# Operational quantification of simultaneous correlations in complementary
bases of two-qubit states via one-sided semi-device-independent steering

Chellasamy Jebarathinam<EMAIL_ADDRESS>Department of Physics and
Center for Quantum Information Science, National Cheng Kung University, Tainan
701, Taiwan  Debarshi Das<EMAIL_ADDRESS>Department of Physics and
Astronomy, University College London, Gower Street, WC1E 6BT London, England,
United Kingdom  Debasis Sarkar<EMAIL_ADDRESS>Department of Applied
Mathematics, University of Calcutta, 92, A.P.C. Road, Kolkata, 700009, India

###### Abstract

Recently, a weak form of quantum steering, i.e., certification of quantum
steering in a one-sided semi-device-independent way, has been formulated
[Jebarathinam et. al. Phys. Rev. A 108, 042211 (2023)]. In this work, for two-
qubit states, we study the relationships between the quantification of one-
sided semi-device-independent steerability and information-theoretic
quantification of simultaneous correlations in mutually unbiased bases [Wu et.
al. Scientific Reports 4, 4036 (2014)]. For two-qubit states that belong to
Bell-diagonal states, quantifying one-sided semi-device-independent
steerability provides an operational characterization of information-theoretic
quantification of simultaneous correlations in mutually unbiased bases. For
another class of two-qubit states, we show that quantifying one-sided semi-
device-independent steerability provides operational quantification of
simultaneous correlations in mutually unbiased bases going beyond the above
information-theoretic quantification. We invoke quantum steering ellipsoid
formalism to shed intuitions on our operational characterization of
simultaneous correlations in complementary bases of two-qubit states that we
consider.

###### pacs:

03.65.Ud, 03.67.Mn, 03.65.Ta

##  I Introduction

Measurements in mutually unbiased or complementary bases have been invoked as
a fundamental resource for optimally observing or quantifying quantum
nonlocality [1, 2]. Quantum steering is a form of quantum nonlocality
introduced by Schrödinger [3]. Quantum steering in composite quantum systems
can be demonstrated by invoking the presence of simultaneous correlations in
complementary bases of the given state [4]. Quantum steering has been
formulated as a one-sided device-independent certification of relevant
resources (incompatible measurements and entanglement in the state) used to
demonstrate the phenomenon [5, 6]. One-sided device-independent quantum
information protocols rely on such certification [7, 8, 9, 10].

Recently, it has been formulated that quantum steering can also be observed
weakly, i.e., in a one-sided semi-device-independent way [11]. Such
formulation implies that quantum steering can also occur in a one-sided semi-
device-independent way for states which are indeed unsteerable in a one-sided
device-independent scenario. In fact, these states have superunsteerability as
formulated in Refs. [12, 13]. Interestingly, superunsteerability also occurs
for separable states with a nonvanishing quantum discord [14, 15].
Superunsteerability is useful for quantum information applications [16, 17].
Such applications motivate the resource theory of one-sided semi-device-
independent steerability. In Ref. [16], the quantification of
superunsteerability of a subclass of states has been formulated.

In Ref. [18], information-theoretic quantities have been introduced to
quantify quantum correlation in mutually unbiased bases directly. Here,
mutually unbiased bases are restricted to those that optimize classical
correlation in the definition of quantum discord. The above information-
theoretic measures can also reveal simultaneous correlations in mutually
unbiased bases in states that do not violate a Bell inequality or steering
inequality but have a nonvanishing quantum discord.

In this work, for a broad class of two-qubit states belonging to Bell-diagonal
states, we study the relationships between the quantification of one-sided
semi-device-independent steerability based on the quantification of
superunsteerability studied in Ref. [16] in two-setting and three-setting
scenarios and information-theoretic measures of simultaneous correlations in
mutually unbiased bases defined in Ref. [18]. As our first main result, the
above-mentioned relationships reveal that the quantification of one-sided
semi-device-independent steerability of Bell-diagonal states provides an
operational characterization of the quantification of simultaneous
correlations in two and three mutually unbiased bases based on the measures
defined in [18].

We next demonstrate that quantifying simultaneous correlations in two mutually
unbiased bases based on the measure in [18] does not fully capture quantum
correlations in certain superunsteerable states. For a class of
superunsteerable states that include these states, we observe that the
quantification of superunsteerability studied in Ref. [16] is insufficient to
fully capture the quantification of one-sided semi-device-independent
steerability. To this end, we formulate two different quantifiers of
steerability which, together with the quantifier in Ref. [16], fully quantify
one-sided semi-device-independent steerability. Consequently, as our second
main result, for the above-mentioned class of superunsteerable states, our
newly defined quantifications of steerability operationally quantifies the
simultaneous correlations in mutually unbiased bases, going beyond the
information-theoretic quantification of simultaneous correlations in mutually
unbiased bases. We further invoke quantum steering ellipsoid formulation to
shed intuitions on quantifying simultaneous correlations in complementary
bases.

##  II Preliminaries

###  II.1 Information-theoretic quantification of correlations in
complementary bases

We present the series of measures that seek to capture the quantumness of
correlations through the persistence of correlations in mutually unbiased
bases corresponding to a bipartite state
$\rho_{AB}\in\mathcal{B}(\mathcal{H}_{A}\otimes\mathcal{H}_{B})$ as proposed
in Ref. [18]. Here $\mathcal{B}(\mathcal{H}_{A}\otimes\mathcal{H}_{B})$ stands
for the set of all bounded linear operators acting on the Hilbert space
$\mathcal{H}_{A}\otimes\mathcal{H}_{B}$. Before that, we present the
definition of classical correlation used in the definition of quantum discord.
For a given basis $\\{\Pi_{i}^{A}\\}$ measured on Alice’s subsystem of a
bipartite state $\rho_{AB}$, the Holevo quantity
$\chi(\rho_{AB},\\{\Pi_{i}^{A}\\})$ of the ensemble $\\{p_{i},\rho_{B|i}\\}$
prepared on Bob’s side is given by

$\displaystyle\chi(\rho_{AB},\\{\Pi_{i}^{A}\\})$
$\displaystyle=S\big{(}\rho_{B}\big{)}-\sum_{i}p_{i}S\big{(}\rho_{B|i}\big{)},$
(1)

which gives an upper bound on Bob’s accessible information about Alice’s
results. The classical correlation $C^{\rightarrow}(\rho_{AB})$ is also
achieved by optimizing $\chi(\rho_{AB},\\{\Pi_{i}^{A}\\})$ over all projective
measurements, i.e.,

$C^{\rightarrow}(\rho_{AB})=\max_{\\{\Pi_{i}^{A}\\}}\chi(\rho_{AB},\\{\Pi_{i}^{A}\\}).$
(2)

For a bipartite system, there may exist more than one basis that gives rise to
$C^{\rightarrow}(\rho_{AB})$. Let us denote the basis
$\left\\{\left|\mathcal{A}_{i}^{1}\right\rangle_{A}|i=1,\cdots,d_{A}\right\\}$
that optimizes $\chi(\rho_{AB},\\{\Pi_{i}^{A}\\})$ in the above definition of
classical correlation as $C^{\rightarrow}_{1}$-basis.

Using the above classical correlation of a bipartite state, quantum discord
[15, 14] from Alice to Bob $D^{\rightarrow}(\rho_{AB})$ can be defined as

$\displaystyle
D^{\rightarrow}(\rho_{AB})=I(\rho_{AB})-C^{\rightarrow}(\rho_{AB}).$ (3)

Here $I(\rho_{AB})=S(\rho_{B})-\tilde{S}(\rho_{B|A})$, is the quantum mutual
information and can be interpreted as the total correlations in $\rho_{AB}$.
$S(\sigma)=-\mbox{tr}(\sigma\log_{2}\sigma)$ is the von Neumann entropy of a
density matrix $\sigma$. $\tilde{S}(\rho_{B|A})=S(\rho_{AB})-S(\rho_{A})$ is
the “unmeasured” quantum conditional entropy [19] (see also [20, 21, 22]).

Quantum discord $D^{\rightarrow}(\rho_{AB})$ as defined above captures quantum
correlation in the state from Alice to Bob. Similarly, quantum discord from
Bob to Alice $D^{\leftarrow}(\rho_{AB})$ can be defined.
$D^{\rightarrow}(\rho_{AB})$ vanishes for a given $\rho_{AB}$ if and only if
it is a classical-quantum (CQ) state of the form,

$\displaystyle\rho_{\texttt{CQ}}=\sum_{i}p_{i}\ket{i}\bra{i}_{A}\otimes\rho^{(i)}_{B},$
(4)

where $p_{i}\geq 0$ and $\sum_{i}p_{i}=1$, $\\{\ket{i}\\}$ forms an
orthonormal basis on Alice’s Hilbert space and $\rho^{(i)}_{B}$ are any
quantum states on Bob’s Hilbert space.

