# Manuscript Title:
Twist-tunable spin control in twisted bilayer bismuthene

Ludovica Zullo
<EMAIL_ADDRESS>Department of Physics, University of Trento, Via
Sommarive 14, 38123 Povo, Italy  Sorbonne Université, CNRS, Institut des
Nanosciences de Paris, UMR7588, F-75252 Paris, France  Dipartimento di Fisica
“E. Pancini”, Università degli Studi di Napoli “Federico II”, Complesso
Universitario M. S. Angelo, via Cintia 21, 80126, Napoli, Italy  Domenico
Ninno  Dipartimento di Fisica “E. Pancini”, Università degli Studi di Napoli
“Federico II”, Complesso Universitario M. S. Angelo, via Cintia 21, 80126,
Napoli, Italy  CNR-SPIN, c/o Complesso Universitario M. S. Angelo, via Cintia
21, 80126, Napoli, Italy  Giovanni Cantele<EMAIL_ADDRESS><EMAIL_ADDRESS>CNR-SPIN, c/o Complesso Universitario M. S. Angelo,
via Cintia 21, 80126, Napoli, Italy

###### Abstract

Twisted bilayer structures have emerged as a fascinating arena in condensed
matter thanks to their highly tunable physics. The role of spin-orbit coupling
(SOC) in twisted bilayers has gained increasing attention due to its potential
for spintronics. Thus, it is appealing to propose new materials for
constructing twisted bilayers with substantial SOC. In this work, the
intriguing effects induced by twisting two layers of two-dimensional
bismuthene are unraveled from large-scale first-principles calculations. We
show that spin-orbit coupling significantly affects the electronic properties
of twisted bilayer bismuthene, even more than in its untwisted counterpart. We
carefully investigate how the interplay between the spin-orbit coupling and
the twist angle impacts the band structure and spin textures of twisted
bilayer bismuthene. We find that the twist angle can be deemed a control knob
to switch from a small-gap semiconductor to a metallic behavior. Most
crucially, the accurate analysis of the energy bands close to Fermi energy
reveals a twist-tunable splitting in the mexican-hat shape of the bands that
can otherwise be obtained only by applying enormous electric fields. Our
predictions provide insight into innovative bismuth-based technologies for
future spintronic devices.

††preprint: APS/123-QED

##  I Introduction

Spin-orbit coupling (SOC) effect is the primary player in the field of
spintronics due to its inner connection with the spin degree of freedom. In
order to achieve spin manipulation, it is compelling to control the SOC-driven
effects such as Dresselhaus [1] and Rashba [2], leading to quantum spin Hall
effect, spin–orbit torque, topological insulators and and many others, for
future two-dimensional (2D) electronics [3].

So far, the most efficient strategies for engineering SOC into 2D devices,
rely on external electric and magnetic fields tuning [4, 5]. The van der Waals
nature of 2D materials offers a wide variety of control knobs such as strain,
stacking, and number of layers, that have been extensively studied to produce
electrical and magnetic tunability [6, 7].

Among them, the twist angle between layers has sparked a high interest in the
2D materials community as a unique platform to control a wide range of
physical phenomena. Several intriguing phenomena arise from the moiré pattern
as a consequence of twist, such as quantum spin Hall effect [8],
unconventional superconductivity and flat bands [9], correlated electronic
phase, and spin-polarized phases [10, 11]. Thus, recently, twistronics has
been proposed as a brand new way to obtain SOC modulation and enforcement in
systems like graphene/transition metal dichalcogenides (TMDs) hetero bi- and
tri-layers [12, 13, 14] and TMDs/CrI3 [15].

However, these promising results depend on proximity effect induced by the
different nature of layers composing the heterostructures. Twisted group-IV
buckled atomic layer have been proposed to obtain current-induced spin
polarization (CISP) [16]. Nevertheless, the search for a homostructure with a
high SOC to provide spin order control via twisting is still ongoing.

Bismuth bulk is the heaviest group-V semimetal. Being a quasi-layered crystal
composed of buckled layers, it can be easily cleaved or grown along the (111)
crystalline plane [17]. The Bi(111) surface displays topologically nontrivial
electronic properties [18] associated with the presence of metallic surface
states [19] that can be tuned by strain [20] and thickness [21].

Bismuthene is the 2D bismuth single layer (1L) obtained from the Bi (111)
surface. This compound has a huge SOC effect leading to Rashba split bands
[22, 23, 24]. In bilayer bismuthene, inversion symmetry is restored by
stacking two layers, while SOC shapes the bands in a mexican hat (MH)-like
dispersion. Rashba and MH dispersion can be tuned in bismuthene by means of
strain, electric field [25] and sublattice half-oxidation (SHO) [26]. A moiré
heterostructure of bismuthene/Black phosphorous has been shown modulating the
topological edge states of bismuthene [27], while a study on bismuthene
twisted homostructure is still missing.

In this work, by means of extensive first principles calculations, we unveil
for the first time the electronic properties of twisted bilayer bismuthene
(TB-Bi). We design the phase diagram of TB-Bi, predicting the twist-driven
semiconductor to metal transition by considering a wide range of big twist
angles. Most crucially, we find the possibility of tune SOC-driven MH
parameters in twisted bilayer bismuthene. Interestingly, in addition to the
flattening process, we find that the twist angle acts similarly to an external
electric field, shaping the bands. Finally, we illustrate how spin textures
evolve with twisting in comparison with the untwisted bilayer case, with and
without the effect of an external electric field. Our findings highlight the
potential of twisted bilayer bismuthene as a candidate for future experimental
realizations of twisted strong SOC materials for spintronics.

This paper is organized as follows: in Sec. II we describe the technical
details of our first principles calculations. In Sec. III we present
geometrical and electronic properties of untwisted bilayer bismuthene. We
discuss the modulation of the MH parameters with the inclusion of an external
electric field. In Sec. IV we present the geometrical and electronic
properties of twisted bilayer bismuthene. We argue how the twist angle behaves
similarly to an external electric field shaping the bands and the spin
textures. Finally, in Sec. V we present our conclusions.

##  II Methods

Figure 1: Geometrical properties of untwisted and twisted bilayer bismuthene.
Red and blue atoms represent the first and the second layer respectively. Both
structures are enlarged to a $3\times 3\times 1$ supercell to better highlight
their pattern. a) Side and top view of AA stacked untwisted bilayer
bismuthene. $\mathbf{a_{1}}$ and $\mathbf{a_{2}}$ are the in-plane lattice
vectors defining the primitive unit cell, while the out-of-plane distances, as
detailed in the text, are labelled as $\Delta_{M}$, $\Delta_{B}$ and
$\Delta_{l}$. b) Moiré pattern generated by a twist angle of
$5.0858$°corresponding to the ($7$,$6$) supercell. The moiré periodicity is
described by the lattice vectors $\mathbf{L_{1}}$ and $\mathbf{L_{2}}$.

We perform first principles calculations in the framework of density
functional theory (DFT), as implemented in the Vienna Ab Initio Package (VASP)
[28], version 6.02.

We select two different projected augmented wave (PAW) pseudopotentials, in
the generalized gradient approximation (GGA) based on the Perdew-Burke-
Ernzerhof (PBE) scheme. The first one contains only the $5$ Bi valence
electrons ($6s^{2}6p^{3}$), while the second one comprises $15$ electrons,
taking into account also the last Bi closed ($5d^{10}$) shell. The kinetic
energy cutoff for the plane-wave basis set is set to be $110$ and $250$ eV
respectively. We include the closed ($5d^{10}$) shell in the valence electrons
during the geometry optimization step in every structure under consideration,
while band structure and density of states (DOS) calculations are carried out
using only the $6s^{2}6p^{3}$ electrons. Indeed, as shown in the Supplemental
Material (SM) [29], neglecting the $5d^{10}$ levels, that lie deep below $-20$
eV, results in no significant change in electronic electronic properties
around the Fermi energy, which are dominated by the $p$-orbital character.
This procedure allows us to include SOC in the electronic properties, even for
lower twisted angles which would be otherwise computationally very demanding.

In order to sample the Fermi surfaces of the structures, a Gaussian smearing
of 0.02 eV is applied. The Brillouin zone of the untwisted films is sampled by
$8\times 8\times 1$ Monkhorst-Pack $k$-point grid, while the one of the
twisted structures is scaled according to the dimension of the supercell to
keep the sampling accuracy nearly constant. Considering the $z$-axis as the
stacking direction for the structure, we keep the supercell size along that
direction fixed for all the analyzed structures. A $\sim 14$ Å vacuum space is
added along the out-of-plane direction, proven to be sufficient to prevent
periodic replicas from interacting with each other. For all the considered
structures, the atomic positions are optimized using the conjugate gradient
algorithm in the PBE [30] scheme, with a convergence criterion such that the
total energy difference between consecutive structural optimization steps is
less than $10^{-4}$ eV and all components of the forces acting on the atoms
must be less than $10^{-3}$ eV/Å.

In order to take into account van der Waals interactions between layers, non
local Hamada’s rev-vdW-DF2 functional [31] is employed in the geometry
optimization step in both bilayer and twisted bilayer structures.

##  III Untwisted Bilayer Bismuthene and the effect of the electric field

Figure 2: (a) Band structure of untwisted 2L bismuthene along the
K-$\Gamma$-M-K path in the two-dimensional hexagonal first Brillouin zone
(BZ). Red dashed line corresponds to the Fermi energy. (b) Same as in (a)
under the effect of an external electric field of $0.6$ eV/Å. (c-d) Spin
textures of the two topmost valence bands shown in (a). (e-f) Spin textures of
the two topmost valence bands shown in (b). In (c-f) the color bar corresponds
to the expectation value of $\hat{S}_{z}$ (yellow spin up and violet spin
down). The direction of the arrows corresponds to the in-plane components of
the spin, while the magnitude of the arrows is proportional to
$\sqrt{S_{x}^{2}+S_{y}^{2}}$. Red lines correspond to constant energy
contours, blue lines highlight the first BZ, blue dots highlight K and M
points at the boundary of the first BZ. Figure 3: Evolution of the MH
parameters under an external electric field (ranging $0.1$-$0.6$ eV/Å). (a)
Sketch of the anisotropic MH. The four edges energies are depicted, labeled as
E${}_{MH1}^{+/-}$ and E${}_{MH2}^{+/-}$, where the superscripts $+$ and $-$
define upper and lower peaks (associated to the two different valence bands)
within the same path, respectively, and the subscripts 1 and 2 identify the
peaks along the $\Gamma-K$ and $\Gamma-M$ paths, respectively. (b) MH edge
energies as a function of the external electric field. (c) MH splitting of
each peak $\Delta
E^{+/-}_{MH1,2}=\left|E^{+}_{\mathrm{MH1,2}}-E^{-}_{\mathrm{MH1,2}}\right|$ as
a function of the external field.

