# Quantum information entropies for solitonic systems

Ramkumar Radhakrishnan<EMAIL_ADDRESS>Department of Physics, North
Carolina State University, Raleigh, NC 27695, USA  Mariyah Ughradar
<EMAIL_ADDRESS>Vikash Kumar Ojha<EMAIL_ADDRESS>(corresponding author) Department of Physics, Sardar Vallabhbhai National
Institute of Technology, Surat 395007, Gujarat, India

###### Abstract

Particle with position-dependent mass is a useful concept in the context of
semiconductor physics. We study a particle with the solitonic mass
distribution in two different forms of potential: the quartic and the
symmetric potential. We estimate the Shannon entropy and Fisher information
associated with the ground state of particle in these two scenarioes by
obtaining the wave-function from Zhu-Kroemer equation. The ground state of the
particle in each case satisfies the Bialynicki-Birula–Mycielski inequality.
Upon comparing all four models under consideration, we have observed that the
Shannon entropy is greater for the solitonic mass distribution when it is
subjected to a quartic potential.

##  I Introduction

Solitons, also known as solitary waves, encompass a range of physical
phenomena, including shock waves in sound waves and ultra-short optical pulses
in optical fibers [1]. From a mathematical perspective, solitons can be
understood as localized solutions to partial differential equations that
describe nonlinear systems with infinite degrees of freedom [2]. The
groundbreaking contributions of Feynman have significantly influenced and
propelled the field of Quantum information processing [3]. Consequently, the
study of solitons holds great importance in the realm of Quantum information
processing. More precisely, the study of solitons are profound in the field of
Quantum communication [4, 5, 6]. Solitons are distinct from elementary
particles that emerge from quantized wave-like excitations of the fields [7].
Their characteristics are primarily determined by classical equations. This
research took place during the early stages of quantum field theory in the
late 1960s. There exist important connections between the solitons of a theory
and the wave-like fields that satisfy the linearized field equations.
Quantizing the latter results in states of elementary particles, with the
interactions between these particles being governed by the nonlinear
components [2]. Initially, in regions far from the soliton, the field
progressively tends towards the vacuum state, adhering to the rate determined
by the linearized field equation. Therefore, if the linearized equation lacks
a mass term, leading to the existence of massless elementary particles, the
impact of the soliton will extend over a considerable distance, diminishing
inversely as distance increases. This comprehension has implications for
multiple facets of quantum field theory within condensed matter setups [8].
Initially, in quantum mechanics, the Schrodinger equation focused on physical
systems with a constant effective mass. However, advancements in condensed
matter physics presented a new challenge: the existence of non-relativistic
particles that exhibit a position-dependent effective mass. Subsequently, the
study of systems with position-dependent effective mass has garnered
significant attention from researchers due to its wide range of applications.

In conjunction with the concept of position-dependent effective mass arises
the theory of communication, formally known as the Information theory proposed
by Claude Shannon around mid-1900s. Entropy, as a measurable physical
property, is commonly associated with a state of disorder, randomness, or
uncertainty. In the context of thermodynamics [9], entropy represents a
measure of irreversibility within a physical system. However, it is important
to note that this concept of entropy differs from information entropy, which
is the focus of our interest. Several parameters highlight the distinctions
between thermodynamic entropy and information entropy. Thermodynamic entropy
considers the number of possible structural arrangements within a system,
while informational entropy pertains to choices made within a communication
system. In recent times, there has been a growing interest in utilizing
information-theoretic measures for quantum systems. Notably, entropic
uncertainty relations have been explored as alternatives to the Heisenberg
uncertainty principle [10]. Furthermore, Shannon entropy [11] and entropic
uncertainty relations find applications in various areas. These include the
study of squeezed localization [12], fractional revivals [13, 14], and
reconstruction of charge and momentum densities in atomic and molecular
systems using maximum entropy procedures [15]. Additionally, Shannon entropy
[11] finds utility in Measure theory [16], the study of open quantum systems
involving Markov chains [17], machine learning for decision tree algorithms,
and Bayesian inference [18].