We now present the measures of quantum correlation as given in Ref. [18] to
capture simultaneous correlations in mutually unbiased bases, one of which is
$C^{\rightarrow}_{1}$-basis. To quantify the simultaneous correlation in two
mutually biased bases, $Q^{\rightarrow}_{2}(\rho_{AB})$, a second basis
$\left\\{\left|a_{j}^{2}\right\rangle_{A}|j=1,\cdots,d_{A}\right\\}$, which is
mutually unbiased to the $C^{\rightarrow}_{1}$-basis
$\left\\{\left|\mathcal{A}_{i}^{1}\right\rangle_{A}|i=1,\cdots,d_{A}\right\\}$,
was considered. These are mutually unbiased bases in the sense that
$\left|\left\langle\mathcal{A}_{i}^{1}|a_{j}^{2}\right\rangle\right|=\frac{1}{\sqrt{d_{A}}}$.
The residual correlation in the above second basis has been defined as to
capture quantum correlation, $Q^{\rightarrow}_{2}(\rho_{AB})$, given by

$Q^{\rightarrow}_{2}(\rho_{AB})\equiv\max_{\\{\left|\mathcal{A}_{i}^{1}\right\rangle_{A}\\}}\max_{\\{\left|a_{j}^{2}\right\rangle_{A}\\}}\chi\\{\rho_{AB}|\\{\left|a_{j}^{2}\right\rangle_{A}\\}\\}.$
(5)

A basis $\\{\left|a_{j}^{2}\right\rangle_{A}\\}$ that achieves the maximum
quantum correlation $Q^{\rightarrow}_{2}$ in (5) is called a
$Q^{\rightarrow}_{2}$-basis, and is denoted as
$\left\\{\left|\mathcal{A}_{j}^{2}\right\rangle_{A}|j=1,\cdots,d_{A}\right\\}$.
If there is only one $C^{\rightarrow}_{1}$-basis, the second maximization over
the $C^{\rightarrow}_{1}$-bases
$\left\\{\left|\mathcal{A}_{i}^{1}\right\rangle_{A}\right\\}$ in (5) is not
necessary. If there is more than one $C^{\rightarrow}_{1}$-basis, and not all
of them achieve the maximum in (5), then the $C^{\rightarrow}_{1}$-bases are
redefined as those that also reach the maximum in (5).

As correlations could be present in the given state in more than two bases,
the authors in Ref. [18] also defined the residual correlations in bases other
than the second mutually unbiased basis. For instance, the residual
correlation in a third mutually unbiased basis has been defined as

$Q^{\rightarrow}_{3}(\rho_{AB})\equiv\max_{\\{\left|\mathcal{A}_{i}^{1}\right\rangle_{A}\\}}\max_{\\{\left|\mathcal{A}_{j}^{2}\right\rangle_{A}\\}}\max_{\\{\left|a_{k}^{3}\right\rangle_{A}\\}}\chi\\{\rho_{AB}|\\{\left|a_{k}^{3}\right\rangle_{A}\\}\\},$
(6)

where $\left\\{\left|a_{k}^{3}\right\rangle_{A}|k=1,\cdots,d_{A}\right\\}$ is
any basis mutually unbiased to both
$\left\\{\left|\mathcal{A}_{i}^{1}\right\rangle_{A}\right\\}$ and
$\left\\{\left|\mathcal{A}_{j}^{2}\right\rangle_{A}\right\\}$. An optimum
basis $\\{\left|a_{k}^{3}\right\rangle_{A}\\}$ to achieve the maximum in (6)
is called a $Q^{\rightarrow}_{3}$-basis, and is denoted as
$\left\\{\left|\mathcal{A}_{k}^{3}\right\rangle_{A}|k=1,\cdots,d_{A}\right\\}$.
Similarly, the $Q^{\rightarrow}_{2}$-bases
$\left\\{\left|\mathcal{A}_{j}^{2}\right\rangle_{A}\right\\}$ are redefined as
those that are optimum in both (5) and (6), and further the $C_{1}$-bases
$\left\\{\left|\mathcal{A}_{i}^{1}\right\rangle_{A}\right\\}$ are redefined as
those that are optimum in (2), (5) and (6).

For a given bipartite state, the authors of Ref. [18] have thus proposed the
series of measures
$\\{C^{\rightarrow},Q^{\rightarrow}_{2},Q^{\rightarrow}_{3},\cdots,Q^{\rightarrow}_{d_{A}+1}\\}$,
provided that there exists at least $d_{A}+1$ mutually unbiased bases for
subsystem $A$, to characterize correlations in the bipartite state. In the
case of Alice’s system is qubit, correlations of the state are described with
the set $\\{C^{\rightarrow},Q^{\rightarrow}_{2},Q^{\rightarrow}_{3}\\}$ as
there are three mutually unbiased bases exist for qubit. A nonzero value of
$Q^{\rightarrow}_{2}$ requires that the bipartite state has a nonvanishing
quantum discord $D^{\rightarrow}$.

###  II.2 Quantum steering

####  II.2.1 One-sided device-independent steering

Let us consider a steering scenario where two spatially separated parties,
Alice and Bob, share an unknown quantum system $\rho_{AB}$. Alice performs a
set of black-box measurements, and the Hilbert-space dimension of Bob’s
subsystem is known. Such a scenario is called one-sided device-independent
since Alice’s measurement operators $\\{M_{a|x}\\}_{a,x}$, which are positive
operator-valued measures (POVM), are unknown. The steering scenario is
completely characterized by the set of unnormalized conditional states on
Bob’s side $\\{\sigma_{a|x}\\}_{a,x}$, which is called an unnormalized
assemblage. Each element in the unnormalized assemblage is given by
$\sigma_{a|x}=p(a|x)\rho_{a|x}$, where $p(a|x)$ is the conditional probability
of getting the outcome $a$ when Alice performs the measurement $x$;
$\rho_{a|x}$ is the normalized conditional state on Bob’s side. Quantum theory
predicts that all valid assemblages should satisfy the following criteria:

$\sigma_{a|x}={\rm Tr}_{A}{(M_{a|x}\otimes\openone\rho_{AB})}\hskip
14.22636pt\forall\sigma_{a|x}\in\\{\sigma_{a|x}\\}_{a,x}.$ (7)

######  Definition 1.

In the above scenario, Alice demonstrates steerability to Bob in a one-sided
device-independent way [5] if the assemblage does not have a local hidden
state (LHS) model, i.e., if, for all $a$, $x$, there is no decomposition of
$\sigma_{a|x}$ in the form,

$\sigma_{a|x}=\sum_{\lambda}p(\lambda)p(a|x,\lambda)\rho_{\lambda},$ (8)

where $\lambda$ denotes the classical random variable which occurs with
probability $p(\lambda)$; $\rho_{\lambda}$ are called local hidden states
which satisfy $\rho_{\lambda}\geq 0$ and ${\rm
Tr}\left[{{\rho}}\right]_{\lambda}=1$.

We can further define the detection of the above phenomenon from the
nosignaling (NS) boxes as follows.

######  Definition 2.

Suppose Bob performs a set of projective measurements $\\{\Pi_{b|y}\\}_{b,y}$
on $\\{\sigma_{a|x}\\}_{a,x}$. Then, the scenario is characterized by the set
of measurement correlations or a NS box shared between Alice and Bob
$P(ab|xy)$:=$\\{p(ab|xy)\\}_{a,x,b,y}$, where $p(ab|xy)$ = ${\rm
Tr}(\Pi_{b|y}\sigma_{a|x})$. The box $P(ab|xy)$ detects one-sided device-
independent steerability from Alice to Bob, if and only if it does not have a
decomposition as follows [5, 23, 24]:

$p(ab|xy)=\sum_{\lambda}p(\lambda)p(a|x,\lambda)p(b|y,\rho_{\lambda})\hskip
8.5359pt\forall a,x,b,y;$ (9)

where, $\sum_{\lambda}p(\lambda)=1$, $p(a|x,\lambda)$ denotes an arbitrary
probability distribution arising from local hidden variable (LHV) $\lambda$
($\lambda$ occurs with probability $p(\lambda)$) and $p(b|y,\rho_{\lambda})$
denotes the quantum probability of outcome $b$ when measurement $y$ is
performed on local hidden state (LHS) $\rho_{\lambda}$.