We start discussing the properties of untwisted bilayer (2L) bismuthene. Let
us preliminary notice that in a material, as time reversal symmetry holds
($E(k,\uparrow)=E(-k,\downarrow)$), any energy band is degenerate with respect
to a simultaneous flipping of $k$-vector and spin. Moreover, if also the
inversion symmetry holds, as it happens in the case of a 2L, an additional
degeneracy with respect of $k$-vector flipping is present at fixed spin
component, i.e. $E(k,\uparrow\downarrow)=E(-k,\uparrow\downarrow)$. In other
words, the combination of the time reversal and inversion symmetry leads to
the Kramer’s spin degeneracy, that is, $E(k,\uparrow)=E(k,\downarrow)$. As a
consequence, spin degenerate bands cannot be split in a 2L.

In the case of huge SOC 2D materials with inversion symmetry breaking, such as
bismuthene, Rashba effect shows up in the monolayer limit. Conversely, bilayer
bismuthene inherits inversion symmetry from the bulk, and thus the band
structure assumes a MH shape. In this configuration, as we will discuss, each
individual band is twice degenerate in terms of spin degree of freedom, even
in the presence of SOC. It has been shown that this degeneracy can be lifted
by application of an external electric field or with a surface charge transfer
[25].

Bilayer bismuthene consists in two buckled single layers stacked along the
[111] rhombohedral crystallographic direction, with four atoms in the unit
cell. Each monolayer possesses the peculiar 2D buckled honeycomb lattice of
group-V pnictogens: as depicted in Fig. 1, the primitive cell is composed of
two atoms alternately in an upper and a lower plane, the spacing between which
is the buckling distance $\Delta_{B}$. The buckling derives from the weak
$\pi-\pi$ bonding making advantageous the dehybridization of $sp^{2}$ planar
bonding and the formation of $sp^{3}$-like hybridization. It must be pointed
out that the buckling conformation is inherited by the bulk counterpart, as
the freestanding bismuthene stabilizes by buckling. Beside that, during the
synthetization procedure, monolayer bismuthene becomes flat as a result of the
interaction with the substrate, whose effect is not considered in this work
[22, 24]. In the 2L we can define two more distances $\Delta_{M}$ and
$\Delta_{I}$ (see Fig. 1) which respectively correspond to the minimum
distance along the $z$-axis between two Bi atoms and the interlayer distance
($\Delta_{I}=\Delta_{B}+\Delta_{M}$) (the optimized geometrical parameters are
reported in the SM [29]).

Previous studies have demonstrated that the stacking influences the dynamic
stability of 2L bismuthene [23], and thus we choose to focus on the most
stable configuration, the AA stacking. As detailed later, in order to build
the twisted bilayer, we started from the AA stacking in which each Bi atoms of
the first layer have the same in-plane coordinates of the Bi atoms of the
second layer (Fig. 1).

To gain insight into the effect of the twist angle in the electronic
properties of bismuthene, we started from the untwisted bilayer, evaluating
the band structure with and without the application of an external electric
field. The band structure of untwisted 2L bismuthene along the $K-\Gamma-M-K$
path in the two-dimensional first Brillouin zone (BZ) is presented in Fig. 2
in the PBE+SOC framework. Untwisted 2L is a perfectly compensated semi-metal:
the Fermi level crosses both the valence band maxima (VBMs) peaks situated
along the $K-\Gamma$ and $\Gamma-M$ paths while the conduction band maximum
(CBM) is along the $\Gamma-M$ path.

As mentioned above, untwisted 2L bismuthene is characterized by a MH shape in
the valence bands. We label the characteristic parameter of the MH dispersion
in the valence band as the energy EMH, corresponding to the energy difference
between each peak and the valence band energy at $\Gamma$ point. Similarly,
kMH labels the distance in reciprocal space of each peak from $\Gamma$ point.
It turns out that, as already reported in the literature [32], 2L bismuthene
shows an anisotropy in the MH shape, since the heights of the peaks are
slightly different in energy along the K-$\Gamma$ and the $\Gamma$-M paths of
the first BZ. As depicted in Fig. 3, we take into account the anisotropy by
defining two different energies, namely EMH1 and EMH2, where the subscripts 1
and 2 refer to the peaks along the K-$\Gamma$ and the $\Gamma$-M paths,
respectively. Moreover, for each peak, a superscript $+$ or $-$ is needed to
label the upper and the lower valence bands, respectively, in the cases in
which their degeneracy is removed. For example, $E_{MH1}^{+}$ will refer to
the energy of the upper valence band at the first peak (along the K-$\Gamma$
segment).

For the untwisted bilayer, the $\Gamma$-M peak is slightly higher then the
K-$\Gamma$ one, namely $E_{MH1}=62$ meV and $E_{MH2}=69$ meV (the two valence
bands are degenerate), while the radius of the MH is k${}_{MH}=0.01599$ Å-1.

By introducing an external electric field, the twofold degeneracy of the band
of the untwisted bilayer bismuthene is lifted, as shown in Fig. 2(b) which
leads to band splitting. We label the four MH edge energies as
E${}_{MH1}^{+/-}$ and E${}_{MH2}^{+/-}$, the apex $+/-$ defining upper/lower
peaks within the same path, while $\Delta E^{+/-}$ is the splitting introduced
by the external field of each peak (Fig. 3(a)).

We study the evolution of the MH parameters with the modulation of the
external electric field for a range of $0.1$-$0.6$ eV/Å. We observe a linear
increase of the splitting for both peaks with the field intensity (Fig. 3(b)).
However, the magnitude of the splitting is different for the two peaks as
$\Delta E_{MH1}^{+/-}>\Delta E_{MH2}^{+/-}$, for each value of the electric
field (Fig. 3(c)). In other words, the applied electric field enhances the
anisotropy of the peaks. The spin-split bands along the K-$\Gamma$ path are
shrunk by a tiny factor of $6\times 10^{-4}$ Å-1 in $k$ space with respect to
the initial $k_{MH}$ value. These results are in good agreement with Liu and
coworkers [25], and represent the starting point to study the effect of twist,
as we will see in the next section.

Spin textures in bismuthene are significant due to their relationship with the
material topological properties and its prospective use in spin-based
electronics. As an example, a recent work found that adjusting bulk-edge
interactions in planar bismuthene results in persistent edge spin textures,
even under heavy external electric fields [33]. In Fig. 2(c-d) we report the
calculated spin textures of the two topmost valence bands in the case of
untwisted bilayer Bi. The modifications of these spin textures under the
effect of an external electric field of 0.6 eV/Å is reported in Fig. 2(e-f).

As we can see from the figure, the spins in the MH shaped bands are bound to
circulate in the in-plane directions in most of the BZ as a result of the
strong spin-orbit coupling, while around $K$ and $K^{{}^{\prime}}$ the spin
are purely out-of-plane. The spin textures in the zero field case show a
vortex-like pattern around $K$ and $K^{\prime}$ with alternating helicity,
whereas an hexagonal pattern shows up around $\Gamma$. Electric field can be
employed to tune the spin textures upon switching the electric polarization
[34]. Indeed, by switching on an electric field of $0.6$ eV/Å, the spin
textures are modified as follows. A swap in helicity occurs between the two
topmost valence bands, together with the formation of triangular slices inside
the $M-\Gamma-M^{{}^{\prime}}$ region of the BZ, each one with a small hole
resembling a petal, forming a flower-shape around $\Gamma$.

In both cases, the two topmost valence bands display the same spin textures,
with flipped spin, and thus no exotic phenomena occurs. This picture is
modified by twisting as reported in the next section.

##  IV Twisted Bilayer Bismuthene

After characterizing the untwisted bilayer, we study twisted bilayer
bismuthene (TB-Bi) systems up to six different twist angles. The method to
build a commensurate cell starting from two equal AA-stacked layers of
hexagonal bismuthene lattices is the following. We rotate around an axis
orthogonal to the layers and passing through two Bi atoms, one on top of the
other, belonging to the two planes (that therefore preserve their initial AA
stacking).

Figure 4: Twist-tunable band structure of TB-Bi along the K-$\Gamma$-M-K path
in the two-dimensional hexagonal first mini BZ (mBZ). The red box corresponds
to the gap for insulating systems, while the red dashed line corresponds to
the Fermi energy for metallic systems.

As shown in literature [35], we can define the moiré lattice primitive vectors
$\mathbf{L_{1}}$ and $\mathbf{L_{2}}$ starting from the lattices of the two
layers as:

$\mathbf{L_{1}}=n\mathbf{a_{1}^{(1)}}+m\mathbf{a_{2}^{(1)}}=n^{\prime}\mathbf{a_{1}^{(2)}}+m^{\prime}\mathbf{a_{2}^{(2)}}$

where $\mathbf{a_{1,2}^{(i)}}$ with $i=1,2$ are the primitive lattice vectors
of the first and the second bismuthene layers respectively. $\mathbf{L_{2}}$
is then obtained by rotating $\mathbf{L_{1}}$ by $60\degree$. The two pairs of
integer indices $(n,m)$ and $(n^{\prime},m^{\prime})$ can be set to be equal,
so that the TB pattern is specified by a single pair of moiré indices $(n,m)$.
The twist angle $\theta$ can be derived as a function of the latter as:

$\cos{\theta}=\frac{1}{2}\frac{n^{2}+m^{2}+4nm}{n^{2}+m^{2}+nm},$

whereas the lattice constant $L=|\mathbf{L_{1}}|=|\mathbf{L_{2}}|$ is:

$L=a\sqrt{n^{2}+m^{2}+nm}=a\frac{|n-m|}{2\sin{\theta/2}}$

and thus, the length of the moiré cell $L$ is inversely proportional to the
twist angle $\theta$. The moiré pattern of the lowest twist angle in this work
is depicted in Fig. 1(b). A summary of the moiré parameters of the system
under consideration is summarized in Table 1.

$(n,m)$ |  $\theta$ (°) | $N$ |  $a$ (Å)
---|---|---|---
(2,1) | 21.7868 | 28 | 11.6942
(3,2) | 13.1736 | 76 | 19.2663
(4,3) | 9.4300 | 148 | 26.8858
(5,4) | 7.3410 | 244 | 34.5213
(6,5) | 6.0090 | 364 | 42.1641
(7,6) | 5.0858 | 508 | 49.8109
Table 1: Calculated geometric parameters for TB-Bi. The integers $n$ and $m$
are the moiré indices, $\theta$ is the twist angle, $N$ is the number of atoms
in the primitive cell, $a$ is the supercell lattice parameter.