The Fisher information, also referred to as Fisher’s entropy, is a fascinating
concept that emerges from communication theory and is recognized as a
precursor to Shannon entropy [19]. In the quantum domain, Fisher information
is inherently connected to the uncertainty connected with measurement, as
explained in references [20]. Essentially, Fisher’s information measures the
information content carried by a specific observable concerning a parameter,
while considering its inherent probability [19].

This article is structured as follows: In section II, we provide an overview
of Shannon entropy, entropic uncertainty, and Fisher’s information measure .
In section III, we provide a detailed explanation of the quantum mechanics of
position-dependent mass. In section IV, V, we calculate Shannon and Fisher’s
information measure for different solitonic systems and discuss the results in
section VI. We have computed all calculations for systems in their ground
state and by considering the convention $\hbar=c=1$.

##  II Shannon entropy and Fisher’s information measure - A bird’s eye view

In the realm of information theory, Shannon entropy, originally introduced by
Claude Shannon in 1948 in his publication ‘A Mathematical Theory of
Communication [11], represents an average rate of information generation from
a stochastic data source. A higher value of Shannon entropy implies that a new
value from the process provides more information. Mathematically, for a given
distribution function $p_{i}$, Shannon entropy can be defined as follows [11]:

$S=-\sum p_{i}\text{ln}(p_{i}).$ (1)

Shannon entropy was not derived from pre-established assumptions but rather
developed as a means to quantify the uncertainty principle. It bears a
resemblance to Gibbs entropy and is occasionally referred to as ”Boltzmann-
Gibbs-Shannon” entropy [21]. Shannon entropy can be conceptualized as the
measure of uncertainty within a probability distribution connected to an
information source. In the context of a given probability distribution in
position space, Shannon entropy in the $n$th state is defined as.

$S_{x}=-\int_{-\infty}^{\infty}dx\bigg{[}(\psi_{n}(x))^{2}\text{ln}(\psi_{n}(x))^{2}\bigg{]}.$
(2)

In the reciprocal space (momentum representation), Shannon entropy in the
$n$th state is defined by

$S_{k}=-\int_{-\infty}^{\infty}dk\bigg{[}(\phi_{n}(k))^{2}\text{ln}(\phi_{n}(k))^{2}\bigg{]},$
(3)

where,

$\phi_{n}(k)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}e^{-ikx}\psi_{n}(x)dx.$
(4)

While the uncertainty principle deals with the limits of measurement in
quantum mechanics, Shannon entropy deals with the quantification of
information in a probabilistic framework. However, there is a conceptual
similarity between the two in terms of the idea of uncertainty and the limits
of knowledge or measurement. In quantum mechanics, the uncertainty principle
implies that there are inherent limits to the simultaneous knowledge of
certain pairs of complementary properties, such as position and momentum, or
energy and time. This uncertainty arises from the wave-particle duality of
quantum systems and the probabilistic nature of quantum measurements. Shannon
entropy, on the other hand, provides a measure of uncertainty or
unpredictability in a probability distribution. It quantifies the average
amount of information needed to describe or predict the outcomes of a random
variable. High entropy implies higher uncertainty or unpredictability, while
low entropy implies lower uncertainty or greater predictability. Although
these concepts are not directly connected, they both reflect the fundamental
limitations on our ability to precisely know or predict certain aspects of
physical systems. The uncertainty principle sets limits on the simultaneous
knowledge of complementary properties in quantum mechanics, while it
quantifies the uncertainty or information content of random variables in a
probabilistic framework. This is been briefly discussed in [22, 23], which can
be represented as

$S_{x}+S_{k}\geq D(1+\text{ln}\pi),$ (5)

where $D$ denotes the dimension of the system. The computation of Shannon
entropy analytically is challenging due to the presence of logarithmic terms
in the integrals, especially considering the system’s dimension. As a result,
it has only been determined for specific lower-energy states of certain
systems such as the Harmonic oscillator [24], the Pöschl–Teller potential
[25], the Morse potential [26], and highly excited states of the Coulomb
potential [27].

