# Pair-matching of radio-loud and radio-quiet AGNs

Dorota Kozieł-Wierzbowska 1,∗, Grażyna Stasińska 2, Natalia Vale Asari 3,
Marek Sikora 4, Elisa Goettems 3 and Anna Wójtowicz 1

###### Abstract

##  1 Keywords:

active galaxies, radio galaxies, jetted and non-jetted AGNs, radio-loudness,
galaxy morphology

Active galactic nuclei (AGNs) are known to cover an extremely broad range of
radio luminosities and the spread of their radio-loudness is very large at any
value of the Eddington ratio. This implies very diverse jet production
efficiencies which can result from the spread of the black hole spins and
magnetic fluxes. Magnetic fluxes can be developed stochastically in the
innermost zones of accretion discs, or can be advected to the central regions
prior to the AGN phase. In the latter case there could be systematic
differences between the properties of galaxies hosting radio-loud (RL) and
radio-quiet (RQ) AGNs. In the former case the differences should be negligible
for objects having the same Eddington ratio. To study the problem we decided
to conduct a comparison study of host galaxy properties of RL and RQ AGNs. In
this study we selected type II AGNs from SDSS spectroscopic catalogues. Our RL
AGN sample consists of the AGNs appearing in the Best and Heckman (2012)
catalogue of radio galaxies. To compare RL and RQ galaxies that have the same
AGN parameters we matched the galaxies in black hole mass, Eddington ratio and
redshift. We compared several properties of the host galaxies in these two
groups of objects like galaxy mass, colour, concentration index, line widths,
morphological type and interaction signatures. We found that in the studied
group RL AGNs are preferentially hosted by elliptical galaxies while RQ ones
are hosted by galaxies of later type. We also found that the fraction of
interacting galaxies is the same in both groups of AGNs. These results suggest
that the magnetic flux in RL AGNs is advected to the nucleus prior to the AGN
phase.

##  2 Introduction

Active galactic nuclei (AGNs) are powered by accretion on a supermassive black
hole (BH). Although the first discovered AGNs were radio-loud (RL), most of
AGNs are radio-quiet (RQ). The radio-loudness of RL AGNs, defined as the ratio
of the radio flux to the optical flux (Kellermann et al., 1989), covers
several orders of magnitude (e.g. Sikora et al., 2007; Lal and Ho, 2010) which
implies very diverse jet production efficiencies. For jets powered by rotating
BHs (Blandford and Znajek, 1977), the efficiency of the jet production can be
related to the spread of the BH spins and the amount of magnetic fluxes
accumulated in the central AGNs.

Magnetic fluxes can be developed stochastically in the innermost zones of
accretion discs (Begelman and Armitage, 2014), or can be advected to the
central regions of a galaxy prior to the AGN phase (Sikora et al., 2013;
Sikora and Begelman, 2013). In the latter case there could be systematic
differences between the properties of galaxies hosting radio-loud and radio-
quiet AGNs. Our aim is to compare the properties of the host galaxies of
radio-loud and radio-quiet AGNs to distinguish which of these two scenarios is
more probable. If we find that RQ and RL are hosted by different galaxies, we
could discard the scenario where radio jets are stochastic.

Radio-loud (jetted) and radio-quiet (non-jetted) AGNs have already been
studied extensively and some systematic differences were found between these
two groups of objects:

  * •

The most radio-loud AGNs are found in galaxies with black hole masses $\geq
10^{8}\,M_{\odot}$ (Laor, 2000; Dunlop et al., 2003; Floyd et al., 2004;
McLure and Jarvis, 2004; Best et al., 2005).

  * •

The fraction of radio-loud objects and radio loudness increases with
decreasing Eddington ratio, $\lambda=$$L_{\rm bol}$$/L_{\rm Edd}$ (e.g.
Kratzer and Richards, 2015; Terashima and Wilson, 2003), but there is a large
scatter in radio loudness for AGNs with similar Eddington ratio.

  * •

The fraction of galaxies with disturbed morphology is larger in RL than in RQ
AGNs (Bessiere et al., 2012; Chiaberge et al., 2015).

  * •

RL AGNs are located in denser environments (e.g. Mandelbaum et al., 2009;
Ramos Almeida et al., 2013).

However, the differences listed above concern entire families of objects, but
not objects that have the same accretion properties. Moreover, in catalogues
of AGNs there are many objects with properties characteristic of RL objects
like very massive black holes, low Eddington ratios, and disturbed
morphologies, but they are radio-quiet. Therefore, we ask the question why is
the efficiency of the jet production very different among otherwise similar
objects?

In our project we concentrate on Type 2 (i.e. obscured) objects, to be able to
study the properties of the host galaxies, with Eddington ratios
$\lambda\,\geq$ 0.003, and we seek the differences between radio-loud and
radio-quiet AGNs to check if there are any generic differences between the
host galaxies of these two groups of objects that can indicate different
evolution histories of jetted and non-jetted AGNs.

##  3 Methods

###  3.1 Selection of the samples

The samples of radio-loud and radio-quiet galaxies were selected from the SDSS
DR7 spectroscopic catalogues (Abazajian et al., 2009). Only galaxies with S/N
$\geq$ 10 Å-1 in the region around $\lambda_{0}$ = 4020 Å were chosen. We also
applied the redshift cuts, low velocity dispersion limit and S/N limits in
emission line fluxes as in Kozieł-Wierzbowska et al. (2017). Galaxies with
faulty pixels in the area of emission lines were eliminated. After this step,
AGNs were selected based on the BPT diagram and the Kewley et al. (2001) line.
Using the WHAN diagram (Cid Fernandes et al., 2011), we removed those galaxies
in which the emission lines could be produced by hot, low-mass, evolved stars
(the retired galaxies defined in Stasińska et al., 2008). This procedure
selected 19883 optical AGNs.

Among this sample of optical AGNs, we searched those which belong to the Best
and Heckman (2012, BH12) catalogue of radio galaxies and whose radio emission
is considered by them to be produced by an AGN. The BH12 catalogue flux limit
is 5 mJy.

After limiting ourselves to AGNs with Eddington ratio $\lambda\geq$
0.003111This step ensures us that we calculate the bolometric luminosity
consistently and correctly for all studied sources., i.e. focusing mostly on
sources with radiatively efficient accretion, we obtained our RL AGN sample of
376 objects, and our RQ AGN sample of 10918 objects.

The host galaxy stellar masses, velocity dispersions (used to calculate
$M_{\rm BH}$), nebular extinction, emission line fluxes and equivalent widths,
and the Eddington ratios, $\lambda$, where obtained from the SDSS data after
applying the STARLIGHT (Cid Fernandes et al., 2005) spectral model-fitting.

Figure 1: Top: BPT and WHAN diagrams for RL and RQ AGNs (red and blue points,
respectively). The solid black line in the BPT diagram is the Kewley et al.
(2001) line, while the dashed line separates galaxies with line emission
dominated by the AGN. In the WHAN diagram the solid lines separate SF
galaxies, strong AGNs, weak AGNs and retired galaxies (Cid Fernandes et al.,
2011). Bottom: $M_{BH}$ versus stellar mass and $M_{BH}$ versus Eddington
ratio. The histograms of plotted parameters for RL and RQ galaxies are also
shown. Updated version published in Kozieł-Wierzbowska et al. (2017).

The upper panels of Fig. 1 show the BPT (Baldwin et al., 1981) and the WHAN
(Cid Fernandes et al., 2011) diagrams for our RL and RQ samples. RL AGNs are
shown in red and RQ AGNs in blue. On both axes the normalized histograms are
presented. The black curve is the Kewley et al. (2001) line. The location of
red and blue points is similar which shows that objects from both samples have
similar ionization properties. The bottom panel in Fig. 1 shows $M_{BH}$ as a
function of the galaxy stellar mass $M_{*}$, and of the Eddington ratio,
$\lambda$. In these diagrams RL and RQ galaxies are shifted relative to each
other. On average, RL AGNs have higher black hole masses and higher stellar
masses compared to RQ galaxies. RL objects have also lower $\lambda$. These
observations are compatible with trends found by previous authors (see
Introduction). These two panels clearly show that in order to compare RL and
RQ AGNs it is necessary to match them in black hole mass and Eddington ratio.

###  3.2 Pair-matching of RL and RQ galaxies

To compare RL and RQ galaxies with the same AGN parameters, we applied a pair-
matching technique. Galaxies were matched in redshift, black hole mass
($M_{BH}$) and Eddigton ratio ($\lambda$). In practice for each RL object we
selected all RQ objects for which: $|\Delta z|\leq 0.01$,
$|\Delta\log\lambda|\leq 0.09$, and $|\Delta\log M_{\rm BH}|\leq 0.1206$. We
computed the distance as $d^{2}_{\rm match}=\sum\Delta^{2}$. For each RL AGN
three RQ galaxies with the smallest $d$ were included into the matched RQ
sample (mRQ sample).

We defined the radio loudness parameter by ${\mathcal{R}}\equiv L_{\rm
1.4}[W\,Hz^{-1}]/L_{{\rm H}\alpha}[L_{\odot}]$, where $L_{\rm 1.4}$ is the
radio luminosity at 1.4 GHz. The classical criterion for radio-loud AGN,
${\mathcal{R}}^{(K)}>10$ (Kellermann et al., 1989), translates into
$\log{\mathcal{R}}>15.8$. From the matched RQ sample we excluded galaxies that
were undetected in radio and for which the radio-loudness (estimated from the
flux limit od 5 mJy in the BH12 catalogue) is larger than 10
($\log{\mathcal{R}}>15.8$).

