# A study of anisotropic spheres in $f(Q)$ gravity with quintessence field

Sanjay Mandal 0000-0003-2570-2335<EMAIL_ADDRESS>Department of
Mathematics, Birla Institute of Technology and Science-Pilani,
Hyderabad Campus, Hyderabad-500078, India. G. Mustafa<EMAIL_ADDRESS>Department of Physics, Zhejiang Normal University, Jinhua, 321004, People’s
Republic of China. Zinnat Hassan 0000-0002-6608-2075<EMAIL_ADDRESS>Department of Mathematics, Birla Institute of Technology and Science-Pilani,
Hyderabad Campus, Hyderabad-500078, India. P.K. Sahoo 0000-0003-2130-8832
<EMAIL_ADDRESS>Department of Mathematics, Birla Institute
of Technology and Science-Pilani,
Hyderabad Campus, Hyderabad-500078, India.

###### Abstract

This manuscript examines the compact stars in gravity $f(Q)$, where non-
metricity $Q$ drives gravitational interactions. To achieve this goal, we
consider a spherically symmetric spacetime with the anisotropic fluid
distribution. In particular, the quintessence field is used in the energy-
momentum tensor to explore the solution of compact stars. In addition, we
select three specific types of compact stars, namely HerX-1, SAXJ1808.4-3658,
and 4U1820-30. We develop the field equations using the quintessence field of
a specific form of $f(Q)$ as
$f(Q)=Q-\alpha\gamma\left(1-\frac{1}{e^{\frac{Q}{\gamma}}}\right)$ in $f(Q)$
gravity. In addition, we consider the Schwarzschild metric for the matching
conditions at the boundary. Then, we calculate the values of all the relevant
parameters by imposing matching conditions. We provide detailed graphical
analyses to discuss the parameters’ physical viability, namely energy density,
pressure, gradient and anisotropy. We also examine the stability of compact
stars by testing energy conditions, equations of state, conditions of
causality, redshift functions, mass functions, and compactness functions.
Finally, we find that the solutions we obtained are physically feasible in
modified $f(Q)$ gravity, and also it has good properties for the compact star
models.

##  I Introduction

The debate over compact stars has raged on since the accelerated expansion of
the universe was discovered. Compact stars are quite different from ordinary
stars, which are come into existence after the death of ordinary stars. All
the nuclear processes that took place within the cores of ordinary stars have
ended. The gravitational force inside the star is more dominant than the
repulsive force because the temperature decreases inside the star. This
phenomena ends with the compact stars formation, and all the masses of the
stars accumulate in a very short radius. Therefore, compact stars are seen as
high-density masses with a compact radius. Due to numerous problems in
astrophysics, the discussion of compact stars has aroused great interest among
cosmological researchers today. There are many reference points that can be
used to study compact stars; however, there are some ambiguities because the
perfect formation of compact stars remains problematic. For example, we can
analyze the balance of compact stars by discussing the TOV equation proposed
by Tollman-Oppenheimer-Volkoff. In fact, the balance characteristics of TOV
forces, namely anisotropy, gravity, and hydrostatic force, indicate that the
dense object system we are studying is stable and physically acceptable. Many
compelling efforts by researchers can be found in the literature [1].

Despite the great success of Newtonian gravity, it completely failed under
certain circumstances considering strong gravitational effects, such as the
progress of Mercury other than the Mickelson Morey experiment [2]. The General
Theory of Relativity (GR) was developed by Einstein in 1915, which made it
possible to solve the Mercury problem [3]. Since then, GR is considered to be
the basic theory of gravitational physics. However, GR has several
shortcomings, which show that GR is not the complete fundamental theory of
gravity, such as the dark energy problem [4]. Furthermore, GR violated the
Chandrasekhar mass restriction of the Super Chandrasekhar white dwarf and the
sub-Chandrasekhar quality restriction [5]. Furthermore, GR shows
inconsistencies in the strong gravitational field and recent observations [6].
Therefore, look for the appropriate modifications to the GRs. It is a
proactive task. The most successful modification of GR is the theory of
higher-order curvature, especially gravity $f(R)$, which successfully explains
the existence of dark matter and uses observations to counter the theory of
gravity [7]. Besides this, teleparallel gravity and symmetric teleparallel
gravity provide an interesting geometric interpretation of gravity with the
spacetime metric. Recently, symmetric teleparallel gravity ($f(Q)$ gravity)
was proposed by Jiménez et al [8], which is a well-motivated theory of
gravity, where the non-metricity drive the gravitational interaction in
spacetime.

Moreover, the stellar structures study in modified theories of gravity is an
important topic for discussion. In the last few decades, several studies
presented on stellar structures. However, recently, the compact star models in
$f(R)$ gravity was studied by Nashed and Capozziello [10], there the stability
of compact star tested using TOV(Tolman-Oppenheimer-Volkoff) equation, and
their solution matches with the observational data of millisecond pulsars with
a withe dwarf companion and pulsars presenting thermonuclear bursts.
Astashenok and Odintsov [11] investigated the rotating newton stars with axion
field in $f(R)$ gravity and discussed stars with different frequencies and
masses. The researchers used an interesting method to derive the solution of
the field equation in the context of compact objects, the Karmarkar condition.
This condition was first proposed by Karmarkar [12], and it is considered a
mandatory condition for type I embedding in spherically symmetric space-time.
It can help us to find the exact solution of the motion equations. Using the
Karmarkar condition, several works have been done in modified gravity to study
the stellar structures (see details in the references here [13, 14, 15]).
Furthermore, some interesting studies have been done in theories of gravity to
explore the stellar structure solutions [16].

Since $f(Q)$ gravity was proposed, the research on it has been increasing
rapidly. At the same time, it is faced with the observational limitation of
using GR’s standard formula to counter it. Lazkoz et al. impose an compelling
set of restrictions on gravity $f(Q)$. [17], where Lagrange $f(Q)$ is
expressed as a polynomial function of redshift $z$. Using observational data
such as Type Ia supernovae, baryon acoustic oscillation data, quasars, gamma-
ray bursts, and cosmic microwave background distance data, the bounds of these
models have been derived successfully. Moreover, some interesting works have
been done in $f(Q)$ gravity using observational measures recently [18].
Besides this, $f(Q)$ gravity is also developed to study the wormhole solutions
[19], its signature [20], Spherically symmetric configuration [21], and so on.
As this theory was developed recently, it is quite successful in describing
the current scenario of the universe. This success of $f(Q)$ gravity motivates
us to test the compact star’s stability and present physically acceptable
models in the presence of quintessence filed in the fluid description.
Therefore, exploring the stellar structures in the $f(Q)$ gravity background
would be interesting as it is a novel approach in $f(Q)$ gravity. Here, we aim
to explore the stellar structures in $f(Q)$ gravity with the presence
quintessence field.

This manuscript is organized as follows: in Section II, we derive the field
equations in the background of $f(Q)$ gravity with the presence of
quintessence field in the anisotropic fluid description. Also, there we
considered the Krori and Barua gravitational potentials and Herrera and
Barreto equation of state. In Section III, we use the matching condition to
find the values of parameters, which are used in our formulation. In Section
IV, we discuss the physical analysis of our constructed star models such as
HerX-1, SAXJ1808.4-3658, and 4U1820-30. The physical analysis such as energy
density and pressure components under quintessence field in IV.1, equation of
state parameters in IV.2, Gradients in IV.3, anisotropy in IV.4, causality and
Abreu stability analysis in IV.5, energy conditions in IV.6, mass function,
compactness function, and redshift function in IV.7 are discussed to present a
physically viable compact star solutions. Finally, combining all the outcomes,
we have concluded our study for the compact star in Section V.

##  II Model of anisotropic quintessence star in $f(Q)$ gravity

We have considered the activity for symmetric teleparallel gravity as [8]

$\mathcal{S}=\int\left(\frac{1}{2}\,f(Q)\,+\mathcal{L}_{m}\,\right)\sqrt{-g}\,d^{4}x\,,$
(1)

where the matter Lagrangian density represented by $\mathcal{L}_{m}$, $g$ is
the determinant of the metric $g_{\mu\nu}$, and $f(Q)$ addresses the arbitrary
function of $Q$,.
The non-metricity tensor and its traces can be written as

$Q_{\lambda\mu\nu}=\bigtriangledown_{\lambda}g_{\mu\nu},$ (2)
$\tilde{Q}_{\alpha}=Q^{\mu}\;_{\alpha\mu},\,\,Q_{\alpha}=Q_{\alpha}\;^{\mu}\;{}_{\mu}.$
(3)

Also, the superpotential in terms of the non-metricity tensor can be written
as

$P^{\alpha}\;_{\mu\nu}=\frac{1}{4}\left[-Q^{\alpha}\;_{\mu\nu}+2Q_{(\mu}\;^{\alpha}\;{}_{\nu)}+Q^{\alpha}g_{\mu\nu}-\tilde{Q}^{\alpha}g_{\mu\nu}-\delta^{\alpha}_{(\mu}Q_{\nu)}\right],$
(4)

here the trace of non-metricity tensor [8] takes the following expression

$Q=-Q_{\alpha\mu\nu}\,P^{\alpha\mu\nu}.$ (5)

