $\displaystyle 0$

Eq.(73) points out that $d^{2}F$ turns out to be negative at all points.
Hence, $K_{1}$ is the maxima of the objective function $f$(Eq.(76)), maximum
value being given by:

$f(\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}})=\sqrt{3}$ (75)

Eqs.(IV.2,75) in turn gives Eq.(19).
Case.ii: One of $t_{11},t_{22},t_{33}$ is negative. $f$(Eq.(67)) being
symmetric function, without loss of any generality, let $t_{11}$$<$$0$ and
$t_{22},t_{33}$$\geq$$0.$ Then

$f(t_{11},t_{22},t_{33})=-t_{11}+t_{22}+t_{33}.$ (76)

Let $\gamma_{2}$ be the lagrange multiplier. Following the same approach as
above, analogue of Eq.(71) is given by:

$t_{11}=\frac{1}{2\gamma_{2}}\,\textmd{\small{and}}\,t_{ii}=-\frac{1}{2\gamma_{2}},\,i=2,3.$
(77)

Using Eq.(77), in Eq.(18), we get the same value of $\gamma_{2}$ as given by
Eq.(72). Now, for this case, $t_{11}$$<$$0,$ and $t_{22}$$=$$t_{33}$$\geq$$0,$
so $\gamma_{2}$$=$$-\frac{\sqrt{3}}{2}.$ Critical point($K_{2}$,say) is thus
given by

$K_{2}=(-\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}})$ (78)

Now second order differential of $F_{1}(t_{11},t_{22},t_{33},\gamma_{2})$
being $<0$(Eq.(74), $K_{2}$ turns out to be the maxima in this case and the
maximum value of $f$ given by:

$f(-\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}})=\sqrt{3}$ (79)

Eqs.(IV.2,79) in turn gives Eq.(19).
Case.iii: One of $t_{11},t_{22},t_{33}$ is positive and other two are negative
Case.iv: Each of $t_{11},t_{22},t_{33}$ is negative.
Both these cases can be analyzed in the same approach as used for the first
case. The critical points will be different in different cases(see Table.4),
but in each case, maximum value of objective function $f$ turns out to be
$\sqrt{3}$ which in turn ensures that the minimum value of
$Q_{min}$(Eq.(IV.2)) is given by Eq.(19).

Table 4:  List of all possible critical points at which the QBER attains the minimum value(Eq.(19)) with respect to all possible quantum states. Signature of | Critical Point
---|---
$t_{11},t_{22},t_{33}$ | $K_{i}$
$t_{11},t_{22},t_{33}$$\geq$$0$ |  $K_{1}$$=$$(\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}})$
$t_{11}$$<$$0,$ $t_{22},t_{33}$$\geq$$0$ |  $K_{2}$$=$$(-\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}})$
$t_{22}$$<$$0,$ $t_{11},t_{33}$$\geq$$0$ |  $K_{3}$$=$$(\frac{1}{\sqrt{3}},-\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}})$
$t_{33}$$<$$0,$ $t_{22},t_{11}$$\geq$$0$ |  $K_{4}$$=$$(\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}},-\frac{1}{\sqrt{3}})$
$t_{11}$$\geq$$0,$ $t_{22},t_{33}$$<$$0$ |  $K_{5}$$=$$(\frac{1}{\sqrt{3}},-\frac{1}{\sqrt{3}},-\frac{1}{\sqrt{3}})$
$t_{22}$$\geq$$0,$ $t_{11},t_{33}$$<$$0$ |  $K_{6}$$=$$(-\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}},-\frac{1}{\sqrt{3}})$
$t_{33}$$\geq$$0,$ $t_{22},t_{11}$$<$$0$ |  $K_{7}$$=$$(-\frac{1}{\sqrt{3}},-\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}})$
$t_{11},t_{22},t_{33}$$<$$0$ |  $K_{8}$$=$$(-\frac{1}{\sqrt{3}},-\frac{1}{\sqrt{3}},-\frac{1}{\sqrt{3}})$