##  4 Results

Figure 2: Histograms of selected parameters of RL (red) and mRQ (blue)
objects. In each panel we show the results of the Anderson-Darling and Mann-
Whitney tests in terms of probabilities p that the samples could be drawn from
the same population. Results with $p<0.003$ are written in black, those with
$p<0.05$ are written in dark grey. Blue and red points at the top of each
histogram indicate the median values and the horizontal lines indicate the
quartiles. Published in Kozieł-Wierzbowska et al. (2017)

We present our results in the form of histograms. Fig. 2 shows histograms of
selected parameters of RL (red) and RQ (blue) galaxies. Results for RL AGNs
and their matched RQ galaxies are presented separately for three bins in the
value of the radio-loudness ${\mathcal{R}}$ of the parent RL object.
${\mathcal{R}}$ bins were chosen to represent classically radio-quiet
(although jetted, $\log{\mathcal{R}}<15.8$) objects, intermediate radio-loud
objects ($15.8<\log{\mathcal{R}}<16.8$), and the most radio-loud ones
($\log{\mathcal{R}}>16.8$). In each bin, the tick mark with horizontal line
shown on the top of the histograms represents the median value and the
quartiles. We used two statistical tests to study the difference between RL
and mRQ samples. These are the Anderson-Darling (AD) and Mann-Whitney (MW)
tests. These tests give the probability that both samples are drawn from the
same population.

Among the studied parameters, the stellar galaxy masses, $M_{*}$, and the
colors, u-r, have very similar distributions for RL and RQ AGNs. In the case
of galaxy mass this result is expected since we matched our objects in
$M_{BH}$ which correlates with $M_{*}$. The concentration index, CI, and the
galaxy axes ratio, $b/a$, the two parameters associated with galaxy shape and
morphology, show significant differences. RL galaxies tend to have larger
concentration index, and larger axes ratios in the two highest ${\mathcal{R}}$
bins. The lower values of CI and $b/a$ ratio in the matched RQ sample indicate
more disky galaxy morphology.

In the next two panels we present a comparison of the nebular extinction,
$A_{V}^{\rm neb}$, and of the [O iii] to H$\beta$ line fluxes ratio. The
values of $A_{V}^{\rm neb}$ decrease with increasing ${\mathcal{R}}$. [O
iii]$/$H$\beta$ differs significantly in the bin of the highest
${\mathcal{R}}$. These results may suggest some contribution from the H ii
regions to the line emission. The last panel shows histograms of [O iii] line
widths. We see that the line widths in RL galaxies tend to be larger then in
RQ AGNs in the lowest ${\mathcal{R}}$ bins. We speculate that this can result
from having at low R less collimated jets. Forming wider outflows such jets
may affect the motion of the gas clouds in the narrow line region, and hence
broaden the emission line profiles.

To eliminate the contribution of H ii regions from our studies we decided to
use “cleaned” samples. In these samples only galaxies that lie above the
dashed curve shown in BPT diagram from Fig. 1 are included. In these galaxies
the contribution from H ii regions can be neglected. The results for the
cleaned samples (see Kozieł-Wierzbowska et al., 2017) confirm our findings for
the whole RL and mRQ samples concerning morphological properties.

Figure 3: Cumulative fraction of RL (shades of red) and RQ (shades of blue)
classified as elliptical, lenticular, distorted, ring, or spiral galaxies.
$R_{1}$ corresponds to $\log{\mathcal{R}}<15.8$, $R_{2}$ to
$15.8<\log{\mathcal{R}}<16.8$, and $R_{3}$ to $\log{\mathcal{R}}>16.8$.
Results are shown for three classifiers separately.

From the comparison of CI and of the $b/a$ ratio we see that there is a
difference in morphology between RL and mRQ galaxies. To confirm this result
we decided to perform a morphological classification of all RL galaxies and
the closest match from the RQ galaxies. Classifiers looked at the color SDSS
images of these objects and attributed to each galaxy a morphological type.
The results for the three classifiers are shown in Fig. 3. The panels in this
figure show in different shades of red (blue) the fraction of RL (mRQ)
galaxies classified as elliptical, lenticular, distorted, ring, or spiral
galaxies. As before, results are presented separately for three
${\mathcal{R}}$ bins. As we can see, the fraction of ellipticals among RL
objects in all three radio-loudness bins is larger than among the matched RQ
galaxies. Note that low-CI galaxies are not spirals, but lenticular or
distorted galaxies. This result was confirmed using Galaxy ZOO data.

The classification scheme also included information about galaxy interactions
like major or minor mergers, tail, or suspected interaction. Fig. 4 shows the
results concerning the fraction of galaxies with interaction signatures
considered as certain. Here, as interacting, we consider major or minor
mergers and galaxies with a tail. In Fig. 4 we do not see any systematic
difference between RL and RQ AGNs suggesting that galaxy interaction has no
special effect on the radio activity of the AGN. Our result is in
contradiction with Chiaberge et al. (2015) who found that all radio-loud
galaxies in their sample are mergers, however their sample consisted of FRII
radio galaxies at redshifts larger than 1, while in our sample we have only
local AGNs mostly with compact morphologies.

Figure 4: Cumulative fraction of RL AGNs in three ${\mathcal{R}}$ bins, and
their closest mRQ galaxies showing signs of the galaxy interaction. The layout
is the same as for Fig. 3.

##  5 Summary

The results presented here clearly show that not only Eddington ratio and
black hole mass determine the jet production efficiency. In the description of
the jet production the accumulation of the magnetic flux in the AGN center and
the BH spin have also to be taken into account. The fact that we find a higher
fraction of ellipticals among RL galaxies than among the matched RQ ones
suggest that elliptical galaxies have already a sufficient amount of magnetic
flux accumulated in the center to produce and collimate radio jets in the
Blandford and Znajek, 1977 scenario. This is in agreement with findings of
Sikora et al. (2013) and Sikora and Begelman (2013) on the RL AGN pre-phase.

## ACKNOWLEDGMENTS

This work was carried out within the framework of the Polish National Science
Centre grant UMO-2013/09/B/ST9/00026. Grażyna Stasińska was partially
supported by the National Research Centre, Poland, DEC-2013/08/M/ST9/00664,
within the framework of the HECOLS International Associated Laboratory. GS and
NVA acknowledge the support from the CAPES CSF–PVE project
88881.068116/2014-01. The Sloan Digital Sky Survey is a joint project of The
University of Chicago, Fermilab, the Institute for Advanced Study, the Japan
Participation Group, the Johns Hopkins University, the Los Alamos National
Laboratory, the Max-Planck-Institute for Astronomy, the Max-Planck-Institute
for Astrophysics, New Mexico State University, Princeton University, the
United States Naval Observatory, and the University of Washington. Funding for
the project has been provided by the Alfred P. Sloan Foundation, the
Participating Institutions, the National Aeronautics and Space Administration,
the National Science Foundation, the U.S. Department of Energy, the Japanese
Monbukagakusho, and the Max Planck Society.

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# Equivalence of the quantumness of sequential correlations and spatial
correlations

C. Jebarathinam<EMAIL_ADDRESS>Department of Physics and Center for
Quantum Frontiers of Research & Technology (QFort), National Cheng Kung
University, Tainan 701, Taiwan  Debarshi Das<EMAIL_ADDRESS>Centre
for Astroparticle Physics and Space Science (CAPSS), Bose Institute, Block EN,
Sector V, Salt Lake, Kolkata 700 091, India

###### Abstract

Quantum information protocols can be realized using the ‘prepare and measure’
setups which do not require sharing quantum correlated particles. In this
work, we study the equivalence between the quantumness in a prepare and
measure scenario involving independent devices, which implements quantum
random number generation, and the quantumness in the corresponding scenario
which realizes the same task with spatially separated correlated particles. In
particular, we demonstrate that quantumness of sequential correlations
observed in the prepare and measure scenario gets manifested as
superunsteerability, which is a particular kind of spatial quantum correlation
in the presence of limited shared randomness. In this scenario consisting of
spatially separated quantum correlated particles as resource for implementing
the quantum random number generation protocol, we define an experimentally
measurable quantity which provides a bound on the amount of genuine randomness
generation. Next, we study the equivalence between the quantumness of the
prepare and measure scenario in the presence of shared randomness, which has
been used for implementing quantum random-access codes, and the quantumness in
the corresponding scenario which replaces quantum communication by spatially
separated quantum correlated particles. In this case, we demonstrate that
certain sequential correlations in the prepare and measure scenario in the
presence of shared randomness, which have quantumness but do not provide
advantage for random-access codes, can be used to provide advantage when they
are realized as spatial correlations in the presence of limited shared
randomness. We point out that these spatial correlations are superlocal
correlations, which are another kind of spatial quantum correlations in the
presence of limited shared randomness, and identify inequalities detecting
superlocality.

###### pacs:

03.65.Ud, 03.67.Mn, 03.65.Ta

##  I Introduction

The fundamental aspects of quantum theory such as quantum coherence and
quantum nonlocality are valuable resources underpinning quantum technologies
Streltsov  _et al._ (2017); Horodecki  _et al._ (2009); Brunner  _et al._
(2014); Uola  _et al._ . Therefore, characterizing these resources Chitambar
and Gour (2019) is important for both the foundations of quantum mechanics and
quantum information science. Recently, a relationship between quantum
coherence and quantum discord has been established Ma  _et al._ (2016a).
Quantum discord captures quantumness even in separable states Ollivier and
Zurek (2001); Henderson and Vedral (2001) and characterizes quantum resource
in certain tasks in quantum information science Modi  _et al._ (2012); Adesso
_et al._ (2016); Bera  _et al._ (2017). Recently, it has been studied that
even certain separable states having quantum discord can be used to generate
superlocal or superunsteerable correlations which represent stronger than
classical correlations in the presence of limited shared randomness (see Fig.
1) Donohue and Wolfe (2015); Jebarathinam (2016); Jebaratnam  _et al._ (2017);
Jebaratnam (2017); Jebaratnam  _et al._ (2018); Das  _et al._ (2018a, b);
Jebarathinam  _et al._ (2019). Thus, quantum discord can be used as a resource
for quantum tasks when the shared randomness is not a free resource.