Let us denote the set of all correlations that belongs to the given steering
scenario $\mathcal{N}_{\Sigma^{S}}$. The set of correlations that have a LHV-
LHS model denoted by $\mathcal{L}_{\Sigma^{US}}$ forms a convex subset of
$\mathcal{N}_{\Sigma^{S}}$ [25], which we call unsteerable set. In particular,
any LHV-LHS model can be decomposed in terms of the extremal points of
$\mathcal{L}_{\Sigma^{US}}$ [26]. That is, we can simplify the decomposition
(9) as follows:

$p(ab|xy)=\sum_{\chi,\zeta}p(\chi,\zeta)D(a|x,\chi)\braket{\psi_{\zeta}}{\Pi_{b}{y}|\psi_{\zeta}}{,}$
(10)

with $D(a|x,\chi)=\delta_{a,f(x,\chi)}$. Here, $\chi$ are variables which
determine all values of $\\{M_{a|x}\\}_{a}$ through the function $f(x,\chi)$
and $\zeta$ determines a pure state $\ket{\psi_{\zeta}}$ for Bob.

####  II.2.2 Superunsteerability

In the given steering scenario, even if a box is unsteerable, the box can have
nonclassicality beyond steering defined in Definition $2$, due to a nonzero
quantum discord of the state. Such nonclassicality has been operationally
captured by the notion called superunsteerability [12] defined as follows:

######  Definition 3.

Suppose we have a quantum state in
$\mathbb{C}^{d_{A}}\otimes\mathbb{C}^{d_{B}}$and measurements which produce a
unsteerable bipartite box $P(ab|xy)$ := $\\{p(ab|xy)\\}_{a,x,b,y}$. Then,
superunsteerability holds if and only if there is no decomposition of the box
in the form,

$p(ab|xy)=\sum^{d_{\lambda}-1}_{\lambda=0}p(\lambda)p(a|x,\lambda)p(b|y,\rho_{\lambda})\hskip
8.5359pt\forall a,x,b,y,$ (11)

with the dimension of the shared randomness/hidden variable $d_{\lambda}\leq
d_{A}$. Here $\sum_{\lambda}p(\lambda)=1$, $p(a|x,\lambda)$ denotes an
arbitrary probability distribution arising from LHV $\lambda$ ($\lambda$
occurs with probability $p(\lambda)$) and $p(b|y,\rho_{\lambda})$ denotes the
quantum probability of outcome $b$ when measurement $y$ is performed on LHS
$\rho_{\lambda}$ in $\mathbb{C}^{d_{B}}$.

In Ref. [27], an example of superunsteerability has been demonstrated in a
steering scenario where Alice performs two dichotomic black-box measurements,
and Bob performs two dichotomic trusted qubit measurements in mutually
unbiased bases. In this example, the unsteerable white noise-BB84 family given
by

$P(ab|xy)=\frac{1+(-1)^{a\oplus b\oplus x.y}\delta_{x,y}V}{4},$ (12)

with $0<V\leq 1/\sqrt{2}$, was analysed. The above box can be produced by a
two-qubit Werner state with an entanglement or a nonzero quantum discord for
appropriate local non-commuting measurements. On the other hand, it cannot be
simulated by an LHV-LHS model with $d_{\lambda}=4$ as shown in Ref. [12].

In Ref. [16], a quantity called Schrödinger strength has been defined to
quantify the nonclassicality of unsteerable boxes which demonstrate
superunsteerability. Note that a box $P(ab|xy)$ in the $n$-setting steering
scenario is the set of joint probabilities $p(ab|xy)$ for all possible $a$,
$b$ and for all $x$ $\in$ $\\{0,1,2,...,n-1\\}$ and $y$ $\in$
$\\{0,1,2,...,n-1\\}$. In this scenario, any box $P(ab|xy)$ can be decomposed
into a convex mixture of a steerable part and an unsteerable part,

$P(ab|xy)=pP_{S}(ab|xy)+(1-p)P_{US}(ab|xy),$ (13)

where $P_{S}(ab|xy)$ is a steerable box and $P_{US}(ab|xy)$ is an unsteerable
box which may be superunsteerable; $0\leq p\leq 1$. The weight of the box
$P_{S}(ab|xy)$ minimized over all possible decompositions of the form (13) is
called steering cost of the box $P(ab|xy)$ furnishing a measure of quantum
steering [24].

On the other hand, the weight of the box $P_{S}(ab|xy)$ maximized over all
possible decompositions of the form (13) is called the Schrödinger strength of
the box $P(ab|xy)$, given by

$S_{SU_{n}}\Big{(}P(ab|xy)\Big{)}:=\max_{\text{decompositions}}p.$ (14)

Here, $0\leq S_{SU_{n}}\leq 1$ (since, $0\leq p\leq 1$). The optimal
decomposition that gives the Schrödinger strength of the box is called the
canonical decomposition in which the steerable part is an extremal steerable
box.

In Ref. [24], it has been demonstrated that the optimal decomposition that
gives the steering cost of the box (12) is given by

$\displaystyle P(ab|xy)$
$\displaystyle=\frac{\sqrt{2}V-1}{\sqrt{2}-1}P_{Ext}(ab|xy)$
$\displaystyle+\left(1-\frac{\sqrt{V}-1}{\sqrt{2}-1}\right)P_{US}(ab|xy),$
(15)

with $V\geq 1/\sqrt{2}$, where

$P_{Ext}(ab|xy)=\frac{1+(-1)^{a+b+xy}\delta_{x,y}}{4},\quad
a,b,x,y\in\\{0,1\\}$ (16)

is an extremal steerable box and $P_{US}$ is an unsteerable box which is the
box (12) with $V=1/\sqrt{2}$.

Even for $0<V\leq 1/\sqrt{2}$, the box (12) can be decomposed as a convex
mixture of the extremal steerable box and an unsteerable box as follows:

$P(ab|xy)=VP_{Ext}(ab|xy)+(1-V)P_{N}(ab|xy),$ (17)

where $P_{N}(ab|xy)$ is the white noise for which $P_{N}(ab|xy)=1/4$ for all
$a,b,x,y$. Note that the unsteerable box in the decomposition given by Eq.
(II.2.2) still can be decomposed as a convex mixture of the extremal steerable
box and the white noise to obtain the decomposition given by Eq. (17). The
above decomposition is optimal implying the Schrödinger strength as $V$. Note
that even when the box (12) has the steering cost equals to zero for $0<V\leq
1/\sqrt{2}$, it has a nonvanishing Schrödinger strength in this range.

In a $n$-setting steering scenario where Alice performs $n$ black-box
dichotomic measurements, and Bob performs $n$ characterized dichotomic
measurements, let Alice’s measurement settings and Bob’s measurement settings
are denoted by $A_{x}$ and $B_{y}$ respectively, where $A_{x}$ $\in$
$\tilde{A}$ = $\\{A_{0},A_{1},A_{2},...,A_{n-1}\\}$ and $B_{y}$ $\in$
$\tilde{B}$ = $\\{B_{0},B_{1},B_{2},...,B_{n-1}\\}$. The sets of Alice’s and
Bob’s measurement settings are denoted by $\tilde{A}$ and $\tilde{B}$,
respectively. We will now present the definition of Schrödinger strength of a
bipartite state.

######  Definition 4.

The Schrödinger strength of a bipartite state $\rho$ is defined as:

$S_{SU_{n}}(\rho)=\max_{\tilde{A},\tilde{B}}S_{SU_{n}}\Big{(}P(ab|xy;\rho)\Big{)}.$
(18)

Here $S_{SU_{n}}\Big{(}P(ab|xy;\rho)\Big{)}$ is the Schrödinger strength of
the box $P(ab|xy;\rho)$ = $\\{p(ab|xy;\rho)\\}_{a,x,b,y}$, where
$p(ab|xy;\rho)$ = Tr$[(M_{a|x}\otimes M_{b|y})\rho]$ is the joint probability
distribution of getting the outcomes $a$ and $b$ when measurements $A_{x}$ and
$B_{y}$ are performed locally by Alice and Bob, respectively, on the shared
bipartite state $\rho$. $M_{a|x}$ is the measurement operator corresponding to
the measurement settings $A_{x}$ and outcome $a$. $M_{b|y}$ is defined
similarly. Here the maximization is taken over all possible sets $\tilde{A}$ =
$\\{A_{0},A_{1},A_{2},...,A_{n-1}\\}$ and $\tilde{B}$ =
$\\{B_{0},B_{1},B_{2},...,B_{n-1}\\}$.