The foundation of the second law of thermodynamics lies in the concept of
entropy [28]. According to this law, the overall entropy of a closed system
increases as time progresses and approaches a maximum value in the limit of
infinite time. In its classical formulation, entropy is applicable to systems
in equilibrium and is defined by Clausius and others (see ref.[29]), can be
demonstrated by

$dS=\frac{\delta Q}{T}.$ (6)

As mentioned earlier, when viewed through the lens of information theory,
entropy serves as a metric for the level of disorder within a system [22].
Consequently, the entropy of a thermodynamic system escalates as the count of
accessible microstates for the system expands. In the realm of information
theory, there exists an alternative form of information known as Fisher
information ($F$) [30]. It is of the form,

$F=\sum_{i=1}^{m}\frac{1}{p_{i}}\bigg{[}\frac{dp_{i}}{di}\bigg{]}^{2},$ (7)

where, $p_{i}$ denotes the probability density for finding the system in
micro-state $i$. In the context of a specific thermodynamic system [9], an
increase in the number of microstates leads to a corresponding increase in
entropy. This occurs because the system becomes more disordered, resulting in
higher entropy. For instance, if the system occupies a single microstate, the
probability density function exhibits a steep slope around that state,
indicating high Fisher information. Conversely, if the system exists in
numerous states with approximately equal probabilities, the probability
density function becomes flat, with a nearly zero slope and Fisher
information. Therefore, Fisher information serves as a measure of order within
the system, representing a scenario where the system exists predictably in one
or a few high-probability microstates. It should be noted that despite the
resemblance of equations (1) and (7), these two quantities do not measure the
same property of the system. Shannon entropy is determined by the probability
density function, while Fisher information depends on the derivative of the
probability density function, representing its slope. Consequently, it can be
regarded as a global property, while Fisher information can be considered a
local property relative to the probability density function. Therefore, Fisher
information and Shannon entropy are not analogous to each other. Fisher’s
information for an observable $\zeta$ is defined as

$F_{\zeta}=\int_{-\infty}^{\infty}d\zeta\bigg{[}\rho(\zeta)\bigg{(}\frac{d}{d\zeta}\text{ln}(\rho(\zeta))\bigg{)}^{2}\bigg{]}>0,$
(8)

We calculate Fisher’s information for an observable $\zeta$ whose probability
density is $|\psi(\zeta,t)|^{2}$ is given by

$F_{\zeta}=\int_{-\infty}^{\infty}d\zeta\bigg{[}(\psi(\zeta,t))^{2}\bigg{(}\frac{d}{d\zeta}\text{ln}(\psi(\zeta,t))^{2}\bigg{)}^{2}\bigg{]}>0,$

However, in one dimensional stationary quantum systems (solitonic systems), at
position space, $\rho(\zeta)=|\psi(\zeta,t)|^{2}\approx|\psi(\zeta)|^{2}$, is
the probability density. It can be obtained by approximating $\psi(\zeta,t)$
to $\psi(\zeta)$.

$F_{\zeta}=\int_{-\infty}^{\infty}d\zeta\bigg{[}(\psi(\zeta))^{2}\bigg{(}\frac{d}{d\zeta}\text{ln}(\psi(\zeta))^{2}\bigg{)}^{2}\bigg{]}.$
(9)

Similarly, we compute Fisher information for the same soliton in the momentum
space.

$F_{k}=\int_{-\infty}^{\infty}dk\bigg{[}(\phi(k,t))^{2}\bigg{(}\frac{d}{dk}\text{ln}(\phi(k,t))^{2}\bigg{)}^{2}\bigg{]},$
(10)

Implementing the same approximation as before we get Fisher information as

$F_{k}=\int_{-\infty}^{\infty}dk\bigg{[}(\phi(k))^{2}\bigg{(}\frac{d}{dk}\text{ln}(\phi(k))^{2}\bigg{)}^{2}\bigg{]}.$
(11)

where,
$\phi(k)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}e^{-ik\zeta}\psi(\zeta)d\zeta$.
Uncertainty, in the context of Fisher information, refers to the degree of
imprecision or lack of knowledge about the true value of a parameter. It
captures the spread or variability in the estimates obtained from different
samples or measurements. It measures the uncertainty or randomness associated
with parameter estimation based on observed data, with higher entropy implying
lower uncertainty and better estimation performance. This is represented as
follows

$F_{\zeta}F_{k}\geq 4\hbar^{2}.$ (12)

where, $\hbar$ is reduced Planck’s constant.