Figure 1: Superlocality or superunsteerability scenario where the dimension of
resource (shared randomness $\lambda$ with the size $d_{\lambda}$ or shared
bipartite state $\rho_{AB}$ in $\mathbb{C}^{d_{A}}\otimes\mathbb{C}^{d_{B}}$)
producing the box $\\{p(a,b|x,y)\\}$ is bounded.

Quantum coherence manifests itself as a resource for genuine random number
generation Ma  _et al._ (2016b); Herrero-Collantes and Garcia-Escartin (2017).
In quantum random number generation protocols, genuine randomness is certified
through the violation of a Bell inequality Pironio  _et al._ (2010) or
uncertainty principle guarantees randomness Vallone  _et al._ (2014). In the
approach based on Bell inequality, the protocol provides security in a device-
independent way, i.e., without the need to have knowledge about the internal
working of the quantum devices while the approach based on the uncertainty
principle requires trusted quantum devices. Intermediate between these two
approaches, an approach based on prepare and measure scenario was proposed to
generate genuine randomness without the need to characterize the devices but
by assuming only the Hilbert-space dimension of the devices Lunghi  _et al._
(2015). Semi-device-independent quantum information protocols using shared
quantum state as a quantum channel have also been studied Law  _et al._
(2014); Passaro  _et al._ (2015); Goh  _et al._ (2016); Woodhead .

In the context of the prepare and measure protocols, analogous to the Bell
inequalities, dimension witnesses have been derived for certifying the
Hilbert-space dimension Gallego  _et al._ (2010). The dimension witnesses have
been used as the quantumness certification for the semi-device-independent
quantum information protocols Pawłowski and Brunner (2011); Li  _et al._
(2012). The relationship between the dimension witnesses and the figure merit
of a quantum communication game called quantum random-access codes Nayak
(1999) enabled to demonstrate quantum random-access codes for secure quantum
key distribution in a semi-device-independent way Pawłowski and Brunner
(2011).

In Ref. Pawlowski  _et al._ (2009), a task closely related to random-access
codes which is implemented by Alice and Bob sharing a nonsignaling box was
studied to distinguish quantum and postquantum correlations. Subsequently, two
of the authors of this work have demonstrated implementation of quantum
random-access codes assisted by shared bipartite states replacing the quantum
communication Pawłowski and Żukowski (2010). In this context, correlations
violating the suitable Bell inequalities assisted by one-bit of communication
enable quantum advantage for the random-access codes. The equivalence of
implementation of quantum random-access codes using the prepare and measure
scenario and the scenario where a bipartite quantum state acts as quantum
channel has led to study the relationships between the dimension witnesses and
the Bell inequalities Li  _et al._ (2013), the Popescu-Rohrlich boxes Popescu
and Rohrlich (1994) and random-access codes Grudka  _et al._ (2014); Pawłowski
and Scarani (2015) and the sequential and spatial correlations used in the two
types of scenarios for implementing the quantum random-access codes Tavakoli
_et al._ (2016).

In this work, we study equivalence of quantumness in a prepare and measure
scenario with independent devices in the corresponding scenario which can be
used to realize the same task of the prepare and measure scenario with shared
correlated particles. In such protocols, nonlinear dimension witnesses
constructed in Ref. Bowles  _et al._ (2014) serve as the certification of
quantumness in the sequential correlations arising in the prepare and measure
scenario in a semi-device-independent way. In Ref. Lunghi  _et al._ (2015),
such quantumness certification has been shown to achieve selftesting quantum
random number generation. We convert this prepare and measure protocol into
the protocol assisted by shared bipartite quantum state and we extend the
quantumness certification used in the prepare and measure scenario for the
spatial correlations arising in this scenario. The quantumness certification
defined for the spatial correlations can be used to provide selftesting
quantum random number generation. We demonstrate that this quantum
certification witnesses superunsteerable correlations and provides the
necessary and sufficient certification of two-qubit nonzero discord states
which are neither a classical-quantum state nor a quantum-classical state and
have maximally mixed marginals on one side while providing sufficient
certfication of nonzero discord states in general. Thus, the quantumness of
the prepare and measure scenario gets manifested as superunsteerable
correlations when the sequential correlations are observed as spatial
correlations.

We also consider the prepare and measure scenario where the quantumness of the
sequential correlations is observed in the presence of shared randomness
through a suitable linear dimension witness inequality Gallego  _et al._
(2010). Such form of quantumness has been exploited for achieving quantum
random-access codes Ambainis  _et al._ . The quantumness of sequential
correlations in the presence of shared randomness gets manifested in the form
of Bell nonlocal correlation in the corresponding scenario assisted by shared
bipartite quantum state achieving the quantum random-access codes Tavakoli
_et al._ (2016). We demonstrate that when certain sequential correlations
which have quantumness and do not violate the dimension witness inequality are
observed as spatial correlations, they may become useful for implementing the
quantum random-access codes if there is a constraint on the amount of shared
randomness. In this context, we point out that quantumness of the sequential
correlations gets manifested as superlocal correlations.

##  II Prepare and measure scenario with independent devices

Figure 2: Prepare and measure scenario with uncorrelated devices. Here,
$x_{0},x_{1},y,b\in\\{0,1\\}$ and
$\rho^{\lambda}_{x_{0}x_{1}}\in\mathcal{B}(\mathbb{C}^{2})$, where
$\mathcal{B}(\mathbb{C}^{2})$ is the set of bounded operators acting on the
two-dimensional Hilbert-space of quantum states $\mathbb{C}^{2}$.

In Ref. Lunghi  _et al._ (2015), a prepare and measure (P$\&$M) protocol was
considered as shown in Fig. 2 for semi-device-independent (SDI) quantum random
number generation (QRNG). In this protocol, Alice has access to the set of
four uncharacterized preparations labelled by $x_{0}x_{1}\in\\{00,01,10,11\\}$
which are in qubit states and sends one of them randomly to Bob through a
quantum channel. Bob has access to the set of two uncharacterized measurements
labelled by $y\in\\{0,1\\}$ and performs one of the measurements on the system
received from Alice. To account for the imperfections, the devices are
associated with an internal state denoted by $\lambda$ for Alice and $\mu$ for
Bob. The devices are assumed to be independent, i.e.,
$p(\lambda,\mu)={q}_{\lambda}{r}_{\mu}$, $p_{\lambda},q_{\mu}\geq 0$,
$\sum_{\lambda}{q}_{\lambda}=\sum_{\mu}{r}_{\mu}=1$. Bob observes the set of
conditional probability of obtaining the outcomes $\\{p(b|x_{0}x_{1},y)\\}$
which captures the correlations between Alice’s preparations and Bob’s
measurements. The correlations are given by

$\displaystyle p(b|x_{0}x_{1},y)$ $\displaystyle=$
$\displaystyle\sum_{\lambda,\mu}{q}_{\lambda}{r}_{\mu}p(b|x_{0}x_{1},y,\lambda,\mu)$
(1) $\displaystyle=$
$\displaystyle{\rm{Tr}}({\rho}_{x_{0}x_{1}}\frac{\openone+(-1)^{b}{M}_{y}}{2})$
$\displaystyle=$
$\displaystyle\frac{1}{2}(1+(-1)^{b}{\overrightarrow{S}_{x_{0}x_{1}}}\cdot{\overrightarrow{T}}_{y}),$

where

$\displaystyle{\rho}_{x_{0}x_{1}}$ $\displaystyle=$
$\displaystyle\sum_{\lambda}{q}_{\lambda}{\rho}_{x_{0}x_{1}}^{\lambda}=\frac{1}{2}(1+{\overrightarrow{S}}_{x_{0}x_{1}}\cdot\overrightarrow{\sigma}),$
(2) $\displaystyle{M}_{y}$ $\displaystyle=$
$\displaystyle\sum_{\mu}{r}_{\mu}{M}_{y}^{\mu}={\overrightarrow{T}}_{y}\cdot\overrightarrow{\sigma}\mathrm{,}$
(3)

are the observed states and measurements denoted in terms of the Bloch vectors
$\overrightarrow{S}_{x_{0}x_{1}}$ and $\overrightarrow{T}_{y}$, where
$\overrightarrow{\sigma}$ is the vector of Pauli matrices.