In this work, we consider the following two scenarios: (i) Alice performs two
black-box dichotomic measurements, and Bob performs projective qubit
measurements corresponding to two mutually unbiased bases and (ii) Alice
performs three black-box dichotomic measurements, and Bob performs projective
qubit measurements corresponding to three mutually unbiased bases. In Ref.
[16], the Schrödinger strengths of two-qubit Bell-diagonal states have been
evaluated in the two-setting and three-setting scenarios mentioned above. The
Bell-diagonal states admit the form,

$\tau=\frac{1}{4}\left(\openone\otimes\openone+\sum_{i}c_{i}\sigma_{i}\otimes\sigma_{i}\right)=\sum_{ab}\lambda_{ab}|\beta_{ab}\rangle\langle\beta_{ab}|,$
(19)

where $\lambda_{ab}$ (with $a,b\in\\{0,1\\}$) denote the eigenvalues of the
Bell-diagonal states which are given by
$\lambda_{ab}=\frac{1}{4}\left[1+(-1)^{a}c_{1}-(-1)^{a+b}c_{2}+(-1)^{b}c_{3}\right]$
and $\ket{\beta_{ab}}=\frac{1}{\sqrt{2}}(\ket{0,b}+(-1)^{a}\ket{1,1\oplus b}$
are the Bell states. Here, we take $|c_{1}|\geq|c_{2}|\geq|c_{3}|$ with local
unitary transformations. For this family of states, the eigenvalues satisfy
the ordering $\lambda_{01}\geq\lambda_{00}\geq\lambda_{10}\geq\lambda_{11}$.
This family of states are entangled if and only if $\lambda_{01}>1/2$. The
quantum discord of this family of states $D^{\rightarrow}(\tau)>0$ if and only
if $|c_{2}|>0$ and $|c_{3}|>0$.

The Schrödinger strength of two-qubit Bell-diagonal states in the two-setting
scenario has been evaluated as follows:

######  Proposition 1.

The Schrödinger strength of the Bell-diagonal states in the two-setting
steering scenario is given by

$\displaystyle S_{SU_{2}}(\tau)=|c_{2}|.$ (20)

On the other hand, in the three-setting scenario, the Schrödinger strength of
two-qubit Bell-diagonal states has been evaluated as follows:

######  Proposition 2.

The Schrödinger strength of the Bell-diagonal states in the three-setting
steering scenario is given by

$\displaystyle S_{SU_{3}}(\tau)=|c_{3}|.$ (21)

####  II.2.3 One-sided semi-device-independent steering

Next, we are going to present the formal definition of a weak form steering,
called “one-sided semi-device-independent steering”, as introduced in Ref.
[11].

######  Definition 5.

In the steering scenario as described before, Alice demonstrates steerability
to Bob in a one-sided semi-device-independent way if the assemblage does not
have an LHS model with a restricted hidden variable dimension, i.e. if for all
$a$, $x$, there is no decomposition of $\sigma_{a|x}$ in the form,

$\sigma_{a|x}=\sum^{d_{\lambda}-1}_{\lambda=0}p(\lambda)p(a|x,\lambda)\rho_{\lambda},$
(22)

with $d_{\lambda}\leq d_{A}$.

As noted in Ref. [11], the detection of one-sided semi-device-independent
steering from an NS box can be introduced as follows.

######  Observation 1.

A bipartite box $P(ab|xy)$:=$\\{p(ab|xy)\\}_{a,x,b,y}$ arising in a steering
scenario detects steerability in a one-sided semi-device-independent way if
and only if there is no decomposition of the box in the form given by

$p(ab|xy)=\sum^{d_{\lambda}-1}_{\lambda=0}p(\lambda)p(a|x,\lambda)p(b|y,\rho_{\lambda})\hskip
8.5359pt\forall a,x,b,y,$ (23)

with $d_{\lambda}\leq d_{A}$.

One-sided semi-device-independent steering also occurs in certain states that
are unsteerable in the one-sided device-independent context. Such states have
superunsteerability in a one-sided device-independent way as mentioned before.

The unsteerable white noise-BB84 family (12) exhibits one-sided semi-device-
independent steerability both ways for any $V>0$. There is another example of
superunsteerability shown in Ref. [12] that exhibits one-sided semi-device-
independent steerability both ways. The difference between these two examples
is that the former exhibits superunsteerability with $d_{\lambda}=4$, while
the latter exhibits superunsteerability with lower hidden variable dimension,
i.e., $d_{\lambda}=3$. The above-mentioned other example of two-way
superunsteerability was demonstrated with a box arising the following
separable two-qubit state:

$\rho_{SU^{d_{\lambda}=3}_{2way}}=\frac{1}{2}\Big{(}|00\rangle\langle
00|+|++\rangle\langle++|\Big{)},$ (24)

where, $|0\rangle$ and $|+\rangle$ are the eigenstates of the operators
$\sigma_{z}$ and $\sigma_{x}$, respectively, corresponding to the eigenvalue
$+1$. The above state has nonzero quantum discord from both Alice to Bob and
Bob to Alice since it is neither a classical-quantum nor a quantum-classical
state. If Alice performs the projective measurements of observables
corresponding to the operators $A_{0}=\sigma_{x}$ and $A_{1}=\sigma_{z}$, and
Bob performs projective measurements of observables corresponding to the
operators $B_{0}=\sigma_{x}$ and $B_{1}=\sigma_{z}$, then the following
correlation is produced from the above quantum-quantum state,

$P(ab|xy)=\begin{tabular}[]{c|cccc}\diagbox[dir={NW}]{{\shortstack[l]{(x,y)}}}{{\shortstack[r]{(a,b)}}}{}&(0,0)&(0,1)&(1,0)&(1,1)\\\
\hline\cr\\\\[1.42271pt]
(0,0)&$\dfrac{5}{8}$&$\dfrac{1}{8}$&$\dfrac{1}{8}$&$\dfrac{1}{8}$\\\\[14.22636pt]
(0,1)&$\dfrac{1}{2}$&$\dfrac{1}{4}$&$\dfrac{1}{4}$&$0$\\\\[14.22636pt]
(1,0)&$\dfrac{1}{2}$&$\dfrac{1}{4}$&$\dfrac{1}{4}$&$0$\\\\[14.22636pt]
(1,1)&$\dfrac{5}{8}$&$\dfrac{1}{8}$&$\dfrac{1}{8}$&$\dfrac{1}{8}$\\\
\end{tabular}$ (25)

Here $x$, $y$ denote the input variables on Alice’s and Bob’s sides,
respectively, and $a$, $b$ denote the outputs on Alice’s and Bob’s sides,
respectively. In Ref. [12], it has been demonstrated that the LHV-LHS
decomposition of the correlation given by Eq.(27) cannot be realized with a
hidden variable having dimension $3$. This implies that the box has
superunsteerability.

Just like one-sided device-independent steerability can occur asymmetrically,
one-side semi-device-independent steerability may also occur asymmetrically as
was shown in Ref. [11]. To demonstrate this asymmetry, the authors of Ref.
[11] presented the following example of superunsteerability asymmetrically.
Consider the box arising from the following separable two-qubit state,

$\rho_{SU^{\rightarrow}_{1way}}=\frac{1}{2}\Big{(}|00\rangle\langle
00|+|+1\rangle\langle+1|\Big{)},$ (26)

where, $|0\rangle$ and $|1\rangle$ are the eigenstates of the operator
$\sigma_{z}$ corresponding to the eigenvalue $+1$ and $-1$ respectively;
$|+\rangle$ is the eigenstate of the operator $\sigma_{x}$ corresponding to
the eigenvalue $+1$. The above state has quantum discord
$D^{\rightarrow}(\rho_{AB})>0$ and $D^{\leftarrow}(\rho_{AB})=0$ since it is
not a classical-quantum state but a quantum-classical state [15, 14]. If Alice
performs the projective measurements of observables corresponding to the
operators $A_{0}=\sigma_{x}$ and $A_{1}=\sigma_{z}$, and Bob performs
projective measurements of observables corresponding to the operators
$B_{0}=\sigma_{x}$ and $B_{1}=\sigma_{z}$, then the following correlation is
produced from the above quantum-classical state,

$P(ab|xy)=\begin{tabular}[]{c|cccc}\diagbox[dir={NW}]{{\shortstack[l]{(x,y)}}}{{\shortstack[r]{(a,b)}}}{}&(0,0)&(0,1)&(1,0)&(1,1)\\\
\hline\cr\\\\[1.42271pt]
(0,0)&$\dfrac{3}{8}$&$\dfrac{3}{8}$&$\dfrac{1}{8}$&$\dfrac{1}{8}$\\\\[14.22636pt]
(0,1)&$\dfrac{1}{4}$&$\dfrac{1}{2}$&$\dfrac{1}{4}$&$0$\\\\[14.22636pt]
(1,0)&$\dfrac{3}{8}$&$\dfrac{3}{8}$&$\dfrac{1}{8}$&$\dfrac{1}{8}$\\\\[14.22636pt]
(1,1)&$\dfrac{1}{2}$&$\dfrac{1}{4}$&$0$&$\dfrac{1}{4}$\\\ \end{tabular}$ (27)

In Ref. [11], it has been demonstrated that the box (25) detects
superunsteerability of the quantum-classical state (26). Moreover, the box has
superunsteerability asymmetrically because it does have superunsteerability
from Bob to Alice.