The fluid description of spacetime can be written in terms of energy-momentum
tensor as

$T_{\mu\nu}=-\frac{2}{\sqrt{-g}}\frac{\delta\left(\sqrt{-g}\,\mathcal{L}_{m}\right)}{\delta
g^{\mu\nu}}.$ (6)

Now, by shifting the action (1) with respect to the metric tensor $g_{\mu\nu}$
one can compose the motions equation, which can be written as

$\frac{2}{\sqrt{-g}}\bigtriangledown_{\gamma}\left(\sqrt{-g}\,f_{Q}\,P^{\gamma}\;_{\mu\nu}\right)+\frac{1}{2}g_{\mu\nu}f\\\
+f_{Q}\left(P_{\mu\gamma i}\,Q_{\nu}\;^{\gamma i}-2\,Q_{\gamma
i\mu}\,P^{\gamma i}\;_{\nu}\right)=-T_{\mu\nu},$ (7)

where $f_{Q}=\frac{df}{dQ}$.
Also, one can obtain the following relation by varying (1) with respect to the
connection,

$\bigtriangledown_{\mu}\bigtriangledown_{\nu}\left(\sqrt{-g}\,f_{Q}\,P^{\gamma}\;_{\mu\nu}\right)=0.$
(8)

The curvature less and torsionless make the affine connection to the following
form

$\Gamma^{\alpha}_{\mu\nu}=\left(\frac{\partial
x^{\alpha}}{\partial\xi^{\lambda}}\right)\partial_{\mu}\partial_{\nu}\xi^{\lambda}$
(9)

Here we use a special coordinate choice, called coincident gauge so that the
affine connection, $\Gamma^{\alpha}_{\mu\nu}=0$
and hence, the non-metricity reduces to

$Q_{\alpha\mu\nu}=\partial_{\alpha}g_{\mu\nu},$ (10)

and thereby largely simplifies the calculation since only the metric is the
fundamental variable.
Since the affine connection in Eq. (9) is purely inertial, one could utilize
the covariant formulation by first determining affine connection in the
absence of gravity [9].
Here we built stellar structures by taking the spherically symmetric
spacetime. This spacetime is conventionally composed as

$ds^{2}=e^{2\,a(r)}dt^{2}-e^{2\,b(r)}dr^{2}-r^{2}\left(d\theta^{2}+\text{sin}^{2}\theta
d\phi^{2}\right),$ (11)

where $a(r)$ and $b(r)$ solely depend on the radial coordinate r. For the
current interest, we consider the matter is depicted by an anisotropic stress-
energy tensor of the structure

$T_{\mu}^{\nu}=\left(\rho+p_{t}\right)u_{\mu}\,u^{\nu}-p_{t}\,\delta_{\mu}^{\nu}+\left(p_{r}-p_{t}\right)v_{\mu}\,v^{\nu},$
(12)

where $\rho$, $P_{r}$ and $P_{t}$ correspond to energy density, radial and
transverse pressures, respectively. The expressions $u^{\mu}$ and $v^{\nu}$
are the four-velocity vector and the unitary space-like vector in the radial
direction, respectively and $u^{\mu}=e^{a}\delta_{0}^{\mu}$,
$v^{\mu}=e^{b}\delta_{0}^{\mu}$. Now, we define the energy momentum tensor for
quintessence field, which is realized as:

$\tilde{T}_{\mu}^{\nu}=T_{\mu}^{\nu}+{\mathcal{D}}_{\mu}^{\nu}$ (13)

where ${\mathcal{D}}_{\mu}^{\nu}$ is a energy momentum tensor for quintessence
field. Further, we get the following relations from the energy momentum tensor
for quintessence field.

${\mathcal{D}}_{t}^{t}={\mathcal{D}}_{r}^{r}=-\rho_{q}$ (14)

and

${\mathcal{D}}_{\theta}^{\theta}={\mathcal{D}}_{\phi}^{\phi}=\frac{(3w_{q}+1)\rho_{q}}{2}$
(15)

where $\rho_{q}$ represents the energy density for quintessence filed $w_{q}$,
with ($-1<w_{q}<-\frac{1}{3}$). For the metric (11),the trace of the non-
metricity tensor $Q$ is given below,

$Q=\frac{2}{r}e^{-2\,b(r)}\left(2\,a^{{}^{\prime}}(r)+\frac{1}{r}\right).$
(16)

Now, we define an exponential type function for $f(Q)$ gravity, which is
expressed as:

$f(Q)=Q-\alpha\gamma\left(1-\frac{1}{e^{\frac{Q}{\gamma}}}\right)$ (17)

where $\alpha\in(0,1)$ and $\gamma=\Omega H^{2}_{0}$ with $\Omega$ is
dimensionless parameter and $H^{2}_{0}$ is a Hubble parameter. By using the
symmetry of the system we can choose the Krori and Barua gravitational
potentials [15], which are defined as:

$a(r)=Ar^{2}\;\;\;\;\;\;\;\;\;\;\;\;\;\;b(r)=Br^{2}+C$ (18)

where $A$, $B$, $C$ are the arbitrary constants. By using Eqs. (18) with
quintessence field we get a following field equations

$\displaystyle\rho-\rho_{q}$ $\displaystyle=$
$\displaystyle\frac{e^{-2Ar^{2}}\left(r^{2}+e^{Ar^{2}}\left(e^{Ar^{2}}\left(\alpha\gamma
r^{2}+2\right)+4Ar^{2}-2\right)-\frac{\alpha\chi_{1}\left(r^{2}\chi_{2}+16\right)}{\gamma}\right)}{2r^{4}},$
(19) $\displaystyle p_{r}+\rho_{q}$ $\displaystyle=$
$\displaystyle\frac{e^{-Ar^{2}}\left(\alpha\chi_{1}\left(e^{Ar^{2}}\left(\gamma
r^{2}+2\right)-8Br^{2}-4\right)-e^{Ar^{2}}\left(\alpha\gamma
r^{2}+2\right)+4Br^{2}++2\right)}{2r^{2}},$ (20) $\displaystyle
p_{t}-\frac{1}{2}(3w_{q}+1)\rho_{q}$ $\displaystyle=$
$\displaystyle\frac{-2r^{4}\chi_{3}\left(\alpha\chi_{1}-1\right)e^{-Ar^{2}}+r^{4}e^{-Ar^{2}}\left(r^{2}\chi_{5}+2\right)-\chi_{4}}{2r^{6}}.$
(21)

where

$\displaystyle\chi_{1}$
$\displaystyle=e^{\frac{e^{-Ar^{2}}\left(4Br^{2}+2\right)}{\gamma r^{2}}},$
$\displaystyle\chi_{2}$ $\displaystyle=4\gamma
e^{Ar^{2}}\left(r^{2}(A-B)-1\right)+16A\left(2Br^{2}+1\right)+\gamma
e^{2Ar^{2}}\left(\gamma r^{2}+2\right),$ $\displaystyle\chi_{3}$
$\displaystyle=Br^{4}(A-B)+r^{2}(A-4B)-1,$ $\displaystyle\chi_{4}$
$\displaystyle=\frac{8\alpha
r^{2}\left(Br^{2}+1\right)\left(A\left(2Br^{4}+r^{2}\right)+1\right)\exp\left(\frac{e^{-Ar^{2}}\left(4Br^{2}+2\right)}{\gamma
r^{2}}-2Ar^{2}\right)}{\gamma},$ $\displaystyle\chi_{5}$
$\displaystyle=4B-\alpha\gamma\left(\chi_{1}-1\right)e^{Ar^{2}}.$

To express the explicit relations between energy density and radial pressure,
known as the equation of state. In the current analysis, the following linear
equation of state is considered:

$p_{r}=\omega\rho,\;\;\;\;\;\;\;\;\;\;\;0<\omega<1.$ (22)

where $\omega$ represents the equation of state parameter. Herrera and Barreto
[22] have introduced the general form of this relation.
By using Eq. (22) in Eqs. (19-21), we get the following expressions:

$\displaystyle\rho$ $\displaystyle=$
$\displaystyle-\frac{2(A+B)e^{-Ar^{2}}-\frac{\exp\left(\frac{e^{-Ar^{2}}\left(4Br^{2}+2\right)}{\gamma
r^{2}}-2Ar^{2}\right)\left(2\alpha\gamma
r^{4}(A+B)e^{Ar^{2}}+8\alpha\left(A\left(2Br^{4}+r^{2}\right)+1\right)\right)}{\gamma
r^{4}}}{-\omega-1},$ (23) $\displaystyle p_{r}$ $\displaystyle=$
$\displaystyle-\frac{\omega\left(2(A+B)e^{-Ar^{2}}-\frac{\exp\left(\frac{e^{-Ar^{2}}\left(4Br^{2}+2\right)}{\gamma
r^{2}}-2Ar^{2}\right)\left(2\alpha\gamma
r^{4}(A+B)e^{Ar^{2}}+8\alpha\left(A\left(2Br^{4}+r^{2}\right)+1\right)\right)}{\gamma
r^{4}}\right)}{-\omega-1},$ (24) $\displaystyle\rho_{q}$ $\displaystyle=$
$\displaystyle-\frac{2(A+B)e^{-Ar^{2}}-\frac{\exp\left(\frac{e^{-Ar^{2}}\left(4Br^{2}+2\right)}{\gamma
r^{2}}-2Ar^{2}\right)\left(2\alpha\gamma
r^{4}(A+B)e^{Ar^{2}}+8\alpha\left(A\left(2Br^{4}+r^{2}\right)+1\right)\right)}{\gamma
r^{4}}}{\omega+1}$ (25) $\displaystyle-$
$\displaystyle\frac{e^{-2Ar^{2}}\left(\frac{\alpha\chi_{1}\left(r^{2}\chi_{2}+16\right)}{\gamma}-r^{2}e^{Ar^{2}}\left(e^{Ar^{2}}\left(\alpha\gamma
r^{2}+2\right)+4Ar^{2}-2\right)\right)}{2r^{4}},$ $\displaystyle p_{t}$
$\displaystyle=$
$\displaystyle-\frac{-2r^{4}\chi_{3}\left(\alpha\chi_{1}-1\right)e^{-Ar^{2}}+r^{4}e^{-Ar^{2}}\left(r^{2}\chi_{5}+2\right)-\chi_{4}}{2r^{6}}-\frac{1}{2}\text{$\rho$q}(3w_{q}+1).$
(26)

##  III Matching Conditions

In order, to find the values of involved unknowns, we shall match the interior
spacetime with the exterior spacetime. For this purpose, we shall use the the
Schwarzschild spacetime, which is defined as:

$ds^{2}=\left(1-\frac{2M}{r}\right)dt^{2}-\left(1-\frac{2M}{r}\right)^{-1}dr^{2}-r^{2}\left(d\theta^{2}+\sin^{2}\theta
d\phi^{2}\right).$ (27)

where $M$ is termed as the total mass confined within the star and
$R_{\varepsilon}$ to be the radius of the compact stellar remnant. The
following expressions are carried at $r=R$ by considering the metric
potentials:

$\displaystyle
g_{tt}^{-}=g_{tt}^{+},\;\;\;\;\;\;\;g_{rr}^{-}=g_{rr}^{+},~{}~{}~{}~{}\frac{\partial
g_{tt}^{-}}{\partial r}=\frac{\partial g_{tt}^{+}}{\partial r}.$ (28)

The signature of the intrinsic and extrinsic are illustrated as $(-)$ and
$(+)$ respectively, i.e. $r=R_{\varepsilon}$. The values of the required
constraints are retrieved by comparing the interior and exterior matric as are
and termed out to be as following:

$\displaystyle
A=-\frac{1}{R_{\varepsilon}^{2}}log(1-\frac{2M}{R_{\varepsilon}}),~{}~{}~{}~{}B=-\frac{M}{R_{\varepsilon}^{2}}log(1-\frac{2M}{R_{\varepsilon}})^{-1},~{}~{}~{}C=log(1-\frac{2M}{R_{\varepsilon}})-\frac{M}{R_{\varepsilon}}(1-\frac{2M}{R_{\varepsilon}})^{-1}.$
(29)

Table 1: Approximated values of involved parameters with $(\gamma=-0.02)$, $\omega=0.003;)$ and $w_{q}=-0.0.5;)$. $StrangeQuarkStar$ | $M$ | $R_{\varepsilon}(km)$ | $\frac{M}{R_{\varepsilon}}$ | $A(km^{-2})$ | $B(km^{-2})$
---|---|---|---|---|---
$HerX-1$ | $0.88M_{\odot}$ | $7.7$ | $0.168$ | $0.0069062764$ | $0.0042673646$
$SAXJ1808.4-3658$ | $1.435M_{\odot}$ | $7.07$ | $0.299$ | $0.018231569$ | $0.014880115$
$4U1820-30$ | $2.25M_{\odot}$ | $10.0$ | $0.332$ | $0.010906441$ | $0.0098809523$

##  IV Physical Analysis

In this section, we shall discuss the necessary conditions with the required
behavior for the stellar system.

###  IV.1 Energy density and pressure components under quintessence filed

Here, we discuss the energy-momentum components for the compact star solutions
under the quintessence field. To have a realistic interior solution of the
compact star, the energy density, transverse, and radial pressure needed to be
positive. Furthermore, all of the energy-momentum components should have
finite values at the center of the compact star. Figures 1, 2, 3 and 4 show
the portraits of the energy density, radial pressure, transverse pressure, and
quintessence energy density, respectively, for HerX-1, SAXJ1808.4-3658 and
4U1820-30 stars. From those Figures, it is observed that all of the energy-
momentum components gradually decrease from the center to the surface of the
compact star. But, the quintessence energy density converges to zero towards
the surface. Furthermore, all of the above quantities have finite values at
the center of the stars (i.e., at $r=0$).

Figure 1:  shows the behavior of energy density.

Figure 2:  shows the behavior of radial pressure.

Figure 3:  shows the behavior of tangential pressure.

Figure 4:  shows the behavior of quintessence energy density.

###  IV.2 Equation of state parameters (EoS)

The EoS for a natural compact star was derived by Das et al. [23] is a linear
function of radial component $r$. However, in our work, the EoS parameters are
nonlinear. The EoS for radial and transverse pressure derived in the
following:

$\displaystyle\omega_{r}=\frac{p_{r}}{\rho},\,\,\
\omega_{t}=\frac{p_{t}}{\rho}.$ (30)

Figures 5 and 6 depict the profiles of the EoS for the radial and transverse
pressure, respectively. It is seen that both of the important ratios lie
between 0 and 1 for three compact star models. Thus, all discussed solutions
are physically acceptable in the framework of $f(Q)$ gravity.

Figure 5:  shows the behavior of $w_{r}$.

Figure 6:  shows the behavior of $w_{t}$.

###  IV.3 Gradients

Now we examine the gradients of the energy density, radial, and tangential
pressure for our compact star models. To construct a real compact star model,
gradients have to satisfy the following conditions [24]

$\displaystyle\frac{d\rho}{dr}<0,\,\,\ \frac{dp_{r}}{dr}<0,\,\,\
\frac{dp_{t}}{dr}<0.$ (31)

The above conditions on gradients can be verified from Figures 7, 8, and 9.
The negative behavior of gradients is the necessary condition for a realistic
compact star model. In addition, the gradients are vanished at $r=0$ i.e.,

$\displaystyle\frac{d\rho}{dr}\bigg{|}_{r=0}=0,\,\,\
\frac{dp_{r}}{dr}\bigg{|}_{r=0}=0,\,\,\ \frac{dp_{t}}{dr}\bigg{|}_{r=0}=0.$
(32)

Figure 7:  shows the behavior of $\frac{d\rho}{dr}$.

Figure 8:  shows the behavior of $\frac{dp_{r}}{dr}$.

Figure 9:  shows the behavior of $\frac{dp_{t}}{dr}$.

###  IV.4 Anisotropy

For massive stellar objects, the radial and traverse pressures may not be
equal. Nevertheless, several proposals have been given for the existence of
stellar anisotropy structures, such as different kinds of phase transitions
[25] and the presence of type 3A superfluid [26]. Furthermore, the anisotropy
helps to understand the strange properties of matter at the core of the
compact star. The anisotropy function can be defined as the difference between
the pressure components, i.e., $\Delta=p_{t}-p_{r}$. Figure 10 displays the
profile of the anisotropy function and reveals that the anisotropy force is
positive. This result suggests that $p_{t}>p_{r}$ and anisotropic force is
repulsive.

Figure 10:  shows the behavior of Anisotropy function.

###  IV.5 Causality and Abreu stability analysis

In this subsection, we check the stability of our models using the well-known
causality condition. We test two types of speeds of sound, i.e., radial and
tangential speeds of sounds, which can be defined by $v_{r}^{2}$ and
$v_{t}^{2}$, calculated as follows:

$\displaystyle v_{r}^{2}=\frac{dp_{r}}{d\rho},\,\,\
v_{t}^{2}=\frac{dp_{t}}{d\rho}.$ (33)

The radial sound speed $v_{r}^{2}$ and tangential sound speed $v_{t}^{2}$
should be less than the speed of light to confirm that the causality condition
is satisfied. To ensure that, we present the sound speeds in Figures 11 and
12. It is seen that the sound speeds are less than the speed of light
(considering the speed of light is unity in relativistic units). Furthermore,
we have tested the cracking condition to avoid gravitational cracking for a
stable anisotropic compact star. The cracking condition or Abreu condition
i.e., $-1\leq v_{t}^{2}-v_{r}^{2}\leq 0$ illustrates in 14. It can also be
observed from the in Figure 13, that the expression $0\leq
v_{t}^{2}-v_{r}^{2}\leq 1$, satisfies the Abrea condition conversely.

Figure 11:  shows the behavior of $v^{2}_{r}$.

Figure 12:  shows the behavior of $v^{2}_{t}$.

Figure 13:  shows the behavior of $v^{2}_{t}-v^{2}_{r}$.

Figure 14:  shows the behavior of $v^{2}_{r}-v^{2}_{t}$.