The band structures of TB-Bi for the six considered twist angles are shown in
Fig. 4 along the $K-\Gamma-M-K$ path of the mini-BZ in the PBE+SOC scheme. At
first glance, we can immediately infer that a semimetal to a narrow band-gap
semiconductor transition occurs. Indeed, going from the untwisted bilayer, in
which Fermi level intersects both the VBM and the CBM, to the first twist
angle (namely $(2,1)$ supercell, $\theta=21.7868$°) a small indirect gap of
$E_{g}=$18 meV opens up, with the VBM located along the $K-\Gamma$ line and
the CBM located at $\Gamma$. The next twist angle, $\theta=13.1736\degree$,
corresponding to the $(3,2)$ supercell, follows the trend, displaying a
semiconductor behaviour with an indirect band gap increased to $E_{g}=30$ meV.
Based on these results, we can conclude that the large-twist angle systems
show a twist-driven indirect band-gap tunability, together with the typical
band flattening mechanism that occurs in twisted bilayers. This becomes
evident comparing the band dispersion of the two topmost valence bands of
panels (a) and (b) in Fig. 4.

An abrupt inversion of the trend then occurs when switching to the third
supercell $(4,3)$ with twist angle $\theta=9.4300$°. There is a crossover
between valence and conduction bands that are now overlapped. Moreover, the
two last flat valence bands greatly split: the higher-in-energy valence band
crosses and exceeds the Fermi level along the $K-\Gamma$ and $M-K$ path, while
the lower-in-energy valence band remains below the Fermi level for every
k-points. We can infer that the third twist angle is the critical one, from
which onwards twisted bilayer bismuthene displays metallic behaviour. Indeed,
the next three twist angles, corresponding to the supercells $(5,4)$, $(6,5)$
and $(7,6)$ are metallic.

We can visualize the results by building a twist-tunable phase diagram. This
can be done by plotting the density of states (DOS) per unit cell at the Fermi
level $D(E_{F})/N_{Atoms}$ as shown in Fig. 5(a). The DOS at the Fermi level
is zero for the first two angles, while a peak appears in the $(4,3)$
supercell. A similar metallic behaviour shows up for the other twisted
bilayers, with smaller twist angles, with the DOS per atom increasing as the
twist angle decreases. As confirmed by comparing the band structures in Fig.
5, this corresponds to the filling of the bottom conduction bands, and should
results in an enhanced electrical conductivity as the twist angle decreases.

Another remarkable feature of the band structures of Fig. 4 is that a band
flattening mechanism occurs upon twisting. This concerns, in particular, the
two topmost (SOC-split) valence bands. However, these two bands maintain the
SOC-driven MH shape in the $K-\Gamma$ and $M-K$ paths. The emergence of flat
bands appears at small twist angles in twisted bilayer graphene, and it is
ascribed to the combined action of moiré superlattices and interlayer
interactions [9, 10, 36, 37, 38]. However, it has been demonstrated that flat
bands can also form at twist angles greater than a few degrees, for example in
twisted bilayer hexagonal boron nitride [39], and two-dimensional indium
selenide [40]. Furthermore, a recent study demonstrated that it is possible to
realize continuously tunable superflat bands due to the presence of localized
states around the AA stacking regions in general twisted bilayer systems [40].

The signature of the tunability of the electronic properties with twisting
does not rely only on the semiconducting or metallic behavior of the
considered structures. To better assess this circumstance, let us recall that
in untwisted $2$L bismuthene, in the absence of SOC, each individual band has
a two-fold spin degeneracy. As the SOC is turned on, the degeneracy is lifted,
as it is evident from the band splitting at all twisted angles depicted in
Fig. 4. Remarkably, the twofold degeneracy lifting arises without any external
applied electric field. A similar mechanism occurs in few-layer Bi2Se3 and
PtSe2 heterostructures as a result of the band-alignment, with layer dependent
spin-splittings [41]. In our case, we can associate this effect to the
combination of a strong SOC and the breaking of the centrosymmetric geometry
of the untwisted $2$L induced by the moiré superlattice. For systems that in
the $2L$ form preserve a centrosymmetric geometry the band splitting in the
presence of SOC can be induced only by an external perturbation, such as an
electric field. In the present case, instead, twisting plays the role of a
“pseudo electric field”, making the two layers not equivalent. The analogy may
be stretched even further since, as we will see in the following, the amount
of splitting can be controlled with the electric field intensity, as well as
with the twist angle. To this purpose, we can compare the evolution of MH
parameters as a function of the twist angle with the electric field-induced
band splitting, previously discussed for the untwisted bilayer. Our results
are shown in Fig. 5(b-c). We observe that for large twist angles (small
supercells), the magnitude of the splitting tends to increase until it reaches
a maximum. Again, we can define the angle $\theta=9.43^{\circ}$ as the
critical angle after which we observe a change in the MH parameters trend.
After that (that is, for smaller twist angles and larger supercells) the
previously discussed metallic behaviour is accompanied with a decrease of the
splitting. A second point that is noteworthy to stress is that, as in the case
of a bilayer under external electric field (compare with Fig. 3(c)), the
electric field-induced change in the amplitude of the splitting is different
for the two peaks. Moreover, an “enhanced” anisotropy effect shows up when
compared with the untwisted bilayer, in that $k_{MH1}\neq k_{MH2}$, that is,
the two peaks arise at different distances from $\Gamma$ point. This can be
attributed to a twist-induced anisotropy effect, because twisting causes both
a rotation and a resizing of the BZ (compared to the untwisted one). In the
untwisted supercell, energy peaks along the $K-\Gamma$ path are lower in
energy than those along the $\Gamma-M$ path. Twisted bilayers display the
opposite behaviour.

Therefore, we can conclude that what mostly distinguishes the twist-induced
changes of the MH peaks from the case of the untwisted bilayer under external
electric field is the following. As the BZ shrinks upon decreasing the twist
angle, the positions of the peaks in $k$ space changes, with a consequent
increasing of the MH anisotropy, as further detailed in the SM [29].
Consequently, not only the twist angle acts as an electric field tuning the
SOC-induced splittings, but also it can be used as a control knob to tune
anisotropy in the $k$ space.

Figure 5: (a) Twist-tunable phase diagram of TB-Bi. Squares corresponds to the
DOS per atom at the Fermi level $D(E_{F})/N$ (with $N$ number of atoms in the
twisted supercell). (b-c) Evolution of the MH parameters, as defined in Fig. 3
with twisting. The MH edge energies as a function of $N$ are shown in (b),
whereas the splitting of each peak $\Delta E^{+/-}_{MH1,2}$ introduced by the
twist angle is depicted in (c).

It must be pointed out that in their work Scammel and Scheurer [42] show that
SOC-split bands can be induced by a displacement field that breaks inversion
symmetry in mirror-symmetric twisted trilayer graphene heterostructure with
TMDs. Here, no external parameter is included, since SOC splitting arises from
twisting the heavy Bi atoms.

Figure 6: Twist-tunable spin textures of TB-Bi in the two-dimensional
hexagonal mBZ. The results are shown for the two topmost valence bands of
(a-b) ($2$,$1$), (c-d) ($3$,$2$) and (e-f) ($4$,$3$) supercells, respectively.
The color bar corresponds to the expectation value of $\hat{S}_{z}$ (yellow
spin up and violet spin down). Direction of the arrows corresponds to the in-
plane components of the spin, while the magnitude of the arrows is
proportional to $\sqrt{S_{x}^{2}+S_{y}^{2}}$. Red lines correspond to constant
energy contours, blue lines highlight the first mBZ, blue dots highlight K and
M points at the boundary of the first mBZ.

We conclude showing the spin textures evolution in TB-Bi upon twisting. It is
known in literature that in the case of twisted bilayer graphene, spin
textures are strongly influenced by the twist chirality and angle, as well as
the doping level, leading to peculiar shapes [43, 44]. In Fig. 6 we show the
calculated spin textures for the first three twist angles of TB-Bi. First of
all, we can clearly see that the spin textures are highly twist-tunable, since
the spin pattern dramatically changes with twisting.

Twisting not only changes spin texture, but also causes considerable
anisotropy compared to electric field-induced splittings in untwisted bilayer.
Indeed, as Figs.2(c-f) reveal, the spin patterns in the previous case show a
perfect symmetry between otherwise (that is, in the absence of the external
field) degenerate bands. These bands exhibit the same spin texture in the
whole BZ, with the only difference being reverting the direction of the spin
at each $k$ point. On the other hand, such a symmetry gets broken in the
twisted bilayers, due to the discussed anisotropy of the valence bands. In
particular, the patterns of Fig. 6 clearly reveal that the paired bands do not
exhibit spin textures that can be exactly obtained from each other by
reverting the sign of the spin vector at each $k$-point. To support such a
circumstance, we can also notice that the “asymmetry” between the bands that
would be paired in the untwisted bilayer changes with the twist angle.
Therefore, we can speak about twist-tunable spin textures, with the twist
offering, with respect to the electric field, the further possibility of
breaking the symmetry between the two valence bands. This effect could be
influenced by the flattening mechanism induced by the twist, and it is not
achievable by the action of an external electric field. This result shows that
it is possible not only to tune the spin textures by twisting but also to
select two different spin configurations within the same twist angle thanks to
the MH modification induced by the flattening, a peculiar behaviour that could
be exploited for spintronic devices.

##  V Conclusions

By performing extensive first-principles electronic structure calculations
based on DFT, we predicted the electronic and spin properties of twisted
bilayer bismuthene (TB-Bi). To allow a better assessment of the effect of
twisting, the untwisted bilayer properties have been carefully investigated,
with a particular focus on the MH shape of the topmost valence bands induced
by the SOC. The twofold spin degeneracy of these bands can be broken after the
introduction of an external electric field. We commented on the evolution of
the MH parameters (peak energies and splittings) as a function of the field.
The analysis of TB-Bi reveals that twisting might act as a “pseudo” electric
field, in that it breaks the symmetry of the topmost valence bands. In
addition, combined with SOC, the twist angle acts as a tool to tune and
control the MH parameters. Remarkably, twisting induces an anisotropy of the
MH, that shows up both in the position in $k$ space of the peaks and in their
energies. These results show that twist angle can be employed in this context
as a control knob for the band splitting, without relaying on any external
control, which is usually achievable only with the application of a huge
electric field. We derived the phase diagram of TB-Bi by evaluating the
evolution of DOS at the Fermi level. We showed that a semimetal to a narrow
band-gap semiconductor transition occurs going from the untwisted case to the
first two larger twisted angles. We identified $\theta=9.43^{\circ}$ as the
critical angle after which the flattening mechanism induces a metal behaviour
for the smaller twist angles.

