# NUV-DOA: NUV Prior-based Bayesian Sparse Reconstruction with Spatial
Filtering for Super-Resolution DoA Estimation

###### Abstract

Achieving high-resolution doa (doa) recovery typically requires high snr (snr)
and a sufficiently large number of snapshots. This paper presents ndoa
algorithm, that augments Bayesian sparse reconstruction with spatial filtering
for super-resolution doa estimation. By modeling each direction on the
azimuth’s grid with the sparsity-promoting nuv prior, the non-convex
optimization problem is reduced to iteratively reweighted ls under Gaussian
distribution, where the mean of the snapshots is a sufficient statistic. This
approach not only simplifies our solution but also accurately detects the doa.
We utilize a hierarchical approach for interference cancellation in multi-
source scenarios. Empirical evaluations show the superiority of ndoa,
especially in low snr, compared to alternative doa estimators.

Index Terms—  doa estimation, sparse recovery.

##  1 Introduction

doa estimation is the task of determining the azimuth of source signals using
multiple measurements from a sensor array [1]. It plays a crucial role in
contemporary applications across various fields [2]. While the resolution of
conventional approaches based on covariance recovery via beamforming [3] and
mvdr [4], is limited by the array geometry, subspace methods (e.g., music [5])
achieve super-resolution. However, they rely on a relatively large number of
snapshots and high snr, and their accuracy degrades when these are not met.

To address some of the limitations of previous approaches, dl-based methods
have emerged. These methods utilize either e2e nn for direct doa recovery [6,
7, 8, 9] or augment classic algorithms [10, 11], with trainable architectures
via mb dl [12]. While these methods often function effectively under
challenging conditions, they are inherently dd, relying on labeled data from
which the estimation mapping is learned.

The unique structure of the doa estimation problem has facilitated the
adoption of ssr (ssr) [13] and cs (cs) [14]. In this approach, the azimuth
range is quantized into a grid, with the steering vectors serving as the
dictionary. Methods such as $\ell_{1}$-SVD [15] and SPICE [16], as well as
others [17, 18], have demonstrated their merit in challenging scenarios,
particularly those with few snapshots or correlated sources [2]. Within the
scope of sparse modeling, the Bayesian framework [19] can be interpreted as a
form of regularization where sparsity is promoted through the adoption of
specific prior distributions. Notably, methods such as rvm, sbl, and bcs [20,
21, 22] present an effective surrogate for addressing $\ell_{0}$ minimization.
This often results in more accurate recovery performance compared to many
$\ell_{1}$ minimization approaches. Consequently, Bayesian-based doa
estimation techniques have been proposed, including [23, 24, 25, 26, 27, 28].
Although these methods all stem from a shared concept, they exhibit distinct
differences in modeling, choice of sparsifying prior, estimation objective,
choice of estimation algorithm, tuning parameters, the handling of mmv (mmv),
and decision rules. These nuances impact resolution, recovery performance,
stability, and complexity.

Despite their potential, ssr-based methods exhibit inherent limitations
arising from the unique structure of the doa estimation problem and the
associated resolution trade-off. Achieving higher resolution necessitates
refining the grid quantization. However, this refinement leads to a more
complex problem and a dictionary with greater mutual coherence, which can, in
turn, impair the recovery performance. A further limitation arises from the
handling of mmv. While some approaches address mmv by combining multiple
single-snapshot (smv) estimators, others tackle the full mmv problem,
frequently incorporating joint sparsity. However, these methods often increase
the problem’s complexity proportional to the number of snapshots, rendering
them less efficient.

To tackle these challenges, we present ndoa, a super-resolution ssr-based doa
estimation algorithm that utilizes the mean of the snapshots as its sufficient
statistic. Despite its simplicity, it offers high accuracy and robustness,
especially in scenarios with low snr and a limited number of snapshots. In
this approach, each quantized cell in the azimuth grid is modeled as a complex
decision variable, adhering to a sparsity-inducing nuv prior [29, 30, 31]. The
nuv, deeply rooted in Bayesian sparsity, facilitates the transformation of the
ssr problem into an iteratively reweighted ls Gaussian estimation problem,
where the mean of the snapshots serves as a sufficient statistic. To further
improve the doa recovery performance, we harness the inherent spatial
correlation of the doa estimation problem. We employ a spatial filtering-based
super-resolution algorithm that segments a single, global ssr problem spanning
the entire grid into a series of localized problems. Finally, we propose a
hierarchical algorithm designed to reduce computational complexity for super-
resolution. This approach is especially vital in scenarios with multiple
sources. By incorporating an interference cancellation step, we simplify the
multi-source doa recovery into $K$ single-source problems without incurring
additional complexity.

The remainder of this paper is structured as follows: Section 2 formulates the
problem of doa estimation and outlines its sparse representation; Sections 3-4
describe and evaluate ndoa, respectively, while Section 5 concludes the paper.

##  2 Problem Formulation and Model

###  2.1 DoA Estimation Problem Formulation

We study the estimation of the doa of $K$ far-field radio signals originating
from directions $\boldsymbol{\theta}\in\Theta^{K}$, where
$\Theta=\left[-\frac{\pi}{2},\frac{\pi}{2}\right)$. The doa are recovered from
$L$ noisy snapshots measured using a ula (ula) consisting of $N$ elements
spaced at half-wavelength intervals. A single snapshot $\mathbf{y}(t)$ can be
expressed as

$\mathbf{y}(t)=\mathbf{A}\left(\boldsymbol{\theta}\right)\cdot\mathbf{s}\left(t\right)+\mathbf{v}\left(t\right)\in\mathbb{C}^{N},\quad
t\in\left\\{1,...,L\right\\}.\vspace{-0.1cm}$ (1)

Here, $\mathbf{A}\left(\boldsymbol{\theta}\right)\in\mathbb{C}^{N\times K}$
represents the steering matrix, which projects the $K$ narrow-band impinging
source signals, denoted by vector $\mathbf{s}\left(t\right)\in\mathbb{C}^{K}$,
onto the ula. The steering matrix is comprised of the $K$ steering vectors,
defined as

$\mathbf{a}\left(\theta_{k}\right)=\left(e^{-i\pi\sin\left(\theta_{k}\right)},\ldots,e^{-i\pi\left(N-1\right)\sin\left(\theta_{k}\right)}\right)^{\top}.\vspace{-0.1cm}$
(2)

The term $\mathbf{v}\left(t\right)$ in (1) is an additive noise following a
complex Gaussian distribution, $\mathcal{CN}\left(0,\mathbf{R}\right)$.
Aggregating the $L$ snapshots as the matrix $\mathbf{Y}$, our objective is to
estimate $\boldsymbol{\theta}$ from $\mathbf{Y}$.

###  2.2 Sparse Modeling

To frame the doa estimation problem as an instance of ssr, we quantize the
azimuth $\Theta$ into $M$ equidistant grid cells

$\boldsymbol{\vartheta}\triangleq\left[\vartheta_{0},\ldots,\vartheta_{M-1}\right],\hskip
2.84544pt\vartheta_{m}=m\cdot\Delta\vartheta-\frac{\pi}{2},\hskip
2.84544pt\Delta\vartheta=\frac{\pi}{M}.\vspace{-0.1cm}$

Using these quantized directions, we can approximate a snapshot as a sparse
linear combination of steering vectors. Specifically, let us denote

$\hat{\mathbf{y}}\left(t\right)=\mathbf{A}\left(\boldsymbol{\vartheta}\right)\cdot\mathbf{x}\left(t\right),\hskip
5.69046pt\mathbf{A}\left(\boldsymbol{\vartheta}\right)\in\mathbb{C}^{N\times
M},\hskip 5.69046pt\mathbf{x}\left(t\right)\in\mathbb{C}^{M},\vspace{-0.1cm}$
(3)

where $\mathbf{A}\left(\boldsymbol{\vartheta}\right)$ is an over-complete
dictionary matrix with the $m\textrm{-th}$ column being a steering vector
represented by $\mathbf{a}\left(\vartheta_{m}\right)$ as in (2). Furthermore,
$\mathbf{x}\left(t\right)$ is a sparse vector in which each non-zero entry
corresponds to an active direction. The smv ssr optimization problem with a
single realization $\mathbf{y}$ (omitting $t$ for brevity), is represented as:

$\mathbf{x}^{\ast}=\arg\min_{\mathbf{x}}\left\\{\left\|\mathbf{y}-\mathbf{A}\left(\boldsymbol{\vartheta}\right)\cdot\mathbf{x}\right\|_{2}^{2}+\gamma\left\|\mathbf{x}\right\|_{0}\right\\}.\vspace{-0.1cm}$
(4)

Problem (4) is challenging to address due to the non-convexity introduced by
the $\ell_{0}$ norm. Several approaches have been proposed in the literature
for solving this problem , including greedy OMP methods [32], optimization-
based methods using $\ell_{1}$ and $\ell_{2}$ relaxations [13], and Bayesian
methods [21]. For mmv [33], the conventional method employs a block of
measurements coupled with a block of sparse vectors, assuming joint sparsity.
Yet, this method sees its complexity increase with the number of snapshots. In
the following, we demonstrate that by adopting a Bayesian approach with the
nuv prior, the mmv problem can be distilled down to an smv. In this context,
$\bar{\mathbf{y}}_{L}$, representing the mean of multiple snapshots, emerges
as a sufficient statistic.

Fig. 1: $\theta^{\circ}=-20$

Fig. 2: $\theta^{\circ}=0$

Fig. 3: $\theta^{\circ}=+20$

Fig. 4: $\theta^{\circ}\leq\left\lvert 75\right\rvert$

Fig. 5: $\sigma^{2}=3\cdot 10^{0}$

Fig. 6: $\sigma^{2}=8\cdot 10^{2}$

Fig. 7: $\sigma^{2}=10^{4}$

##  3 NUV-DoA High-Resolution Estimation

Our ndoa algorithm spans three key ingredients: 1) the nssr, a general
Bayesian-based ssr algorithm leveraging the nuv prior to promote sparsity that
is versatile enough to be applied beyond doa estimation to any dictionary
matrix; 2) a super-resolution algorithm that capitalizes on the inherent
spatial correlation of the DoA setting through spatial filtering over
overlapping sub-bands; and 3) a hierarchical strategy for multi-source
interference cancellation.