###  II.3 Quantum steering ellipsoids for two-qubit states

Consider an experimental situation in which Alice and Bob share a two-qubit
state $\rho_{AB}$, which can be written in the form

$\displaystyle\rho_{AB}=\frac{1}{4}\bigg{(}$
$\displaystyle\mathbbm{1}_{A}\otimes\mathbbm{1}_{B}+{\mathbf{a}}\cdot\bm{\sigma}\otimes\mathbbm{1}_{B}+\mathbbm{1}_{A}\otimes{\mathbf{b}}\cdot\bm{\sigma}$
$\displaystyle+\sum^{3}_{j,k=1}T_{jk}\sigma_{j}\otimes\sigma_{k}\bigg{)},$
(28)

where $\bm{\sigma}\equiv(\sigma_{1},\sigma_{2},\sigma_{3})$ denotes the vector
of Pauli spin operators and $\mathbbm{1}_{A},\mathbbm{1}_{B}$ are identity
operators. Here ${\mathbf{a}}$ and ${\mathbf{b}}$ are the Bloch vectors of
Alice’s and Bob’s reduced states and $T$ is the spin correlation matrix, i.e.,

$\displaystyle a_{j}$ $\displaystyle:={\rm
Tr}\left[{{\rho_{AB}\sigma_{j}\otimes\mathbbm{1}_{B}}}\right],~{}~{}~{}b_{k}:={\rm
Tr}\left[{{\rho_{AB}\mathbbm{1}_{A}\otimes\sigma_{k}}}\right],$ $\displaystyle
T_{jk}$ $\displaystyle:={\rm
Tr}\left[{{\rho_{AB}\sigma_{j}\otimes\sigma_{k}}}\right],\qquad(j,k=1,2,3).$
(29)

Let Alice performs a general POVM which has an element
$M_{a}=a_{0}\left(\mathbbm{1}_{A}+\mathbf{m}_{a}\cdot\bm{\sigma}\right)$, with
$a_{0}\geq 0$ and $|\mathbf{m}_{a}|\leq 1$ corresponding to an outcome $a$.
This outcome is obtained with probability

$p_{M_{a}}={\rm
Tr}\left[{{\rho_{AB}\,M_{a}\otimes\mathbbm{1}_{B}}}\right]=a_{0}\left(1+{\mathbf{a}}\cdot\mathbf{m}_{a}\right),$
(30)

leading to the steered state

$\rho^{M_{a}}_{B}=\frac{{\rm
Tr}_{A}[\rho_{AB}\,E\otimes\mathbbm{1}_{B}]}{p_{M_{a}}}=\frac{1}{2}\left(\mathbbm{1}_{B}+\frac{({\mathbf{b}}+T^{\top}\mathbf{m}_{a})\cdot\bm{\sigma}}{1+{\mathbf{a}}\cdot\mathbf{m}_{a}}\right)$
(31)

for Bob’s qubit.

Considering all possible local measurements by Alice, it follows that the
corresponding set of Bob’s steered states is represented by the set of Bloch
vectors

$\mathcal{E}_{B|A}=\left\\{\frac{{\mathbf{b}}+T^{\top}\mathbf{m}_{a}}{1+{\mathbf{a}}\cdot\mathbf{m}_{a}}:|\mathbf{m}_{a}|\leq
1\right\\}.$ (32)

This set forms a (possibly degenerate) ellipsoid, called a quantum steering
ellipsoid [28]. The subscript $B|A$ indicates the steering ellipsoid for Bob
that is generated by Alice’s local measurements. Similarly, a steering
ellipsoid for Alice is generated by Bob’s local measurements, denoted by
$\mathcal{E}_{A|B}$. The ellipsoid $\mathcal{E}_{B|A}$ is fully determined by
its centre,

${\mathbf{c}}_{B|A}=\frac{{\mathbf{b}}-T^{\top}{\mathbf{a}}}{1-a^{2}},$ (33)

and its orientation matrix,

$Q_{B|A}=\frac{1}{1-a^{2}}\left(T-{\mathbf{a}}{\mathbf{b}}^{\top}\right)^{\top}\left(I+\frac{{\mathbf{a}}{\mathbf{a}}^{\top}}{1-a^{2}}\right)\left(T-{\mathbf{a}}{\mathbf{b}}^{\top}\right),$
(34)

where the eigenvalues and corresponding eigenvectors of $Q_{B|A}$ describe the
squared lengths of the ellipsoid’s semiaxes and their orientations [28].

The quantum steering ellipsoid $\mathcal{E}_{B|A}$, together with the reduced
Bloch vectors ${\mathbf{a}}$ and ${\mathbf{b}}$, provides a faithful
representation of any two-qubit state up to local unitary operations on
Alice’s qubit [28]. Additionally, its various geometric properties encode
useful information about the state. For example, the state is separable if and
only if its steering ellipsoid can be nested in a tetrahedron that is, in
turn, nested in the Bloch sphere [28].

Furthermore, separable two-qubit states can be classified into states with a
nonzero volume for their steering ellipsoids and states with degenerate
steering ellipsoids. A nonzero volume for quantum steering ellipsoids of the
two-qubit states is stronger than nonvanishing discord. A degenerate steering
ellipsoid can be $2$-dimensional (a steering pancake) or $1$-dimensional (a
steering needle) or trivially $0$-dimensional. Within quantum steering
ellipsoid formalism, a two-qubit state has zero discord from Alice to Bob if
and only if her ellipsoid degenerates to a segment of a diameter.

In cases of steering pancakes and steering needles, steering can be either
complete or incomplete. Complete steering of Bob occurs if and only if for any
convex decomposition of the reduced state of Bob in terms of the states in
$\mathcal{E}_{B|A}$, there exists a POVM on Alice’s side that steers Bob to
this ensemble.

##  III Results

In the following, we study the quantification of one-sided semi-device-
independent steerability in two classes of two-qubit states and their
relationships with the quantification of simultaneous correlations in mutually
unbiased bases of these states. For one of the classes of these states that
belong to Bell-diagonal states, we use the Schrödinger strengths mentioned
earlier to quantify one-sided semi-device-independent steerability in the two-
setting and three-setting scenarios and show that they provide an operational
characterization of $Q^{\rightarrow}_{2}$ and $Q^{\rightarrow}_{3}$,
respectively.

We then observe that

  * •

the Schrödinger strength does not fully quantify one-sided semi-device-
independent steerability in the two-setting scenario and

  * •

the information-theoretic quantification of the simultaneous correlations in
two mutually unbiased bases captured by $Q^{\rightarrow}_{2}$ does not capture
quantum correlation in certain superunsteerable states that have vanishing
Schrödinger strength.

In the two-setting scenario, for the other class of two-qubit states mentioned
above, which are superunsteerable, the Schrödinger strength and
$Q^{\rightarrow}_{2}$ do not capture one-sided semi-device-independent
steerability and the simultaneous correlations in two mutually unbiased bases,
respectively, fully. To study the relationship between the quantification of
steerability and the simultaneous correlations in two mutually unbiased bases
of such classes of states, we introduce two other quantifiers of
superunsteerability, two-way superunsteerable and one-way superunsteerable
strengths, to quantify steerability in superunsteerable boxes having vanishing
Schrödinger strength. With these two quantities, we then show that

  * •

the Schrödinger strength together with two-way superunsteerable and one-way
superunsteerable strengths provide full quantification of one-sided semi-
device-independent steerability of the above-mentioned class of
superunsteerable states and

  * •

such total quantification of steerability provides quantification of the
simultaneous correlations in two mutually unbiased bases beyond the
information-theoretic quantification of quantum correlation given by
$Q^{\rightarrow}_{2}$.