##  III Quantum mechanics with position dependent mass

In standard quantum mechanics, the mass of a particle is typically treated as
a constant. However, there are certain physical systems where the mass of a
particle can depend on its position within the system. This introduces an
additional complexity in the mathematical formalism of quantum mechanics, but
it can be accommodated using position-dependent mass quantum mechanics. The
Schrodinger equation, which describes the behavior of quantum systems, needs
to be modified to account for the position-dependent mass. The general form of
the Schrodinger equation for a particle with position-dependent mass is

$\frac{-\hbar^{2}}{2m(x)}\frac{d^{2}\psi}{dx^{2}}+V(x)\psi=E\psi,$ (13)

where, $E$ is the energy, $\psi(x)$ is a wave function and $V(x)$ is the
potential energy. The position-dependent mass function m(x) introduces a
spatial dependence in the kinetic energy term of the Schrodinger equation.
This means that the behavior of the particle can vary depending on its
position within the system. Different choices for $m(x)$ can lead to a variety
of interesting physical phenomena. Throughout the sections we use the mass
distribution to be

$m(x)=\frac{m_{0}}{1+x^{2}}.$ (14)

Thus, for a position dependent mass, the kinetic energy term ($\hat{T}$) of
Schrodinger equation eq. (13) is modified as follows

$\hat{T}=\frac{-\hbar^{2}}{2\sqrt{m(x)}}\frac{d^{2}}{dx^{2}}\frac{1}{\sqrt{m(x)}}.$
(15)

The Hamiltonian operator ($\hat{H}$), which describes the energy of a
stationary system (solitonic system) in quantum mechanics, can be expressed in
the following manner

$\hat{H}=\frac{-\hbar^{2}}{2\sqrt{m(x)}}\frac{d^{2}}{dx^{2}}\frac{1}{\sqrt{m(x)}}+V(x).$
(16)

The Schrodinger equation modifies as

$\frac{-\hbar^{2}}{2m}\frac{d^{2}\psi(x)}{dx^{2}}+\frac{-\hbar^{2}}{2}\frac{m^{\prime}(x)}{m(x)^{2}}\frac{d\psi(x)}{dx}+\Bigg{[}\frac{\hbar^{2}}{4}\frac{m^{\prime\prime}(x)}{m(x)^{2}}-\\\
\frac{3\hbar^{2}}{8}\frac{m^{\prime}(x)^{2}}{m(x)^{3}}+V(x)\Bigg{]}\psi(x)=E\psi(x).$
(17)

This differential equation [eq.(17)] is derived by Zhu and Kroemer [31] and is
used in the context of semiconductor physics to explore the electronic
properties of semiconductor devices. In the next section, we shall discuss
Shannon and Fisher’s information measure for different solitonic systems. The
motive of this particular work is from the context of solitons in field
theory, where we look for time-independent, finite energy, and localized
solutions. The majority of research on solitons focuses on theoretical
analyses of soliton solutions in quantum field theories that are not
applicable to describing our physical universe. When considering particle
physics models that accurately represent our world, solitons often possess
unconventional properties such as exotic magnetic monopole charges [2], making
them relatively heavy. These applications of solitons represent only a small
portion of the extensive literature on solitons. Moreover, there are
exceptional cases, like the description of quarks and leptons through a dual
electromagnetic gauge group [2] using a magnetic monopole framework, which is
even rarer in the soliton literature.

Figure 1: Solitonic mass distribution for constant $m_{0}$

##  IV Solitons in a Quartic Potential

The double well potential also known as the Quartic potential, holds
significant importance in quantum mechanics, quantum field theory, and other
other fields for investigating different physical phenomena and mathematical
properties. The symmetric well potential is ubiquitous as it serves as a model
to illustrate the concept of instantons [2] and Feynman path integral
formulation in quantum mechanics [32]. Solitons in quartic potential are
formed by the interplay between self-phase modulation and anomalous second-
and fourth-order dispersion [33]. This has a widespread application in the
field of optical communication [34, 35], such as shape in variance in
nonlinear wave packets which enhances optical communication. Recent notable
achievements in this area have shown promising outcomes in terms of the
potential observation of localized quartic solitons within specifically
engineered slot wave guides based on silicon [36]. This is been observed in
the case of quartic solitons with a mass distribution of
$\frac{m_{0}}{1+x^{2}}$, thus studying the quantum information entropies for
the same will help us to understand it much better [37]. The given solitonic
mass distribution and well-known quartic potential with parameters $a$ and $b$
can be written as

$V(x)=ax^{2}+bx^{4},b>0,$ (18)

Figure 2: Quartic potential for different values of $a$ and $b$.