###  3.1 NUV-SSR

In our model (3), it holds that the vectors
$\left\\{\mathbf{x}\left(t\right)\right\\}_{t=1}^{L}$ are jointly sparse; that
is, they all share the same support. Moreover, every snapshot is considered
independent and of equal importance. Consequently, we represent $\mathbf{x}$
as a vector of complex decision variables, where each component
$\mathrm{x}_{m}\in\mathbb{C}$ corresponds to a potential hypothesis, such as a
direction $\vartheta_{m}$, and follows a complex nuv prior, specifically:

$\mathrm{x}_{m}\sim\mathcal{CN}\left(0,\mathrm{q}_{m}^{2}\right),\quad\mathbf{p}\left(\mathrm{q}_{m}^{2}\right)\propto
1,\quad\mathrm{q}^{2}_{m}\in\mathbb{R}_{+}.\vspace{-0.1cm}$ (5)

The joint likelihood of the snapshots, given the decision vector, can be
decomposed into conditionally independent and identical Gaussian
distributions. Given that our prior adheres to a Gaussian distribution, the
posterior is Gaussian as well. Thus, the temporal mean
$\bar{\mathbf{y}}_{L}=\frac{1}{L}\sum_{t=1}^{L}\mathbf{y}(t)$, emerges as a
sufficient statistic, significantly simplifying inference.

The conventional approach for recovery involves a Type I method: specifically,
a map (map) estimation of $\mathbf{x}$ given the sufficient statistic
$\bar{\mathbf{y}}_{L}$. However, studies have shown that Type II offers
superior recovery performance [34]. In Type I, the vector of unknown variances
$\mathbf{q}^{2}=\left(\mathrm{q}^{2}_{1},\ldots,\mathrm{q}^{2}_{M}\right)^{\top}$
is integrated out, while here, in Type II, it is estimated through evident
maximization:

$\hat{\mathbf{q}}^{2}=\arg\max_{\mathbf{q}^{2}\in\mathbb{R}_{+}^{M}}\left\\{\mathbf{p}\left(\mathbf{q}^{2}\left|\bar{\mathbf{y}}_{L}\right.\right)\right\\}.$
(6)

To solve this optimization problem, we employ a two-step em (em) algorithm
[35]. By considering $\mathbf{x}$ as a latent vector dependent on
$\mathbf{q}^{2}$ and maximizing the log-posterior lower bound difference, the
algorithm reduces to an iteratively reweighted ls estimation of the unknown
variance, based on the following update rule:

$\hat{\mathrm{q}}^{2}_{m,\left[i+1\right]}=\mathds{E}_{\left[i\right]}\left(\left\lvert\mathrm{x}_{m}\right\rvert^{2}\right)=\left\lvert\mathds{E}_{\left[i\right]}\left(\mathrm{x}_{m}\right)\right\rvert^{2}+\mathds{V}_{\left[i\right]}\left(\mathrm{x}_{m}\right).$
(7)

Here, $\mathds{E}_{\left[i\right]}$, and $\mathds{V}_{\left[i\right]}$ are the
expectation and variance, respectively, computed under the posterior
distribution
$\mathbf{p}\left(\mathbf{x}\left|\bar{\mathbf{y}}_{L};\hat{\mathbf{q}}^{2}_{\left[i\right]}\right.\right)$,
with the nuv set to their values from the preceding iteration. These values
arise from minimizing a ls cost function, for which the closed-form solution
is

$\displaystyle\hat{\mathbf{x}}_{\left[i\right]}=\mathds{E}_{\left[i\right]}\left(\mathbf{x}\right)=\Gamma\left(\hat{\mathbf{q}}^{2}_{\left[i\right]}\right)\cdot\mathbf{A}^{\mathsf{H}}\cdot\mathbf{W}_{\left[i\right]}\cdot\bar{\mathbf{y}}_{L},$
(8a)
$\displaystyle\mathds{V}_{\left[i\right]}\left(\mathbf{x}\right)=\hat{\mathbf{q}}^{2}_{\left[i\right]}-\Gamma\left(\hat{\mathbf{q}}^{4}_{\left[i\right]}\right)\cdot\operatorname*{\text{diag}}\left(\mathbf{A}^{\mathsf{H}}\cdot\mathbf{W}_{\left[i\right]}\cdot\mathbf{A}\right),$
(8b)

where $\mathbf{W}_{\left[i\right]}$ is a precision matrix of size $n\times n$
given by:

$\mathbf{W}_{\left[i\right]}=\left(\mathbf{A}\cdot\Gamma\left(\hat{\mathbf{q}}^{2}_{\left[i\right]}\right)\cdot\mathbf{A}^{\mathsf{H}}+\frac{\sigma^{2}}{L}\cdot\mathbf{I}\right)^{-1}.$
(9)

In the above, $\left(\boldsymbol{\vartheta}\right)$ was omitted to indicate
that $\mathbf{A}$ can serve as a general dictionary.
$\Gamma\left(\mathbf{u}\right)$ is a diagonal matrix with $\mathbf{u}$ being
its main diagonal, and $\operatorname*{\text{diag}}\left(\mathbf{B}\right)$
denotes an operator of extracting the diagonal of matrix $\mathbf{B}$ and
formatting it as a vector. The nssr is initialized with
$\hat{\mathbf{q}}^{2}_{\left[0\right]}$ as a non-zero random value, and
continues until a predefined convergence criterion is met. A notable benefit
of this method is its single-sparsity tuning parameter $\sigma^{2}$, further
elaborated upon in Section 4.

A key feature of the nuv representation is that when $\mathrm{q}^{2}_{m}=0$,
it implies that $\left\lvert\mathrm{x}_{m}\right\rvert=0$ as well. To select
the $K$ active hypotheses, we introduce
$\Omega=\left(\Omega_{m}=\left\lvert\mathrm{x}_{m}\right\rvert\right)^{\top}$
as the nus from which we choose the $K$ most dominant peaks. If $K$ is not
pre-defined, we select the most dominant peaks that exceed a threshold $\eta$.

Fig. 8: $L=100$, $\theta^{\circ}\leq\left\lvert 75\right\rvert$

Fig. 9: $L=100$, $\left\lvert 75\right\rvert\leq\theta^{\circ}\leq\left\lvert
85\right\rvert$

Fig. 10: $L=2,10$, $\theta^{\circ}\leq\left\lvert 75\right\rvert$

Fig. 11: $K=2$, $L=100$, $\theta^{\circ}\leq\left\lvert 75\right\rvert$

###  3.2 Super-Resolution by Spatial Filtering

The outlined nssr is powerful and robust, and we successfully employed it to
address an ssr problem encompassing $M=3000$ atoms. Despite the computational
complexity of the problem, the performance is commendable. In doa recovery,
$M=3000$ translates to a resolution of $\Delta\vartheta=0.06^{\circ}$.
However, aiming for finer resolutions like $\Delta\vartheta=0.01^{\circ}$
dramatically inflates the problem size to $M=18,000$ atoms, presenting
substantial challenges to all ssr algorithms. The unique structure of the doa
recovery problem further complicates matters: the steering vectors exhibit
spatial correlation, thus increasing the dictionary’s mutual coherence which
can potentially reduce performance.

To further enhance doa recovery performance, we augment nssr with super-
resolution by leveraging the inherent spatial correlation, leading to the
implementation of ndoa by spatial filtering. Here, we segment the single,
global ssr problem spanning the entire azimuth grid into a series of smaller
localized overlapping tasks. Each task targets a sub-band of the azimuth
around $\vartheta_{m}$ of size $2\alpha$, represented as
$\vartheta^{\sqcap}_{m}=\left[\vartheta_{m}-\alpha,\vartheta_{m}+\alpha\right]$.
For every sub-band $\vartheta^{\sqcap}_{m}$, an ssr problem is solved based on
the steering sub-matrix
$\mathbf{A}^{\sqcap}_{m}\left(\boldsymbol{\vartheta}\right)$ using nssr.
Within each of these sub-bands, only the center element from the sub-spectrum
$\Omega^{\sqcap}_{m}\left(0\right)$ is extracted. Following this, these
elements are combined to produce the full spectrum that spans the entire
azimuth, after which peak detection is executed.

This strategy excels in delivering high resolution, precision, and a universal
tuning principle. When targeting a resolution of
$\Delta\vartheta=0.01^{\circ}$ with $\alpha=0.5$, each sub-band spans
$1^{\circ}$. This results in $M^{\sqcap}=101$ decision variables for each sub-
task. This method proves more effective and efficient than other alternatives.
Yet, its efficiency can be further enhanced by adopting a hierarchical
approach, as we show next.

###  3.3 Hierarchical Approach and Multi-Source Estimation

While the spatial filtering strategy significantly enhances resolution, it can
also introduce an off-grid effect. Energy from sources located outside the
band might leak into the sub-band of interest, causing interference. To
address this, we employ a hierarchical approach for adjacent channel
interference cancellation, transforming multi-source doa recovery into $K$
single-source problems, without adding extra complexity.

We start with a low-resolution preprocessing phase to coarsely estimate the
$K$ directions, using nssr in low snr and Root-MUSIC in high snr. For each
single estimated source, interference from its $K-1$ neighboring sources is
canceled by subtracting a linear combination of their steering vectors from
the observations. This leaves a single source doa estimation, to which our
high-resolution algorithm can be applied. Given our reliable direction
estimates, high-resolution processing is confined to a narrow azimuth sub-band
of size $6\cdot\epsilon$ around the low-resolution estimate, where $\epsilon$
represents the empirical standard deviation of the low-resolution algorithm’s
error at the given snr. This hierarchical method is also adaptable for single-
source estimation, offering a significant reduction in complexity.

###  3.4 Discussion

While the sparsifying nuv prior is deeply rooted in sbl, its potential in the
Gaussian setting has not been fully harnessed for general ssr and doa
estimation tasks. The fact that the mean of the snapshots is a sufficient
statistic not only simplifies the problem but also enhances the robustness of
our ndoa, particularly in low snr and with a limited number of snapshots.
Additionally, since our method does not rely on the covariance of the
snapshots, it can effectively manage scenarios with coherent sources. Having
only a single tuning parameter, $\sigma^{2}$ makes ndoa easy to implement.

Here, estimating a positive variance directly correlates with the recovery of
an atom from the support and with detecting a source signal in a particular
direction. From a statistical perspective, our parameter estimation algorithm
translates into a simultaneous statistical test spanning a family of $M$
latent hypotheses, of which only $K$ are active. Owing to the sparsity-
enforcing nature of the nuv, we attain high detection rates while maintaining
low false alarms.