###  IV.6 Energy conditions

Energy conditions are important tools for the non-vacuum solution as well as
in relativistic cosmology. In modified theories of gravity, commonly, four
types of energy conditions are considered. These conditions can be written as

  * •

Null energy condition (NEC): $\rho+p_{r}\geq 0,\,\,\ \rho+p_{t}\geq 0$.

  * •

Weak energy condition (WEC): $\rho\geq 0$ and $\rho+p_{r}\geq 0,\,\,\
\rho+p_{t}\geq 0$.

  * •

Dominant energy condition (DEC): $\rho>|p_{r}|,\,\,\ \rho>|p_{t}|$.

  * •

Strong energy condition (SEC): $\rho+p_{r}+2p_{t}\geq 0$.

Moreover, the dominant energy condition suggests that the speed of light
should be greater than equal to the speed of energy. To ensure the DEC, we
have shown the profiles of $\rho-p_{r}\geq 0,\,\,\ \rho-p_{t}\geq 0$ in Figure
17 and 18, respectively. It is observed that DEC is validated. Furthermore,
NEC, WEC, and SEC are satisfied, and their profiles are presented in Figures
15, 16, and 19.

Figure 15:  shows the behavior of $\rho+p_{r}$.

Figure 16:  shows the behavior of $\rho+p_{t}$.

Figure 17:  shows the behavior of $\rho-p_{r}$.

Figure 18:  shows the behavior of $\rho-p_{t}$.

Figure 19:  shows the behavior of $\rho+p_{r}+2p_{t}$.

###  IV.7 Mass function, compactness function, and redshift function

In the compact stellar object, the mass function $m(r)$, compactness function
$u(r)$, and redshift function $Z_{s}$ play an important role in understanding
its physical properties. The mass function for the stellar object can be
written as

$m(t)=\int_{0}^{r}(4\pi r^{2}\rho)dr.$ (34)

The compactness function can be defined as

$u(r)=\frac{2m(r)}{r}.$ (35)

The gravitational redshift function for the stellar object can be written as

$Z_{s}=\frac{1}{\sqrt{1-2u(r)}}\left(1-\sqrt{1-2u(r)}\right)$ (36)

Figure 20 illustrates the behavior of mass function $m(r)$, from which it is
seen that the mass function increases monotonically from the center to the
surface of compact star for three models. The compactness function’s evolution
is shown in Figure 21. We observed that $u(r)\leq 0.30$, which satisfied the
Buchdahl condition [27] for the current study. The gravitational redshift
function $Z_{s}$ is depicted in Figure 22 for three models. The values of
$Z_{s}$ are less than 0.5 i.e., $Z_{s}<0.5$ for all considered models. These
results are compatible with the results proposed in literature such as
$Z_{s}\leq 5$ by Bohmer and Harko [28], $Z_{s}\leq 5.211$ by Ivanov [29],
$Z_{s}\leq 2.400$ Mustafa, Shamir, and Tie-Cheng [13] for anisotropy fluid
sphere. All of the above results suggest that our compact star models are
validated in the framework of modified $f(Q)$ gravity.

Figure 20:  shows the behavior of $m(r)$.

Figure 21:  shows the behavior of $u(r)$.

Figure 22:  shows the behavior of $z(r)$.

##  V Conclusion

In this manuscript, we studied the compact stellar objects in well-known
modified $f(Q)$ gravity with the presence of a quintessence field. We focused
on exploring three compact star models, namely, HerX-1, SAXJ1808.4-3658, and
4U1820-30, with the Schwarzchild spacetime and anisotropic source of fluid. To
do this, we have considered an exponential form of $f(Q)$ as
$f(Q)=Q-\alpha\gamma\left(1-\frac{1}{e^{\frac{Q}{\gamma}}}\right)$ and
formulated the field equations in $f(Q)$ gravity. To our best of knowledge,
this is the first attempt to study compact stars in the background of $f(Q)$
gravity. Furthermore, the matching condition is used to find the different
values of parameters, i.e., the Schwarzchild metric as an exterior solution
with the interior spacetime metric at the boundary. Besides this, the model
parameter $\alpha$ act an important role in our analysis as the rest of our
study depends on it, and we choose $\alpha$ in $[0.020,0.040]$ to proceed our
study. It is worthy to note here that we calculated the values of parameters
by imposing the matching condition. In our further study, we have analyzed the
profiles of energy-momentum components i.e., $\rho,\,\ p_{r},\,\ p_{t},\,\
\rho_{q}$, equation of state parameters ($\omega_{r}$ and $\omega_{t}$),
gradients ( $\rho^{\prime}(r),\,\ p_{r}^{\prime}(r),\,\ p_{t}^{\prime}(r)$),
anisotropy function $\Delta=p_{t}-p_{r}$, causality and Abreu stability
analysis, energy conditions, and mass function, compactness function, and
redshift function. Some of the important results of this manuscripts are
discussed in the following:

  * •

The graphical representation of energy density $\rho$ is depicted in Figure 1
for three compact star models in $f(Q)$ gravity. It is observed that $\rho$ is
positive throughout the configuration of all parameters, whereas the
quintessence energy density is negative for the same configuration (see Figure
4). The radial and tangential pressures are positive for $r<R$, and their
profiles are shown in Figures 2 and 3.

  * •

The Equation of state parameter for radial and tangential pressure should be
lied in between $0$ and $1$ to present a stable stellar object. From Figures 5
and 6, we have seen that the EoS satisfied the setup for the stable stellar
model.

  * •

The gradient of energy density, radial, and tangential pressures are also
important for our analysis. The gradient must satisfy the following conditions
for $0<r\leq R$, we must have $\frac{d\rho}{dr}<0,\,\,\
\frac{dp_{r}}{dr}<0,\,\,\ \frac{dp_{t}}{dr}<0$ and at the origin
$\frac{d\rho}{dr}\bigg{|}_{r=0}=0,\,\,\
\frac{dp_{r}}{dr}\bigg{|}_{r=0}=0,\,\,\ \frac{dp_{t}}{dr}\bigg{|}_{r=0}=0.$
The profiles of all of the above conditions are presented in Figures 7, 8, and
9.

  * •

The behavior of the anisotropic function is illustrated in Figure 10 for three
stellar models. The positive behavior of anisotropic function suggests that
the anisotropic force is repulsive because $p_{t}>p_{r}$.

  * •

From Figures 11 and 12, we depicted the radial and tangential velocities for
three models. It is seen that the velocities are satisfied the condition
$0\leq v_{r}\,\ \&\,\ v_{t}\leq 1$. Also, the Abrea condition is presented in
Figures 13 and 14.

  * •

Moreover, all the energy conditions such as NEC, WEC, DEC, and SEC are
satisfied for three stellar models. Furthermore, their behavior is shown in
Figures 1, 15, 16, 17, 18, and 19.

  * •

Furthermore, the graphical representation of mass function $m(r)$, compactness
function $u(r)$, and redshift function $Z_{s}$ have been shown in Figures 20,
21, and 22, respectively. In this study, we observed that the mass function
monotonically increases towards the surface of stellar structure. The
compactness function also satisfied the Buchdahl condition i.e., $u(r)\leq
0.30$. In addition, we analyze the redshift function for $Z\leq 0.15$, which
is a good agreement with the results presented by [27, 28, 29, 13].

Finally, combining all the outcomes, it is concluded that the discussed
stellar models are physically acceptable in the framework of modified $f(Q)$
gravity. It would be interesting to extend the analysis of compact stars in
$f(Q)$ gravity in the near future.

## Acknowledgments

S.M. acknowledges Department of Science & Technology (DST), Govt. of India,
New Delhi, for awarding Senior Research Fellowship (File No. DST/INSPIRE
Fellowship/2018/IF180676). Z.H. acknowledges Department of Science and
Technology (DST), Government of India, New Delhi, for awarding a Junior
Research Fellowship (File No. DST/INSPIRE Fellowship/2020/IF190911). We are
very much grateful to the honorable referee and the editor for the
illuminating suggestions that have significantly improved our work in terms of
research quality and presentation.

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# Towards Human-Agent Communication via the Information Bottleneck Principle

Mycal Tucker Appeared in Robotics Science and Systems Workshop on Social
Intelligence in Humans and Robots. New York City, USA 2022<EMAIL_ADDRESS>Julie Shah<EMAIL_ADDRESS>Roger Levy<EMAIL_ADDRESS>Noga
Zaslavsky<EMAIL_ADDRESS>
###### Abstract

Emergent communication research often focuses on optimizing task-specific
utility as a driver for communication. However, human languages appear to
evolve under pressure to efficiently compress meanings into communication
signals by optimizing the Information Bottleneck tradeoff between
informativeness and complexity. In this work, we study how trading off these
three factors — utility, informativeness, and complexity — shapes emergent
communication, including compared to human communication. To this end, we
propose Vector-Quantized Variational Information Bottleneck (VQ-VIB), a method
for training neural agents to compress inputs into discrete signals embedded
in a continuous space. We train agents via VQ-VIB and compare their
performance to previously proposed neural architectures in grounded
environments and in a Lewis reference game. Across all neural architectures
and settings, taking into account communicative informativeness benefits
communication convergence rates, and penalizing communicative complexity leads
to human-like lexicon sizes while maintaining high utility. Additionally, we
find that VQ-VIB outperforms other discrete communication methods. This work
demonstrates how fundamental principles that are believed to characterize
human language evolution may inform emergent communication in artificial
agents.