.

###  IV.1 Constant Mass

We start by calculating wave function for this system for constant mass (i.e.,
$m=m_{0}$), Schrodinger equation for this system is simplified as follows

$\frac{d^{2}\psi(x)}{dx^{2}}+\bigg{[}A-Bx^{2}-Cx^{4}\bigg{]}\psi(x)=0,$ (19)

where,
$A=\frac{2m_{0}E}{\hbar^{2}},B=\frac{2m_{0}a}{\hbar^{2}},C=\frac{2m_{0}b}{\hbar^{2}}$.
The wave function is computed (briefly discussed in VII.1) and is given by

$\psi(x)=\sqrt{\frac{A}{\pi}}\text{exp}\bigg{[}\frac{-A}{2}x^{2}\bigg{]},$
(20)

In reciprocal space we can write the wave function as

$\phi(k)=\frac{1}{\sqrt{\pi}}\text{exp}\bigg{[}\frac{-k^{2}}{2A}\bigg{]}.$
(21)

####  IV.1.1 Shannon’s entropy

According to Bonn [38], the statistical interpretation of the quantum system,
which describes the probability of finding the particle in the state
$\psi(x,t)$ between the spatial interval $x$ and $x+dx$ describes as
$|\psi(x,t)|^{2}\approx|\psi(x)|^{2}$. Shannon’s entropy in position space
($S_{x}$) is given by

$S_{x}=-\int_{-\infty}^{\infty}|\psi(x)|^{2}\text{ln}[|\psi(x)|^{2}]dx,$ (22)

and we obtain the value for $S_{x}$ as

$S_{x}=\sqrt{\frac{A}{\pi}}\bigg{(}\frac{1}{2}-\text{ln}\bigg{[}\frac{A}{\pi}\bigg{]}\bigg{)}.$
(23)

In reciprocal space, it is given by $S_{k}$

$S_{k}=-\int_{-\infty}^{\infty}|\phi(k)|^{2}\text{ln}[|\phi(k)|^{2}]dk,$ (24)

where $D$ is the dimension of the spatial coordinates of the system. and we
obtain the value for $S_{k}$ as

$S_{k}=\sqrt{\frac{A}{\pi}}\bigg{(}\frac{1}{2}+\text{ln}\pi\bigg{)},$ (25)

such that,

$S_{x}+S_{k}\geq D(1+\text{ln}\pi).$ (26)

where $D$ is the dimension of the spatial coordinates of the system. The
numerical study of Shannon entropy was carried out considering the
eigenfunctions at the position and momentum space, as well as the definition
of Shannon entropy, i.e. eq. (24) and eq. (22). The numerical result of
Shannon entropy for constant mass is presented in the following table:

$A$ |  $S_{x}$ |  $S_{k}$ |  $S_{x}+S_{k}$ |  $1+\text{ln}\pi$
---|---|---|---|---
3.5 |  0.4137 |  1.7359 |  2.1496 |  2.1447
3.6 |  0.3894 |  1.7605 |  2.1499 |  2.1447
3.7 |  0.3650 |  1.7848 |  2.1498 |  2.1447
3.8 |  0.3406 |  1.8088 |  2.1494 |  2.1447
3.9 |  0.3161 |  1.8324 |  2.1485 |  2.1447
4.0 |  0.2915 |  1.8558 |  2.1473 |  2.1447
4.2 |  0.2424 |  1.9016 |  2.1448 |  2.1447
Table 1:  Numerical results of the Shannon entropy

####  IV.1.2 Fisher’s Information measure

For an observable $x$, Fisher information ($F_{x}$) is given by

$F_{x}=\int_{-\infty}^{\infty}|\psi(x)|^{2}\bigg{[}\frac{d}{dx}\text{ln}|\psi(x)|^{2}\bigg{]}^{2}dx>0,$
(27)

and we obtain the value for $F_{x}$ as

$F_{x}=\frac{2A^{\frac{3}{2}}}{\sqrt{\pi}}.$ (28)