###  III.1 Bell-diagonal states

For the Bell-diagonal states (19), the steering ellipsoid of Bob is given by

$\displaystyle\frac{x^{2}}{\ell_{1}^{2}}+\frac{y^{2}}{\ell_{2}^{2}}+\frac{z^{2}}{\ell_{3}^{2}}=1,$
(35)

where $\ell_{1}=c_{1}$, $\ell_{2}=c_{2}$, and $\ell_{3}=c_{3}$. The center of
the ellipsoid is at the origin. The $x$, $y$ and the $z$ semi-axes lengths of
the ellipsoid is given by

$\displaystyle|l_{1}|$
$\displaystyle=|c_{1}|,\quad|l_{2}|=|c_{2}|,\quad|l_{3}|=|c_{3}|$ (36)

The reduced state $\rho^{B}$ corresponds to the point $B=(0,0,0)$, which is
the origin. The above steering ellipsoid’s normalized volume as defined in
Ref. [29] is given by $v_{B|A}=|c_{1}c_{2}c_{3}|$. Note that if
$c_{1}-c_{2}=c_{1}+c_{2}$ or $(1+c_{3})^{2}=(1-c_{3})^{2}$, the ellipsoid
degenerates to an ellipse or a line segment.

The optimal decomposition of the reduced state $\rho_{B}$ that achieves the
classical correlation $C^{\rightarrow}(\tau)$ [30] is given by

$\displaystyle\rho_{B}$
$\displaystyle=\frac{1}{2}\rho_{B|0}+\frac{1}{2}\rho_{B|1},$ (37)

where,

$\displaystyle\rho_{B|0/1}=\frac{1}{2}\left(\openone\pm
c_{1}\sigma_{x}\right).$ (38)

It has been checked that the ensemble corresponding to this decomposition can
be achieved by Alice performing measurement in the basis of $\sigma_{x}$. We
also note that this is the only basis that achieves
$C_{1}^{\rightarrow}(\tau)$ given by

$C_{1}^{\rightarrow}(\tau)=1-h\left(\frac{1+|c_{1}|}{2}\right).$ (39)

From this fact, it follows that a second basis, which is mutually unbiased to
the basis of $\sigma_{x}$, gives rise to $Q_{2}(\tau)$. The second basis which
is mutually unbiased to this basis is either the basis of $\sigma_{y}$ or
$\sigma_{z}$. The Holevo quantity $\chi(\tau,\sigma_{y})$ pertaining to the
basis of $\sigma_{y}$ is given by

$\chi(\tau,\sigma_{x})=1-h\left(\frac{1+|c_{2}|}{2}\right),$ (40)

where $h(\cdot)$ is the binary entropic function; on the other hand, the
Holevo quantity $\chi(\tau,\sigma_{z})$ pertaining to the basis of
$\sigma_{z}$ is given by

$\chi(\tau,\sigma_{z})=1-h\left(\frac{1+|c_{3}|}{2}\right).$ (41)

Since $|c_{2}|\geq|c_{3}|$, it follows that $Q_{2}(\tau)$ is achieved by the
basis of $\sigma_{y}$. On the other hand, $Q_{3}(\tau)$ is achieved by the
basis of $\sigma_{z}$. Therefore, $Q_{2}(\tau)$ and $Q_{3}(\tau)$ are given by

$Q^{\rightarrow}_{2}(\tau)=1-h\left(\frac{1+|c_{2}|}{2}\right)$ (42)

and

$Q^{\rightarrow}_{3}(\tau)=1-h\left(\frac{1+|c_{3}|}{2}\right),$ (43)

respectively.

Now, note that if a function $f(x)$ is continuous and differentiable in any
given interval $a\leq x\leq b$, then $f(x)$ is a monotonically increasing
function of $x$ provided the derivative of $f(x)$ with respect to $x$ is not
negative, i.e., $\frac{df(x)}{dx}\geq 0$ for all $x\in[a,b]$. Since
$\frac{dC^{\rightarrow}(\tau)}{d|l_{1}|}\geq 0$,
$\frac{dQ^{\rightarrow}_{2}(\tau)}{d|l_{2}|}\geq 0$ and
$\frac{dQ^{\rightarrow}_{3}(\tau)}{d|l_{3}|}\geq 0$ for all values of
$|l_{i}|$, with $i=1,2,3$, respectively, we have obtained the following
observation.

######  Observation 2.

$C^{\rightarrow}(\tau)$, $Q^{\rightarrow}_{2}(\tau)$ and
$Q^{\rightarrow}_{3}(\tau)$ are a monotonically increasing function of the
length of the semi-axis of the steering ellipsoid in descending order,
respectively.

Also, note that $Q^{\rightarrow}_{3}(\tau)>0$ if and only if the quantum
steering ellipsoid of Bob has a nonvanishing volume.

We now proceed to derive analytical relationships between the measures of
complementary correlations and Schrödinger strengths of $\tau$. We first
observe the analytical relationship between $Q^{\rightarrow}_{2}(\tau)$ and
$S_{SU_{2}}(\tau)$. Using Eqs. (20) and (42), it is seen that
$S_{SU_{2}}(\tau)$ and $Q^{\rightarrow}_{2}(\tau)$ are related to each other
as follows:

$Q^{\rightarrow}_{2}(\tau)=1-h\left(\frac{1+S_{SU_{2}}(\tau)}{2}\right).$ (44)

By differentiating the left hand side of Eq. (44) with respect to
$S_{SU_{2}}(\tau)$, it can be checked that
$\frac{dQ^{\rightarrow}_{2}}{dS_{{SU}_{2}}(\tau)}\geq 0$ for all values of
$S_{SU_{2}}(\tau)$ which implies that $Q^{\rightarrow}_{2}$ is a monotonically
increasing function of $S_{SU_{2}}(\tau)$.

Next, we observe the analytical relationship between $Q_{3}(\tau)$ and
$S_{SU_{3}}(\tau)$. The following relationship can be easily obtained using
Eqs. (21) and (43):

$Q^{\rightarrow}_{3}(\tau)=1-h\left(\frac{1+S_{SU_{3}}(\tau)}{2}\right).$ (45)

Similar to the earlier two-setting case, it can be checked from Eq. (45) that
the first derivative of $Q^{\rightarrow}_{3}$ with respect to
$S_{SU_{3}}(\tau)$ is non-negative, i.e.,
$\frac{d\mathcal{Q}_{3}}{dS_{SU_{3}}(\tau)}\geq 0$ for all values of
$S_{SU_{3}}(\tau)$; i.e., in the three-setting case too, $Q^{\rightarrow}_{3}$
is a monotonically increasing function of $S_{SU_{3}}(\tau)$.

From the above two analytical relationships, we have obtained the following
result.

######  Result 1.

The Schrödinger strengths $S_{SU_{2}}(\tau)$ and $S_{SU_{3}}(\tau)$ provide
operational characterization of $Q^{\rightarrow}_{2}(\tau)$ and
$Q^{\rightarrow}_{3}(\tau)$, respectively.

In Ref. [31], a different version of information-theoretic quantification of
simultaneous correlations in complementary bases has been studied. Such
quantification is stronger than the quantification studied in Ref. [18] as the
former may capture more simultaneous correlations in the given state. For the
Bell-diagonal states, Ref. [4] has studied the relationships between this
quantification of simultaneous correlations and the measures of one-sided
device-independent steerability studied in Ref. [32]. Such relationships
reveal that this quantification of one-sided device-independent steerability
provides an operational characterization of the simultaneous correlations in
complementary using the measures in Ref. [31] for the steerable Bell-diagonal
states.

###  III.2 A family of separable states

Note that the Schrödinger strength vanishes for the superunsteerable states
given by Eq. (24) and Eq. (26). This has been found by checking whether the
boxes given by Eqs. (25) and (27) admits a convex decomposition as an extremal
steerable box given by

(52)

and an unsteerable box. Since such decomposition does not exist for the boxes
Eq. (25) and Eq. (27), they do not have a nonvanishing Schrödinger strength.
Therefore, in the two-setting steering scenario, the Schrödinger strength of
boxes arising from certain two-qubit states does not fully capture the
quantification of one-sided semi-device-independent steerability.

To capture the quantification of superunsteerability fully, we define the
following two other strengths of superunsteerable parts of the most general
superunsteerable box. To this end, we note that the steerable part of the
decomposition in the definition of Schrödinger strength lies on the boundary
of the set of all boxes in the given steering scenario. Such a box lying on
the boundary has zero probabilities in the correlation table. Note that the
superunsteerable boxes given by Eqs. (25) and (27), which have vanishing
Schrödinger strength lie on the boundary of the set of all boxes. These two
boxes exhibit two different types of superunsteerablility: the former exhibits
two-way superunsteerability, whereas the latter exhibits one-way
superunsteerability.