##  I Introduction

Good communication is a critical component of successful teams of humans and
artificial agents, but differing notions of “good” makes training agents to
develop such communication challenging. One view of communication focuses on
utility by framing languages as successful to the extent they enable high task
performance. Emergent communication literature, wherein agents learn to
communicate while optimizing a task-specific reward or utility, emphasizes
this view [1, 2, 3, 4, 5, 6]. However, this approach may lead to communication
that is too complex for humans to understand [7] or has slow convergence rates
[8], implying that additional constraints are needed in order to guide agents
toward human-like communication.

Here, we use constraints from cognitive science by building upon a recent line
of work that argues that human languages evolve under pressure to efficiently
compress meanings into communicative signals [9, 10, 11, 12, 13]. This notion
of efficiency is formulated in terms of the Information Bottleneck (IB)
principle [14], which is a general information-theoretic principle with broad
scope in machine learning [15, 16, 17, 18, 19], and in this context, it can be
interpreted as a tradeoff between the complexity and informativeness of
communication [9]. Intuitively, complexity corresponds to the number of bits
that are needed for communication, and informativeness corresponds to how well
a listener can infer the speaker’s intended meaning regardless of a specific
task (e.g., humans understand what “blue” means regardless of any specific
task, which may be “avoid the blue box” or “find a blue cup”). We hypothesize
that taking into account complexity and informativeness, in addition to
optimizing a task-specific utility function, may help guide artificial agents
toward more natural, human-like communication systems.

We test this idea by training teams of artificial agents to optimize a
tradeoff between maximizing task utility, maximizing communicative
informativeness, and minimizing communicative complexity, as depicted in
Figure 1. To this end, we extend prior work [20, 21] and propose Vector-
Quantized Variational Information Bottleneck (VQ-VIB), a method for training
neural agents to compress inputs into signals, while maintaining a discrete
(symbolic) signal representation that is embedded in a continuous space. In
experiments, we find that training to increase informativeness improves the
convergence rate of emergent communication to high-performance communication,
while annealing the complexity loss generates a spectrum of learned
communication, which we show aligns with a variety of human languages.
Additionally, we find that VQ-VIB appears better able to achieve high utility
for the same complexity and informativeness than other discrete communication
methods. This suggests a promising avenue for supporting human-robot
communication by instilling inductive biases inspired by cognitive science.

(a) Theoretical Framework

\begin{overpic}[abs,unit=1mm,trim=0.0pt 0.0pt 199.16928pt
0.0pt,clip={true},width=173.44534pt]{figures/arch_decoder_nograd.png}
\put(55.0,20.0){Informativeness $:=-\mathbb{E}[d(X,\hat{X})]$}
\put(55.0,12.0){Complexity $:=I(X,C)$} \put(55.0,4.0){Utility $:=U(X,Y)$}
\end{overpic} (b) Simplified Architecture. A speaker, $S$, maps an input, $x$,
to a communication signal, $c$. A listener, $L$, takes an action, $y$, and a
decoder, $D$, reconstructs $x$. Informativeness is negatively related to the
expected distortion between $x$ and $\hat{x}$; complexity is defined as the
mutual information between $X$ and $C$; utility can be any given task-specific
reward function or loss.

Figure 1: Our theoretical framework and training losses allow us to weigh
tradeoffs between utility ($\lambda_{U}$), informativeness ($\lambda_{I}$),
and complexity ($\lambda_{C}$) to guide properties of emergent communication.

##  II Related Work

Here, we describe related work in emergent communication, including in a
specific color domain which we use; we include approaches for measuring
informativeness and complexity in Section III. Generally, in emergent
communication settings, researchers train agents to accomplish a cooperative
task in which communication should enable better task performance [3, 5, 22,
23, 24]. Some works simplify communication by discretizing communication, by
adding noise to the environment, or passing communication through a
centralized message processor; these methods all correspond to limiting
complexity [21, 25, 26]. At the same time, other works find that auxiliary
losses designed to create “meaningful” communication improve convergence rates
[8, 27, 28], often inspired by notions adjacent to informativeness. Color
reference games have been widely used in emergent communication literature [1,
29, 30, 31], mirroring the centrality of color naming for studying humans’
semantic categorization and language evolution [32]. Most relevant to our work
is Zaslavsky et al. [9]’s finding that color naming systems across languages
(from a large-scale, cross-linguistic dataset, the Word Color Survey [WCS;
33]) attain complexity–informativeness tradeoffs that are near-optimal in the
Information Bottleneck [IB; 14] sense. Chaabouni et al. [31] further showed
that AI agents can learn IB-efficient color naming systems by playing color
reference games (and see [30] for a related study using different notions of
efficiency). While emergent communication literature has begun to rediscover
the importance of complexity and informativeness in communication, these terms
have rarely been explicitly recognized, whereas we include terms for utility,
informativeness, and complexity in training agents.

##  III Technical Approach

Here, we show how to adapt standard neural architectures to enable
informativeness- and complexity-based objectives. We then introduce a new
neural architecture, Vector-Quantized Variational Information Bottleneck (VQ-
VIB), for learning complexity-limited discrete representations in a continuous
space. Lastly, we combine utility, informativeness, and complexity terms into
an overall objective.

###  III-A Information Bottleneck for Semantic Communication

We adopt and extend Zaslavsky et al. [9]’s widely-supported IB framework for
semantic communication [10, 11, 12, 13]. In this framework, speakers and
listeners jointly optimize the IB tradeoff between the complexity and
informativeness of their communication system. A speaker is characterized as a
probabilistic encoder $q(c|x)$ that, given an input, $x\sim p(x)$, generates a
signal, $c$. A listener is characterized as a probabilistic decoder
$q(\hat{x}|c)$, that given $c$ aims to reconstruct the speaker’s input.
Complexity is measured by the mutual information between the speaker’s inputs
and signals, $I(X,C)$. Informativeness is measured by the degree to which the
speaker’s mental representation of $x$ matches the listener’s mental
representation of $\hat{x}$. In general, these mental representations are
defined by probability distributions, $m$ and $\hat{m}$ respectively, over
some feature space. In that case, minimizing the expected Kullback-Leibler
(KL) divergence $\mathbb{E}[D[m\|\hat{m}]]$ amounts to maximizing the
informativeness of the communication. This yields the following optimization
principle: $\min I(X,C)-\beta\mathbb{E}[D[m\|\hat{m}]]$, where $\beta$ is a
tradeoff parameter. It has been shown that this principle is equivalent to the
IB principle [9].

Solving this optimization problem directly is challenging in large scale
domains and with agents that are parameterized by neural networks. To address
this, we take a Variational Information bottleneck [VIB; 18] approach, in
which an upper bound is optimized, similar to ELBO in Variational Autoencoders
[VAEs, 34] and $\beta$-VAEs [35]. We also approximate informativeness by an
autoencoding loss, $d(x,\hat{x})$, defined by the mean squared error (MSE)
loss of the speaker’s input and the listener’s reconstruction.

###  III-B Vector Quantized Variational Information Bottleneck

Our proposed neural architecture, the Vector Quantized Variational Information
Bottleneck (VQ-VIB), learns information-constrained discrete communication in
a continuous space. Our technique combines prior work in discrete
representation learning [VQ-VAE, 20] with $\beta$-VAE [35].

####  III-B1 Vector-Quantized Variational Autoencoder

First, consider a simplified Vector-Quantized Variational Autoencoder (VQ-VAE)
architecture [20]. As in standard VAEs, an encoder is a conditional
distribution over latent representations, given an input: $q(z|x)$. A
deterministic network maps from an input, $x$, to a latent, $z$: we dub such
encodings, $z(x)$. Next, for a VQ-VAE parametrized with $K$ embedding vectors,
$\boldsymbol{\zeta}$, in the latent space, $z(x)$ is discretized to a
particular embedding vector, $\zeta_{i}$, by mapping to the nearest embedding
vector.

####  III-B2 $\beta$ Variational Autoencoder ($\beta$-VAE)

We also use notions from $\beta$-VAE, a variational approach for generating
information-constrained continuous representations with neural nets. Following
Higgins et al. [35], an encoder network maps from an input to $\mu$ and
$\sigma$, which are used as parameters to a Gaussian distribution from which a
latent representation, $z$, is sampled. $\beta$-VAE networks are trained via a
weighted combination of a reconstruction loss and ($\beta$ times) the KL
divergence of $\mu(x)$ and $\sigma(x)$ from a unit Gaussian; this KL loss
corresponds to an upper bound of $I(X,Z)$.

Figure 2: VQ-VIB architecture blends Vector Quantization and Variational
Information Bottleneck. The speaker encodes an input, $x$, as parameters to a
Gaussian, from which a communication vector is sampled and then discretized by
mapping to the nearest encoding, to produce communication, $c$.