In reciprocal space, Fisher information for an observable $k$, ($F_{k}$) is
given by

$F_{k}=\int_{-\infty}^{\infty}|\phi(k)|^{2}\bigg{[}\frac{d}{dk}\text{ln}|\phi(k)|^{2}\bigg{]}^{2}dk>0,$
(29)

and we obtain the value for $F_{k}$ as

$F_{k}=\frac{2}{\sqrt{A\pi}}.$ (30)

Let us also remember that the standard deviation of the position and momentum
measures is given respectively by

$\sigma^{2}_{x}=<x^{2}>-<x>^{2},$ (31) $\sigma^{2}_{k}=<k^{2}>-<k>^{2}.$ (32)

where, $<x>,<k>,<x^{2}>,<k^{2}>$ are the expected values of $x,k,x^{2},k^{2}$
respectively. Using the definitions of Fisher information given in eq. (8) and
eq. (10) and standard deviations given in eq. (31) and eq. (32), we present
the numerical result of Fisher’s information measure for mass distribution is
presented in the following table:

$A$ |  $F_{x}$ |  $F_{k}$ |  $\sigma^{2}_{x}$ |  $\sigma^{2}_{k}$ |  $F_{x}\cdot F_{k}$
---|---|---|---|---|---
3.5 |  7.386 |  0.6029 |  0.1507 |  1.846 |  4.453
3.6 |  7.704 |  0.5945 |  0.1486 |  1.926 |  4.580
3.7 |  8.028 |  0.5864 |  0.1466 |  2.007 |  4.707
3.8 |  8.356 |  0.5786 |  0.1447 |  2.089 |  4.834
3.9 |  8.688 |  0.5711 |  0.1428 |  2.172 |  4.962
4.0 |  9.024 |  0.5640 |  0.1410 |  2.256 |  5.089
4.2 |  9.709 |  0.5504 |  0.1376 |  2.427 |  5.344
Table 2:  Results for the uncertainty relation and Fisher information measure
Figure 3: Fisher’s information measure for soliton in position space (constant
mass) in quartic potential. Figure 4: Fisher’s information measure for soliton
in momentum space (constant mass) in quartic potential.

Consequently, by examining the results of the standard deviation, the
Heisenberg uncertainty principle was explored, leading to the derivation of
the following relationships,

$F_{x}=4\sigma^{2}_{k},$ (33) $F_{k}=4\sigma^{2}_{x},$ (34)

Heisenberg uncertainty principle is written as

$F_{x}F_{k}\geq 4\hbar^{2}.$ (35)

###  IV.2 Mass distribution

We consider the mass distribution to be $m(x)=\frac{m_{0}}{1+x^{2}}$. For the
sake of simplicity, we take the values of $a$ and $b$ to be unity. Thus the
modified potential can be read as,

$V(x)=x^{2}+x^{4}.$ (36)

The Schrodinger equation for the given mass distribution is simplified as,

$\frac{d^{2}\psi}{dx^{2}}-\frac{2m_{0}}{\hbar^{2}}x^{2}\psi=\frac{-2m_{0}E}{\hbar^{2}}\frac{1}{1+x^{2}}\psi,$
(37)

The wave function is computed (briefly discussed in VII.1) and is given by

$\psi(x)=\frac{A}{(1+x^{2})^{1/4}}\text{exp}\bigg{[}\frac{-x^{2}}{2(1+x^{2})}\bigg{]}.$
(38)

In reciprocal space, we can write the wave function as

$\phi(k)=\frac{A}{\sqrt{2}}\exp\left(\frac{-k^{2}}{2}-ik\right).$ (39)

where $A$ is a constant. This computation is carried out by assuming the
values of $m_{0}$ to be $1$.