##  4 Empirical Evaluation

Here, we empirically evaluate the properties and performance of our proposed
ndoa.111The source code can be found at
https://github.com/MengyuanZha0/ICASSP24-NUV-DoA We begin by highlighting the
operation of sub-spectrums combining. For a single source at
$\theta=0^{\circ}$, the intermediate sub-spectrum centered on this direction,
as shown in Fig. 7, achieves a gain of $\approx 17\left[\textrm{dB}\right]$
over those centered at $\theta=-20^{\circ}$ and $\theta=+20^{\circ}$, depicted
in Figs. 7 and 7, respectively. The combined spectrum, illustrated in Fig. 7
is both sharp and accurate.

Next, we examine the impact of the tuning parameter $\sigma^{2}$. A low
$\sigma^{2}$ means the nuv does not promote sparsity, leading to an imprecise
spectrum with numerous false alarms (Fig. 7). Conversely, a high $\sigma^{2}$
aggressively enforces sparsity, yielding a spectrum that is accurate but wide
and of low magnitude (Fig. 7). An optimal $\sigma^{2}$ produces a spectrum
that is precise, narrow, and of high magnitude, as seen in Fig. 7.

We conclude by comparing the performance of ndoa, across multiple snr
conditions, with benchmark algorithms. Fig. 11 reveals that our method, with a
resolution of $\Delta\theta^{\circ}=0.01$, significantly surpasses its
counterparts in both high and low snr, especially when focusing on the mid-
interval of the azimuth $\left(\theta^{\circ}\leq\left\lvert
75\right\rvert\right)$. This performance differential is even more pronounced
at the azimuth boundaries $\left(\left\lvert
75\right\rvert\leq\theta^{\circ}\leq\left\lvert 85\right\rvert\right)$ as seen
in Fig. 11. Notably, while competing algorithms falter even in medium snr or
with limited snapshots (e.g., $L=10$), ndoa retains efficacy even at an
extreme low of $L=2$, as depicted in Fig. 11. Lastly, Fig. 11 highlights the
efficacy of our ic mechanism, particularly when contending with two non-
coherent sources spaced as closely as $15^{\circ}$ apart.

##  5 Conclusions

We proposed ndoa, a high precision doa estimation algorithm. ndoa combines
Bayesian sparse recovery using nuv priors, alongside spatial filtering for
super-resolution and hierarchical multi-source estimation. Our numerical
results show the superiority of ndoa in robustness and accuracy, particularly
in low snr.

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  *[doa]:
  *[snr]:
  *[nuv]:
  *[mvdr]:
  *[music]:
  *[ssr]:
  *[cs]:
  *[rvm]:
  *[sbl]:
  *[bcs]:
  *[mmv]:
  *[smv]:
  *[ula]:
  *[map]:
  *[em]:
  *[ndoa]:


# Quantum Advantage in Shared Randomness Processing

Tamal Guha  Physics and Applied Mathematics Unit, Indian Statistical
Institute, 203 B.T. Road, Kolkata 700108, India.  Mir Alimuddin  Physics and
Applied Mathematics Unit, Indian Statistical Institute, 203 B.T. Road, Kolkata
700108, India.  Sumit Rout  Integrated Science Education and Research Centre,
Visva Bharati University, Santiniketan 731235, India.  Amit Mukherjee  S.N.
Bose National Center for Basic Sciences, Block JD, Sector III, Salt Lake,
Kolkata 700098, India.  Some Sankar Bhattacharya  Department of Computer
Science, The University of Hong Kong, Pokfulam Road, Hong Kong.  Manik Banik
School of Physics, IISER Thiruvanathapuram, Vithura, Kerala 695551, India.

###### Abstract

Randomness appears both in classical stochastic physics and in quantum
mechanics. A number of communication models and other information protocols
entail sharing randomness among distant parties. Here we report a
computational scenario of shared randomness processing where quantum sources
manifest clear-cut precedence over the corresponding classical counterparts.
The advantage is established in a resource theoretic set-up which we formalize
in this work. The classical mutual information turns out to be a faithful
resource quantifier of shared randomness although it does not sufficiently
characterize all possible transitions among the resources. Quantum theory
exhibits advantage in shared randomness processing in the sense that the set
of resourceful states obtained under free operation from a bipartite quantum
system is strictly larger than the set obtained from an analogous classical
system. Interestingly, the quantum advantage can be manifested in the optimal
payoff of a two players co-operative game – the ‘non-monopolize social
subsidy’ game. We also show that the quantum states resulting to the desired
advantage necessitates nonclassicality in the form of quantum discord. We then
address the task of distributing shared randomness between two parties. Quite
interestingly, it turns out that noisy quantum channel can exhibit advantage
even when it has zero quantum capacity and classical capacity much less than
the corresponding classical channel. The imperfect channel examples
facilitates and welcomes experiment to achieve the reported quantum advantage
with presently available quantum devices.

The nascent quantum technology is getting increasingly sophisticated with each
passing days with the aim to control individual quantum systems for enabling
them in different practical tasks that otherwise are not possible in classical
world. It already finds several applications, such as secure communication
Jiang  _et al._ (2007); Dixon  _et al._ (2008); Wengerowsky  _et al._ (2019);
Yin  _et al._ (2020), quantum imaging Tan  _et al._ (2008); Schneider  _et
al._ (2017); Asban  _et al._ (2019); Gregory  _et al._ (2020), quantum
metrology Roos  _et al._ (2006); Appel  _et al._ (2009); Giovannetti  _et al._
(2011); Kurizki  _et al._ (2015), and more excitingly promises opportunities
for Near-Term Quantum Computing Systems Li  _et al._ (2001); Larsen  _et al._
(2019); Kandala  _et al._ (2019); Flühmann  _et al._ (2019); Arute  _et al._
(2019); García-Patrón  _et al._ (2019). These developments urge for
identifying more and more practical instances where quantum theory can exhibit
advantage over the corresponding classical counterpart. Such advantages,
however, are notoriously hard to find and even harder to prove. For instance
the set of functions computable quantum mechanically is the same as the set
computable in classical physics. However, there is a strong believe that
quantum computing is efficient for a range of problems, such as factoring Shor
, as it yields exponential speed-up over the best known classical algorithms.
But, from a strictly theoretical point the advantages is not yet proven, it is
rather supported on some assumptions in complexity theory.

In this work we report an interesting quantum advantage which is not only
probable but also robust in noisy condition and hence achievable with present
day imperfect quantum devices. We consider the computational scenario of
shared randomness processing. Shared randomness is known to be an important
resource in a number of applications, viz privacy amplification Bennett  _et
al._ (1988, 1995); Newman and Szegedy (1996), simultaneous messages passing
Babai and Kimmel , secret sharing and secret key generation protocol Gavinsky
_et al._ (2012); Ahlswede and Csiszar (1993), classical simulation of quantum
nonlocal statistics Brassard  _et al._ (1999); Toner and Bacon (2003); Bowles
_et al._ (2015), Bayesian game theory Aumann (1987); Brunner and Linden
(2013); Roy  _et al._ (2016); Banik  _et al._ (2019), and communication
complexity Canonne  _et al._ (2017). However, among spatially separated
parties, shared randomness can not be established free of cost – it requires
communication channel among them which in Shannon theory is considered as an
expensive resource Shannon (1948). In a physical scenario shared randomness
can be obtained from some correlated systems - either classical or quantum -
shared between the distant parties. Naturally the question arises whether
application of quantum rules provides any advantage over the classical sources
in shared randomness processing and its distribution. Interestingly, our work
provides affirmative answer to these questions and hence finds a novel
instance where quantum theory yields provable advantage over its classical
counterpart.

The obvious question arises: how to comprehend the obtained quantum advantage?
At this point we find the language of resource theory quite pertinent. During
the recently past, researchers in quantum information community successfully
apply this technique to identify, characterize, and quantify different useful
resources Braunstein and van Loock (2005); Bartlett  _et al._ (2007);
Horodecki  _et al._ (2009); Modi  _et al._ (2012); Brandão  _et al._ (2013);
Grudka  _et al._ (2014); Rivas  _et al._ (2014); Veitch  _et al._ (2014);
Gallego and Aolita (2015); Winter and Yang (2016); Chitambar and Gour (2019);
Wolfe  _et al._ (2020). Such a framework is operationally motivated - firstly
it identifies the free states that contain no resource and specifies the free
operations that are unable to produce any resource from a free state and hence
are allowed to be implemented at no cost. Once this is done, the theory aims
to find the resource conversion conditions (either necessary or sufficient,
sometimes both), commonly phrased as monotones, that characterize possible
transformation among the resourceful states under free operations. In case of
SR, classical mutual information turns out to be a faithful quantifier of the
resource which takes zero value for all the free states while nonzero for any
resourceful state. For a clear acquaintance of the framework we elaborately
discuss the state space of shared randomness with two spatially separated
parties holding a pair of 2-faced classical shared coins (two-2-coin). Within
the proposed resource theoretic framework, every two-2-coin state can be
freely obtained from its ts quantum analogue,namely two-2-quoin which
corresponds to a two-qubit quantum system with Hilbert space
$\mathbb{C}^{2}\otimes\mathbb{C}^{2}$. Quantum advantage is then established
by identifying two-d-coin states that can be obtained from a two-2-quoin under
free operation but cannot be obtained from any two-2-coin state.