Finally, we showed that it is possible to tune the spin textures of bismuthene
by twisting. We unveil an exotic behaviour of the spin patterns induced by the
asymmetry of the MH shape, as a result of the twisted-induced flattening
mechanism. This work propose bismuthene as a new candidate for twistronics,
being a system with potentially high built-in spin control for future
spintronic devices.

## Acknowledgments

We acknowledge the CINECA award under the ISCRA initiative, for the
availability of high-performance computing resources and support. Financial
support and computational resources from MUR, PON "Ricerca e Innovazione
2014-2020", under Grant No. PIR01_00011 (I.Bi.S.Co.) are acknowledged.

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# Revisiting Common Randomness, No-signaling and Information Structure in
Decentralized Control

Apurva Dhingra11footnotemark: 1 Center of Machine Intelligence and Data
Science, CMInDS, Indian Institute of Technology Bombay, Mumbai 400076
<EMAIL_ADDRESS>Ankur A. Kulkarni22footnotemark: 233footnotemark:
344footnotemark: 4<EMAIL_ADDRESS>Systems and Control Engineering,
Indian Institute of Technology Bombay, Mumbai 400076  Center of excellence in
Quantum Information, Computing Science, and Technology, QuICST, Indian
Institute of Technology Bombay, Mumbai 400076

###### Abstract

This work revisits the no-signaling condition for decentralized information
structures. We produce examples to show that within the no-signaling polytope
exist strategies that cannot be achieved by passive common randomness but
instead require agents to either share their observations with a mediator or
communicate directly with each other. This poses a question mark on whether
the no-signaling condition truly captures the decentralized information
structure in the strictest sense.

###### keywords:

no-signaling, decentralized control, common randomness, information structure

###### MSC:

[2010] 00-01, 99-00

††journal: System and Control letters

##  1 Introduction

We consider a decentralized team of agents bound by a static information
structure. These agents have access only to their own information, without in-
game access to a central coordinating device. The static information structure
demands that agents cannot affect the observations of other agents, implying
that agents cannot communicate with each other either. Such scenarios arise,
for example, in sensor networks like fire alarms.

In such a setting, the agents must devise a strategy – a conditional
probability of choosing an action profile given an observation profile – for
optimizing a certain cost. Three main types of strategies exist:
deterministic, behavioral, and local. Under a deterministic or pure strategy,
agents choose their actions with certainty given their observations. In a
behavioral strategy agents locally and independently randomize over actions,
conditioned on their observations. Lastly, in a local strategy agents have
access to an external passive common randomness along with their own
observation, and using these the agents choose their actions by locally and
independently randomizing. All these strategies respect the information
structure, since there is no in-game communication.

Another way of thinking of the information structure is as follows: the
essence of the static information structure is that the information of the
agent must be a sufficient statistic for its action. Consequently, any
strategy must respect that an agent’s action given its information must be
independent of the other agents’ information. In physics, this mathematical
condition is called the ’no-signaling condition’, and is a universal law
requiring that no information can travel instantaneously through space. There
seems to be a belief in the control and physics communities that this no-
signaling condition is equivalent to the requirement of the information
structure, i.e., that the agents do not communicate during the game.

A seminal work of Anantharam and Borkar in decentralized control [1] takes
this belief forward. It asks if every no-signaling strategy can be generated
using passive common randomness. They show that there exist strategies within
the no-signaling polytope that cannot be created with any amount of common
randomness. In other words, passive common randomness has limitations in
accessing all distributions in the no-signaling polytope. Their finding
emphasizes that specific strategies within no-signaling polytope are
infeasible using mechanisms like behavioral and local strategies. This opens
the question of whether there are more general mechanisms that respect no-
signaling and information structure requirement; indeed there are quantum
strategies, [2, 3, 4],that do meet this requirement.

Our contribution in this paper is a re-look at the no-signaling condition, and
asking if it does indeed capture the information structure. It is quite clear
that satisfaction of the no-signaling condition is necessary for the
information structure to hold. Our focus is on the converse. We demonstrate
that within the no-signaling polytope, strategies exist that do not respect
the information structure. In particular there are no-signaling strategies
that can only be created using active common randomness, i.e., a coordinating
device that is dependent on the observations of both agents. Such coordination
allows for indirect exchange of information between agents, and is a violation
of the information structure. We show also that the above no-signaling
strategies that cannot be created using passive common randomness, can be
created if the passive common randomness is combined with one-way
communication between the agents. In other words, if the information structure
allows agents to play such strategies, it is effectively allowing them access
to either an active coordinating device, or one-way communication. In summary,
these results show that all strategies within the no-signaling polytope are
not necessarily feasible as per the ‘no communication’ requirement of the
information structure. A strategy that satisfies no-signaling conditions can
still violate the static information structure.

We release this work with a tinge of diffidence: our results are shown by
numerical construction of counterexamples and protocols, and the mathematics
involved is not at all deep. However, to the best of our understanding, our
results expose that the control community must either define the information
structure requirements more broadly to match up with the no-signaling
condition, or accept that the no-signaling conditions capture much more than
the requirement of ‘no communication’ in the strictest sense. They also
highlight the need for further examination and a deeper understanding of the
no-signaling condition.

This article has been organized as follows. Section 2 provides the background
on the decentralized control setting. Section 3 includes the results of
Anantharam and Borkar [1]. It also features a schematic representation of the
geometry of no-signaling. Section 4 presents the our main contributions, while
Section 5 presents a ‘posterior’ perspective on no-signaling. We conclude in
Section 6 with a short discussion.

##  2 Background

###  2.1 Notations

We use $(\Omega,\mathcal{F},P)$ to denote a probability space, with $\Omega$
as the sample space, $\mathcal{F}$ as event space and $P$ as the probability
measure. All random variables in this paper are discrete and all nonempty
events take positive probability. We use capital letters $X,Y,$ and so on, to
denote random variables, small letters $x,y$ to denote their realizations and
the calligraphic letters $\mathcal{X},\mathcal{Y}$ to denote the space where
they take their values. We let ${\cal P}({\cal X})$ denote the set of
probability distributions on ${\cal X}$. For $P\in{\cal P}({\cal X})$ we use
$P(x)$ to denote the probability $P(X=x)$. When we want to make a statement
for all realizations $x$ of $X$, we use $P(X)$. With slight abuse of notation,
we use $P(X)$ to also denote the probability distribution of $X$. Similarly,
$P(X|A)$ denotes conditional probability of $X$ given $A$, and $P(x|a)$ is a
particular realization with $X=x$ and $A=a$. The notation
‘$X\perp\\!\\!\\!\\!\perp Y\ |A$’ stands for ‘$X$ and $Y$ are independent
given $A$’. We use $I(W;A,B)$ to denote the mutual information between $W$ and
$(A,B)$. Similarly, $I(X;B|A)$ denotes the conditional mutual information
between $X$ and $B$ given $A$ (more details on properties of mutual
information in [7]). For $a,b\in\\{0,1\\}$, $a\oplus b$ denotes addition
modulo 2, and $a.\ b$ denotes logical AND (multiplication) operation.

###  2.2 Common randomness and decentralized control

Consider a decentralized system consisting of two agents, Venkat and Vivek.
Each agent observes a random variable, denoted as $A\in\mathcal{A}$ and
$B\in\mathcal{B}$ respectively, distributed according to a joint distribution
$P(A,B)$. The agents are required to choose their actions, denoted as
$X\in\mathcal{X}$ and $Y\in\mathcal{Y}$ based on their private observations. A
strategy for Venkat and Vivek is defined as a conditional joint distribution
$P(X,Y|A,B)$ of their actions $X,Y$ given their observations $A,B$.

These strategies satisfy the positivity and normalization conditions of any
conditional probability distribution, and must also respect the information
structure of the problem. The decentralized information structure demands that
Venkat and Vivek do not communicate with each other after receiving their
observations $A,B$. For some strategies, Venkat and Vivek may be helped by
common randomness $W$ drawn from some finite set ${\cal W}$ provided by an
agent Hans555Through the choice of agent names, we wish to pay tribute to
three of our scientific heroes: Venkat Anantharam, Vivek Borkar and Hans
Witsenhausen.. Notice that we have not defined what it means to “communicate”,
and indeed as we shall see in this paper, the root of our main observations
lies precisely in defining what amounts to communication. For now, we shall
stick to the colloquial meaning of communication.

This leads to various strategic classes for Venkat and Vivek defined by
specific operational mechanisms that they can employ.

####  2.2.1 Behavioural strategies $\mathscr{B}$:

In a behavioural strategy, Venkat and Vivek choose $X,Y$ upon observing $A,B$
by locally and independently randomizing. In other words, $P$ satisfies,

$\displaystyle P(X,Y|A,B)$ $\displaystyle=P(X|A)P(Y|B).$ (1)

The set of all such strategies is denoted by $\mathscr{B}$. One can see that
(1) is equivalent to the requirements

$\displaystyle X$ $\displaystyle\perp\\!\\!\\!\\!\perp Y\ |A,B,$
$\displaystyle X$ $\displaystyle\perp\\!\\!\\!\\!\perp B\ |A,$ $\displaystyle
Y$ $\displaystyle\perp\\!\\!\\!\\!\perp A\ |B.$

The last two requirements above comprise the no-signaling condition, a subject
of our focus in this paper. We discuss them further in Section 2.3.