####  IV.2.1 Shannon Entropy

Shannon’s entropy in position space ($S_{x}$) is given by

$S_{x}=-\int_{-\infty}^{\infty}|\psi(x)|^{2}\text{ln}|\psi(x)|^{2}dx,$ (40)

and we obtain the form for Shannon entropy $(S_{x})$ as

$S_{x}=-\int_{-\infty}^{\infty}\Bigg{(}\frac{2A^{2}\text{ln}A}{\sqrt{1+x^{2}}}\text{exp}\bigg{[}\frac{-x^{2}}{1+x^{2}}\bigg{]}-\frac{A^{2}}{2\sqrt{1+x^{2}}}\text{ln}(1+x^{2})\\\
\text{exp}\bigg{[}\frac{-x^{2}}{1+x^{2}}\bigg{]}-\frac{A^{2}x^{2}}{(1+x^{2})^{3/2}}\text{exp}\bigg{[}\frac{-x^{2}}{1+x^{2}}\bigg{]}\Bigg{)}dx,$

Similarly, Shannon entropy in momentum space is given by

$S_{k}=-\int_{-\infty}^{\infty}|\phi(k)|^{2}\text{ln}|\phi(k)|^{2}dk.$ (41)

and is obtained as follows

$S_{k}=\frac{\sqrt{\pi}A^{2}}{4}-\frac{A^{2}\sqrt{\pi}}{2}\text{ln}\bigg{[}\frac{A^{2}}{2}\bigg{]}.$
(42)

The numerical result of Shannon entropy for mass distribution is presented in
the following table:

$A$ |  $S_{x}$ |  $S_{k}$ |  $S_{x}+S_{k}$ |  $1+\text{ln}\pi$
---|---|---|---|---
0.01 |  20.458 |  0.00092 |  20.459 |  2.1447
0.02 |  81.523 |  0.0032 |  81.526 |  2.1447
0.05 |  506.964 |  0.0159 |  506.98 |  2.1447
0.1 |  2020.11 |  0.0514 |  2020.16 |  2.1447
0.2 |  8049.46 |  0.1564 |  8049.62 |  2.1447
0.3 |  18070.50 |  0.2872 |  18070.79 |  2.1447
0.4 |  32073.9 |  0.4290 |  32074.33 |  2.1447
Table 3:  Numerical results of the Shannon entropy

####  IV.2.2 Fisher’s Information measure

For an observable $x$, Fisher information ($F_{x}$) is given by

$F_{x}=\int_{-\infty}^{\infty}|\psi(x)|^{2}\bigg{[}\frac{d}{dx}\text{ln}|\psi(x)|^{2}\bigg{]}^{2}dx>0,$
(43)

and we obtain the form for the fisher’s information measure $(F_{x})$ as

$F_{x}=\int_{-\infty}^{\infty}\Bigg{(}\frac{A^{2}}{\sqrt{1+x^{2}}}\text{exp}\bigg{[}\frac{-x^{2}}{1+x^{2}}\bigg{]}\Bigg{)}\Bigg{[}\frac{d}{dx}\Bigg{(}2\text{ln}A-\\\
\frac{1}{2}\text{ln}(1+x^{2})-\frac{x^{2}}{1+x^{2}}\Bigg{)}\Bigg{]}^{2}dx,$

In reciprocal space, Fisher information for an observable $k$, ($F_{k}$) is
given by

$F_{k}=\int_{-\infty}^{\infty}|\phi(k)|^{2}\bigg{[}\frac{d}{dk}\text{ln}|\phi(k)|^{2}\bigg{]}^{2}dk>0.$
(44)

and we obtain the value for $F_{k}$ as

$F_{k}=A^{2}\sqrt{\pi}$ (45)

$A$ |  $F_{x}$ |  $F_{k}$ |  $\sigma^{2}_{x}$ |  $\sigma^{2}_{k}$ |  $F_{x}\cdot F_{k}$
---|---|---|---|---|---
1.2 |  2.215 |  2.552 |  0.6380 |  0.5538 |  5.652
1.4 |  3.014 |  3.474 |  0.8685 |  0.7535 |  10.470
1.6 |  3.937 |  4.538 |  1.134 |  0.9842 |  17.866
1.8 |  4.982 |  5.742 |  1.436 |  1.246 |  28.607
2.0 |  6.152 |  7.090 |  1.772 |  1.538 |  43.618
2.2 |  7.444 |  8.579 |  2.145 |  1.861 |  63.862
2.4 |  8.858 |  10.209 |  2.552 |  2.214 |  90.431
Table 4:  Results for the uncertainty relation and Fisher information measure
Figure 5: Fisher’s information measure for soliton in position space (mass
distribution) in quartic potential. Figure 6: Fisher’s information measure for
soliton in momentum space (mass distribution) in quartic potential.