To certify the quantumness of preparations and measurements in the above
scenario, Lunghi et. al. Lunghi  _et al._ (2015) considered the following
nonlinear dimension witness introduced in Ref. Bowles  _et al._ (2014):

$\displaystyle
W=\left|\begin{array}[]{cc}p(\mathrm{0|00,0})-p(\mathrm{0|01,0})&p(\mathrm{0|10,0})-p(\mathrm{0|11,0})\\\
p(\mathrm{0|00,1})-p(\mathrm{0|01,1})&p(\mathrm{0|10,1})-p(\mathrm{0|11,1})\end{array}\right|.$
(6)

The witness takes the value $0$ for any strategy involving a classical bit
while it takes the value $0\leq W\leq 1$ for any strategy involving a qubit.
In case either the qubit preparations or the measurements are classical, i.e.,
when either the qubit preparations or the measurements are commuting, the
above witness $W$ takes the value $0$. Thus, $0<W\leq 1$ serves as
certification of quantumness of the correlations $\\{p(b|x_{0}x_{1},y)\\}$ in
a SDI way since no assumption is required on the devices except that Alice and
Bob prepare and measure on systems of the given dimension. Moreover, Lunghi
et. al. demonstrated that a nonzero $W$ quantifies the incompatibility of
Bob’s measurements. The witness takes the maximal value of $W=1$ for the
following preparations:

$\displaystyle\rho_{00}$
$\displaystyle=\frac{1}{2}\left(\openone+\sigma_{z}\right)$
$\displaystyle\rho_{01}$
$\displaystyle=\frac{1}{2}\left(\openone-\sigma_{z}\right)$
$\displaystyle\rho_{10}$
$\displaystyle=\frac{1}{2}\left(\openone+\sigma_{x}\right)$
$\displaystyle\rho_{11}$
$\displaystyle=\frac{1}{2}\left(\openone-\sigma_{x}\right)$ (7)

and measurements:

$\displaystyle M_{0}$ $\displaystyle=\sigma_{z},\quad M_{1}=\sigma_{x},$ (8)

which correspond to the BB84 protocol Bennett and Brassard (1984).

###  II.1 Relating the quantumness of sequential correlations with that of
spatial correlations

Figure 3: A semi-device-independent scenario where a shared bipartite state
acts as the quantum channel to replicate the P$\&$M scenario in Fig. 2.

To observe the quantumness of sequential correlations in the above scenario as
spatial correlations, let us consider a SDI scenario as shown in Fig. 3 where
Alice and Bob share a bipartite quantum state $\rho_{AB}$ of dimension
$d\times 2$, where $d$ is arbitrary. The spatial correlations that can be
observed in this scenario can be seen to produce the sequential correlation in
the P$\&$M scenario as follows. In the SDI scenario as shown in Fig. 3, Alice
has access to the set of two black-box measurements labelled by
$x_{0}\in\\{0,1\\}$. The outcome of each measurement is denoted by
$x_{1}\in\\{0,1\\}$. These two measurements of Alice prepare a set of four
qubit states on Bob’s side $\\{\rho_{x_{1}|x_{0}}\\}$. The set of unnormalized
conditional states prepared on Bob’s side $\\{\sigma_{x_{1}|x_{0}}\\}$ is
given by

$\sigma_{x_{1}|x_{0}}=\operatorname{Tr}_{A}\left(M_{x_{1}|x_{0}}\otimes\openone\rho_{AB}\right)=p(x_{1}|x_{0})\rho_{x_{1}|x_{0}},$
(9)

where $M_{x_{1}|x_{0}}$ denotes the measurements operator of Alice’s
measurement and $p(x_{1}|x_{0})$ is the probability of obtaining the outcome
$x_{1}$ on Alice’s side. On the conditional states prepared by Alice’s
measurement, Bob performs one of two uncharacterized measurements labelled by
$y\in\\{0,1\\}$ which is chosen randomly.

Such a SDI scenario has been considered in Ref. Moroder  _et al._ (2016) for
detecting dimension-bounded quantum steering where only the Hilbert-space
dimension of the trusted party is characterized and no other assumption, such
as which measurements the trusted party performs, is made. This SDI scenario
demonstrates quantum steering if $\\{\sigma_{x_{1}|x_{0}}\\}$ does not have a
local-hidden-state (LHS) model Wiseman  _et al._ (2007); Pusey (2013). The set
of normalized qubit states prepared on Bob’s side in the Bloch representation
is given by

$\rho_{x_{1}|x_{0}}=\frac{1}{2}(1+{\overrightarrow{S}}_{x_{0}x_{1}}\cdot\overrightarrow{\sigma}).$
(10)

Comparing Eqs. (2) and (10), it follows that the sequential correlations
produced in the P$\&$M scenario can be reproduced as spatial correlations in
the SDI scenario as in Fig. 3 with the following differences. In the case of
scenario depicted in Fig. 2, Alice uses two random bits as the input to the
device, on the other hand, in the case of scenario depicted in Fig. 3, Alice
uses only one random bit as the input to the device. In the scenario depicted
in Fig. 3, Bob receives the random preparations which are the conditional
states prepared by Alice’s measurements by a priory sharing a correlated
quantum system with Alice, while in the scenario depicted in Fig. 2, Bob
receives the random preparations through quantum communication. In the case of
scenario depicted in Fig. 3, the non-signaling conditions from Alice to Bob
Pusey (2013) takes into account of all imperfections on the prepared states
$\rho_{x_{1}|x_{0}}$ in a device-independent way, but the non-signaling
condition is not satisfied in the SDI scenario as in Fig. 2 as Alice can
signal to Bob by sending the qubit. In the scenario depicted in Fig. 2, we
assume qubit dimension on Alice’s side and associate an internal state
$\lambda$ to account for noise/imperfection in the prepared states of the
given dimension. On the other hand, in the SDI scenario depicted in Fig. 3,
imperfection in Alice’s device to prepare the relevant conditional states is
taken into account in a device-independent way because of the nonsignaling
conditions from Alice to Bob:

$\sum_{x_{1}}\sigma_{x_{1}|0}=\sum_{x_{1}}\sigma_{x_{1}|1},\quad\forall x_{1}$
(11)

Here device-independence is applicable to Alice’s device only while Bob’s
device works at the semi-device-independent level. Therefore, we do not need
to associate an internal state $\lambda$ to Alice’s device to account for the
imperfection in the conditional states prepared on Bob’s side, but we
associate an internal state $\mu$ on Bob’s device to account for the
imperfection in Bob’s device.

In the scenario depicted in Fig. 3, as we model the imperfection in Bob’s
measuring device by an internal state $\mu$ which occurs with the probability
$r_{\mu}$, $\sum_{\mu}r_{\mu}=1$, the spatial correlations as captured by the
set of conditional probabilities $\\{p(b|x_{1};x_{0},y)\\}$ are given by

$\displaystyle p(b|x_{1};x_{0},y)$ $\displaystyle=$
$\displaystyle\sum_{\mu}{r}_{\mu}p(b|x_{1};x_{0},\mu)$ (12) $\displaystyle=$
$\displaystyle{\rm{Tr}}({\rho}_{x_{1}|x_{0}}\frac{1+b{M}_{y}}{2})$
$\displaystyle=$
$\displaystyle\frac{1}{2}(1+(-1)^{b}{\overrightarrow{S}}_{x_{0}x_{1}}\cdot{\overrightarrow{T}}_{y}),$

where ${M}_{y}$ are given as in Eq. (3).

To certify the quantumness of the preparations $\\{\rho_{x_{1}|x_{0}}\\}$ and
measurements $\\{M_{y}\\}$ in the SDI scenario depicted in Fig. 3, we extend
the dimension witness given by Eq. (34) for this scenario as follows:

$\displaystyle
Q=\left|\begin{array}[]{cc}p(\mathrm{0|0;0,0})-p(\mathrm{0|1;0,0})&p(\mathrm{0|0;1,0})-p(\mathrm{0|1;1,0})\\\
p(\mathrm{0|0;0,1})-p(\mathrm{0|1;0,1})&p(\mathrm{0|0;1,1})-p(\mathrm{0|1;1,1})\end{array}\right|.$
(15)

Let us now demonstrate that $Q=0$ for any set of joint behaviours
$\\{P(x_{1},b|x_{0},y)=P(b|x_{1};x_{0},y)P(x_{1}|x_{0})\\}$ which has the form
$P(x_{1},b|x_{0},y)=P(x_{1}|x_{0})P(b|y)$ (i.e., there is no correlation
between the outcomes $x_{1}$ and $b$). For any set of joint behaviours which
does not have correlation, the conditional probability of observing Bob’s
outcome takes the form $P(b|x_{1};x_{0},y)=P(b|y)$. With this form of
$P(b|x_{1};x_{0},y)$, evaluating $Q$ as given by Eq. (15) gives
$Q=\braket{M_{0}}\braket{M_{1}}-\braket{M_{1}}\braket{M_{0}}$, where
$\braket{M_{y}}=P(0|y)-P(1|y)$. This implies that $Q=0$ for the set of joint
behaviours which do not have correlation. On the other hand, $Q>0$, in
general, even for the unsteerable correlations. However, a nonzero $Q$ for the
set of conditional probability distributions observed on Bob’s side
$\\{p(b|x_{1};x_{0},y)\\}$ implies that both the preparations and measurements
have quantumness as we demonstrate now. Comparing Eqs. (1) and (12), it
follows that any set of preparations and measurements giving rise to nonzero
$Q$ in the scenario depicted in Fig. 3 also gives rise to nonzero $W$ in the
scenario depicted in Fig. 2 and vice versa. Now, $W>0$ implies randomness
certification in the SDI scenario depicted in Fig. 2. Therefore, any set of
conditional probability distributions observed on Bob’s side
$\\{p(b|a;x,y)\\}$ in the scenario depicted in Fig. 3 has intrinsic randomness
if it gives rise to $Q>0$.