To make this advantage more interesting we design a two-player co-operative
game, namely the ‘non-monopolize social subsidy’ game. While the players can
achieve perfect payoff when assisted with two-2-quoin, their payoff always
remain sub-optimal with two-2-coin states. Interestingly, we show that non-
classical payoff necessitates the two-2-quoin to be a non-zero discordant
state – an intriguing feature that captures quantumness in bipartite state
even when it is not entangled Ollivier and Zurek (2001); Henderson and Vedral
(2001). We then consider the scenario where one wish to establish shared
randomness with a distant fellow. To this aim she considers a bipartite
system, either classical or quantum, prepared in some correlated state and
sends one of the subsystems through some communication channel to the other
fellow. Quite interestingly,in such a scenario quantum channel can exhibit
advantage over the corresponding classical counterpart. Such an advantage is
quite remarkable as there exist no-go results Holevo (1973); Frenkel and
Weiner (2015) the limit the utilities of quantum systems in different
information protocols. While a classical channel transfer only classical
information and its efficacy is characterize by channel capacity Shannon
(1948), a quantum channel can be used for transmitting both classical
information and quantum information and consequently its utility is measured
through classical capacity Holevo (1998); Schumacher and Westmoreland (1997)
and quantum capacity Lloyd (1997); Shor (2002); Devetak (2005) respectively.
In distributing SR, a quantum channel can surpasses its classical version even
when its classical capacity is much less than the corresponding classical
channel. At this point, it seems natural to think that such advantage then
necessitates the noisy quantum channel to possess non-zero quantum capacity.
However, quite surprisingly, it turns out that the quantum advantage persists
even when the quantum channel has no quantum capacity. Importantly, these
imperfect channel examples facilitate noise robust empirical setups to verify
the obtained quantum advantage with the present day quantum devices.

## RESULTS

###  .1 Resource theory of shared randomness

The framework of resource theory provides a novel approach to quantify the
resources of shared randomness. A generic framework of any resource theory
consists of the following main ingredients: class of non-resources or free
states, the set of free operations, and resource conversion conditions (either
necessary or sufficient, sometimes both) that are commonly phrased as
monotones Brandão and Gour (2015).

#### Free resources

A source of shared randomness is specified by a bipartite probability
distribution $P(\mathcal{X},\mathcal{Y})\equiv\\{p(x,y)\leavevmode\nobreak\
|\leavevmode\nobreak\ x\in\mathcal{X},\leavevmode\nobreak\
y\in\mathcal{Y}\\}$, where $\mathcal{X}$ and $\mathcal{Y}$ are the parts of
the shared variable accessible by spatially separated parties Alice and Bob,
respectively. Probability distributions of the product form
$P(\mathcal{X},\mathcal{Y})=P(\mathcal{X})Q(\mathcal{Y})$ are considered as
free resources/states as each of the shared variables follows an independent
probability distribution and consequently information of one does not provide
any knowledge about the other. Unlike the resource theories of quantum
entanglement Horodecki  _et al._ (2009) or quantum coherence Winter and Yang
(2016) the set $\mathcal{F}_{SR}$ of free states does not form a convex set.

Figure 1: [Color on-line] Resource theory of shared randomness processing. By
performing free operations (local stochastic operations) on two-2-coin states
$\mathfrak{C}(2)$ one can obtain only a proper subset
$\mathfrak{S}_{C}(2\mapsto d)$ of two-d-coin state space $\mathfrak{C}(d)$.
Such a transformation can never increase classical mutual information of the
coin state. For instance, the transformation
$\mathcal{C}_{1/2}(2)\mapsto\mathcal{C}_{1/6}(6)$ is not allowed under free
operations, where
$\mathcal{C}_{1/6}(6):=1/6\sum_{f=1}^{6}\mathbf{ff}\in\mathfrak{C}(6)$.

In an operational theory, shared randomness between Alice and Bob can be
obtained from a shared bipartite system by performing local measurement on
their respective parts. The state space of such a system, in a convex
operational theory, is given by $\Omega_{A}\otimes\Omega_{B}$, where
$\Omega_{K}$ be the marginal state space which is a convex compact subset
embedded in some real vector space $V_{K}$; $K\in\\{A,B\\}$ Hardy (2001);
Barrett (2007); Chiribella  _et al._ (2011). For instance, state space of
$d$-level classical system is the $d$-simplex embedded in $\mathbb{R}^{d-1}$,
whereas for $d$-dimensional quantum system it is
$\mathcal{D}(\mathbb{C}^{d})\subset\mathbb{R}^{d^{2}-1}$;
$\mathcal{D}(\mathcal{H})$ denotes the set of density operators acting on the
Hilbert space $\mathcal{H}$ associated with the system. For simplexes the
choice of tensor product is unique (results in another higher level simplex)
which is not the case for other convex sets Namioka and R.Phelps (1969);
Barker (1976, 1981); Aubrun  _et al._ (2019).

#### Free operations

The set of free operations for shared randomness processing consists of all
possible local product operations $L_{A}\otimes L_{B}$ applied by Alice and
Bob on their respective parts of the joint system. For classical systems such
operations are most generally described by tensor product of local stochastic
matrices $\mathcal{S}_{A}\otimes\mathcal{S}_{B}$, where $\mathcal{S}_{A}$ maps
Alice’s local probability vector $P(\mathcal{X})$ into a new probability
vector $P^{\prime}(\mathcal{X}^{\prime})$ and $\mathcal{S}_{B}$ does the
similar on Bob’s part. Note that cardinality of $\mathcal{X}$ and
$\mathcal{X}^{\prime}$ can be different in general (see Fig.1). In quantum
scenario, the allowed operations are local unitary operations and/or local
measurements generally described by a positive operator valued measure (POVM)
Kraus (1983). At this point a comparison with the resource theory of quantum
entanglement is worth mentioning. In entanglement theory classical
communication is considered free, but it bears a cost in the present scenario
as it can create a non-product joint distribution, i.e., a resourceful state,
starting from a product one. In any operational theory if Alice and Bob
initially share a joint state $\omega_{AB}\in\Omega_{A}\otimes\Omega_{B}$ of
the product from, i.e., $\omega_{AB}=\omega_{A}\otimes\omega_{B}$, then a free
operation on it always results in a free state.

#### Resource monotone

A necessary condition of state conversion from a distribution
$P(\mathcal{X,Y})$ to another $Q(\mathcal{X}^{\prime},\mathcal{Y}^{\prime})$
is given by $I(Q)\leq I(P)$, where $I(P)$ is the classical mutual information
defined as $I(P):=H(\mathcal{X})+H(\mathcal{Y})-H(\mathcal{X,Y})$, with
$H(\mathcal{X})$ being the Shannon entropy,
$H(\mathcal{X}):=-\sum_{x\in\mathcal{X}}p(x)\log_{2}p(x)$. Importantly, mutual
information is a faithful resource quantifier as it takes zero value for every
free state while non-zero for all the resourceful states. In the subsequent
section, however, we will see that it can not sufficiently characterize the
possible resource conversions.

Figure 2: [Color on-line] Two-2-coin state space $\mathfrak{C}(2)$. All the
four vertices are the free states. Green (red) line denotes the
$\alpha$-correlated ($\alpha$-anti-correlated) edges. The remaining four edges
consist of free states only. Dots in the left [right] figure denote the states
obtained from $\mathcal{C}_{1/2}(2)$ [$\mathcal{C}_{1/3}(2)$] by applying
randomly generated local stochastic maps. Action of such maps on
$\alpha$-correlated (or $\alpha$-anti-correlated) edge generate the whole
state space $\mathfrak{C}(2)$ (see Lemma 1).

#### Two-2-coin state space

Consider that Alice and Bob share a pair of 2-faced classical coins
(two-2-coin), i.e., $\mathcal{X}\equiv\\{\mbox{head}(\mathtt{h}),$
$\mbox{tail}(\mathtt{t})\\}\equiv\mathcal{Y}$. A generic state of this system
is described by a column vector
$\mathcal{C}(2)\equiv(p(\mathtt{hh}),p(\mathtt{ht}),p(\mathtt{th}),p(\mathtt{tt}))^{\intercal}\in\mathfrak{C}(2)$;
with $\mathfrak{C}(2)$ denoting the set of all two-2-coin states. A state
$\mathcal{C}(2)\equiv(x,y,z,1-x-y-z)^{\intercal}$ is isomorphic to the vector
$\mathbb{R}^{3}\ni\mathcal{V}\equiv(x,y,z)^{\intercal}$ with $x,y,z\geq
0\leavevmode\nobreak\ \&\leavevmode\nobreak\ x+y+z\leq 1$, forming a convex
subset $\mathbf{T}$ in the positive octant (see Fig.2). All the four vertices
($0$-faces) $\mathcal{C}^{\mathtt{hh}}(2)$, $\mathcal{C}^{\mathtt{ht}}(2)$,
$\mathcal{C}^{\mathtt{th}}(2)$, and $\mathcal{C}^{\mathtt{tt}}(2)$ are free
states. We call the states
$\mathcal{C}_{\alpha}(2):=(\alpha,0,0,1-\alpha)^{\intercal}\equiv\alpha\leavevmode\nobreak\
\mathtt{hh}+(1-\alpha)\mathtt{tt}$ as $\alpha$-correlated. Whenever
$\alpha\notin\\{0,1\\}$, $\mathcal{C}_{\alpha}(2)$ contains shared randomness
even though they are obtained by convex mixing of two free states, hence
implies non-convexity of $\mathcal{F}_{SR}$. The $\alpha$-correlated states
live in one of the edges ($1$-faces) of $\mathbf{T}$ and we call it
$\alpha$-correlated edge which will be denoted as
$\mathcal{E}_{[\alpha]}(2):=\\{\mathcal{C}_{\alpha}(2);\leavevmode\nobreak\
\alpha\in[0,1]\\}$. Under free operation this edge can be transferred into the
$\alpha$-anti-correlated edge
$\tilde{\mathcal{E}}_{[\alpha]}(2)\equiv\\{\tilde{\mathcal{C}}_{\alpha}(2):=\alpha\leavevmode\nobreak\
\mathtt{ht}+(1-\alpha)\mathtt{th};\leavevmode\nobreak\ \alpha\in[0,1]\\}$. In
fact, every $\mathcal{C}_{\alpha}(2)$ is connected to the corresponding
$\tilde{\mathcal{C}}_{\alpha}(2)$ by local permutation, a free operation that
keeps the mutual information invariant. The remaining four edges of
$\mathbf{T}$ contain only free states. Except these states no other state
residing on any of the four $2$-faces of $\mathbf{T}$ is free. However, the
volume ($3$-face) of $\mathbf{T}$ contains both free and resourceful states.