####  III-B3 VQ-VIB Architecture

Here, we integrate VQ-VAE and $\beta$-VAE for a method of generating discrete,
information-bounded representations. Our architecture, Vector-Quantized
Variational Information Bottleneck (VQ-VIB), corresponds to first using an
encoder to sample a representation $z$ in a continuous space, followed by a
quantization layer that discretizes the sample by mapping to the nearest
encoding $\zeta$. This is depicted in Figure 2 and is formally defined in
Equation 1. The likelihood of a communication vector, $c$, given an input,
$x$, corresponds to the likelihood that an encoding $z$, sampled with
parameters $\mu(x)$ and $\sigma(x)$, is closest to an embedding vector equal
to $c$ (Eq. 1). This intuitively corresponds to notions of uncertainty over
which “word” to use. One may compute an upper bound on $I(X,C)$ based on this
uncertainty by using the standard KL divergence used for Gaussians in a
$\beta$-VAE, using $\mu(x)$ and $\sigma(x)$.

$\begin{split}&z\sim\mathcal{N}(\mu(x),\sigma(x))\\\
&q(c|x)=\mathbb{P}(c=\text{argmin}_{\zeta}[(z-\zeta)^{2}]\\\ \end{split}$ (1)

(a) 9 Points

(b) Uniform

Figure 3: In particle-world environments, a speaker (clear) communicated to a
listener (gray) to reach a desired landmark. Figure 4: In the color reference
game, agents observed colors from the WCS Dataset [33].

Figure 5: Top: In the 9 points environment, greater $\lambda_{I}$ led to
faster convergence for VQ-VIB; Bottom: Annealing $\lambda_{C}$ decreased
reward and complexity for all methods. Figure 6: Color reference game results.
Left: By annealing complexity, we recreated a range of near-optimal color-
naming schemes (black curve is the theoretical IB bound from [9]). Middle: 2D
PCA of VQ-VIB Before communication shows relationships between different
clusters in the communication space. Right: mode maps of the agent (no box)
and human (dashed box) color naming systems, where each color chip from Figure
4 is colored by the centroid of its modal category.

###  III-C Overall Optimization

Here, we combine losses into an idealized objective function to maximize,
shown in Equation 2. This objective function is the weighted combination of
three terms, wherein we seek to maximize a tradeoff between high utility, high
informativeness (low expected distortion $d$), and low complexity.
Intuitively, these three scalar weights form a simplex that describe a range
of communication from silence (minimizing complexity) to task-optimized
communication that does not generalize (maximizing utility) to complete
descriptions (maximizing informativeness).

$\text{maximize}\quad\lambda_{U}U(X,Y)-\lambda_{I}\mathbb{E}[d(X,\hat{X})]-\lambda_{C}I(X,C)$
(2)

In practice, we train neural agents via gradient descent to maximize
approximations of these terms. Utility is represented via the policy gradient
(for RL settings) or a supervised loss. Informativeness is computed via the
negative MSE between the speaker’s input and a decoder’s output, given the
communication. Lastly, complexity is bounded by the KL divergence between the
speaker’s distribution over communication and a fixed marginal (either a
uniform distribution for onehot communication, or a unit Gaussian for others).

##  IV Results

###  IV-A Particle Worlds

We trained agents in two types of 2D particle worlds, as depicted in Figure 3.
In each environment, the speaker observed a “target” landmark (in 9 points,
there were 9 landmarks at fixed locations, whereas in Uniform, the landmarks
were spawned a random locations) and communicated to the listener. The team
reward was the negative distance from the listener to the target, so agents
learned to communicate about the target location. We trained agents with
Multi-Agent Deep Deterministic Policy Gradient [MADDPG; 3] and compared
continuous (real-valued), onehot, VQ-VIB, and prototype-based [Proto; 21]
communication. All except continuous communication are fundamentally discrete.
For all environments and architectures, training agents with higher
$\lambda_{I}$ improved training convergence, as depicted for VQ-VIB in the
9-points environment in Fig. 5 (top). Furthermore, as we increased
$\lambda_{C}$, all architectures learned less complex, but also lower-utility,
communication (Fig. 5 bottom). However, VQ-VIB achieved higher utility, for
the same complexity, than other discrete communication methods. These trends
also appeared in the Uniform environment: 1) higher $\lambda_{I}$ improved
convergence, 2) increasing $\lambda_{C}$ created a spectrum of communication
complexity and 3) VQ-VIB outperformed other discrete methods.

###  IV-B Color Reference Games

We also trained agents in a color reference game in which a speaker agent
observed a target color (drawn from WCS data, as shown in Fig. 4), and a
listener agent had to distinguish between the target and a distractor color.
As in the previous experiments, we found that increasing $\lambda_{I}$, for a
fixed $\lambda_{C}$, improved convergence rates, and VQ-VIB agents typically
learned more complex communication than onehot agents.

We tested agents for $\lambda_{I}=1.0$ while linearly annealing $\lambda_{C}$
to penalize complexity. As $\lambda_{C}$ increased, the communication
complexity decreased (Figure 6 left). Over 5 random seeds, the points in the
plot, generated at each training epoch, closely approximated the IB upper
bound for maximum informativeness for a given complexity. The learned color
naming schemes at different complexity levels exhibited important patterns
(Figure 6 right). As complexity decreased, the number of communication
clusters emitted by VQ-VIB agents decreased (visualized via 2-dimensional
principle component analysis). Clustering in the communication space was
reflected in the color space (e.g., “yellow” and “orange” merging). More
generally, a least-squares linear correlation between the distance between
pairs of communication vectors and the distance between associated colors (in
perceptually-based CIELAB space) showed $r^{2}=0.97\pm(0.01)$ for VQ-VIB,
whereas for onehot communication $r^{2}=0.76\pm(0.03)$. That is, for VQ-VIB,
if two communication vectors were far apart in communication space, they
described colors that were far apart. This shows a fundamental connection
between the form and meaning of communication used by VQ-VIB agents, which
onehot-based communication is architecturally unable to support.

Lastly, we compared the agent color naming schemes to human language data. For
each of the 110 languages in the WCS Dataset [33], we found the best-matching
agent as determined by gNID, a measure of dissimilarity of naming
distributions introduced by Zaslavsky et al. [9]. Example nearest languages
are shown in Figure 6 in dashed boxes. Annealing complexity of artificial
communication was necessary to generate similar naming conventions for each
human language. Furthermore, while onehot and VQ-VIB Before achieved similar
informativeness for the same complexity, VQ-VIB Before reached higher utility,
indicating an architectural advantage.

##  V Conclusion

In this work, we explore the interplay of utility, informativeness, and
complexity in emergent communication. To the best of our knowledge, we are the
first to directly optimize agents to communicate according to a combination of
all three terms. Rewarding informativeness led to faster communication
convergence, and penalizing complexity created more human-like communication.
While we focused on training agents entirely in “self-play,” natural
extensions of this work would include supporting human-robot communication,
wherein robots could learn partner-specific communication by appropriately
constraining complexity and a small amount of supervised data.

## Acknowledgments

M.T. is supported by an Amazon Science Hub Fellowship. N.Z. is supported by a
K. Lisa Yang Integrative Computational Neuroscience (ICoN) Postdoctoral
Fellowship.

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XX X Year

RESEARCH ARTICLE

INFORMATION SCIENCE

Corresponding author. Email<EMAIL_ADDRESS>
# Quantum-Inspired Analysis of Neural Network Vulnerabilities: The Role of
Conjugate Variables in System Attacks

Jun-Jie Zhang1 Deyu Meng2,∗ 1Division of Computational physics and Intelligent
modeling, Northwest Institute of Nuclear Technology, Shaanxi, Xi’an 710024,
China
E-mail<EMAIL_ADDRESS>2School of Mathematics and Statistics and
Ministry of Education Key Lab of Intelligent Networks and Network Security,
Xi’an Jiaotong University, Shaanxi, P. R. China.
Email<EMAIL_ADDRESS>
(XX XX Year; XX XX Year; XX XX Year)

###### Abstract

[ABSTRACT]Neural networks demonstrate inherent vulnerability to small, non-
random perturbations, emerging as adversarial attacks. Such attacks, born from
the gradient of the loss function relative to the input, are discerned as
input conjugates, revealing a systemic fragility within the network structure.
Intriguingly, a mathematical congruence manifests between this mechanism and
the quantum physics’ uncertainty principle, casting light on a hitherto
unanticipated interdisciplinarity. This inherent susceptibility within neural
network systems is generally intrinsic, highlighting not only the innate
vulnerability of these networks but also suggesting potential advancements in
the interdisciplinary area for understanding these black-box networks.

###### doi:

10.1093/nsr/XXXX

###### keywords:

Neural Networks, Adversarial Attacks, Accuracy-robustness Trade-off,
Uncertainty Principle, Quantum Physics

##  1 Introduction

Despite the widely demonstrated success across various domains — from image
classification[1] and speech recognition[2] to predicting protein
structures[3], playing chess[4] and other games[5], etc. — deep neural
networks have recently come under scrutiny for an intriguing vulnerability[6,
7]. The robustness of these intricately trained models is being called into
question, as they seem to falter under attacks that are virtually
imperceptible to human senses.