To quantify the amount of randomness generated in our SDI scenario 3, let us
define the average guessing probability of the events $\\{x_{0},x_{1},y,b\\}$
as
${p}^{g}:=\frac{1}{4}\sum_{x_{0},x_{1},y,\mu}{r}_{\mu}\mathop{{\rm{\max}}}\limits_{b}p(b|x_{1};x_{0},y,\mu)$.
Then the randomness rate in our protocol can be quantified by the min-entropy
of ${p}^{g}$. By adopting the procedure given in Ref. Fei  _et al._ (2017)
which has been used to provide a tighter upper bound on the guessing
probability, one can derive the following upper bound on the guessing
probability in our SDI scenario too:

$\begin{array}[]{rcl}{p}^{g}&=&\frac{1}{4}\sum_{x_{0},x_{1},y,\mu}{r}_{\mu}\mathop{{\rm{\max}}}\limits_{b}p(b|x_{1};x_{0},y,\mu)\\\
&&\leq\,\frac{1}{2}(1+\sqrt{\frac{2-Q}{2}})\equiv f(Q)\end{array}$ (16)

Thus, the min-entropy has a lower bound as
$H_{\min}=-\log_{2}p_{g}=-\log_{2}f(Q)$ which implies that $H_{\min}$ is a
monotonic function of the quantity $Q$ which quantifies the quantumness of the
spatial correlations in our SDI scenario depicted in Fig. 3. In the following,
we identify what kind of quantum correlation can act as a resource for
randomness certification in our SDI scenario.

###  II.2 Quantum correlation beyond steering as a resource for certifying
randomness

In the context of a steering scenario where Alice performs a set of black-box
measurements labelled by $x$ and Bob performs a set of quantum measurements of
fixed dimension denoted by $M_{y}$, the set of joint behaviours
$\\{p(a,b|x,y)\\}$ detecting steerability from Alice to Bob does not have a
decomposition of the form given by Wiseman  _et al._ (2007),

$p(a,b|x,y)=\sum^{d_{\lambda}-1}_{\lambda=0}p_{\lambda}p(a|x,\lambda)p(b|y;\rho_{\lambda}).$
(17)

Here, $\\{p(a|x,\lambda)\\}_{a,x}$ is the set of arbitrary probability
distributions $p(a|x,\lambda)$ conditioned upon shared randomness/hidden
variable $\lambda$ occurring with the probability $p_{\lambda}$;
$\sum^{d_{\lambda}-1}_{\lambda=0}p_{\lambda}=1$. On the other hand,
$\\{p(b|y;\rho_{\lambda})\\}_{b,y}$ is the set of quantum probability
distributions
$p(b|y;\rho_{\lambda})=\operatorname{Tr}(\Pi_{b|y}\rho_{\lambda})$, arising
from some local hidden state $\rho_{\lambda}$ due to projective measurement
associated with the projector $\Pi_{b|y}$, and $d_{\lambda}$ denotes the
dimension of the shared randomness/hidden variable $\lambda$. The above
decomposition is called a local hidden variable-local hidden state (LHV-LHS)
model. If the correlation arising from the given steering scenario does not
have steerability, it can still have quantumness in the form of
superunsteerability Das  _et al._ (2018a, b) when there is a restriction on
the amount of shared randomness. Superunsteerability Das  _et al._ (2018a, b)
is defined as the requirement for a larger dimension of the classical variable
that the steering party (Alice) preshares with the trusted party (Bob) for
simulating the given unsteerable correlations, than that of the quantum state
which reproduces them. Suppose we have a quantum state in
$\mathbb{C}^{d_{A}}\otimes\mathbb{C}^{d_{B}}$ and measurements which produce a
unsteerable bipartite box $\\{p(a,b|x,y)\\}_{a,x,b,y}$. Then,
superunsteerability holds if and only if (iff) there is no decomposition of
the box in the form given by Eq. (17), with $d_{\lambda}\leq d_{A}$.
Superunsteerability provides an operational characterization to the
quantumness of unsteerable boxes Das  _et al._ (2018a). Note that
superunsteerability is defined in the standard steering scenario. The only
difference is that there is a constraint on the dimension of the resources
producing the correlations in the context of superunsteerability.

Here we should emphasize that the above definition of superunsteerability is
also applicable in the ‘dimension-bounded quantum steering scenario, where
Bob’s measurements are uncharacterized POVMs acting on a fixed dimensional
Hilbert space. Hence, in this case, the probability distributions
$p(b|y;\rho_{\lambda})$ appeared in Eq.(17) is given by,
$p(b|y;\rho_{\lambda})=\operatorname{Tr}(M_{b|y}\rho_{\lambda})$, where
$M_{b|y}$s are POVM elements with $M_{b|y}\geq 0$ $\forall$ $b,y$ and
$\sum_{b}M_{b|y}=\mathbb{I}$ $\forall$ $y$.

We now demonstrate the following proposition.

######  Proposition 1.

Any nonzero value of the quantity $Q$ given by Eq. (15) arising from any given
two-qubit state certifies superunsteerability or steering.

###### Proof.

In the context of the SDI scenario depicted in Fig. 3, any two-qubit state
which is either a classical-quantum state,

$\rho_{CQ}=\sum_{i=0}^{1}p_{i}|i\rangle\langle i|\otimes\chi_{i}$ (18)

or a quantum-classical (QC) state,

$\rho_{QC}=\sum_{j=0}^{1}p_{j}\phi_{j}\otimes|j\rangle\langle j|,$ (19)

with $\\{|i\rangle\\}$ and $\\{|j\rangle\\}$ being orthonormal sets, and,
$\chi_{i}$ and $\phi_{j}$ being arbitrary quantum states, cannot be used to
demonstrate superunsteerability or steering Das  _et al._ (2018a). Next, it
will be demonstrated that any such two-qubit state cannot be used to give rise
to nonzero value of the quantity $Q$ given by Eq. (15).

The classical-quantum states given by Eq. (18) can be decomposed as follows

$\displaystyle\rho_{CQ}$ $\displaystyle=$
$\displaystyle\frac{1}{4}\big{[}\openone\otimes\openone+(p_{0}-p_{1})\hat{r}\cdot\vec{\sigma}\otimes\openone+\openone\otimes\left(p_{0}\vec{s}_{0}+p_{1}\vec{s}_{1}\right)\cdot\vec{\sigma}$
(20) $\displaystyle+$
$\displaystyle\hat{r}\cdot\sigma\otimes\left(p_{0}\vec{s}_{0}-p_{1}\vec{s}_{1}\right)\cdot\vec{\sigma}\big{]},$

where $\hat{r}$ is the Bloch vector of the projectors $|i\rangle\langle i|$
and $\vec{s}_{i}$ are the Bloch vectors of the quantum states $\chi_{i}$.
Since we only make qubit assumption on Alice’s side, her measurements are a
priory POVM with elements given by

$M_{x_{1}|x_{0}}=\gamma^{x_{1}}_{x_{0}}\openone+(-1)^{x_{1}}\frac{\eta_{x_{0}}}{2}\hat{u}_{x_{0}}\cdot\vec{\sigma},$
(21)

where $x_{0}\in\\{0,1\\}$, $x_{1}\in\\{0,1\\}$,
$\gamma^{0}_{x_{0}}+\gamma^{1}_{x_{0}}=1$ $\forall$ $x_{0}$ and
$0\leq\gamma^{x_{1}}_{x_{0}}\pm\frac{\eta_{x_{0}}}{2}\leq 1$ $\forall$
$x_{0},x_{1}$. On the other hand, Bob’s measurements are also POVMs with
elements given by

$M_{b|y}=\gamma^{b}_{y}\openone+(-1)^{b}\frac{\eta_{y}}{2}\hat{u}_{y}\cdot\vec{\sigma},$
(22)

where $y\in\\{0,1\\}$, $b\in\\{0,1\\}$, $\gamma^{0}_{y}+\gamma^{1}_{y}=1$
$\forall$ $y$ and $0\leq\gamma^{b}_{y}\pm\frac{\eta_{y}}{2}\leq 1$ $\forall$
$y,b$. Now, It can be easily checked that the above measurement settings
always lead to $Q=0$ for the classical-quantum states $\rho_{CQ}$.

The quantum-classical states given by Eq. (18) can be decomposed as follows

$\displaystyle\rho_{QC}$ $\displaystyle=$
$\displaystyle\frac{1}{4}\big{[}\openone\otimes\openone+\left(p_{0}\vec{s}_{0}+p_{1}\vec{s}_{1}\right)\cdot\vec{\sigma}\otimes\openone+\openone\otimes(p_{0}-p_{1})\hat{r}\cdot\vec{\sigma}$
(23) $\displaystyle+$
$\displaystyle\left(p_{0}\vec{s}_{0}-p_{1}\vec{s}_{1}\right)\cdot\vec{\sigma}\otimes\hat{r}\cdot\sigma\big{]},$

where $\vec{s}_{i}$ are the Bloch vectors of the quantum states $\phi_{j}$ and
$\hat{r}$ is the Bloch vector of the projectors $|j\rangle\langle j|$. For the
measurement operators given by Eqs. (21) and (22), the above quantum-classical
states always give $Q=0$.

Note that in our SDI scenario, any unsteerable box having the decomposition
(17) can be reproduced by a classical-quantum state of the form
$\sum^{d_{\lambda}-1}_{\lambda=0}p_{\lambda}|\lambda\rangle\langle\lambda|\otimes\rho_{\lambda}$
Das  _et al._ (2018a), where $\\{|\lambda\rangle\langle\lambda|\\}$ forms an
orthonormal basis in $\mathbb{C}^{d_{\lambda}}$, with $d_{\lambda}\leq 4$
111Here the dimension of the hidden variable is upper bounded by $4$ since any
local as well as unsteerable correlation corresponding to this scenario can be
simulated by shared classical randomness of dimension $d_{\lambda}\leq 4$
Donohue and Wolfe (2015); Das  _et al._ (2018a).. This implies that any
unsteerable box produced from a two-qubit state that requires a hidden
variable of dimension $d_{\lambda}=2$ for providing a LHV-LHS model (i.e., any
unsteerable box produced from a two-qubit state that is not superunsteerable)
can be simulated by a two-qubit state which admits the form of the classical-
quantum state given by Eq.(18). Thus, for any such unsteerable box, $Q=0$. On
the other hand, $Q>0$ produced from a two-qubit state certifies that the box
does not arise from a classical-quantum or quantum-classical state. Hence,
that box either requires the hidden variable of dimension $d_{\lambda}>2$ for
providing a LHV-LHS model (i.e., the box is superunsteerable) or that box is
steerable.