Consider a state
$\mathcal{C}_{\Delta}(2):=\left(1/3,0,1/3,1/3\right)^{\intercal}$ residing on
one of the $2$-faces of $\mathbf{T}$. The state $\mathcal{C}_{\Delta}(2)$ can
be obtained from $\mathcal{C}_{1/2}(2)$ under free operation. The two possible
free operations allowing this transformation are given by,

$\left\\{\begin{pmatrix}0&2/3\\\
1&1/3\end{pmatrix}\otimes\begin{pmatrix}1/3&1\\\
2/3&0\end{pmatrix};\leavevmode\nobreak\ \leavevmode\nobreak\
\begin{pmatrix}2/3&0\\\ 1/3&1\end{pmatrix}\otimes\begin{pmatrix}1&1/3\\\
0&2/3\end{pmatrix}\right\\}.$

The reverse transformation
$\mathcal{C}_{\Delta}(2)\mapsto\mathcal{C}_{1/2}(2)$ is not possible under
free operations as the former has lesser mutual information than the latter.
Importantly, such a transformation may not be possible even if the initial
state has more mutual information than the targeted one. For instance, none of
the states $\mathcal{C}_{\alpha}(2)$ can be obtained from
$\mathcal{C}_{1/2}(2)$ whenever $\alpha\not\in\left\\{0,1/2,1\right\\}$,
though $I(\mathcal{C}_{1/2}(2))\geq I(\mathcal{C}_{\alpha}(2))$, with strict
inequality holding for $\alpha\in[0,1/2)\cup(1/2,1]$. It establishes
insufficiency of mutual information in characterizing the possible state
conversions. It furthermore proves non-convexity of the set of states obtained
from a given resource under the free operations.

###  .2 Quantum advantage

In this section we will present our main result which establishes quantum
advantage in shared randomness processing. To this aim we first introduce
quantum analogue of the two-2-coin.

#### Two-2-quoin state space.–

The quantum analogue of two-2-coin state, which we call two-2-quoin and denote
as $\mathcal{Q}(2)$, corresponds to the states of a two-qubit quantum system.
The state space is given by
$\mathfrak{Q}(2)\equiv\mathcal{D}(\mathbb{C}^{2}_{A}\otimes\mathbb{C}^{2}_{B})$,
where subsystems $A$ and $B$ are held by Alice and Bob, respectively. From the
two-2-quoin states Alice and Bob can prepare any states of $\mathfrak{C}(2)$
by applying local POVMs on their respective parts of the joint system.
Therefore, the former can always replace the latter for any shared randomness
processing task. From these paired classical and quantum 2-level coins one can
obtain paired d-level classical coin states. Let $\mathfrak{S}_{C}(2\mapsto
d)$ denotes the subset of two-d-coin states $\mathfrak{C}(d)$ that are freely
simulable (i.e., can be obtained under allowed free operations) with
$\mathfrak{C}(2)$. Similarly, $\mathfrak{S}_{Q}(2\mapsto d)$ denotes the
subset of $\mathfrak{C}(d)$ freely simulable from $\mathfrak{Q}(2)$. Both
these subsets are non-convex, and for any state $\mathcal{C}(2\mapsto
d)\in\mathfrak{S}_{J}(2\mapsto d)$, with $J\in\\{C,Q\\}$, we have
$I(\mathcal{C}(2\mapsto d))\leq 1$. The following theorem establishes quantum
advantage in shared randomness processing through a proper set inclusion
relation between these subsets.
Theorem 1. $\mathfrak{S}_{C}(2\mapsto d)\subset\mathfrak{S}_{Q}(2\mapsto d)$,
for $d>2$.
Proof (outline).– The proof follows in two steps. First we prove that any
correlation in $\mathfrak{S}_{C}(2\mapsto d)$ also lies in
$\mathfrak{S}_{Q}(2\mapsto d)$. For that, please note that a $2-$coin state
$\mathcal{C}(2)\equiv(p,q,r,1-p-q-r)^{\intercal}$ can be obtained from a two-
qubit state, $\rho_{AB}=p\leavevmode\nobreak\ |00\rangle\langle
00|+q\leavevmode\nobreak\ |01\rangle\langle 01|+r\leavevmode\nobreak\
|10\rangle\langle 10|+(1-p-q-r)\leavevmode\nobreak\ |11\rangle\langle 11|$ by
performing local measurement in computational basis. Furthermore,
corresponding to every $2\times d$ stochastic matrix, applied locally on
$\mathcal{C}(2)$ there is a $d-$outcome POVM acting locally on the part of
$\rho_{AB}$, which implies that $\mathfrak{S}_{C}(2\mapsto
d)\subseteq\mathfrak{S}_{Q}(2\mapsto d)$. In second step we will show that
there exists a correlation in $\mathfrak{S}_{Q}(2\mapsto d)$ that does not
belong to $\mathfrak{S}_{C}(2\mapsto d)$. To make the advantage practically
more interesting and robust against the noisy in the following we introduce a
two-player game.

#### Non-monopolizing social subsidy game

The game $\mathbb{G}(n)$ involves two employees Alice & Bob working in an
organization and $n$ different restaurants $r_{1},\cdots,r_{n}$. On every
working day each of the employees buys beverage from the restaurant chosen at
her/his will. The organization has a reimbursement policy to pay back the
beverage bill. For this purpose each day’s bill is accounted for a long time
to calculate the probability $P(ij)$ of Alice visiting $r_{i}$ restaurant and
Bob $r_{j}$ restaurant. Events $(ij)$ where each employee ends up in different
restaurants $(i\neq j)$ are considered for reimbursement / payoff111This
condition encourages the physical distancing norm which people need to follow
during the unfortunate pandemic of COVID-19.. Now it may be the case that the
employees pick their favorite restaurants which happen to be different and
become regular visitors. But this will leave the other restaurants out of
business. To circumvent this situation a sub-clause is added to the subsidy
rule which says that the payoff will be defined as
$\$\mathcal{R}(n)=\$\min_{i\neq j}P(ij)$ assuming per day expense $\$1$ for
each of the employees. Since the reimbursement policy encourages total trade
to be distributed among all the restaurants we call it ‘non-monopolizing
subsidy’ rule. The employees are non-communicating and possess no SR. However,
they may be assisted with some shared coin state (either classical or quantum)
along with the local strategies chosen from the set of free operations.
Following result bounds their achievable payoff.
Theorem 2. $\frac{1}{n^{2}}\leq\mathcal{R}(n)\leq\frac{1}{n(n-1)}$.
Proof.– Lower bound is achieved following a uniformly randomized local
strategy by each of the employees. For maximal payoff, there are $n(n-1)$
different cases where the employees’ bills get reimbursed. Since minimum
probability of these events will be considered for reimbursement, the optimal
payoff will be achieved if they choose these cases with equal probability,
i.e., with probability $\frac{1}{n(n-1)}$.
Consider a coin state $\mathcal{C}_{\neq\alpha}(d)\in\mathfrak{C}(d)$ such
that $p(\mathbf{ff})=0\leavevmode\nobreak\ \&\leavevmode\nobreak\
p(\mathbf{ff^{\prime}})\neq 0,\leavevmode\nobreak\ \forall\leavevmode\nobreak\
\mathbf{f},\mathbf{f^{\prime}}\in\\{1,\cdots,d\\},\leavevmode\nobreak\
\&\leavevmode\nobreak\ \mathbf{f}\neq\mathbf{f^{\prime}}$; and we call it
‘not-$\alpha$-correlated’ state. The maximum achievable payoff in
$\mathbb{G}(n)$ is assured if the employees share a particular
not-$\alpha$-correlated coin state $\mathcal{C}^{eq}_{\neq\alpha}(n)$, where
$p(\mathbf{ff^{\prime}})=1/n(n-1),\leavevmode\nobreak\
\forall\leavevmode\nobreak\
\mathbf{f},\mathbf{f^{\prime}}\in\\{1,\cdots,n\\},\leavevmode\nobreak\
\&\leavevmode\nobreak\ \mathbf{f}\neq\mathbf{f^{\prime}}$. What follows next
is the two Lemmas regarding simulability of different sets of classical coin
states.
Lemma 1. Under the action of free operations, any coin state of
$\mathfrak{C}(2)$ can be obtained from the $\alpha$-correlated edge
$\mathcal{E}_{[\alpha]}(2)$, i.e., $\mathcal{E}_{[\alpha]}(2)$ freely
simulates the state space $\mathfrak{C}(2)$.
Proof.– A state $\mathcal{C}(2)\in\mathfrak{C}(2)$ can most generally be
expressed as,

$\displaystyle\mathcal{C}(2)$ $\displaystyle=$
$\displaystyle\left(x,y,z,1-x-y-z\right)^{\intercal},$ (1) $\displaystyle
0\leq x\leq 1;$ $\displaystyle 0\leq y\leq 1-x;\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ 0\leq z\leq 1-x-y.$

The range of $y$ is determined by the value of $x$, i.e.,
$\forall\leavevmode\nobreak\ x\in[0,1]$ the value of $y$ lies within
$[0,1-x]$. Similarly the range of $z$ is specified by $x$ and $y$. Except
specifying the range their values are mutually random, i.e., the variables fix
range of each other but not the exact value. We wish to show that by applying
local stochastic operations on
$\mathcal{C}_{\alpha}(2)\in\mathcal{E}_{[\alpha]}(2)$ Alice and Bob can
prepare any vector of the form of Eq.(1). We therefore can wright

$\displaystyle\mathcal{K}:=\mathcal{S}^{2\mapsto
2}_{A}\otimes\mathcal{S}^{2\mapsto 2}_{B}\leavevmode\nobreak\
\times\leavevmode\nobreak\ \mathcal{C}_{\alpha}(2)$
$\displaystyle=\begin{pmatrix}a_{1}&1-a_{1}\\\
a_{2}&1-a_{2}\end{pmatrix}^{\intercal}\otimes\begin{pmatrix}b_{1}&1-b_{1}\\\
b_{2}&1-b_{2}\end{pmatrix}^{\intercal}\times\begin{pmatrix}\alpha\\\ 0\\\ 0\\\
1-\alpha\end{pmatrix}$
$\displaystyle=\begin{pmatrix}a_{1}b_{1}\alpha+a_{2}b_{2}(1-\alpha)\leavevmode\nobreak\
\leavevmode\nobreak\ [:=k_{1}]\\\
a_{1}\alpha+a_{2}(1-\alpha)-k_{1}\leavevmode\nobreak\ [:=k_{2}]\\\
b_{1}\alpha+b_{2}(1-\alpha)-k_{1}\leavevmode\nobreak\ \leavevmode\nobreak\
[:=k_{3}]\\\ 1-\sum_{i=1}^{3}k_{i}\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \end{pmatrix}=\begin{pmatrix}k_{1}\\\ k_{2}\\\ k_{3}\\\
1-\sum k_{i}\end{pmatrix},$

where $a_{1},a_{2},b_{1},b_{2}\in[0,1]$. Since action of local stochastic
matrix $\mathcal{S}_{A}\otimes\mathcal{S}_{B}$ on $\mathcal{C}_{\alpha}(2)$
always results in a probability vector, therefore constraints as of Eq.(1)
among $k_{1},k_{2},$ and $k_{3}$ are always satisfied. Thus we only need to
prove that $k_{1},k_{2},$ and $k_{3}$ are mutually random. First note that,
for every fixed values of $a_{2},b_{2}\in[0,1]$, $\exists\leavevmode\nobreak\
\alpha,a_{1},b_{1}\in[0,1]$ s.t. $k_{1}$ takes all values in $[0,1]$.
Therefore $a_{2}$ and $b_{2}$ are mutually random with $k_{1}$ and with each
other. Consequently, $k_{2}$ and $k_{3}$ are mutually random with $k_{1}$ and
with each other. This completes the proof.
Lemma 2. None of the coin state $\mathcal{C}_{\neq\alpha}(n)$ is freely
simulable from $\mathcal{E}_{[\alpha]}(2)$, whenever $n>2$.
Proof.– A generic stochastic operation $\mathcal{S}^{2\mapsto n}$ mapping a
two dimensional probability vector into an $n$ dimensional probability vector
is of the form