A growing body of both empirical[8, 9, 10, 11, 12, 13, 14, 15, 16] and
theoretical[17, 18, 19, 20] evidences suggests that these sophisticated
networks can be tripped up by minor, non-random perturbations, producing high-
confidence yet erroneous predictions — a striking and quite succinct example
being the Fast Gradient Sign Method (FGSM) attack[18]. These findings raise
significant concerns about the vulnerabilities of such neural networks. If
their performance can indeed be undermined by such slight disruptions, the
reliability of technologies that hinge on state-of-the-art deep learning could
potentially be at risk.

A natural question emerges concerning the vulnerability of deep neural
networks. Despite the classical approximation theorems[21, 22, 23, 24]
promising that a neural network can approximate a continuous function to any
desired level of accuracy, is the observed trade-off between accuracy and
robustness an intrinsic and universal property of these networks?

This query stems from the intuition that stable problems, described by stable
functions, should intrinsically produce stable solutions. The debate within
the scientific community is still ongoing. If this trade-off is indeed an
inherent feature, then a comprehensive exploration into the foundations of
deep learning is warranted. Alternatively, if this phenomenon is merely an
outcome of approaches to constructing and training neural networks, it would
be beneficial to concentrate on enhancing these processes, as have already
been undertaken, e.g., the certified Adversarial Robustness via Randomized
Smoothing[25, 26, 27], and the concurrent training strategy[28, 29, 30, 19,
31, 32, 33, 34], etc.

In this study, we uncover an intrinsic characteristic of neural networks:
their vulnerability shares a mathematical equivalence with the uncertainty
principle in quantum physics[35, 36]. This is observed when gradient-based
attacks[18, 37, 38, 39, 40, 41] on the inputs are identified as conjugate
variables, in relation to these inputs.

Figure 1: Illustration of $\Delta x$ and $\Delta p$ in a three-layer
convolutional neural network trained on the MNIST dataset over 50 epochs. The
data’s high-dimensional feature space was reduced to two dimensions using the
t-SNE (t-Distributed Stochastic Neighbor Embedding) algorithm for easy
visualization. (A) Shaded regions indicate the class predictions obtained by
the finally trained network, and the colors imposed on individual points
indicate the true labels of corresponding test samples. (B) All test samples
were subjected to the Projected Gradient Descent (PDG) adversarial attack
method [37, 38] with $\epsilon=0.1$ and $\alpha=0.1/4$ over four iterative
steps. It is seen that these adversarially perturbed samples are evidently
deviated from class regions they should be located. (C) The prediction region
evolution for the digit ’8’ is displayed at epochs 1, 21, and 41. More deeper
the color is, more confident the prediction is by the network. (D) The shaded
area is similar to (C), but with points representing the adversarial
predictions of the attacked images, illustrating the temporal impact of the
PDG attack on model accuracy.

Taking into account a trained neural network model, denoted as $f(X,\theta)$,
where $\theta$ signifies the parameters and $X$ represents the input variable
of the network, we observe a consistent pattern. The network cannot achieve
arbitrary levels of measuring certainties on two factors simultaneously: the
conjugate variable $\nabla_{X}l(f(X,\theta),Y)$ (where $Y$ denotes the
underlying groundtruth label of $X$) and the input $X$, leading to the
observed accuracy-robustness trade-off. This phenomenon, similar to the
quantum physics’ uncertainty principle, offers a nuanced understanding of the
limitations inherent in neural networks.

##  2 Results

###  2.1 Conjugate variables as attacks

In quantum mechanics, the concept of conjugate variables plays a critical role
in understanding the fundamentals of particle behavior. Conjugate variables
are a pair of observables, typically represented by operators, which do not
commute. This non-commutativity implies that the order of their operations is
significant and it is intrinsically tied to Heisenberg’s uncertainty
principle[35, 36]. A prime example of such a pair is the position operator,
$\hat{x}_{\text{qt}}$, and the momentum operator,
$\hat{p}_{\text{qt}}=-i\frac{\partial}{\partial x_{\text{qt}}}$. Here, the
order of operations matters such that $\hat{x}_{\text{qt}}\hat{p}_{\text{qt}}$
is not equal to $\hat{p}_{\text{qt}}\hat{x}_{\text{qt}}$, indicating the
impossibility of simultaneously determining the precise values of both
position and momentum. This inherent uncertainty is quantitatively expressed
in Heisenberg’s uncertainty relation: $\Delta x_{\text{qt}}\Delta
p_{\text{qt}}\geq\frac{1}{2}$, where $\Delta x_{\text{qt}}$ and $\Delta
p_{\text{qt}}$ represent the standard deviations of position and momentum
measurements, respectively.

Drawing an analogy from quantum mechanics, we can formulate the concepts of
conjugate variables within the realm of neural networks. Specifically, the
features of the input data provided to a neural network can be conceptualized
as feature operators, denoted as $\hat{x}_{i}$, while the gradients of the
loss function with respect to these inputs can be viewed as attack operators,
denoted as $\hat{p}_{i}=\frac{\partial}{\partial x_{i}}$. Here, the subscript
$i$ refers to the $i$-th feature of the entire input feature vector. The
attack operators, corresponding to the gradients on inputs, hold a clear
relationship with gradient-based attacks, such as the FGSM attack (the
application of such attacks often involves a sign function, although this is
not strictly necessary[42, 43]).

This analogy leads us to an inherent uncertainty relation for neural networks,
mirroring the Heisenberg’s uncertainty principle in quantum mechanics.
Providing a trained neural network with properly normalized loss functions,
the relation reads: $\Delta x_{i}\Delta p_{i}\geq\frac{1}{2}$ (see derivations
in Methods). This relation, relying on both the dataset and the network
structure, suggests that there exists an intrinsic limitation in precisely
measuring both features and attacks simultaneously. This intrinsically reveals
an inherent vulnerability of neural networks, echoing the uncertainty we
observe in the quantum world.

To intuitively visualize the manifestation of $\Delta x=(\sum{\Delta
x_{i}})^{1/2}$ and $\Delta p=(\sum{\Delta p_{i}})^{1/2}$ within neural
networks, we use the MNIST dataset as a representative example. The neural
network is trained and subsequently subjected to attacks at each training
epoch.

In this scenario, a trained network partitions the hyperspace (the space
inhabited by the samples) into distinct regions. A given input, represented as
a point in this space, is classified based on the label of the region it falls
within. After 50 epochs of training, the shaded areas encapsulate most
correctly labeled data points (Fig. 1A). Conversely, the attacks shift these
input points slightly, leading to misclassification. The shifted points do not
overlap with the regions defined by the trained network (Fig. 1B).

We pay particular attention to class number 8, which exhibits the most
interconnections with other classes. This class is further illustrated in Fig.
1C and D. As the training epochs progress, the "effective radius" of the
shaded area shrinks, causing the area to gradually coincide with the correctly
labeled data points (Fig. 1C). Simultaneously, the "effective radius" of the
attacked points begins to deviate further from the shaded regions, and thus
from the correctly labeled data (Fig. 1D).

This visualization reveals an inherent trade-off: a reduction in the effective
radius of the trained class corresponds to an increase in the effective radius
of the attacked points. These two radii can be conceptualized as the visual
representations of the uncertainties, $\Delta x$ and $\Delta p$, highlighting
the delicate balance of precision and vulnerability in neural networks.

In addition to the adversarial attacks explored in this study, there exist
analogous effective conjugates in other types of adversarial attacks as
well[37, 39, 40, 41]. While we are currently unable to explicitly define the
conjugates associated with black-box attacks as referenced in [44, 45], it is
plausible that these methods may adhere to the same underlying principle.

###  2.2 Manifestation of the uncertainty principle in neural networks

Figure 2: Results of the three different types of neural networks: a three-
layer convolutional network running on the MNIST dataset, a four-layer
convolutional network on the CIFAR-10 dataset, and a residual network[46] with
eight convolutional layers on the CIFAR-10 dataset. The term "feature" in the
labels represents the results obtained by attacking the features of the input
images, while "pixel" corresponds to attacks directed at the pixels
themselves. Each neural network underwent training for a span of 50 epochs.
The quantities $\Delta x$ and $\Delta p$ were determined through high-
dimensional Monte-Carlo integrations. Subfigures (A), (C), (E), (G), (I), and
(K) depict the test and robust accuracy metrics, with the robust accuracy
evaluated on images perturbed by the PDG adversarial attack method, using
parameters $\epsilon=8/255$ and $\alpha=2/255$ across four iterative steps.
Subfigures (B), (D), (F), (H), (J), and (L) illustrate the trade-off
relationship between $\Delta x$ and $\Delta p$.