####  2.2.2 Local strategies $\mathscr{L}$:

A local strategy $P(X,Y|A,B)$ is any strategy for which there exists
$W\in{\cal W}$, independent of $(A,B)$ with probability distribution $P(W)$,
such that

$\displaystyle P(X,Y,A,B,W)$ $\displaystyle=\ P(W)P(A,B)P(X|A,W)\ P(Y|B,W).$
(2)

The above factorization is equivalent to,

$\displaystyle X$ $\displaystyle\perp\\!\\!\\!\\!\perp Y\ |A,B,W,$
$\displaystyle X$ $\displaystyle\perp\\!\\!\\!\\!\perp B\ |A,W,$ (3)
$\displaystyle Y$ $\displaystyle\perp\\!\\!\\!\\!\perp A\ |B,W,$
$\displaystyle W$ $\displaystyle\perp\\!\\!\\!\\!\perp(A,B).$ (4)

We thus get the strategy,

$P(X,Y|A,B)=\sum_{W\in{\cal W}}P(W)P(X|A,W)P(Y|B,W).$ (5)

The mechanism that creates a local strategy is as follows. Before heading to
the field to observe $A,B$, Venkat and Vivek are given access to $W$, drawn
according to $P(W)$. When in the field, they make use of their knowledge of
$W$ and their observations to choose their actions by locally and
independently randomizing (as in strategies in $\mathscr{B}$). Hans produces
$W$ without reference to $(A,B)$; hence $W$ is independent of $(A,B)$. The set
of all local strategies forms a polytope, called the local polytope, denoted
as $\mathscr{L}$. In fact $\mathscr{L}$ is the convex hull of $\mathscr{B}$
(see [8], [2] for more details).

The random variable $W$ is called common randomness. If, as above, $W$ is
independent of $(A,B)$, then $W$ is called passive, otherwise we refer to $W$
as active. For passive common randomness $W$,

$\displaystyle P(W,A,B)$ $\displaystyle=P(W)P(A,B),$ (6)

while for active common randomness, this equation does not hold for some
$a\in{\cal A},b\in{\cal B},w\in{\cal W}$.

###  2.3 No-signaling strategies $\mathscr{NS}$

We now come to the question of information structure, or equivalently
communication, and its meaning. Concretely, we ask: what is the set of all
strategies that are allowed by the information structure of our problem? In
stochastic control, it is common to define the allowed strategy space by means
of explicit mechanisms, as in $\mathscr{B}$ and $\mathscr{L}$. One can of
course define specific mechanisms that respect the information structure, but
what is to say that more general mechanisms cannot be devised? Indeed more
general mechanisms are known. A case in point are those that deploy entangled
quantum systems to create randomness, see e.g., [2]. These too respect the
information structure and describe a set strictly larger than $\mathscr{L}$.
While that does push the boundary, it still fails to answer what is the
universal set of strategies that Venkat and Vivek can play?

This brings us to one of the central topics of this paper, the no-signaling
condition.

######  Definition 2.1.

A strategy distribution $P(X,Y|A,B)$ is said to be no-signaling if Venkat’s
action $X$ is independent of $B$ given $A$, and Vivek’s action $Y$ is
independent of $A$ given $B$, i.e.,

$\displaystyle P(X|A,B)=P(X|A),$ $\displaystyle P(Y|A,B)=P(Y|B).$ (7)

These are called no-signaling conditions and we denote by $\mathscr{NS}$ the
set of all strategies that satisfy (7).

When players cannot communicate with each other, each player’s observation
must be a sufficient statistic for its action. This is precisely the condition
demanded by (7). It appears plausible that $\mathscr{NS}$ is in fact the set
of strategies that respect the information structure. Indeed in [1]666p. 570,
first column at the bottom of the page, Anantharam and Borkar say,

> The salient characteristic of the distributed creation of the pair $(X,Y)$
> from $(A,B)$ is that $X$ is created with access to $A$ but without reference
> to $B$ and $Y$ is created with access to $B$ but without reference to $A$.
> Thus it is natural to conjecture that for every $(X,Y,A,B)$ satisfying the
> conditions,
>
> $\displaystyle(A,B)$ $\displaystyle\sim P(A,B),$ $\displaystyle I(X;B|A)$
> $\displaystyle=0,$ (8) $\displaystyle I(Y;A|B)$ $\displaystyle=0,$
>
> there must exist a $W$ (on a possibly augmented sample space) such that
> $(X,Y,A,B,W)$ satisfy conditions (2)777From the present paper.

Note that (8) is equivalent to (7). Anantharam and Borkar proceed to give a
counterexample to the above conjecture. We discuss this counterexample in the
next section. For now we note that Anantharam and Borkar are implicitly
implying that every strategy in $\mathscr{NS}$ is allowed by the information
structure; indeed it is worthwhile seeking a $W$ as above only if one believes
that every strategy in $\mathscr{NS}$ is a valid strategy for the problem. Any
strategy that respects the information structure of the problem is indeed no-
signaling (it is easy to verify that
$\mathscr{B}\subseteq\mathscr{L}\subseteq\mathscr{NS}$; quantum strategies
from [2] are also no-signaling). However, Anantharam and Borkar seem to have
in mind that the converse also holds.

The term ‘no-signaling’ originated in physics as a universal law that cannot
be violated by any physical system. In physics too there seems to be a similar
belief. In [9, p. 80, first paragrpah of Section 4.4], Laloe says,

> The theory of relativity requires that it should be fundamentally impossible
> to transmit signals containing information between two distant points faster
> than the speed of light (relativistic causality); suppressing this absolute
> limit would lead to serious internal inconsistencies in the theory.
>
> $\ldots$
>
> From a general point of view, and without restricting the discussion to
> quantum mechanics, what are the general mathematical conditions ensuring
> that a theory is “non-signaling” (NS conditions), meaning that it does not
> allow Alice to transmit instantaneous signals to Bob (and conversely)?
>
> $\ldots$
>
> When the experiment is repeated, since Bob does not have access to Alice’s
> results, the only thing that he can measure is the occurrence frequency of
> his own results. This corresponds to the probabilities obtained by a
> summation over $X$ of the preceding probabilities (sum of probabilities
> associated with exclusive events):
>
> $\sum_{X}P(X,Y|A,B)$
>
> The NS condition amounts to assuming that this probability is independent of
> $a$; we therefore obtain the condition:
>
> $\sum_{X}P(X,Y|a,b)=\sum_{X}P(X,Y|a^{\prime},b)\ \ \forall\ b.$

Note that the above equation is in fact Eq (7).

From the above quotes it is apparent that the view in the scientific community
at large is that $\mathscr{NS}$ is the set of strategies allowed by the
information structure. Our main finding is that things are not as simple.
There are strategies that allow Venkat and Vivek to exchange information that
are also included in $\mathscr{NS}$. These include strategies where active
common randomness is provided by Hans, and it is exploited by Venkat and Vivek
in choosing their actions. Another possible protocol is using passive common
randomness and classical one-way communication from Venkat to Vivek.

##  3 No-signaling with passive common randomness

###  3.1 Anantharam and Borkar’s conjecture and counterexample

As mentioned in the previous section, Anantharam and Borkar seem to take the
view that every strategy in $\mathscr{NS}$ is allowable in the information
structure. Taking this thought forward, they next ask what mechanism may the
agents use to achieve strategies in $\mathscr{NS}$. They make (the very
reasonable) conjecture that every point in $\mathscr{NS}$ is achievable
through passive common randomness.

######  Conjecture 3.1.

[1] Given any strategy $P(X,Y|A,B)$ that satisfies the no-signaling condition
(7), there exists some passive common randomness $W$ and a joint distribution
on $(X,Y,A,B,W)$ such that $P(X,Y|A,B)$ satisfies (5).

Anantharam and Borkar present a counterexample to this conjecture. We recount
it below.

Example 3.1.  This is a counterexample to Conjecture 3.1.

$(a,b)$ |  |  | $P(x,y|a,b)$ |  |  | $(a,b)$ |  |  | $P(x,y|a,b)$ |
---|---|---|---|---|---|---|---|---|---|---
|  | $\frac{1}{3}$ | 0 | 0 |  |  |  | 0 | $\frac{1}{3}$ | 0
(1,1) |  | 0 | $\frac{1}{3}$ | 0 |  | (1,0) |  | $\frac{1}{3}$ | 0 | 0
|  | 0 | 0 | $\frac{1}{3}$ |  |  |  | 0 | 0 | $\frac{1}{3}$
|  | 0 | $\frac{1}{3}$ | 0 |  |  |  | 0 | 0 | $\frac{1}{3}$
(0,1) |  | 0 | 0 | $\frac{1}{3}$ |  | (0,0) |  | 0 | $\frac{1}{3}$ | 0
|  | $\frac{1}{3}$ | 0 | 0 |  |  |  | $\frac{1}{3}$ | 0 | 0

In this table, the rows of $P(X,Y|A,B)$ are for index $x\in{\cal
X}=\\{0,1,2\\}$ and columns for $y\in{\cal Y}=\\{0,1,2\\}$. For this example
$P(a,b)=\frac{1}{4}\ \forall a,b$.

The distribution $P(X,Y|A,B)$ shown in Table 3.1 follows the no-signaling
conditions (7). Specifically, $P(x|a,b)=\frac{1}{3}\ \forall\ x,a,b$ whereby
$P(X|A,B)\equiv P(X|A)$ and similarly $P(Y|A,B)=P(Y|B)$. However, it is
impossible to find a common randomness $W$ such that the joint distribution
factorizes as (2). Details of this can be found in [1]. $\square$

This counterexample implies that there are strategies within $\mathscr{NS}$
that can not be accessed using common randomness. Anantharam and Borkar view
this result as a limitation of common randomness.

Below we present another counterexample to Conjecture 3.1, showing that the
conjecture is false even for binary spaces.

Example 3.2.  This is another counterexample to Conjecture3.1 with strategy
distribution $P(X,Y|A,B)$ shown as in 3.1.

$(a,b)$ |  | $P(x,y|a,b)$ |  |  | $(a,b)$ |  | $P(x,y|a,b)$ |
---|---|---|---|---|---|---|---|---
|  | $\frac{1}{2}$ | $0$ |  |  | $\frac{1}{2}$ | $0$ |
$(0,0)$ |  | $\frac{1}{3}$ | $\frac{1}{6}$ |  | $(0,1)$ | $0$ | $\frac{1}{2}$ |
|  | $\frac{1}{2}$ | $0$ |  |  | $0$ | $\frac{1}{2}$ |
$(1,0)$ |  | $\frac{1}{3}$ | $\frac{1}{6}$ |  | $(1,1)$ | $\frac{1}{2}$ | $0$ |

Here, $x\in\\{0,1\\}$ is represented row-wise and $y\in\\{0,1\\}$ are indexed
column-wise in the strategy distribution $P(X,Y|A,B)$, where
$a,b\in\\{0,1\\}$. Note that the no-signaling condition is satisfied
$P(x|a,b)\ =\ \frac{1}{2}\ \forall\ x,a,b\ $ and $\ P(y|b,a)=p(y|b)$.