∎

Now, in order to illustrate proposition 1, let us give an example where non-
zero $Q$ arising from two-qubit state certifies superunsteerability or
steering. Consider the following white-noise BB84 family,

$P_{BB84}(x_{1}b|x_{0}y)=\frac{1+(-1)^{x_{1}\oplus b\oplus
x_{0}.y}\delta_{x_{0},y}V}{4},$ (24)

where $V$ is a real number such that $0<V\leq 1$; $x_{0}$, $y$ denote the
input variables on Alice’s and Bob’s sides respectively; and $x_{1}$, $b$
denote the outputs on Alice’s and Bob’s sides respectively,
$x,y,a,b\in\\{0,1\\}$. This family of correlations can be produced from the
two-qubit Werner state,

$\rho_{V}=V|\psi^{-}\rangle\langle\psi^{-}|+\frac{1-V}{4}\mathbb{I}_{4},$ (25)

with $|\psi^{-}\rangle=\frac{1}{\sqrt{2}}(|01\rangle-|10\rangle)$ and $0<V\leq
1$ if Alice performs the projective measurements of observables corresponding
to the operators $A_{0}=-\sigma_{z}$ and $A_{1}=\sigma_{x}$, and Bob performs
projective measurements of observables corresponding to the operators
$B_{0}=\sigma_{z}$ and $B_{1}=\sigma_{x}$ Das  _et al._ (2018a).

The above family of correlations is superunsteerable for
$0<V\leq\frac{1}{\sqrt{2}}$ and steerable for $\frac{1}{\sqrt{2}}<V\leq 1$ Das
_et al._ (2018a). It can be easily checked that for the white-noise BB84
family given by Eq.(24), $Q=|V^{2}|$. Hence, for this family of correlations,
$Q>0$ for $V>0$. Non-zero value of $Q$ arising from the white-noise BB84
family, therefore, certifies superunsteerability or steering.

The above proposition 1 demonstrates that superunsteerability or steering is
linked with non-zero values of $Q$ in the context of two-qubit states. Next,
we will address which two-qubit states can demonstrate superunsteerability or
steering. It is known that entanglement is necessary for demonstrating
steering Uola  _et al._ . Next, we will address which two-qubit unsteerable
states can demonstrate superunsteerability.

In Ref. Luo (2008) it was shown that any two-qubit state, up to local unitary
transformations, can be reduced to the following form:

$\zeta=\frac{1}{4}\left(\openone\otimes\openone+\vec{a}\cdot\vec{\sigma}\otimes\openone+\openone\otimes\vec{b}\cdot\vec{\sigma}+\sum^{3}_{i=1}c_{i}\sigma_{i}\otimes\sigma_{i}\right),$
(26)

where $\openone$ is the $2\times 2$ identity matrix and
$\\{\vec{a},\vec{b},\vec{c}\\}\in\mathbf{R}^{3}$ are vectors with norm less
than or equal to unity and $\vec{a}^{2}+\vec{b}^{2}+\vec{c}^{2}\leq 3$. Let
Alice’s projective qubit measurements are given by the measurement operators

$M_{x_{1}|x_{0}}=\frac{1}{2}\left(\openone+(-1)^{x_{1}}\hat{u}_{x_{0}}\cdot\vec{\sigma}\right).$
(27)

For such measurements, the conditional states prepared on Bob’s side are given
by

$\rho_{x_{1}|x_{0}}=\frac{1}{2}\left(\openone+\frac{\sum^{3}_{i=1}\left(b_{i}+(-1)^{x_{1}}{u_{i}}_{x_{0}}c_{i}\right)\sigma_{i}}{1+(-1)^{x_{1}}\hat{u}_{x_{0}}\cdot\vec{a}}\right).$
(28)

For simplicity, without lose of generality, let us consider the two-qubit
states given by Eq. (26) with $|c_{1}|\geq|c_{2}|\geq|c_{3}|$ and let Alice
performs measurements along the directions $\hat{u}_{0}=\hat{x}$ and
$\hat{u}_{1}=\hat{y}$ and Bob performs the measurement along the directions
$\vec{T}_{0}=\hat{x}$ and $\vec{T}_{1}=\hat{y}$. Note that for this choice of
measurement settings, the correlations arising from certain two-qubit states
violate the two-setting linear steering inequality and the correlations
arising from maximally entangled two-qubit states violate the two-setting
linear steering inequality maximally Costa and Angelo (2016). With this choice
of the two-qubit states and measurements, $Q$ as given by Eq. (15) has been
evaluated to be of the form given by

$Q=\frac{|(c_{1}-a_{1}b_{1})(c_{2}-a_{2}b_{2})-a_{1}b_{1}a_{2}b_{2}|}{(1-a^{2}_{1})(1-a^{2}_{2})}.$
(29)

The right hand side of the above quantity is nonzero if and only if the state
has a nonzero discord from Alice to Bob as well as from Bob to Alice Dakić
_et al._ (2010) for the two-qubit states with either Alice’s or Bob’s marginal
being maximally mixed, i.e., $\vec{a}=0$ or $\vec{b}=0$.

Therefore, we arrive at the following:

######  Proposition 2.

There exist suitable measurements, which can be used to demonstrate the
maximal violation of the linear steering inequality by the maximally entangled
state, for which the quantity $Q$ is nonzero for any two-qubit state which is
neither a classical-quantum state nor a quantum-classical state and has one of
the marginals maximally mixed.

Let us now analyze which nonzero discord states having nonmaximally mixed
marginals on both the sides can lead to nonzero $Q$. Consider the two-qubit
nonzero discord state with $a_{1}=a_{2}=b_{1}=b_{2}=c_{1}=c_{2}=1/2$ and
$a_{3}=b_{3}=0=c_{3}=0$. For this state, $Q$ as given by Eq. (29) takes the
value $0$. In Ref. Das  _et al._ (2018a), it has been demonstrated that such
separable state can be used to demonstrate superunsteerability.

Let now demonstrate that $Q=0$ for the above such superunsteerable state even
if Alice and Bob perform measurements which optimize the violation of the
linear steering inequality Costa and Angelo (2016). For this choice of
measurements which have the directions given by
$\hat{u}_{0}=\frac{c_{1}\hat{x}+c_{2}\hat{y}}{\sqrt{c^{2}_{1}+c^{2}_{2}}}$ and
$\hat{u}_{1}=\frac{c_{1}\hat{x}-c_{2}\hat{y}}{\sqrt{c^{2}_{1}+c^{2}_{2}}}$ on
Alice’s side and $\vec{T}_{0}=\frac{\hat{x}+\hat{y}}{\sqrt{2}}$ and
$\vec{T}_{1}=\frac{\hat{x}-\hat{y}}{\sqrt{2}}$ on Bob’s side, $Q$ for the two-
qubit states (26) with $|c_{1}|\geq|c_{2}|\geq|c_{3}|$ has been evaluated to
be of the form given by,

$\displaystyle
Q=\left|\frac{4c_{1}c_{2}(a_{2}b_{2}c_{1}+a_{1}b_{1}c_{2}-c_{1}c_{2})}{a_{1}^{4}c_{1}^{4}+(-2+a_{2}^{2}c_{2}^{2})^{2}-2a_{1}^{2}c_{1}^{2}(2+a_{2}^{2}c_{2}^{2})}\right|.$
(30)

Note that $Q$ as given above takes the value zero for the above-mentioned
superunsteerable state with $a_{1}=a_{2}=b_{1}=b_{2}=c_{1}=c_{2}=1/2$ and
$a_{3}=b_{3}=0=c_{3}=0$.

In fact, it can be checked that when Alice and Bob perform arbitrary POVMs
given by Eqs. (21) and (22), respectively, on the above-mentioned
superunsteerable state with $a_{1}=a_{2}=b_{1}=b_{2}=c_{1}=c_{2}=1/2$ and
$a_{3}=b_{3}=0=c_{3}=0$, one will get $Q=0$. Thus, we arrive at the following
proposition.

######  Proposition 3.

There exist superunsteerable states which cannot be used to demonstrate
nonzero $Q$

As discussed earlier, the genuine randomness produced in our SDI scenario is
quantified by $Q$. Hence, quantum correlation as captured by
superunsteerability or steering acts as a resource for genuine randomness
generation in the SDI depicted in Fig. 3.

##  III Prepare and measure scenario with correlated devices

Figure 4: Prepare and measure scenario with correlated devices. Here,
$x_{0},x_{1},y,b\in\\{0,1\\}$ and
$\rho^{\lambda}_{x}\in\mathcal{B}(\mathbb{C}^{2})$.

Let us consider a SDI scenario as in Fig. 4 which is the P$\&$M scenario as in
Fig. 2 but with the correlated devices, i.e., $p(\lambda,\mu)\neq
q_{\lambda}\cdot r_{\mu}$. In such a SDI scenario where the devices are not
independent, it has been demonstrated that the set of distributions
$\\{p(b|x_{0}x_{1},y)\\}$ that achieve the maximal value $1$ for the witness
$W$ as given by Eq. (6) can also achieved by a classical bit Pawłowski and
Brunner (2011). Thus, in the presence of shared randomness, the P$\&$M
scenario cannot be used to certify the randomness by observing nonzero $W$.