$\begin{pmatrix}u_{11}&u_{21}&\cdots&1-\sum_{i=1}^{n-1}u_{i1}\\\
u_{12}&u_{22}&\cdots&1-\sum_{i=1}^{n-1}u_{i2}\end{pmatrix}^{\intercal},$

where $u_{ij}\in[0,1]$ and $\sum_{i=1}^{n-1}u_{ij}\leq 1$. Action of a local
such operations by Alice and Bob on their respective parts of the coin state
$\mathcal{C}_{\alpha}(2)$ yields a two-$n$-coin state,

$\displaystyle\mathcal{C}(n)$ $\displaystyle=$
$\displaystyle\mathcal{S}^{2\mapsto n}_{A}\otimes\mathcal{S}^{2\mapsto
n}_{B}\leavevmode\nobreak\ \times\leavevmode\nobreak\ \mathcal{C}_{\alpha}(2)$
$\displaystyle=$
$\displaystyle\begin{pmatrix}a_{11}&a_{21}&\cdots&1-\sum_{i=1}^{n-1}a_{i1}\\\
a_{12}&a_{22}&\cdots&1-\sum_{i=1}^{n-1}a_{i2}\end{pmatrix}^{\intercal}\otimes$
$\displaystyle\begin{pmatrix}b_{11}&b_{21}&\cdots&1-\sum_{i=1}^{n-1}b_{i1}\\\
b_{12}&b_{22}&\cdots&1-\sum_{i=1}^{n-1}b_{i2}\end{pmatrix}^{\intercal}\times\begin{pmatrix}\alpha\\\
0\\\ 0\\\ 1-\alpha\end{pmatrix}.$

Whenever $\alpha\in\\{0,1\\}$, the initial state is free and hence the final
one. The resulting state to be $\mathcal{C}_{\neq\alpha}(n)$, we require
$\alpha a_{i1}b_{i1}+(1-\alpha)a_{i2}b_{i2}=0,\leavevmode\nobreak\
\forall\leavevmode\nobreak\ i\in\\{1,...,n\\}$. Since $\alpha\in(0,1)$,
therefore $a_{ij}b_{ij}=0,\leavevmode\nobreak\ \forall\leavevmode\nobreak\
i\in\\{1,...,n\\}\leavevmode\nobreak\ \&\leavevmode\nobreak\
\forall\leavevmode\nobreak\ j\in\\{1,2\\}$. Presence of anti-correlated terms
in $\mathcal{C}_{\neq\alpha}(n)$ demands, $\alpha
a_{i1}b_{k1}+(1-\alpha)a_{i2}b_{k2}\neq 0,\leavevmode\nobreak\
\forall\leavevmode\nobreak\ i,k\in\\{1,...,n\\}\leavevmode\nobreak\
\&\leavevmode\nobreak\ i\neq k$. Therefore for every $(i,k\neq i)$ pair
$\exists$ at-least one $j\in\\{1,2\\}$ s.t. $a_{ij}b_{kj}\neq 0\implies
a_{ij}\neq 0$ and $b_{kj}\neq 0$. Similarly, for the corresponding reverse
pair $(k,i\neq k)\leavevmode\nobreak\ \exists$ at-least one
$j^{\prime}\in\\{1,2\\}$ s.t. $a_{kj^{\prime}}b_{ij^{\prime}}\neq 0\implies
a_{kj^{\prime}}\neq 0$ and $b_{ij^{\prime}}\neq 0$. Now $j$ and $j^{\prime}$
should be different, otherwise a correlated term of the resulting coin state
will become non-vanishing. Since $j,j^{\prime}\in\\{1,2\\}$, the requirement
$j\neq j^{\prime}$ can not be satisfied whenever $i,k\in\\{1,...,n\\}$, with
$n>2$. This completes the proof.
These two Lemmas lead us to the following result describing limitation of the
paired classical coins in achieving the maximum payoff in the game
$\mathbb{G}(n)$.
Theorem 3. Given any coin state from $\mathfrak{C}(2)$ the payoff
$\mathcal{R}(n)$ is always suboptimal for all $n>2$.
Proof.– Contrary to the hypothesis, let us assume that there exist a
two-2-coin state $\mathcal{C}^{n}_{win}(2)$ that provides perfect success in
$\mathbb{G}(n)$. Since perfect success of $\mathbb{G}(n)$ requires the
two-$n$-coin state $\mathcal{C}^{eq}_{\neq\alpha}(n)$, this implies that
$\mathcal{C}^{eq}_{\neq\alpha}(n)$ can be obtained from
$\mathcal{C}^{n}_{win}(2)$ under free operation. Invoking Lemma 1 we can say
that the state $\mathcal{C}^{eq}_{\neq\alpha}$ can be obtained freely from
$\mathcal{E}_{[\alpha]}(2)$. This, however, contradicts Lemma 2.
At this point one can ask for maximum payoff
$\mathcal{R}^{\mathfrak{C}(m)}_{\max}(n)$ that can be achieved in
$\mathbb{G}(n)$ given an assistance from $\mathfrak{C}(m)$. This can be casted
into an optimization problem. For that first note that, given a two-$m$-coin
$\mathcal{C}(m)\equiv\left(p(\mathbf{11}),\cdots,p(\mathbf{1m}),\cdots,p(\mathbf{mm})\right)^{\intercal}$
Alice and Bob can obtain some two-$n$-coin states
$\mathcal{C}(n)\equiv\left(q(\mathbf{11}),\cdots,q(\mathbf{1n}),\cdots,q(\mathbf{nn})\right)^{\intercal}$
by applying local stochastic maps (free operation), i.e.,
$\mathcal{C}(n)=\mathcal{S}^{m\mapsto n}_{A}\otimes\mathcal{S}^{m\mapsto
n}_{B}\cdot\mathcal{C}(m)$. We therefore have

$\displaystyle\mathcal{R}^{\mathfrak{C}(m)}_{\max}(n)$ $\displaystyle=$
$\displaystyle\underset{\underset{\mathcal{S}^{m\mapsto
n}_{A}\otimes\mathcal{S}^{m\mapsto
n}_{B}}{\mathcal{C}(m)\in\mathfrak{C}(m)}}{\mbox{maximize}}\leavevmode\nobreak\
q(\mathbf{i}\neq\mathbf{j})$ (2)
$\displaystyle\mbox{subject\leavevmode\nobreak\ to}\leavevmode\nobreak\
\leavevmode\nobreak\ q(\mathbf{i}\neq\mathbf{j})\leq\leavevmode\nobreak\
q(\mathbf{i}^{\prime}\neq\mathbf{j}^{\prime})$
$\displaystyle\mathbf{i}\neq\mathbf{i}^{\prime}\leavevmode\nobreak\
\mbox{and/or}\leavevmode\nobreak\ \mathbf{j}\neq\mathbf{j}^{\prime}.$

Here $\mathcal{S}^{m\mapsto n}_{A/B}$ is a stochastic maps mapping
$m$-dimensional probability vectors into $n$-dimensional ones. While
calculating $\mathcal{R}^{\mathfrak{C}(m)}_{\max}(n)$, for $m=2$, Lemma 1
allows us to restrict the optimization over the edge
$\mathcal{E}_{[\alpha]}(2)$, instead of the full two-$2$-coin state space
$\mathfrak{C}(2)$. In Table 1 we list maximum payoffs for few cases. There we
also provide the optimal coin state of $\mathfrak{C}(m)$ an the applied free
operation on it that maximize $\mathcal{R}^{\mathfrak{C}(m)}_{\max}(n)$. Our
next result establishes quantum advantage of shared randomness processing in
non-monopolize social subsidy game.
Theorem 4. The optimum payoff $\mathcal{R}(n)$ for $n=3,4$ can be obtained
with a coin state from $\mathfrak{Q}(2)$.
Proof.– Let the players share the two-2-quoin state
$\mathcal{Q}_{\mathtt{singlet}}(2):=|\psi^{-}\rangle_{AB}=\frac{1}{\sqrt{2}}\left(|01\rangle_{AB}-|10\rangle_{AB}\right)$.
Both of them perform the same three outcome Trine-POVM
$\mathcal{M}^{\mathrm{T}}\equiv\left\\{\Pi_{k}:=\frac{2}{3}|\psi_{k}\rangle\langle\psi_{k}|\right\\}$,
where
$|\psi_{k}\rangle:=\cos(k-1)\theta_{3}|0\rangle+\sin(k-1)\theta_{3}|1\rangle$;
$k\in\\{1,2,3\\},\leavevmode\nobreak\ \theta_{3}=2\pi/3$. This strategy leads
to the coin state $\mathcal{C}^{eq}_{\neq\alpha}(3)$ yielding the optimum
payoff in $\mathbb{G}(3)$. To obtain the optimum payoff in $\mathbb{G}(4)$,
they consider the SIC-measurement
$\mathcal{M}^{\mathrm{S}}\equiv\\{\frac{1}{2}|0\rangle\langle
0|,\frac{1}{2}|\psi_{k}\rangle\langle\psi_{k}|\leavevmode\nobreak\
|\leavevmode\nobreak\ k=0,1,2\\}$, where
$|\psi_{k}\rangle=\sqrt{\frac{1}{3}}|0\rangle+e^{i\frac{2k\pi}{3}}\sqrt{\frac{2}{3}}|1\rangle$.
This leads to the coin state $\mathcal{C}^{eq}_{\neq\alpha}(4)$ resulting
optimum payoff in $\mathbb{G}(4)$. This completes the proof.