This example satisfies the no-signaling condition (7), as shown below,

$\displaystyle P(x|a,b)$ $\displaystyle=P(x|a)=\frac{1}{2}\forall a,b,x$ (9)
$\displaystyle P(y|a,b)$ $\displaystyle=\ \begin{cases}\frac{1}{2}\
\;\textrm{if}\;b=1\ \forall\ y,a\\\ \frac{1}{6}\ \;\textrm{if}\;y=1,b=0\
\forall\ a\\\ \frac{5}{6}\ \;\textrm{if}\;y=0,b=0\ \forall\ a\end{cases}$ (10)
$\displaystyle=P(y|b)\ \forall\ y,b$

A proof that this counterexample lies in $\mathscr{NS}\backslash\mathscr{L}$
is shown in Section 7. $\square$

In the following section we discuss the geometry of no-signaling distributions
[10] and non-local quantum correlations [6] as described below.

####  3.1.1 Geometry of no-signaling and Counter-example 3.1

The geometry of the $\mathscr{NS}$ polytope becomes significantly more
complicated with increasing size of the spaces ${\cal A},{\cal B},{\cal
X},{\cal Y}$. We give a description of this geometry for the case where these
sets are binary. For such action and observation spaces the extreme points of
$\mathscr{NS}$ are well understood and classified as local and non-local [6],
as described below.

  * 1.

Local vertices: These refer to the extreme points of the local polytope,
$\mathscr{L}$, or deterministic strategies in $\mathscr{L}$ . There are a
total of 16 local extreme points expressed as,

$\displaystyle P(X,Y|A,B)$ $\displaystyle=\begin{cases}&1\ ,\ X=\alpha\ A\
\oplus\ \beta\ ,\ \ Y=\gamma\ B\ \oplus\ \delta\\\ &0\
,\text{else}\end{cases}$ (11)

where $\alpha,\beta,\gamma,\delta,\in\\{0,1\\}$ and $x\in{\cal X},y\in{\cal
Y},a\in{\cal A},b\in{\cal B}$.

  * 2.

Non-local vertices: These refer to the extreme points of the no-signaling
polytope, $\mathscr{NS}$, excluding the local vertices. There are total 8 non-
local vertices that can be expressed in the form,

$\displaystyle P(X,Y|A,B)$ $\displaystyle=\begin{cases}&\frac{1}{2}\ ,\ X\
\oplus Y=A\ .\ B\oplus\alpha\ A\oplus\beta\ B\ \oplus\ \gamma\\\ &0\
,\text{else}\end{cases}$ (12)

where $\alpha,\beta,\gamma,\in\\{0,1\\}$ and $x\in{\cal X},y\in{\cal
Y},a\in{\cal A},b\in{\cal B}$.

Note, all of the local vertices are also extreme points of the $\mathscr{NS}$,
and $\mathscr{L}\subset\mathscr{NS}$. This geometry is shown in the schematic
diagram in Figure 1 as per [6],[2].

Figure 1: A schematic representation of no-signaling polytope. L and NL label
the vertices for local and non-local points. The set bounded by local vertices
is $\mathscr{L}$ and overall is the no-signaling polytope $\mathscr{NS}$.
Example 3.1 represents a distribution
$P(X,Y|A,B)\in\mathscr{NS}\backslash\mathscr{L}$.

##  4 No-signaling with active common randomness

As evident so far, there are no-signaling strategies that lie outside
$\mathscr{L}$. This begs the following question: if Venkat and Vivek were
allowed access to active common randomness, what strategies would they
achieve? We investigate this in the following section.

###  4.1 No-signaling with active common randomness

Consider the following mechanism involving active common randomness. Hans is
allowed to view the realized value of the pair $(A,B)$ and generates a $W$
whose distribution depends on this value. The realized value of $W$ is
informed to Venkat and Vivek, but they are not informed of the realized value
of $(A,B).$ Venkat and Vivek then set out into the field where they receive
realized values of $A,B$ respectively. Thus in the field, Venkat has access to
$A,W$ and Vivek has access to $B,W.$ They now generate $X,Y$ by independently
and locally randomizing using this information, without any communication with
each other. Under this mechanism, their strategy $P(X,Y|A,B)$ takes the form

$P(X,Y|A,B)=\sum_{W}P(W|A,B)P(X|A,W)P(Y|B,W).$ (13)

This factorization satisfies the first three conditions from (3). Indeed, it
appears that access to a $W$ that depends on $(A,B)$ would have allowed Venkat
and Vivek to gain access to some bits of each other’s information and thereby
the no-signaling condition would now be violated. Or will it? We show below
that there are strategies for which the no-signaling condition still holds,
even when players have access to active common randomness. In fact, we show
something stronger: there are strategies in $\mathscr{NS}$ for which active
common randomness is essential, i.e., these strategies can be generated with
access to active common randomness but cannot be generated with any passive
common randomness.

In other words, we show that the following conjecture is false.

######  Conjecture 4.1.

If $P(X,Y|A,B)$ satisfies (13) with active common randomness $W$ and
$P(X,Y|A,B)\notin\mathscr{L}$, then $P(X,Y|A,B)$ does not belong to
$\mathscr{NS}$.

####  4.1.1 Counterexample of Conjecture 4.1:

The strategy $P(X,Y|A,B)$ shown in Example[3.1] serves as our counterexample.
Note that $P(X,Y|A,B)\in\mathscr{NS}\backslash\mathscr{L}$, whereby it
suffices to construct a $W$ such (13) holds for this strategy.

We consider active common randomness $W$ with $\mathcal{W}=\\{1,2,3,4\\}$. We
take $P(w|a,b)$ as

$\displaystyle P(w|a=0,b=1)$ $\displaystyle=\frac{1}{4}=P(w|a=1,b=1)\ \forall\
w$ $\displaystyle P(w|a=1,b=0)$ $\displaystyle=\
\begin{cases}&{\small\frac{1}{2}},\ w=1\\\ &{\small\frac{1}{3}},\ w=2\\\
&{\small 0},\ w=3\\\ &{\small\frac{1}{6}},\ w=4\end{cases}$ $\displaystyle
P(w|a=0,b=0)$ $\displaystyle=\ \begin{cases}&{\small\frac{1}{3}},\ w=1\\\
&{\small\frac{1}{2}},\ w=2\\\ &{\small\frac{1}{6}},\ w=3\\\ &{\small 0},\
w=4\end{cases}\ $ (14)

whereby $W$ is clearly active. As we want to satisfy (3), that can also be
written as,

$P(x,y|a,b,w)=P(x|a,b,w)P(y|a,b,w)=P(x|a,w)P(y|b,w),$

it is enough to define the marginal distributions $P(x|a,w),P(y|b,w)$. For
ease of exposition, we represent $P(x|a,w)$ as a column vector where rows are
indexed as $x=\\{0,1\\}.$

$\displaystyle P(x|a=0,w=1,3)=\begin{bmatrix}0\\\
1\end{bmatrix}=P(x|a=1,w=2,4)$ $\displaystyle
P(x|a=1,w=1,3)=\begin{bmatrix}1\\\ 0\end{bmatrix}=P(x|a=0,w=2,4)$ (15)

For each fixed $x,a,w$, we define $P(x|a,b,w)=P(x|a,w)\ \forall\ b.$
Similarly, we define $P(y|a,b,w)=P(y|b,w)\ \forall\ a$. The marginal
distribution $P(y|b,w)$ is shown below, represented as a row vector with each
column relating to $y\in\\{0,1\\}$.

$\displaystyle P(y|b=0,w=1,2)=\begin{bmatrix}1&0\end{bmatrix}=P(y|b=1,w=2,4)$
$\displaystyle P(y|b=1,w=1,3)=\begin{bmatrix}0&1\end{bmatrix}=P(y|b=0,w=3,4)$
(16)

We now check that (13) holds. For any fixed $a,b$ we express $P(x,y|a,b)$ as a
matrix where rows represent $x\in\\{0,1\\}$ and columns represent
$y\in\\{0,1\\}$. Note, the product of $P(y|b,w)P(x|a,w)$ is treated as an
outer product between a column and row vector, for the sake of representation.
Consider $a=1$ and $b=1$, using (4.1.1) - (16), we have

$\displaystyle\sum_{w\in{\cal W}}P(x$
$\displaystyle|a=1,w)P(y|b=1,w)P(w|a=1,b=1)$
$\displaystyle=\frac{1}{4}\sum_{w\in{1,2,3,4}}P(x|a=1,w)P(y|b=1,w),$
$\displaystyle=\frac{1}{4}\left(\begin{bmatrix}1\\\
0\end{bmatrix}\begin{bmatrix}0&1\end{bmatrix}+\begin{bmatrix}0\\\
1\end{bmatrix}\begin{bmatrix}1&0\end{bmatrix}+\begin{bmatrix}1\\\
0\end{bmatrix}\begin{bmatrix}0&1\end{bmatrix}+\begin{bmatrix}0\\\
1\end{bmatrix}\begin{bmatrix}1&0\end{bmatrix}\right)$
$\displaystyle=\frac{1}{2}\begin{bmatrix}0&1\\\ 1&0\end{bmatrix}$
$\displaystyle=P(x,y|a=1,b=1).$

The last equality can be inferred by inspection with the top-left matrix in
the table in Example 3.1. Similarly, for $a=0$ and $b=1$,

$\displaystyle\footnotesize\sum_{w\in{\cal W}}P(x$
$\displaystyle|a=0,w)P(y|b=1,w)P(w|a=0,b=1)$
$\displaystyle=\frac{1}{4}\sum_{w\in{1,2,3,4}}P(x|a=0,w)P(y|b=1,w),$
$\displaystyle=\frac{1}{4}\left(\begin{bmatrix}0\\\
1\end{bmatrix}\begin{bmatrix}0&1\end{bmatrix}+\begin{bmatrix}1\\\
0\end{bmatrix}\begin{bmatrix}1&0\end{bmatrix}+\begin{bmatrix}0\\\
1\end{bmatrix}\begin{bmatrix}0&1\end{bmatrix}+\begin{bmatrix}1\\\
0\end{bmatrix}\begin{bmatrix}1&0\end{bmatrix}\right)$
$\displaystyle=\frac{1}{2}\begin{bmatrix}1&0\\\
0&1\end{bmatrix}=P(x,y|a=0,b=1),$

as required. Now for $a=0$ and $b=0$ we have,

$\displaystyle\sum_{w\in{\cal W}}P(x$ $\displaystyle|a=0,w)P(y|b=0,w)P(w|a,b)$
$\displaystyle=\sum_{i\in\\{1,2,3\\}}P(x|a=0,w=i)P(y|b=0,w=i)P(w=i|a,b)$
$\displaystyle=\frac{1}{3}\begin{bmatrix}0\\\
1\end{bmatrix}\begin{bmatrix}1&0\end{bmatrix}+\frac{1}{2}\begin{bmatrix}1\\\
0\end{bmatrix}\begin{bmatrix}1&0\end{bmatrix}+\frac{1}{6}\begin{bmatrix}0\\\
1\end{bmatrix}\begin{bmatrix}0&1\end{bmatrix}$
$\displaystyle=\begin{bmatrix}\small\frac{1}{2}&0\\\
\small\frac{1}{3}&\small\frac{1}{6}\end{bmatrix}=P(x,y|a=0,b=0).$