In the context of scenario as in Fig. 4, quantum advantage of $2$-to-$1$
random-access code was studied Ambainis  _et al._ and a linear dimension
witness was derived Gallego  _et al._ (2010). In a $2$-to-$1$ quantum random-
access code which is implemented using the protocol given in Fig. 4, Bob’s
goal is to guess Alice’s $y$th bit. The average success probability of Bob to
guess Alice’s $y$th bit is given by

$P_{B}:=\frac{1}{8}\sum_{x_{0},x_{1},y}P(b=x_{y}|x_{0}x_{1},y).$ (31)

If Alice sends a classical bit to Bob, the optimal average success probability
is upper bounded by $3/4$. On the other hand, if Alice sends a qubit to Bob,
then the average success probability of Bob can beat this classical bound. The
quantum strategy that gives the optimal average success probability of
$\frac{1}{2}\left(1+\frac{1}{\sqrt{2}}\right)$ is given by

$\displaystyle\rho_{00}$
$\displaystyle=\frac{1}{2}\left(\openone+\frac{\sigma_{x}+\sigma_{y}}{\sqrt{2}}\right)$
$\displaystyle\rho_{01}$
$\displaystyle=\frac{1}{2}\left(\openone+\frac{\sigma_{x}-\sigma_{y}}{\sqrt{2}}\right)$
$\displaystyle\rho_{10}$
$\displaystyle=\frac{1}{2}\left(\openone-\frac{\sigma_{x}-\sigma_{y}}{\sqrt{2}}\right)$
$\displaystyle\rho_{11}$
$\displaystyle=\frac{1}{2}\left(\openone-\frac{\sigma_{x}+\sigma_{y}}{\sqrt{2}}\right)$
(32)

and

$\displaystyle M_{0}$ $\displaystyle=\sigma_{x},\quad M_{1}=\sigma_{y}.$ (33)

In Ref. Pawłowski and Brunner (2011), a relationship between the average
success probability of $2$-to-$1$ random-access code and the linear dimension
witness has been demonstrated. The linear dimension witness is given by

$\displaystyle W_{L}$ $\displaystyle:=P(0|00,0)+P(0|00,1)+P(0|01,0)-P(0|01,1)$
$\displaystyle-P(0|10,0)+P(0|10,1)-P(0|11,0)-P(0|11,1)\leq 2.$ (34)

The violation of the above inequality certifies that the system is a qubit in
a device-independent way. On the other hand, by assuming qubit dimension, the
violation of the linear dimension witness inequality (34) certifies the
quantumness of preparations and measurements. The quantum strategy as given by
Eqs. (32) and (33) gives the maximal quantum violation of $2\sqrt{2}$ for the
inequality given by Eq. (34). This is related to the maximal quantum violation
of the CHSH inequality Clauser  _et al._ (1969) as discussed in the next
section. The average success probability of $2$-to-$1$ random-access code is
related to the linear dimension witness as $P_{B}=\frac{W_{L}+4}{8}$. From
this relationship, it follows that the violation of the inequality $P_{B}\leq
3/4$, which certifies the quantum advantage of the random-access code, implies
the violation of the inequality given by (34) and vice versa. In the P$\&$M
scenario as in Fig. 4, the quantum advantage of the random-access codes
implied by the violation of the inequality $P_{B}\leq 3/4$ or the quantumness
as certified by the violation of the linear dimension witness inequality has
been used to demonstrate secure quantum key distribution Pawłowski and Brunner
(2011) and genuine randomness generation Li  _et al._ (2012).

###  III.1 Relating the quantumness of sequential correlations with that of
spatial correlations

Figure 5: A device-independent scenario where a shared bipartite state acts as
the quantum channel to implement the $2$-to-$1$ random-access code. Here,
$\\{\rho_{0|0}=\rho_{00},\rho_{1|0}=\rho_{11},\rho_{0|1}=\rho_{01},\rho_{1|1}=\rho_{10}\\}$.

To observe the quantumness of sequential correlations in the above scenario
depicted in Fig. 4 as the quantumness of spatial correlations, let us now
consider the scenario as in Fig. 5, which is a fully device-independent
scenario where Alice and Bob share a bipartite quantum state and performs two
dichotomic black box measurements. Here we consider the fully device-
independent scenario since shared randomness is used in the corresponding
prepare and measure scenario. In this fully device-independent scenario, let
us observe the spatial correlations through the set of conditional
probabilities $\\{p(b|a;x,y)\\}$ given by

$p(b|a;x,y)=\frac{\operatorname{Tr}\left(M^{(1)}_{a|x}\otimes
M^{(2)}_{b|y}\rho_{AB}\right)}{\operatorname{Tr}\left(M^{(1)}_{a|x}\otimes\openone\rho_{AB}\right)}$
(35)

where $\rho_{AB}$ is a bipartite quantum state of arbitrary local Hilbert
space dimension and $M^{(1)}_{a|x}$ and $M^{(2)}_{b|y}$ are the measurement
operators of Alice and Bob’s measurements respectively. The scenario in Fig. 5
supplemented with one bit of classical communication from Alice to Bob can be
used to implement the $2$-to-$1$ random-access code Pawłowski and Żukowski
(2010). Therefore, the violation of the inequality $P_{B}\leq 3/4$ which can
be used to certify the quantum advantage of the $2$-to-$1$ random-access code
assisted by shared bipartite state implies that the set of joint behaviours
$\\{P(a,b|x,y)=p(b|a;x,y)p(a|x)\\}$ violates a Bell inequality. Thus, we can
conclude that the quantumness in the P$\&$M scenario with the correlated
devices is manifested in the form of Bell-nonlocal correlations in the
corresponding scenario where a shared bipartite state is used as the quantum
channel. Further, observing the violation of the inequality (34) in the
context of producing the distributions $\\{P(b|x_{0}x_{1},y)\\}$ using a
shared bipartite state witnesses Bell nonlocality. Thus, the linear dimension
witnesess provide Bell inequalities in terms of the conditional probabilities
222In Refs. Pramanik  _et al._ (2014); Maccone  _et al._ (2015), steering
inequalities and separability inequalities are derived respectively using the
conditional probabilities. To our knowledge, Bell inequalities in terms of the
conditional probabilities have not been studied so far..

###  III.2 Quantum correlations beyond Bell-nonlocality as a manifestation of
quantumness of sequential correlations

In a bipartite Bell scenario, superlocality Donohue and Wolfe (2015) is
defined as follows. Suppose we have a quantum state in
$\mathbb{C}^{d_{A}}\otimes\mathbb{C}^{d_{B}}$ and measurements which produce a
local bipartite box $\\{p(a,b|x,y)\\}$. Then, superlocality holds iff there is
no decomposition of the box in the form,

$p(a,b|x,y)=\sum^{d_{\lambda}-1}_{\lambda=0}p_{\lambda}p(a|x,\lambda)p(b|y,\lambda)\hskip
8.5359pt\forall a,x,b,y,$ (36)

with dimension of the shared randomness/hidden variable $d_{\lambda}\leq$
min($d_{A}$, $d_{B}$). Here $\sum_{\lambda}p_{\lambda}=1$, $p(a|x,\lambda)$
and $p(b|y,\lambda)$ denotes arbitrary probability distributions arising from
LHV $\lambda$ ($\lambda$ occurs with probability $p_{\lambda}$). Superlocality
provides an operational characterization to the quantumness of local boxes
Jebaratnam  _et al._ (2017, 2018).

In Ref. Bobby and Paterek (2014), Bobby and Paterek studied quantum advantage
of $2$-to-$1$ and $3$-to-$1$ random-access codes assisted by two correlated
qubits in the presence of two-bits of shared randomness. As a figure merit of
the task, Bobby and Paterek considered worst-case success probability of Bob’s
correct guess on Alice’s $y$th bit defined as ${{P}_{{\rm min}}}={{{\rm
min}}_{x_{0},x_{1},y}}p(b=x_{y}|x_{0}x_{1},y)$. Bobby and Paterek derived the
following inequalities: A classical $n$-to-$1$ random-access code assisted
with two bits from a common source has

${{P}_{{\rm min}}}\leqslant\frac{1}{2}\quad\texttt{if}\quad n>2;$ (37)
${{P}_{{\rm min}}}\leqslant\frac{2}{3}\quad\texttt{if}\quad n=2;$ (38)

and

${{P}_{{\rm min}}}\leq\frac{1}{2}$ (39)

for all $n>1$ if the assisting bits have maximally mixed marginals for Bob. By
using these inequalities, Bobby and Paterek have shown that even certain
separable two-qubit state may become useful for implementing $2$-to-$1$ and
$3$-to-$1$ quantum random-access codes.

Note that for $n=2$ in Eqs. (37), (38) and (39), the inequalities correspond
to the Bell-Clauser-Horne-Shimony-Holt (Bell-CHSH) scenario Clauser  _et al._
(1969) while for $n=3$, the inequalities are related to the bipartite Bell
scenario corresponding to Gisin’s elegant Bell inequality Gisin (2009) 333In
Ref. Ghorai and Pan (2018), in the context of $n$-bit parity-oblivious
multiplexing task, the suitable Bell expressions corresponding to quantum
success probability of this task has been identified. The Bell inequalities
corresponding to these Bell expressions, in turn, are related to quantum
advantage of $n$-to-$1$ random-access codes.. The violation of the
inequalities given by Eqs. (37) and (38) by the local boxes implies that the
simulation of these boxes by using shared classical randomness requires the
hidden variable of dimension $d_{\lambda}>2$. Therefore, these inequalities
serve as sufficient certification of superlocality.

Previously, it was shown that the quantity $Q$ given by Eq. (15) of any two-
qubit state is linked with superunsteerability in the scenario depicted in
Fig. 3. Now, we will show that $Q$ can be linked with superlocality as well in
the present context.