Table 1: Maximum payoff in $\mathbb{G}(n)$ given a coin state from $\mathfrak{C}(m)$. Coin states $\mathcal{C}(m)$ and the free operations $\mathcal{S}^{m\mapsto n}_{A}\otimes\mathcal{S}^{m\mapsto n}_{B}$ yielding maximum success $\mathcal{R}^{\mathfrak{C}(m)}_{\max}(n)$ is not unique in general. $\leavevmode\nobreak\ \mathcal{R}^{\mathfrak{C}(m)}_{\max}(n)\leavevmode\nobreak\ $ | $\leavevmode\nobreak\ \mathcal{C}(m)\leavevmode\nobreak\ $ | $\leavevmode\nobreak\ \mathcal{S}^{m\mapsto n}_{A}\leavevmode\nobreak\ $ | $\leavevmode\nobreak\ \mathcal{S}^{m\mapsto n}_{B}\leavevmode\nobreak\ $
---|---|---|---
$\mathcal{R}^{\mathfrak{C}(2)}_{\max}(3)=\frac{1}{8}$ | $\mathcal{C}_{1/2}(2)$ | $\begin{pmatrix}0&\frac{1}{2}\\\ \frac{1}{2}&0\\\ \frac{1}{2}&\frac{1}{2}\end{pmatrix}$ | $\begin{pmatrix}\frac{1}{2}&0\\\ 0&\frac{1}{2}\\\ \frac{1}{2}&\frac{1}{2}\end{pmatrix}$
$\mathcal{R}^{\mathfrak{C}(2)}_{\max}(4)=\frac{1}{15}$ | $\mathcal{C}_{1/2}(2)$ | $\begin{pmatrix}\frac{1}{5}&\frac{1}{3}\\\ \frac{1}{5}&\frac{1}{3}\\\ \frac{2}{5}&0\\\ \frac{1}{5}&\frac{1}{3}\end{pmatrix}$ | $\begin{pmatrix}\frac{1}{3}&\frac{1}{5}\\\ \frac{1}{3}&\frac{1}{5}\\\ 0&\frac{2}{5}\\\ \frac{1}{3}&\frac{1}{5}\end{pmatrix}$
$\mathcal{R}^{\mathfrak{C}(3)}_{\max}(4)=\frac{2}{27}$ | $\mathcal{C}^{eq}_{\neq\alpha}(3)$ | $\begin{pmatrix}0&\frac{2}{3}&0\\\ 0&0&\frac{2}{3}\\\ \frac{2}{3}&0&0\\\ \frac{1}{3}&\frac{1}{3}&\frac{1}{3}\end{pmatrix}$ | $\begin{pmatrix}0&\frac{2}{3}&0\\\ 0&0&\frac{2}{3}\\\ \frac{2}{3}&0&0\\\ \frac{1}{3}&\frac{1}{3}&\frac{1}{3}\end{pmatrix}$

Remark: Theorem 3 and Theorem 4 together provide a proof for the second part
of the Theorem 1 for $d=3,4$. According to Theorem 3,
$\mathcal{C}^{eq}_{\neq\alpha}(d)\notin\mathfrak{S}_{C}(2\mapsto d)$ whenever
$d>2$. In fact, from Lemma 2 we can say that
$\mathcal{C}_{\neq\alpha}(d)\notin\mathfrak{S}_{C}(2\mapsto d)$. On the other
hand, Theorem 4 tells that
$\mathcal{C}^{eq}_{\neq\alpha}(d)\in\mathfrak{S}_{Q}(2\mapsto d)$ for $d=3,4$
and hence proves the second part of the Theorem 1. For higher values of $d$,
consider the two-2-quoin state $\mathcal{Q}_{\mathtt{singlet}}(2)$ and
consider same $d$ outcome POVM
$\mathcal{M}^{(d)}\equiv\left\\{\Pi_{k}:=\frac{2}{d}|\psi_{k}\rangle\langle\psi_{k}|\right\\}$
for Alice and Bob, where
$|\psi_{k}\rangle:=\cos(k-1)\theta_{d}|0\rangle+\sin(k-1)\theta_{d}|1\rangle$;
$k\in\\{1,\cdots,d\\}$ and $\theta_{d}=2\pi/d$. This leads to a state
$\mathcal{C}_{\neq\alpha}(d)$ and completes the proof of Theorem 1 for
arbitrary $d$. Note that for higher $d$’s the state
$\mathcal{C}^{eq}_{\neq\alpha}(d)$ can not be obtained from
$\mathcal{Q}_{\mathtt{singlet}}(2)$ and hence perfect payoff in
$\mathbb{G}(d)$ can not be obtained even when a coin state from
$\mathfrak{Q}(2)$ is given as an assistance. Optimal classical vs quantum
payoff(s) for the generic case $\mathbb{G}(d)$ we leave here as an open
question.

###  .3 Noise-robust quantum advantage

The quantum advantage established above considers a two-qubit perfect
entangled state. However, entanglement is extremely fragile under noise and
hence the advantage obtained with such an perfect state deemed impossible to
archive in practical scenario. Thus a more realistic question is whether the
advantage persists even under noisy consideration. To this aim let us consider
a noisy two-2-quoin
$\mathcal{Q}_{p}(2):=p|\psi^{-}\rangle_{AB}\langle\psi^{-}|+(1-p)\frac{\mathbb{I}}{2}\otimes\frac{\mathbb{I}}{2}$.
Please note that $\mathcal{R}^{\mathfrak{C}(2)}_{\max}(3)=1/8$ and
$\mathcal{R}^{\mathfrak{C}(2)}_{\max}(4)=1/15$ (see Table 1). Any quantum
strategy providing a greater payoff can be considered advantageous over the
classical resources. With the two-2-quoin state $\mathcal{Q}_{p}(2)$ if we
follow the same strategy as discussed in Theorem 4, they it becomes
advantageous for $p>1/4$ and $p>1/5$ for the cases $\mathbb{G}(3)$ and
$\mathbb{G}(4)$. At this point, it is noteworthy that the state
$\mathcal{Q}_{p}(2)$ not even an entangled state whenever $p\leq 1/3$. This
raises another fundamentally important question: what quantum feature results
to the desired advantage in shared randomness processing? Our next result
addresses this question.
Theorem 5. Classically correlated bipartite states will not provide any
advantage in shared randomness processing.
Proof.– A bipartite state
$\rho_{AB}\in\mathcal{D}(\mathcal{H}_{A}\otimes\mathcal{H}_{B})$ is called
classically correlated (CC) if it has a diagonal representation in some
orthogonal product basis, i.e. $\rho_{AB}=\sum_{a,b}p_{ab}|a\rangle_{A}\langle
a|\otimes|b\rangle_{B}\langle b|$, where $\\{|a\rangle_{A}\\}$ is an
orthogonal basis for $\mathcal{H}_{A}$ & $\\{|b\rangle_{B}\\}$ for
$\mathcal{H}_{B}$ and $p_{ab}\geq 0,\leavevmode\nobreak\
\&\leavevmode\nobreak\ \sum_{ab}p_{ab}=1$. Some state has zero quantum discord
- a quantity that captures quantumness present in the correlation of bipartite
state Ollivier and Zurek (2001); Henderson and Vedral (2001). Interestingly,
unentangled state may also possess quantumness as they may have non-zero
quantum discord. For instance, the two-2-quoin $\mathcal{Q}_{p}(2)$ has non-
zero quantum discord for $p\in(0,1]$, whereas it is unentangled/separable
whenever $p\leq 1/3$. For our purpose, without loss of any generality we can
consider the computational basis and hence can represent a two-qubit CC state
as $\rho_{AB}=\sum_{u,v}p_{uv}\leavevmode\nobreak\ |uv\rangle_{AB}\langle
uv|;\leavevmode\nobreak\ u,v\in\\{0,1\\},\leavevmode\nobreak\ p_{uv}\geq
0\leavevmode\nobreak\ \&\leavevmode\nobreak\ \sum_{u,v}p_{uv}=1$. To obtain a
two-d-coin state $\mathcal{C}(d)$, Alice and Bob perform some $d$-outcome
POVMs $\\{\mathcal{M}_{i}^{A}|\sum_{i=1}^{d}\mathcal{M}_{i}^{A}=\mathbb{I}\\}$
and $\\{\mathcal{M}_{i}^{B}|\sum_{i=1}^{d}\mathcal{M}_{i}^{B}=\mathbb{I}\\}$
on their respective subsystems. Probability of clicking the POVM elements
$\mathcal{M}_{i}^{A}\otimes\mathcal{M}_{j}^{B}$ on the state $\rho_{AB}$ is
given by,

$\displaystyle p(ij)$ $\displaystyle=$
$\displaystyle\sum_{u,v=0}^{1}p_{uv}\leavevmode\nobreak\ \langle
u|\mathcal{M}_{i}^{A}|u\rangle\langle v|\mathcal{M}_{j}^{B}|v\rangle.$ (3)

Obviously, $\langle\psi|\mathcal{M}_{i}^{X}|\psi\rangle\geq
0,\leavevmode\nobreak\ \&\leavevmode\nobreak\
\sum_{i=1}^{d}\langle\psi|\mathcal{M}_{i}^{X}|\psi\rangle=1,\leavevmode\nobreak\
\forall\leavevmode\nobreak\ |\psi\rangle;$ $X\in\\{A,B\\}$. This fact leads us
to construct stochastic matrices $\mathcal{S}_{X}^{2\to d}$, with the
elements,

$\left[\mathcal{S}_{X}^{2\to d}\right]_{kl}=\langle
l|\mathcal{M}_{k}^{X}|l\rangle,\text{where}\leavevmode\nobreak\
l\in\\{0,1\\}.$ (4)

Evidently, action of $\mathcal{S}_{A}^{2\to d}\otimes\mathcal{S}_{B}^{2\to d}$
on a classical coin $\mathcal{C}_{p}(2)\equiv p_{00}\leavevmode\nobreak\
\mathtt{hh}+p_{01}\leavevmode\nobreak\ \mathtt{ht}+p_{10}\leavevmode\nobreak\
\mathtt{th}+p_{11}\leavevmode\nobreak\ \mathtt{tt}$ will produce the same
probability statistics of Eq.[3]. Therefore, any $\mathcal{C}(d)$ coin state
generated from any zero-discord two-qubit state, can be freely simulated from
a properly chosen $\mathcal{C}(2)$ coin state. Hence, the quantum advantage in
shared randomness processing necessitates the $\mathcal{Q}(2)$ state to be a
non-zero discordant state.
The above theorem is quite important as it establishes a fundamentally new
application of quantum discord Ollivier and Zurek (2001); Henderson and Vedral
(2001). It would be interesting to explore further quantitative connections
between the measure of discord with the quantum advantage obtained in the
noisy scenario. Noisy scenario becomes even more interesting if we consider
distribution of shared randomness between two parties.