Similarly for $a=1$ and $b=0$, we have

$\displaystyle\sum_{w\in{\cal W}}P(x$ $\displaystyle|a=0,w)P(y|b=1,w)P(w|a,b)$
$\displaystyle=\sum_{i\in\\{1,2,4\\}}P(x|a=0,w=i)P(y|b=1,w=i)P(w=i|a,b)$
$\displaystyle=\frac{1}{2}\begin{bmatrix}1\\\
0\end{bmatrix}\begin{bmatrix}1&0\end{bmatrix}+\frac{1}{3}\begin{bmatrix}0\\\
1\end{bmatrix}\begin{bmatrix}1&0\end{bmatrix}+\frac{1}{6}\begin{bmatrix}0\\\
1\end{bmatrix}\begin{bmatrix}0&1\end{bmatrix}$
$\displaystyle=\begin{bmatrix}\footnotesize\small\frac{1}{2}&0\\\
\small\frac{1}{3}&\small\frac{1}{6}\end{bmatrix}=P(x,y|a=0,b=1).$

Thus, (13) holds for the marginal distributions (15), (16), creating the
Example 3.1. Hence, this counterexample shows that even after giving active
common randomness to the agents the resulting strategy still satisfies the no-
signaling condition. Moreover, this strategy can only be generated using
active common randomness.

####  4.1.2 Where is the communication?

Let us now understand what form of information exchange or communication has
taken place between the agents. To answer this question, one needs to be
precise about the parties involved in the communication and the timeline
according to which parties access information. Recall that we assumed Hans is
informed of $A,B$. How does Hans get access to this information? There are two
possibilities that we can think of:

  1. (a)

In the first possibility, $A,B$ need to be sensed in the field, and the only
agents that can do this sensing are Venkat and Vivek. Communication causally
follows the sensing and it is between Venkat and Hans, and Vivek and Hans,
respectively. $W$ is then realized and is communicated back from Hans to
Venkat and Vivek, respectively. This is in the same spirit as confidential
communication with a mediator in incomplete information games [11]; in this
case $P(W|A,B)$ can be thought of as a Bayesian correlated strategy.

  2. (b)

Another possibility is that Hans may know of $A,B$ through a separate channel.
Here there is communication between the environment that generates $(A,B)$ and
Hans, and one-way communication from Hans to Venkat and Vivek. Venkat and
Vivek do not communicate with each other after they are informed of $W$ and
their observations.

In both cases Venkat and Vivek do not communicate with each other directly. In
case (a) there is two-way private communication with Hans, whereas in case
(b), Hans has separate access to $A,B$. If case (b) is the view taken, then
one must ask why it does not degenerate the decentralized nature of the
problem. Indeed if Hans could access both $A,B$, he may as well signal the
optimal centralized action to the players via $W$. On the other hand if case
(a) is the view of the mechanism, we are compelled to ask if such
communication is permitted as part of the definition of the information
structure. Literally read, the information structure only says that players
“cannot communicate with each other”, leaving a loophole where players could
communicate confidentially with a mediator. The no-signaling condition allows
for strategies where players exploit this loophole. That $\mathscr{NS}$ also
contains such strategies is our first main finding of this paper.

To us, this finding highlights the following:

  1. 1.

In defining the information structure of the problem, one must have clarity on
what amounts to communication and what exact communication is permitted by the
information structure.

  2. 2.

One must accept that the no-signaling condition is not tantamount to “no
communication” in the wide sense of the word “communication”.

Our leaning is that confidential communication as in case (a) is also a
violation of the information structure. For instance, consider that the set of
$W$’s is as large as the set of all observations, i.e., $|\mathcal{W}|=|{\cal
A}\times{\cal B}|$, then when Hans could encode $W$ to include all information
about the pair $(A,B)$, whereby Venkat and Vivek would indirectly know the
observation of the other via $W$, and when they choose their respective
actions $X,Y$, using their own observation and $W$, they have effectively
referenced the observation of other agent. We feel this should not be
permitted by the information structure.

A natural question arises: how does the no-signaling condition not get
violated in spite of this information exchange? The answer to this lies in the
nature of the condition itself. No-signaling is a statistical condition,
because it talks of the probability measure of the random variables involved.
In a frequentist view, this says that on average $X$ must be independent of
$B$ given $A$ (and similarly for $Y$). This does not preclude that there are
sample-paths for which some form of communication has happened. Contrast this
with specific mechanisms which are expressed in terms of random variables
instead.

###  4.2 Strategy within $\mathscr{NS}$ with one-way communication

Consider the same strategy $P(X,Y|A,B)$ as given in Example 3.1, considering
$x,y\in\\{0,1\\}$ and $a,b\in\\{0,1\\}$. We now show a protocol where this
distribution is created by Venkat and Vivek using one-way communication and
passive common randomness. The process is as follows,

  * 1.

Hans produces passive common randomness $W\in\\{1,2,3\\}$, uniformly at
random. He shares the realization of $W$ with Venkat and Vivek.

  * 2.

During the game, Venkat makes his observation $A$. He generates his action $X$
as per the following equation, for $W=1,2$,

$\displaystyle X=\ \begin{cases}A\oplus 1,&\;\textrm{if}\;W=1\\\
A,&\;\textrm{if}\;W=2\end{cases}$ (17)

For $W=3$, the action $X$ is generated uniformly at random, i.e
$X\in\\{0,1\\}$ is equally likely (and hence not dependent on $A$).

  * 3.

Venkat shares the realization of his observation $A$ and the action $X$ with
Vivek when $W=3$, otherwise there is no communication or sharing.

  * 4.

Vivek makes his observation $B$. Based on his available information he
generates his action $Y$ as per the following equation,

$\displaystyle Y$ $\displaystyle=\ \begin{cases}B&\;\textrm{if}\;W=1\\\
0&\;\textrm{if}\;W=2\\\ X\oplus A.B&\;\textrm{if}\;W=3\end{cases}$ (18)

Note: For $W=3$, Venkat has shared $X,A$ with Vivek, so this is the case of
one-way communication.

We now show that this process creates the strategy $P(X,Y|A,B)$ from Example
3.1. We have,

$\displaystyle P(X,Y|A,B)$
$\displaystyle=\sum_{W\in\\{1,2,3\\}}P(X,Y|A,B,W)P(W)$
$\displaystyle=\frac{1}{3}\sum_{W\in\\{1,2,3\\}}P(X,Y|A,B,W).$ (19)

Comparing (17) and (18) with marginal distribution (15), (16) for $w=1,2$, we
note that $P(X,Y|A,B,W)$ for $W=1,2$ here takes the form as shown in Table 1.

Table 1: Distribution $P(x,y|a,b,w=1,2)$ with common randomness $W$

$P(x,y|a,b,w=1)$ | $P(x,y|a,b,w=2)$
---|---
$a=0,b=0$ | $a=0,b=1$ | $a=0,b=0$ | $a=0,b=1$
$\begin{bmatrix}0&0\\\ 1&0\end{bmatrix}$ | $\begin{bmatrix}0&0\\\ 0&1\end{bmatrix}$ | $\begin{bmatrix}1&0\\\ 0&0\end{bmatrix}$ | $\begin{bmatrix}1&0\\\ 0&0\end{bmatrix}$
$a=1,b=0$ | $a=1,b=1$ | $a=1,b=0$ | $a=1,b=1$
$\begin{bmatrix}1&0\\\ 0&0\end{bmatrix}$ | $\begin{bmatrix}0&1\\\ 0&0\end{bmatrix}$ | $\begin{bmatrix}0&0\\\ 1&0\end{bmatrix}$ | $\begin{bmatrix}0&0\\\ 1&0\end{bmatrix}$

Here, $x\in\\{0,1\\}$ represented with row-wise elements and $y\in\\{0,1\\}$
are indexed column-wise in the strategy distribution $P(x,y|a,b,w=1,2)$, where
$a,b\in\\{0,1\\}$

For the case of $W=3$ in this section, $P(X,Y|A,B,W=3)$ can be created as
follows,

  * 1.

For the pair $(a,b)\in\\{(0,0),(0,1),(1,0)\\}$, the term $A.B=0$, so, for
these three cases, $Y=X$ as per (18). Thus for these three cases,

$\displaystyle P(x,y|a,b,w=3)$
$\displaystyle=\mathbb{I}_{\\{x=y\\}}P(x|a,b,w=3)=\mathbb{I}_{\\{x=y\\}}P(x|w=3)$
$\displaystyle=\frac{1}{2}\begin{bmatrix}1&0\\\ 0&1\end{bmatrix}.$ (20)

  * 2.

For the pair $(a,b)=(1,1)$, the term $A.B=1$, so $Y\neq X$.

$\displaystyle P(x,y|a,b,w=3)$ $\displaystyle=\mathbb{I}_{\\{x\neq
y\\}}P(x|a,b,w=3)=\mathbb{I}_{\\{x\neq y\\}}P(x|w=3)$
$\displaystyle=\frac{1}{2}\begin{bmatrix}0&1\\\ 1&0\end{bmatrix}.$ (21)

Using (4.2), we want to verify that the $P(X,Y|A,B)$ matches Example (3.1).
Consider $(a,b)=(0,0)$, and using the matrix representation again,

$\displaystyle P(x,y|a=b=0)$
$\displaystyle=\sum_{w\in\\{1,2,3\\}}P(x,y|a=b=0,w)P(w)$
$\displaystyle=\frac{1}{3}\left(\begin{bmatrix}0&0\\\
1&0\end{bmatrix}+\begin{bmatrix}1&0\\\
0&0\end{bmatrix}+\frac{1}{2}\begin{bmatrix}1&0\\\ 0&1\end{bmatrix}\right)$
$\displaystyle=\begin{bmatrix}\frac{1}{2}&0\\\
\frac{1}{3}&\frac{1}{6}\end{bmatrix},$ (22)

which is same as top-left matrix of Example (3.1). Next, consider
$(a,b)=(1,1)$, as below,

$\displaystyle P(x,y|a=b=0)$
$\displaystyle=\frac{1}{3}\left(\begin{bmatrix}0&1\\\
0&0\end{bmatrix}+\begin{bmatrix}0&0\\\
1&0\end{bmatrix}+\frac{1}{2}\begin{bmatrix}0&1\\\ 1&0\end{bmatrix}\right)$
$\displaystyle=\begin{bmatrix}0&\frac{1}{2}\\\ \frac{1}{2}&0\end{bmatrix}.$
(23)

Similarly, it can be verified for the remaining $(a,b)$ pairs, using (4.2),
along with Table (1) and (1).