The shaded areas in Fig. 1A are actually representative of wave functions in
quantum physics. Specifically, for the MNIST dataset, we have ten
corresponding wave functions corresponding to ten digit number classes.
Therefore, the uncertainty relation $\Delta x\Delta p\geq\frac{1}{2}$ shown in
Fig. 1C and D should be reinterpreted as $\Delta x[\text{class 8}]\Delta
p[\text{class 8}]\geq\frac{1}{2}$, indicating that we are concentrating on the
class of number 8. This equation is a clear depiction of the trade-off between
$\Delta x[\text{class 8}]$ and $\Delta p[\text{class 8}]$, as depicted in Fig.
2B, accompanied by the associated trade-off between accuracy and robustness
(Fig. 2A). The Cifar-10 dataset, having a higher complexity than MNIST, poses
a potential indeterminacy in identifying a specific class that has more
connectivity with other classes. In this case, the average values $\Delta
x=\text{Mean}(\Delta x[\text{All classes}])$ and $\Delta p=\text{Mean}(\Delta
p[\text{All classes}])$ are employed instead. The similar results obtained on
Cifar-10 underscore the inherent uncertainty relation that drives the
accuracy-robustness trade-off, as demonstrated in Fig. 2C and D.

##  3 Discussions

Table 1: Comparison of the uncertainty principle between quantum physics and
neural networks. The subscript $i$ represents the $i$-th dimension. For
physics, $i$ stands for the spatial coordinates ($x$, $y$, and $z$), whereas
in the context of neural networks, $i$ refers to the $i$-th feature. When we
consider pixels, $i$ simply pertains to the $i$-th pixel. Additionally, we
utilize Dirac notation, for instance,
$\langle\hat{x}_{i,\text{qt}}\rangle=\int\psi^{*}(X)x_{i,\text{qt}}\psi(X)dX$,
where $\langle\hat{x}_{i,\text{qt}}\rangle$ is the expectation value of the
$i$-th dimension. Similarly,
$\langle\hat{x}_{i}\rangle=\int\psi_{Y}(X)x_{i}\psi_{Y}(X)dX$ for neural
networks.

Quantum physics | Neural networks
---|---
position | $X=(x,y,z)$ | $X=(x_{1},...,x_{i},...,x_{M})$ | image/feature (input)
| momentum
---
(conjugate of position)
$P=(p_{x},p_{y},p_{z})$ | $P=(p_{1},...,p_{i},...,p_{M})$ |  | attack
---
(conjugate of input)
wave function | $\psi(X)$ | $\psi_{Y}(X)$ |  | normalized loss function
---
(neural packet)
normalize condition | $\int|\psi(X)|^{2}dX=1$ | $\int|\psi_{Y}(X)|^{2}=1$ | normalize condition
position operator | $\hat{x}_{i,\text{qt}}\psi(X)=x_{i,\text{qt}}\psi(X)$ | $\hat{x}_{i}\psi_{Y}(X)=x_{i}\psi_{Y}(X)$ | feature operator
momentum operator | $\hat{p}_{i,\text{qt}}\psi(X)=-i\frac{\partial}{\partial x_{i,\text{qt}}}\psi(X)$ | $\hat{p}_{i}\psi_{Y}(X)=\frac{\partial}{\partial x_{i}}\psi_{Y}(X)$ | attack operator
| standard deviation for
---
measuring position
$\sigma_{x_{i,\text{qt}}}=\langle(\hat{x}_{i,\text{qt}}-\langle\hat{x}_{i,\text{qt}}\rangle)^{2}\rangle^{1/2}$ | $\Delta{x_{i}}=\langle(\hat{x}_{i}-\langle\hat{x}_{i}\rangle)^{2}\rangle^{1/2}$ |  | standard deviation for
---
resolving pixel
| standard deviation for
---
measuring momentum
$\sigma_{p_{i,\text{qt}}}=\langle(\hat{p}_{i,\text{qt}}-\langle\hat{p}_{i,\text{qt}}\rangle)^{2}\rangle^{1/2}$ | $\Delta{p_{i}}=\langle(\hat{p}_{i}-\langle\hat{p}_{i}\rangle)^{2}\rangle^{1/2}$ |  | standard deviation for
---
resolving attack
uncertainty relation | $\sigma_{x_{i,\text{qt}}}\sigma_{p_{i,\text{qt}}}\geq\frac{1}{2}$ | $\Delta{x_{i}}\Delta{p_{i}}\geq\frac{1}{2}$ | uncertainty relation

###  3.1 Attacking features is more effective than attacking pixels

The pixels in our dataset serve as the raw, unprocessed data, gathered
directly from the detectors. These pixels carry the features that serve as an
accurate representation of the real world. While there is a possibility of
manipulating these features, it is more common and practical to focus on the
pixels themselves. By doing so, we can observe the accuracy-robustness trade-
off (Fig. 2E and G), a fundamental concept that is underpinned by the
uncertainty relation, as seen in Fig. 2F and H.

However, it is important to note, as evidenced by the testing accuracy results
from the MNIST dataset, that there is an initial learning curve or ‘kick’ that
is encountered (Fig. 2E). This is to be expected as the neural network must
first familiarize itself with, or ‘learn’, the features before it can
effectively classify the images.

While processing the initial learning stages, it is also worth noting the
fluctuation in both $\Delta x$ and $\Delta p$ for input pixels. This
fluctuation is more pronounced than that seen in the features, highlighting
the random exploration nature of the learning algorithm. As illustrated in
Fig. 2H, these fluctuations could be attributed to the inherent randomness of
the learning process, a factor that is crucial to potentially uncover more
optimal weight configurations.

###  3.2 Phenomenon in attacking well designed neural networks

Typically, network structures are scrupulously architected to fit the demands
of specific tasks. Take Fig. 2C,D,G,H as an example. In the figure, the
network only achieves a test accuracy of around 65% due to the relatively
simple network architecture. To address this, we introduce a more advanced
network structure that incorporates residual networks and additional
convolutional layers. This refined structure increases the accuracy to nearly
90%111Given that the quantities $\Delta x$ and $\Delta p$ are approximately
computed through high-dimensional Monte Carlo integrations, a process that is
exceedingly time-consuming, we can only feasibly perform these computations
for the network with such complexity. If they could be calculated more
accurately under more complex and accurate networks with stronger
computational resources, we believe the calculated patterns will better
conform to the expected regularities.. One can still observe a clear pattern
in the trade-off between $\Delta x$ and $\Delta p$ for both features and
pixels (Fig. 2I-L). Besides, this trade-off is also more pronounced for
features than for pixels. Understanding this trade-off allows for a more
effective optimization of the network structure. In closing, constructing a
network structure that best fits the task at hand is pivotal in delivering
optimal performance.

###  3.3 Neural network as a complex physical system

As scientific research and engineering become increasingly reliant on
artificial intelligence (AI) methods, questions about the future role of human
beings in these fields naturally arise. Whether guiding AI or being guided by
it, understanding the fundamental principles underpinning these sophisticated
structures is paramount. One approach to glean this understanding is to treat
neural networks as complex physical systems, thereby applying principles of
physics to elucidate the inner mechanisms of AI.

In the study at hand, it is posited that neural networks, much like quantum
systems, are subject to a form of the uncertainty principle. This connection
potentially uncovers intrinsic vulnerabilities within the neural networks. A
comparison of formulas from these distinct fields is presented in Table 1.
Here, concepts from quantum physics such as position, momentum, and wave
function are juxtaposed with their counterparts in neural networks: image,
attack, normalized loss function, and so on. This comparison not only reveals
striking similarities but also indicates that the methodologies employed in
physical sciences could potentially be harnessed to investigate the properties
of neural networks.

The intersection of AI and physics has the potential to provide novel insights
into the intricate complexities of neural networks. For instance, the emergent
capabilities exhibited by large language models might be correlated with
principles found in statistical physics. Moreover, phenomena such as small
data learning could be linked to concepts from Noether’s theorem and gauge
transformations[47]. By drawing inspiration from physical processes such as
weak interactions, we can devise innovative generative models, such as "Yukawa
Generative Models"[48]. Viewing neural networks through the lens of physics
can give us a deeper understanding of their structure and functionality from
an entirely new perspective.

The synergy between AI and physics, two seemingly distinct fields, could lead
to advancements in both domains. It’s a two-fold benefit: AI could gain from
the structured, universal laws of physics, and in return, physics could
possibly leverage the predictive and analytical power of AI.

##  4 Conclusion

This study reveals the remarkable link between quantum physics and neural
networks, demonstrating that these artificial systems, like quantum systems,
are subject to the uncertainty principle. This principle, often associated
with precision and vulnerability trade-offs, provides new insights into the
potential frailties inherent in neural networks.

Our findings also indicate that attacking the features of a neural network can
be more effective than focusing on its pixels. This insight could possibly
influence the optimization of network structures for better performance.

Meanwhile, viewing neural networks as complex physical systems allows us to
apply principles from physics to understand the behaviour of these AI systems
better. This interdisciplinary approach not only enhances our comprehension of
AI systems but also suggests a wealth of potential applications and
advancements in both fields.

As we move forward, further exploration of this accuracy-robustness trade-off
and its influence on the design of neural networks will be crucial. While this
study provides a valuable perspective on the relationship between quantum
physics and AI, additional research is still needed to more comprehensively
understand how these principles can be applied to improve neural network
robustness and design.

##  5 Methods

Detailed methods and materials are given in the online supplementary data.

##  6 Data Availability

All data are available in the main text or the supplementary materials.
Additional data related to this paper are available at
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https://doi.org/10.48550/arXiv.2205.01493.

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