####  .3.1 Quantum advantage in distributing SR

In the non-monopolize social subsidy game we have considered that both the
shared randomness and the strategies of the players are assisted by the
referee. Let us consider now a scenario where the a paired-coin is prepared by
one of the players who wish to distribute, through some communication channel,
its one part to the other player to maximize their payoff. Distribution of the
coin state $\mathcal{C}_{1/2}(2)$ in its exact form requires a perfect binary
channel of capacity $1$-bit. In quantum scenario, a communication channel cab
be most generally describe by completely positive trace preserving maps Kraus
(1983). Let Alice prepares some two-$2$-quoin state
$\mathcal{Q}(2)=\rho_{AB}\in\mathcal{D}(\mathbb{C}^{2}\otimes\mathbb{C}^{2})$
in her laboratory and sends the $B$ part to Bob through a qubit channel
$\Lambda$. They ends up with a two-$2$-quoin state
$\mathcal{Q}^{\prime}(2)=\mathbf{I}\otimes\Lambda\left[\rho_{AB}\right]$,
where $\mathbf{I}$ denotes the identity map (i.e. noiseless process) on the
$A$ part. Then they obtain a classical shared coin from
$\mathcal{Q}^{\prime}(d)$ by applying allowed free operations as suggested by
the referee. In the following we analyze two familiar noisy qubit channels for
achieving better payoff in $\mathbb{G}(n)$.

Qubit phase-flip channel: Its action on an arbitrary state
$\rho\in\mathcal{D}(\mathbb{C}^{2})$ is given by,
$\Lambda^{z}_{p}(\rho):=p\leavevmode\nobreak\ \rho+(1-p)\leavevmode\nobreak\
\sigma_{z}\rho\sigma_{z}$, where $p\in[0,1]$. If Alice sends one part of the
coin state $\mathcal{Q}_{\psi^{-}}(2)=|\psi^{-}\rangle\langle\psi^{-}|$ to Bob
through $\Lambda^{z}_{p}$ then they end up with sharing the state
$\mathcal{Q}^{z}_{p}(2)=\mathbf{I}_{2}\otimes\Lambda^{z}_{p}\left[\mathcal{Q}_{\psi^{-}}(2)\right]=p\leavevmode\nobreak\
\mathcal{Q}_{\psi^{-}}(2)+(1-p)\leavevmode\nobreak\
\mathcal{Q}_{\psi^{+}}(2)$. Applying the same Trine-POVM as in Theorem 4, the
probabilities
$p(ij)=\mbox{Tr}\left[\left(\Pi_{i}\otimes\Pi_{j}\right).\mathcal{Q}^{z}_{p}(2)\right]$
can be represented as a $3\times 3$ matrix

$\mathcal{P}^{z}_{p}(3)\equiv\begin{pmatrix}0&\mu&\mu\\\
\mu&(1-p)\leavevmode\nobreak\ \mu&p\leavevmode\nobreak\ \mu\\\
\mu&p\leavevmode\nobreak\ \mu&(1-p)\leavevmode\nobreak\ \mu\end{pmatrix},$

where $\mu=1/6$. Since the maximum payoff for $\mathbb{G}(3)$ with a perfect
classical channel is $1/8$, the $\Lambda^{z}_{p}$ channel is advantageous
whenever $\beta>3/4$. For the $\mathbb{G}(4)$ case, following the same SIC-
POVM strategy we have,

$\mathcal{P}^{z}_{p}(4)\equiv\begin{pmatrix}0&3\nu&3\nu&3\nu\\\
3\nu&\nu^{\prime}&(1+2p)\nu&(1+2p)\nu\\\
3\nu&(1+2p)\nu&\nu^{\prime}&(1+2p)\nu\\\
3\nu&(1+2p)\nu&(1+2p)\nu&\nu^{\prime}\end{pmatrix},$

where $\nu=1/36$ and $\nu^{\prime}=(1-p)/9$. In this case maximum classical
payoff is $1/15$ which makes $\Lambda^{z}_{p}$ advantageous for more noisy
cases, i.e. whenever $p>7/10$.

Qubit de-polarizing channel: Its action is given by,
$\Lambda^{D}_{p}(\rho):=p\leavevmode\nobreak\ \rho+(1-p)\leavevmode\nobreak\
\frac{\mathbb{I}}{2}$. If Alice prepares $\mathcal{Q}_{\psi^{-}}(2)$ then end
up with sharing the state $\mathcal{Q}^{D}_{p}(2)=p\leavevmode\nobreak\
\mathcal{Q}_{\psi^{-}}(2)+(1-p)\leavevmode\nobreak\
\frac{\mathbf{I}}{2}\otimes\frac{\mathbf{I}}{2}$. A straightforward
calculation, in this case, yields

$\mathcal{P}^{D}_{p}(3)=\begin{pmatrix}\eta&\eta^{\prime}&\eta^{\prime}\\\
\eta^{\prime}&\eta&\eta^{\prime}\\\
\eta^{\prime}&\eta^{\prime}&\eta\end{pmatrix}\leavevmode\nobreak\
\leavevmode\nobreak\ \&\leavevmode\nobreak\ \leavevmode\nobreak\
\mathcal{P}^{D}_{p}(4)=\begin{pmatrix}\delta&\delta^{\prime}&\delta^{\prime}&\delta^{\prime}\\\
\delta^{\prime}&\delta&\delta^{\prime}&\delta^{\prime}\\\
\delta^{\prime}&\delta^{\prime}&\delta&\delta^{\prime}\\\
\delta^{\prime}&\delta^{\prime}&\delta^{\prime}&\delta\end{pmatrix},$

where $\eta=(1-p)/9$, $\eta^{\prime}=(2+p)/18$, $\delta=(1-p)/16$. and
$\delta^{\prime}=(3+p)/48$. Therefore, the channel $\Lambda^{D}_{p}$ is
advantageous in $\mathbb{G}(3)$ [$\mathbb{G}(4)$] whever $p>1/4$ [$p>1/5$].
Importantly $\Lambda^{D}_{p}$ is an entanglement breaking channel whenever
$\beta\leq 1/3$ Werner (1989). Therefore the channel exhibits advantage in
shared randomness distribution even when its quantum capacity is zero Lloyd
(1997); Shor (2002); Devetak (2005). Also recall that classical capacity of
qubit de-polarizing channel is given by
$\chi(\Lambda^{D}_{p})=1-H\left(\frac{1+p}{2}\right)$, where
$H(x):=-x\log_{2}x-(1-x)\log_{2}(1-x)$ is the binary entropy King (2003).
Therefore, quantum advantage is tangible even when the classical capacity of
the quantum channel is much less than unity.

## Discussion

Advent of quantum information theory identifies list of practical instances
where application of quantum rules provide advantage over the classical
physics. Our work lengthens this list by establishing quantum advantage in
shared randomness processing. To capture this advantage we formulate the
resource theoretic framework for shared randomness processing. We also show
precedence of quantum channel over its classical counterpart in distributing
shared randomness. Such advantage is quite remarkable if we recall look back
into some fundamental no-go results that limits the power of quantum system in
classical information processing. For instance, Holevo theorem limits the
classical capacity of a quantum channel Holevo (1973) whereas the recent no-go
result by Frenkel-Weiner Frenkel and Weiner (2015) limits classical
information storage capacity in a quantum system. However, our work establish
that a quantum channel can surpasses a classical channel in shared randomness
processing even when its classical capacity is much less than the
corresponding classical counterpart. More surprisingly, the quantum advantage
endures for extreme noisy channels that lag its most prominent quantum
signature, the quantum capacity. A discussion of our Theorem 1 in connection
with the seminal Bell’s theorem is worth mentioning. Recall that Bell’s
theorem establishes that there exists quantum correlation that cannot have a
classical like (local realistic) description BELL (1966). Quite analogously
our theorem shows that a two-$d$-quoin can yield shared randomness that cannot
be obtained from a two-$d$-coin. Importantly, while Bell’s theorem involves
more than one measurements on the each part of the spatially separated systems
and hence requires the assumption of ‘measurement independence’ Hall (2010);
Barrett and Gisin (2011), our result only invokes one measurement on each side
and thus free of this particular assumption.

Our work addresses and raises a number of important questions regarding
utility of nonclassical origin of randomness which will be of interest to the
broader community of quantum foundations and quantum information. First of
all, the class of monotones, completely characterizing the (im)possibility of
conversion between two shared randomness resources, is still missing.
Advantage of two-2-quoins and noisy qubit channels in shared randomness
processing and its distribution, demonstrated in this work necessitates
further characterization of quantum resources providing such preeminence. Our
work also serves as a stepping stone towards the rich potentiality of
accomplishing quantum advantage in randomness processing for higher
dimensional and multipartite scenarios.

Acknowledgement: We gratefully acknowledge the discussions with G. Kar. MA
acknowledges support from the CSIR project 09/093(0170)/2016- EMR-I. SSB is
supported by the National Natural Science Foundation of China through grant
11675136, the Foundational Questions Institute through grant FQXiRFP3-1325,
the Hong Kong Research Grant Council through grant 17300918, and the John
Templeton Foundation through grant 60609, Quantum Causal Structures. The
opinions expressed in this publication are those of the authors and do not
necessarily reflect the views of the John Templeton Foundation. MB
acknowledges research grant of INSPIRE-faculty fellowship from the Department
of Science and Technology, Government of India.

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