######  Proposition 4.

Any nonzero value of the quantity $Q$ given by Eq. (15) arising from any given
two-qubit state certifies superlocality or Bell-nonlocality.

###### Proof.

Note that in our device-independent scenario, any local box having the
following LHV-LHV decomposition,

$p(a,b|x,y)=\sum^{d_{\lambda}-1}_{\lambda=0}p_{\lambda}p(a|x,\lambda)p(b|y,\lambda)\hskip
8.5359pt\forall a,x,b,y,$ (40)

can be reproduced by performing appropriate measurements on the quantum-
classical state of the form
$\sum^{d_{\lambda}-1}_{\lambda=0}p_{\lambda}\rho_{\lambda}\otimes|\lambda\rangle\langle\lambda|$,
or on the classical-quantum state of the form
$\sum^{d_{\lambda}-1}_{\lambda=0}p_{\lambda}|\lambda\rangle\langle\lambda|\otimes\rho_{\lambda}$
where $\\{|\lambda\rangle\langle\lambda|\\}$ forms an orthonormal basis in
$\mathbb{C}^{d_{\lambda}}$, with $d_{\lambda}\leq 4$ 444Here the dimension of
the hidden variable is upper bounded by $4$ since any local correlation
corresponding to this scenario can be simulated by shared classical randomness
of dimension $d_{\lambda}\leq 4$ Donohue and Wolfe (2015). This implies that
any local box arising from a two-qubit state that requires a hidden variable
of dimension $d_{\lambda}=2$ for providing a LHV-LHV model (i.e., any local
box arising from a two-qubit state that is not superlocal) can be simulated by
a two-qubit state which admits the form of the classical-quantum state or
quantum-classical state. Now, such state cannot give non-zero $Q$ (follows
from the proof of propoasition 1). Thus, for any local box arising from a two-
qubit state that is not superlocal, $Q=0$. On the other hand, $Q>0$ produced
from a two-qubit state certifies that the box does not arise from a classical-
quantum or quantum-classical state. Hence, that box either requires the hidden
variable of dimension $d_{\lambda}>2$ for providing a LHV-LHV model (i.e., the
box is superlocal) or that box is Bell-nonlocal. ∎

Note that for the measurements that has been used to demonstrate
superunsteerability of any two-qubit nonzero discord state in Proposition 2,
the correlations arising from certain two-qubit states violate the two-setting
linear steering inequality and the correlations arising from maximally
entangled two-qubit states violate the two-setting linear steering inequality
maximally Costa and Angelo (2016). We now demonstrate that for the
measurements that give rise to the maximal Bell-CHSH inequality violation or
maximal quantum advantage of $2$-to-$1$ random-access code by the maximally
entangled state, the two-qubit states given by Eq. (26) with
$|c_{1}|\geq|c_{2}|\geq|c_{3}|$ give rise to non-zero $Q$. Let Alice performs
measurements along the directions
$\hat{u}_{0}=\frac{\hat{x}+\hat{y}}{\sqrt{2}}$ and
$\hat{u}_{1}=\frac{\hat{x}-\hat{y}}{\sqrt{2}}$ and Bob performs the
measurement along the directions $\vec{T}_{0}=\hat{x}$ and
$\vec{T}_{1}=\hat{y}$. For this choice of measurement directions which can be
used to provide the maximal quantum advantage of $2$-to-$1$ random-access
codes by the maximally entangled state Pawłowski and Żukowski (2010), the two-
qubit states given by Eq. (26) with $|c_{1}|\geq|c_{2}|\geq|c_{3}|$ give rise
to the following expression of $Q$,

$Q=\frac{2|(c_{1}-a_{1}b_{1})(c_{2}-a_{2}b_{2})-a_{1}b_{1}a_{2}b_{2}|}{a_{1}^{4}+(-2+a_{2}^{2})^{2}-2a_{1}^{2}(2+a_{2}^{2})}.$
(41)

The right hand side of the above quantity is nonzero if and only if the state
has a nonzero discord from Alice to Bob as well as from Bob to Alice Dakić
_et al._ (2010) provided that either Alice’s or Bob’s marginal of that two-
qubit state is being maximally mixed, i.e., $\vec{a}=0$ or $\vec{b}=0$.
Therefore, we arrive at the following.

######  Proposition 5.

Nonzero $Q$ provides necessary and sufficient certification of superlocal two-
qubit states which belong to the two-qubit states with one of the marginals
being maximally mixed and provides sufficient certification of superlocality
in general.

Here it should be noted that for the set of quantum correlations that gives
$Q$ value as given by Eq. (41), $Q$ is upper bounded by $Q\leq 1$. On the
other hand, for the Popescu-Rohrlich boxes which are nonsignaling
correlations, but are stronger than quantum correlations violating a Bell-CHSH
inequality to its algebraic maximum of $4$ Barrett  _et al._ (2005), $Q$ takes
the algebraic maximum of $2$.

For the protocol given in Ref. Bobby and Paterek (2014) which implements the
$2$-to-$1$ quantum random-access code using the Bell-diagonal states which are
the two-qubit states given by Eq. (26) with $\vec{a}=\vec{b}=0$, the separable
state with $c_{1}=c_{2}=1/2$ and $c_{3}=0$ optimizes the quantum advantage
within the separable Bell-diagonal states. In this case, the protocol prepares
the following conditional states on Bob’s side,

$\displaystyle\rho_{0|0}$
$\displaystyle=\rho_{00}=\frac{1}{2}\left(\openone+\frac{\sigma_{x}+\sigma_{y}}{2\sqrt{2}}\right)$
$\displaystyle\rho_{0|1}$
$\displaystyle=\rho_{01}=\frac{1}{2}\left(\openone+\frac{\sigma_{x}-\sigma_{y}}{2\sqrt{2}}\right)$
$\displaystyle\rho_{1|1}$
$\displaystyle=\rho_{10}=\frac{1}{2}\left(\openone-\frac{\sigma_{x}-\sigma_{y}}{2\sqrt{2}}\right)$
$\displaystyle\rho_{1|0}$
$\displaystyle=\rho_{11}=\frac{1}{2}\left(\openone-\frac{\sigma_{x}+\sigma_{y}}{2\sqrt{2}}\right)$
(42)

and Bob performs the following measurements:

$\displaystyle M_{0}$ $\displaystyle=\sigma_{x},\quad M_{1}=\sigma_{y}$ (43)

It has been checked that the above preparations and measurements do not
violate the dimension witness inequality given by Eq. (34). Thus, there exist
quantum strategies in the P$\&$M scenario which have quantumness and cannot be
used to provide quantum advantage for the random-access code. However, the
spatial correlations realized using such quantum strategies in the scenario as
in Fig. 5 violate the inequality given by Eq. (38) with $n=2$ and give rise to
nonzero $Q$. Therefore, these quantum strategies provides advantage for the
random-access codes in the presence of limited shared randomness, i.e., two
bits of shared classical randomness. Thus, quantumness of certain sequential
correlations gets manifested in the form of superlocal correlations which
represent the stronger than classical correlations in the presence of limited
shared randomness.

##  IV Conclusions

In this work, we have studied the equivalence between the quantumness of
sequential correlations in the prepare and measure scenario with independent
devices, which implements quantum random number generation, and the
corresponding scenario which replaces quantum communication by sharing
correlated particles. In this context, we have demonstrated that quantumness
of sequential correlations gets manifested as superunsteerable correlations
which are stronger than classical correlations in the presence of limited
shared randomness. We have introduced an experimentally measurable quantity to
bound the genuine randomness generation in the scenario using shared
correlated particles as resource. We have shown that this quantity provides
certification of superunsteerable correlations and necessary and sufficient
certification of any two-qubit nonzero discord state which is neither a
classical-quantum state nor a quantum-classical state and has one of the
marginals maximally mixed while providing sufficient certfication of nonzero
discord states in general.

In the case of prepare and measure scenario with correlated devices, the
quantumness of sequential correlations providing quantum advantage for random-
access codes gets manifested as Bell nonlocal correlations in the
corresponding scenario assisted by shared correlated particles. We have
demonstrated that when certain sequential correlations which have quantumness
but do not violate the dimension witness inequality are realized as spatial
correlations, they violate an inequality detecting superlocal correlations
which are stronger than the classical correlations in the presence of limited
shared randomness.

It would be interesting to check whether the quantity proposed in this work to
certify superlocaliy and superunsteerability provides necessary and sufficient
certification of superlocality and superunsteerability with unsteerable and
local boxes respectively having minimum hidden variable dimension $4$ (the
examples presented in this work indicate this). It would be interesting to
study whether this quantity is upper bouded by $1$ for quantum correlations.
This will be useful to discriminate quantum and post-quantum correlations
Pawłowski and Scarani (2015) since for the Popescu-Rohrlich boxes, this
quantity takes the algebraic maximum of $2$. It would be interesting to
generalize the present work to quantum correlations with more number of
outputs or inputs as well as with more number of parties. In Refs. Law  _et
al._ (2014); Passaro  _et al._ (2015), it has been demonstrated that genuine
randomness can be certified in the presence of local-hidden-state models. It
would be interesting to study implications of the present work to these
previous works. Finally, in the light of the present work, we plan to
formulate a resource theory of superlocality and superunsteerability just like
the resource theory of Bell nonlocality and quantum steering, respectively
Wolfe  _et al._ ; Chitambar and Gour (2019).

## Acknowledgement

C.J. acknowledges the financial support from the Ministry of Science and
Technology, Taiwan (Grant No. MOST 108-2811-M-006-516).

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