We can once again reflect on the communication involved in this example.
Venkat and Vivek were provided access to only passive common randomness by
Hans, which we noted was insufficient to generate the strategy $P(X,Y|A,B)$.
However they can combine this passive common randomness with explicit, direct
communication to generate this strategy. It is known in games that equilibria
of noncooperative games that are equipped with a noisy communication system
can be simulated by a correlated strategy involving one way communication with
a mediator, and vice-versa [11]. Something analogous to this seems to have
played out here: instead of communicating confidentially with Hans, the
players have used the passive common randomness from Hans as noise and coupled
that with direct communication to produce the same strategy as they did with
confidential communication.

This interchangeability of direct noisy communication and confidential
mediated communication poses a dilemma for ascertaining if an information
structure is respected. Even if one takes the view that confidential mediated
communication is permissible under a decentralized information structure, the
above example shows that such indirect communication is statistically
indistinguishable from certain forms of direct communication (both the
underlying mechanisms produce the same conditional distribution or strategy).
Thus from the strategy alone one cannot say if players have in fact
communicated. This further underlines the need to have a careful (possibly
sample-path based) definition of the information structure.

##  5 No-signaling and posterior distributions

We now consider the no-signaling conditions, i.e., (7) again in an attempt to
interpret them in a somewhat different way.

######  Proposition 5.1.

A strategy $P(X,Y|A,B)$ satisfies the no-signaling conditions (7) if and only
if following posterior condition is satisfied

$P(A|B,Y)=P(A|B),\ \ P(B|A,X)=P(B|A)$ (24)

###### Proof.

Consider forward direction, i.e., no-signaling condition implies posterior,

$P(A|B,Y)=\frac{P(Y|A,B)P(A,B)}{P(B,Y)}=\frac{P(Y|B)\ P(A|B)}{P(Y|B)}=P(A|B)$

Here, we used the no-signaling condition in obtaining the second equation.
Similarly it can be shown for $P(B|A,X)=P(B|A)$.

Conversely, the condition (24) on the posteriors implies the no-signaling
condition,

$P(X|A,B)=\frac{P(B|A,X)P(A,X)}{P(A,B)}=\frac{P(B|A)}{P(B|A)}P(X|A)=P(X|A)$

Similarly it can be shown for $P(Y|A,B)=P(Y|B)$.

We know that the no-signaling condition (7) says that action $X$ given $A$ is
chosen without reference to observation $B$, and similarly for $Y$. The
posterior condition, (24) says that the belief of Vivek about Venkat’s
observation $A$ is independent of Vivek’s action $Y$ given his observation
$B$. In other words, Vivek’s action has no additional “information” to provide
about Venkat’s observation than Vivek’s observation already provides.

Notice in (18) in Section 4.2, the action $Y$ is created using both $A,B$ and
also action $X$. Despite this the overall distribution $P(X,Y|A,B)$ follows
the no-signaling condition and also the posterior condition (24). In this
case, Vivek knows the realization of observation $A$ of Venkat and uses it in
constructing $Y$, and yet, posterior condition (24) holds.

##  6 Discussion

It has been shown through simple counterexample to Conjecture 4.1 that our
current interpretation of the no-signaling condition insofar as describing
information structure in decentralized control, is incomplete. From the above
examples we have constructed, we have seen that the no-signaling condition
allows strategies where direct and indirect communication are performed. A
statistical condition such as no-signaling seems to us to be insufficient to
capture the presence or absence of communication, and to thereby serve as a
boundary on permissible strategies under a decentralized information
structure.

In summary, these counterexamples challenge our assumptions about the scope
and limitations of no-signaling. They call for deeper investigations into the
nature of no-signaling and a more careful definition of information structure
in decentralized control. The question “what is the set of all strategies that
adhere to the information structure” still remains unanswered. Alongside,
another question remains unanswered: what is the meaning of no-signaling if it
is not “no-communication”?

## References

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URL https://link.aps.org/doi/10.1103/PhysRevLett.23.880

##  7 Appendix

We now show that the distribution $P(X,Y|A,B)$ in Example 3.1 is another
counterexample of Conjecture3.1. We wish to prove the claim : ‘the
distribution $P(X,Y|A,B)$ in Example 3.1 lies in
$\mathscr{NS}\backslash\mathscr{L}$.’ The proof of this claim uses the CHSH
inequality [12], that is described in short below.

Consider two agents Venkat and Vivek, with observations $a,b\in\\{0,1\\}$ and
actions $x,y\in\\{0,1\\}$. Then by $\langle(-1)^{x}(-1)^{y}\rangle_{a,b}$
denote,

$\langle(-1)^{x}(-1)^{y}\rangle_{a,b}=\sum_{y\in\\{0,1\\}}\sum_{x\in\\{0,1\\}}P(x,y|a,b)(-1)^{x}(-1)^{y}.$
(25)

The CHSH inequality is given by,

$\displaystyle\langle(-1)^{x}(-1)^{y}\rangle_{0,0}$
$\displaystyle+\langle(-1)^{x}(-1)^{y}\rangle_{0,1}+$
$\displaystyle\langle(-1)^{x}(-1)^{y}\rangle_{1,0}-$
$\displaystyle\langle(-1)^{x}(-1)^{y}\rangle_{1,1}\ \ \leq 2$ (26)

######  Lemma 7.1.

If a distribution $P(X,Y|A,B)$ lies in $\mathscr{L}$, then it must satisfy the
CHSH inequality (26).

###### Proof.

Suppose there exists a passive common randomness $W$, such that $P(X,Y|A,B,W)$
satisfies (3). Then the correlation (25) can be rewritten as,

$\displaystyle\langle(-1)^{x}$
$\displaystyle(-1)^{y}\rangle_{a,b}=\sum_{y\in\\{0,1\\}}\sum_{x\in\\{0,1\\}}\sum_{w}P(x,y|a,b,w)P(w)(-1)^{x}(-1)^{y}$
$\displaystyle=\sum_{w}P(w)\sum_{y\in\\{0,1\\}}(-1)^{y}P(y|a,b,w)\sum_{x\in\\{0,1\\}}P(x|a,b,w)(-1)^{x}$

The LHS of (26) simplifies as

$\displaystyle\sum_{w}P(w)\sum_{y=0}^{1}(-1)^{y}\sum_{x=0}^{1}(-1)^{x}$
$\displaystyle\quad(P(y|0,0,w)P(x|0,0,w)+P(y|1,0,w)P(x|1,0,w)+$
$\displaystyle\quad P(y|0,1,w)P(x|0,1,w)-P(y|1,1,w)P(x|1,1,w))$ (27)
$\displaystyle=\sum_{w}P(w)\sum_{x,y=0}^{1}(-1)^{x+y}(\
P(x|0,0,w)\left(P(y|0,0,w)+P(y|1,1,w)\right)+$ $\displaystyle
P(x|1,1,w)\left(P(y|0,0,w)-P(y|1,1,w)\right)\ )$
$\displaystyle=\sum_{w}P(w)\sum_{x,y=0}^{1}(-1)^{x+y}\left(P(x,y|0,0,w)+P(x,y|1,1,w)\right)$
$\displaystyle\leq\sum_{w}P(w)\sum_{x,y=0}^{1}\left(P(x,y|0,0,w)+P(x,y|1,1,w)\right)=2.$

Here, (27) is simplified using the property that $X\perp\\!\\!\\!\\!\perp
B|A,W$ and $Y\perp\\!\\!\\!\\!\perp A|B,W$. This gives us the CHSH
inequality(26). Note, this proposition and proof is a manifestation of
classical CHSH property as originally studied in [12].

Now coming back to the claim about our Example 3.1.

######  Lemma 7.2.

The distribution $P(X,Y|A,B)$ in Example 3.1 lies in
$\mathscr{NS}\backslash\mathscr{L}$.

###### Proof.

Using (9)-(10), we know that $P(x|a,b)=P(x|a)\ \forall b$ and
$P(y|a,b)=P(y|b)\ \forall a$, so we know that this Example (3.1) lies in
$\mathscr{NS}$. It suffices to show that $P(X,Y|A,B)$ violates the condition
(26). Consider the correlation for $(a,b)=(0,0)$,

$\displaystyle\langle(-1)^{x}(-1)^{y}\rangle_{0,0}$
$\displaystyle=\sum_{y\in\\{0,1\\}}\sum_{x\in\\{0,1\\}}P(x,y|a=0,b=0)(-1)^{x}(-1)^{y}$
$\displaystyle=\frac{1}{2}(1)+\frac{1}{6}(1)+\frac{1}{3}(-1)=\frac{1}{3}.$

Similarly for $(a,b)=(1,0)$, the correlation is
$\langle(-1)^{x}(-1)^{y}\rangle_{1,0}=\frac{1}{3}$. For $(a,b)=(1,1)$,

$\displaystyle\langle(-1)^{x}(-1)^{y}\rangle_{1,1}$
$\displaystyle=\sum_{y\in\\{0,1\\}}\sum_{x\in\\{0,1\\}}P(x,y|a=1,b=1)(-1)^{x}(-1)^{y}$
$\displaystyle=\frac{1}{2}(-1)+\frac{1}{2}(-1)=-1.$

For $(a,b)=(0,1)$,

$\displaystyle\langle(-1)^{x}(-1)^{y}\rangle_{0,1}$
$\displaystyle=\sum_{y\in\\{0,1\\}}\sum_{x\in\\{0,1\\}}P(x,y|a=0,b=1)(-1)^{x}(-1)^{y}$
$\displaystyle=\frac{1}{2}(1)+\frac{1}{2}(1)=1.$

So, the LHS of (26) is

$\frac{1}{3}+1+\frac{1}{3}-(-1)=2+\frac{2}{3}>2,$ (28)

showing that $P(X,Y|A,B)$ in Example 3.1 lies in
$\mathscr{NS}\backslash\mathscr{L}.$