In the following, we define the quantification of superunsteerability of
unsteerable boxes having vanishing Schrödinger strength but having
superunsteerablity due to a different superunsteerable box exhibiting two-way
superunsteerability and lying on the boundary of the set of boxes. We say that
a superunsteerable box that lies on the boundary of the set of boxes is an
extremal superunsteerable box in the sense that it cannot be decomposed as a
steerable or superunsteerable box, and an unsteerable box which does not have
superunsteerability. Let $P_{SU^{d_{\lambda}=3}_{2way}}(ab|xy)$ is a
superunsteerable box exhibiting two-way superunsteerability with the minimal
$d_{\lambda}=3$. An unsteerable box $P(ab|xy)$ having vanishing Schrödinger
strength can be decomposed into a convex mixture of such an extremal
superunsteerable box and an unsteerable box,

$\displaystyle P(ab|xy)$
$\displaystyle=p_{SU^{d_{\lambda}=3}_{2way}}P_{SU^{d_{\lambda}=3}_{2way}}(ab|xy)$
$\displaystyle+(1-p_{SU^{d_{\lambda}=3}_{2way}})P_{US}(ab|xy),$ (53)

where $P_{US}(ab|xy)$ is an unsteerable box which may be superunsteerable due
to one-way superunsteerability; $0\leq p_{SU^{d_{\lambda}=3}_{2way}}\leq 1$.
The weight of the box $P_{SU^{d_{\lambda}=3}_{2way}}(ab|xy)$ maximized over
all possible decompositions of the form (III.2) is called two-way
superunsteerable strength of the box $P(ab|xy)$, given by

$S_{SU^{d_{\lambda}=3}_{2way}}\Big{(}P(ab|xy)\Big{)}:=\max_{\text{decompositions}}p_{SU^{d_{\lambda}=3}_{2way}}.$
(54)

Here, $0\leq S_{SU^{d_{\lambda}=3}_{2way}}\leq 1$ (since, $0\leq
p_{SU^{\rightarrow}_{1way}}\leq 1$). A unsteerable box $P(ab|xy)$ having
vanishing Schrödinger strength and two-way superunsteerable strength can be
decomposed into a convex mixture of a superunsteerable part and an unsteerable
part,

$P(ab|xy)=p_{SU^{\rightarrow}_{1way}}P_{SU^{\rightarrow}_{1way}}(ab|xy)+(1-p)P_{US}(ab|xy),$
(55)

where $P_{SU^{\rightarrow}_{1way}}$ is a superunsteerable box exhibiting one-
way superunsteerability and $P_{US}(ab|xy)$ is an unsteerable box which is not
superunsteerable; $0\leq p_{SU^{\rightarrow}_{1way}}\leq 1$. The weight of the
box $P_{SU^{\rightarrow}_{1way}}$ maximized over all possible decompositions
of the form (55) is called one-way superunsteerable strength of the box
$P(ab|xy)$, given by

$S_{SU^{\rightarrow}_{1way}}\Big{(}P(ab|xy)\Big{)}:=\max_{\text{decompositions}}p_{SU^{\rightarrow}_{1way}}.$
(56)

Here, $0\leq S_{SU^{\rightarrow}_{1way}}\leq 1$ (since, $0\leq
p_{SU^{\rightarrow}_{1way}}\leq 1$).

A general box in the two-setting scenario may have all of three strengths:
Schrödinger strength, two-way and one-way superunsteerable strengths. The
three strengths above indicate total quantification of one-sided semi-device-
independent steerability for such a box.

Any separable two-qubit state can be written as a convex mixture of at most
four product states [28]. In the following, we consider the nonproduct
separable states with rank-$4$ and $2$. These states are given by

$\displaystyle\rho_{SU^{d_{\lambda}=4}_{2way}}$
$\displaystyle=\frac{1}{4}\Big{(}|00\rangle\langle 00|+|++\rangle\langle++|$
$\displaystyle+|11\rangle\langle 11|+|--\rangle\langle--|\Big{)},$ (57)

$\rho_{SU^{d_{\lambda}=3}_{2way}}$ and $\rho_{SU^{\rightarrow}_{1way}}$ given
by Eqs. (24) and (26), respectively. Such states have characterization in the
quantum steering ellipsoid formalism as follows. The quantum steering
ellipsoid of the Bell-diagonal state $\rho_{SU^{d_{\lambda}=4}_{2way}}$
degenerates to a pancake that has dimension $2$ and exhibits complete steering
both ways. Next, the quantum steering ellipsoid of the two-way
superunsteerable state $\rho_{SU^{d_{\lambda}=3}_{2way}}$ is also a pancake
but exhibits incomplete steering both ways. Finally, the quantum steering
ellipsoid of the one-way superunsteerable state
$\rho_{SU^{\rightarrow}_{1way}}$ degenerates to a needle which is
$1$-dimensional and exhibits one-way incomplete steering. We now state the
following observation.

######  Observation 3.

Since the volume of the quantum steering ellipsoid of the states,
$\rho_{SU^{d_{\lambda}=4}_{2way}}$, $\rho_{SU^{d_{\lambda}=3}_{2way}}$ and
$\rho_{SU^{\rightarrow}_{1way}}$, is zero, they do not exhibit simultaneous
correlations in the three mutually unbiased bases. These states, each
producing a different type of superunsteerability, have distinguishing quantum
steering ellipsoid characterization in how they exhibit complete or incomplete
steering.

We consider the following three-parameter family of separable states:

$\displaystyle\rho=p\rho_{SU^{d_{\lambda}=4}_{2way}}+q\rho_{SU^{d_{\lambda}=3}_{2way}}+(1-p-q)\rho_{SU^{\rightarrow}_{1way}}.$
(58)

The above state can be written in the form

$\displaystyle\rho=\frac{1}{4}\bigg{(}$
$\displaystyle\mathbbm{1}_{A}\otimes\mathbbm{1}_{B}+\frac{1-p}{2}(\sigma_{x}+\sigma_{z})\otimes\mathbbm{1}_{B}$
$\displaystyle+\mathbbm{1}_{A}\otimes\frac{q}{2}(\sigma_{x}+\sigma_{z})$
$\displaystyle+\frac{p+q}{2}\sigma_{x}\otimes\sigma_{x}-\frac{1-p-q}{2}\sigma_{x}\otimes\sigma_{z}+\frac{1}{2}\sigma_{z}\otimes\sigma_{z}\bigg{)}.$
(59)

######  Proposition 3.

The Schrödinger, two-way superunsteerable and one-way superunsteerable
strengths of the family of states given by Eq. (58) are given by

$\displaystyle S_{St}(\rho)$ $\displaystyle=\frac{p}{2}$ $\displaystyle
S_{SU^{d_{\lambda}=3}_{2way}}(\rho)$ $\displaystyle=q$ $\displaystyle
S_{SU^{\rightarrow}_{1way}}(\rho)$ $\displaystyle=1-p-q,$ (60)

respectively, where $S_{St}(\rho)$, is the Schrödinger strength of the state.

###### Proof.

If Alice performs the projective measurements of observables corresponding to
the operators $A_{0}=\sigma_{x}$ and $A_{1}=\sigma_{z}$, and Bob performs
projective measurements of observables corresponding to the operators
$B_{0}=\sigma_{x}$ and $B_{1}=\sigma_{z}$, then the correlation is produced
from the above state is given by

$\displaystyle P(ab|xy)$
$\displaystyle=\begin{tabular}[]{c|cccc}\diagbox[dir={NW}]{{\shortstack[l]{(x,y)}}}{{\shortstack[r]{(a,b)}}}{}&(0,0)&(0,1)&(1,0)&(1,1)\\\
\hline\cr\\\\[1.42271pt]
(0,0)&$\dfrac{3+2q}{8}$&$\dfrac{3-2p-2q}{8}$&$\dfrac{1}{8}$&$\dfrac{1+2p}{8}$\\\\[14.22636pt]
(0,1)&$\dfrac{1+q}{4}$&$\dfrac{2-p-q}{4}$&$\dfrac{1}{4}$&$\dfrac{p}{4}$\\\\[14.22636pt]
(1,0)&$\dfrac{3-p+q}{8}$&$\dfrac{3-p-q}{8}$&$\dfrac{1+p+q}{8}$&$\dfrac{1+p-q}{8}$\\\\[14.22636pt]
(1,1)&$\dfrac{4-p+q}{8}$&$\dfrac{2-p-q}{8}$&$\dfrac{p+q}{8}$&$\dfrac{2+p-q}{8}$\\\
\end{tabular}$ (67)

and can be decomposed as follows:

$\displaystyle P(ab|xy)$
$\displaystyle=\frac{p}{2}P_{S}(ab|xy)+\frac{p}{2}P_{N}+qP_{SU^{d_{\lambda}=3}_{2way}}(ab|xy)$
$\displaystyle+(1-p-q)P_{SU^{\rightarrow}_{1way}}(ab|xy)$ (68)

where $P_{S}(ab|xy)$ is an extremal steerable box given by Eq. (52),
$P_{{SU^{d_{\lambda}=3}_{2way}}}(ab|xy)$ and
$P_{SU^{\rightarrow}_{1way}}(ab|xy)$ are the superunsteerable boxes given by
Eqs. (25) and (27). The above decomposition provides an optimal decomposition
to quantify all three strengths optimally. This follows from the fact that the
Schrödinger, two-superunsteerable and one-way superunsteerable strengths of
the box (III.2) are proportional to the fractions of
$\rho^{d_{\lambda}=4}_{SU_{2way}}$, $\rho^{d_{\lambda}=3}_{SU_{2way}}$ and
$\rho_{SU^{\rightarrow}_{1way}}$, respectively, that give rise to the
different types of extremal boxes in the decomposition given by Eq. (III.2). ∎

Let us now proceed to study the relationship between the quantification of
one-sided semi-device-independent steerability of the state given by Eq. (58)
and quantification of its simultaneous correlations in two mutually unbiased
bases via $Q^{\rightarrow}_{2}$. To this end, we note that the optimal basis
that gives rise to the amount of classical correlation $C^{\rightarrow}_{1}$
of the states $\rho_{SU^{d_{\lambda}=3}_{2way}}$ and
$\rho_{SU^{\rightarrow}_{1way}}$ was studied in Ref. [33]. We now show the
following result.

######  Proposition 4.

$Q^{\rightarrow}_{2}$ of the states $\rho_{SU^{d_{\lambda}=3}_{2way}}$ and
$\rho_{SU^{\rightarrow}_{1way}}$ do not quantify simultaneous correlations in
two mutually unbiased bases of these states.

###### Proof.

As shown in Ref. [33], the optimal bases that give rise to
$C^{\rightarrow}_{1}$ of such rank-$2$ states are the bases of the observables
$\frac{\mp\sigma_{z}\pm\sigma_{x}}{\sqrt{2}}$. For such basis, the classical
correlations of the states are obtained as
$C^{\rightarrow}_{1}(\rho_{SU^{d_{\lambda}=3}_{2way}})=h\left(\frac{1+\sqrt{3}/2}{2}\right)$
and
$C^{\rightarrow}_{1}(\rho_{SU^{\rightarrow}_{1way}})=h\left(\frac{1+1/\sqrt{2}}{2}\right)$,
respectively. On the other hand, the Holevo quantity of the ensemble obtained
from these states for a basis which is mutually unbiased to the abovementioned
$C^{\rightarrow}_{1}$-bases of the state vanishes. For instance, for the basis
associated with the observable $\frac{\sigma_{z}+\sigma_{x}}{\sqrt{2}}$,
$\chi(\rho_{SU^{d_{\lambda}=3}_{2way}},\frac{\sigma_{z}+\sigma_{x}}{\sqrt{2}})=\chi(\rho_{SU^{\rightarrow}_{1way}},\frac{\sigma_{z}+\sigma_{x}}{\sqrt{2}})=0$.
This implies that the simultaneous correlations in two mutually unbiased bases
of the states $\rho_{SU^{d_{\lambda}=3}_{2way}}$ and
$\rho_{SU^{\rightarrow}_{1way}}$ are not captured by $Q^{\rightarrow}_{2}$. ∎

The above result implies that a nonvanishing $Q^{\rightarrow}_{2}$ is stronger
than a nonvanishing quantum discord $D^{\rightarrow}_{2}$.

While $Q^{\rightarrow}_{2}$ does not capture the simultaneous correlations in
two mutually unbiased bases of the states $\rho_{SU^{d_{\lambda}=3}_{2way}}$
and $\rho_{SU^{\rightarrow}_{1way}}$, the observation of one-sided semi-
device-independent steerability using an extremal superunsteerable box such as
Eqs. (25) and (27) implies the presence of simultaneous correlations in two
mutually unbiased bases. We also note the following with the following
assemblages producing these boxes: In the case of the box (25), it is given by
$\sigma_{a|0}=\\{\frac{3}{8}\left(\openone+\frac{2}{3}\sigma_{x}+\frac{1}{3}\sigma_{z}\right),\frac{1}{8}\left(\openone+\sigma_{z}\right)\\}$
and
$\sigma_{a|1}=\\{\frac{3}{8}\left(\openone+\frac{1}{3}\sigma_{x}+\frac{2}{3}\sigma_{z}\right),\frac{1}{8}\left(\openone+\sigma_{x}\right)\\}$
and in the case of the box (27), it is given by
$\sigma_{a|0}=\\{\frac{3}{8}\left(\openone-\frac{1}{4}\sigma_{z}\right),\frac{1}{8}\left(\openone+\frac{1}{4}\sigma_{z}\right)\\}$
and
$\sigma_{a|1}=\\{\frac{3}{8}\left(\openone+\sigma_{z}\right),\frac{1}{8}\left(\openone-\sigma_{z}\right)\\}$.

######  Observation 4.

The quantum steering ellipsoids of the states,
$\rho_{SU^{d_{\lambda}=3}_{2way}}$ and $\rho_{SU^{\rightarrow}_{1way}}$, which
are a pancake and a needle, respectively, contain two steered states that
touch the Bloch sphere, i.e., a pure state and these steered states included
in the assemblages that lead to the extremal superunsteerable boxes detecting
the simultaneous correlations of the states optimally.

For the box 67, the presence of three different types of steerability is
signified by a nonzero Schrödinger strength, two-way and one-way
superunsteerable strength. A nonzero Schrödinger strength implies a strong
form of superunsteerability. Since it arises due to superunsteerability with
the minimal $d_{\lambda}=4$, which is higher than that of the minimal
$d_{\lambda}=3$ corresponding to the superunsteerable boxes given by Eqs. (25)
and (27). While superunsteerabiltiy of the boxes Eqs. (25) and (27) is weaker
than that of a box with superunsteerability with the minimal $d_{\lambda}=4$,
the superunsteerabiltiy of the box Eq. (25) is stronger than that of the box
(27). This stronger superunsteerability can be attributed to the box Eq. (25)
exhibiting two-way superunsteerability, which is stronger than one-way
superunsteerability. And also, the assemblage producing the box Eq. (25) has
two pure steered states for two different measurements. In contrast, the
assemblage producing the box (27) has two pure steered states for the same
measurement. We now state our second main result on quantifying the
simultaneous correlation in two mutually unbiased bases of the family of
states given by Eq. (58).

######  Result 2.

The total quantification of one-sided semi-device-independent steerability of
the family of states, Eq. (58), via Schrödinger strength, two-way and one-way
superunsteerable strengths quantify quantum correlation beyond
$Q^{\rightarrow}_{2}$ and serves to quantify operationally different types of
simultaneous correlations in two mutually unbiased bases due to the presence
of three different types of steerability.

##  IV Conclusions

In this work, for a broad class of two-qubit states that belong to the Bell-
diagonal states, we have studied the relationships between the quantification
of one-sided semi-device-independent steerability in two and three-settings
scenarios and information-theoretic measures of simultaneous correlations in
mutually unbiased bases defined in Ref. [18]. Such relationships reveal that
the quantification of one-sided semi-device-independent steerability of Bell-
diagonal states provides an operational characterization of the quantification
of simultaneous correlations in two and three mutually unbiased bases of the
Bell-diagonal states based on the measures defined in [18].

We have then demonstrated that quantifying simultaneous correlations in two
mutually unbiased bases based on the measure in [18] does not capture quantum
correlation in certain superunsteerable states. For a class of
superunsteerable states that include these states, the quantification of
superunsteerability studied in Ref. [16] does not fully capture the
quantification of one-sided semi-device-independent steerability. To fully
capture the quantification of one-sided semi-device-independent steerabillity,
we have formulated two different quantifiers of steerability together with the
quantifier in Ref. [16]. The total quantification of steerability based on
these three quantifiers operationally quantifies the simultaneous correlations
in mutually unbiased bases comprehensively, which goes beyond the information-
theoretic quantification of simultaneous correlations in mutually unbiased
bases on the measure in Ref. [18]. We have also invoked quantum steering
ellipsoid formulation to shed intuitions on quantifying simultaneous
correlation in complementary bases in the two-qubit states we have studied.

## Acknowledgements

CJ is grateful to Prof. Wei-Min Zhang for valuable discussions and would like
to thank the National Science and Technology Council (formerly the Ministry of
Science and Technology), Taiwan (Grant No. MOST 111-2811-M-006-040-MY2). DD
acknowledges the Royal Society (United Kingdom) for the support through the
Newton International Fellowship (NIF$\backslash$R1$\backslash$212007).

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