Based on the findings, it is observed that there is a phenomenon of
”information propagation” of the solitonic mass distribution when it is
exposed to the confining potential $V(x)=ax^{2}+bx^{4}$.

##  V Solitons in a Symmetric well

The prevalence of solitons is primarily attributed to a limited set of
adaptable nonlinear equations that govern various physical and biological
systems. Despite the distinct nature of these systems and their nonlinear
characteristics, the mathematical equations describing them can exhibit
significant similarities or even identical forms. Consequently, the study and
analysis of these generic nonlinear models hold significant importance in
diverse physical contexts, allowing for a deeper understanding of a wide range
of phenomena. We shall consider ultra-fast pulsed fiber lasers as an example.
It finds its application in the field of optical and quantum communication.
One of the major factors that affect communication in this system is
dispersion. Dispersion management is a key route for manipulating optical
pulses for some desired output in ultra-fast optics. Minimal dispersion is
achieved in the case of symmetric potential. So it’s crucial for us to
understand its behavior in case of symmetric potential [39]. In this section,
we shall discuss solitons in symmetric potential. We know the well-known
symmetric potential as

$V(x)=V_{0}\bigg{(}\frac{1-\lambda xcot(\lambda x)}{(\lambda x)^{2}}\bigg{)},$
(46)

where, $V_{0}$ is a constant parameter, $\lambda$ corresponds to a parameter
in the potential which has a unit of $[Length]^{-1}$.

Figure 7: Symmetric potential for several values of parameter $\lambda$ and
$V_{0}$ constant.

###  V.1 Constant mass

We start by calculating the wave function for this system for constant mass
(i.e., $m=m_{0}$), Schrodinger equation for this system is simplified as
follows

$\frac{d^{2}\psi(x)}{dx^{2}}+\frac{2m_{0}}{\hbar^{2}}\Bigg{[}E-V_{0}\bigg{(}\frac{1-\lambda
xcot(\lambda x)}{(\lambda x)^{2}}\bigg{)}\Bigg{]}\psi(x)=0,$ (47)

The wave function is computed (briefly discussed in VII.2) and is given by

$\psi(x)=\bigg{(}\frac{1}{\pi}\sqrt{\frac{2}{45}}\bigg{)}^{1/4}e^{\frac{-\lambda
x^{2}}{2}}\bigg{[}1-0.087(4\lambda x^{2}-2)\bigg{]}.$ (48)

In reciprocal space, the wave function $(\phi(k))$ can be written as

$\phi(k)=\bigg{(}\frac{1}{\pi}\sqrt{\frac{2}{45}}\bigg{)}^{1/4}e^{\frac{-k^{2}}{2\lambda}}\bigg{[}\frac{1.174}{\sqrt{\lambda}}-\frac{0.348}{\lambda^{3/2}}(\lambda-k^{2})\bigg{]}.$
(49)

####  V.1.1 Shannon Entropy

Shannon’s entropy in position space ($S_{x}$) is given by

$S_{x}=-\int_{-\infty}^{\infty}|\psi(x)|^{2}\text{ln}|\psi(x)|^{2}dx,$ (50)

and we obtain the form for Shannon entropy $(S_{x})$ as

$S_{x}=-\int_{-\infty}^{\infty}\Bigg{(}\bigg{(}\frac{1}{\pi}\sqrt{\frac{2}{45}}\bigg{)}^{1/2}e^{-\lambda
x^{2}}\bigg{[}1-0.087(4\lambda x^{2}-2)\bigg{]}^{2}\Bigg{)}\\\
\Bigg{(}\frac{1}{2}\text{ln}\bigg{(}\frac{1}{\pi}\sqrt{\frac{2}{45}}\bigg{)}-\lambda
x^{2}+2\text{ln}\bigg{[}1-0.087(4\lambda x^{2}-2)\bigg{]}\Bigg{)}dx,$

Similarly, Shannon entropy in momentum space is given by

$S_{k}=-\int_{-\infty}^{\infty}|\phi(k)|^{2}\text{ln}|\phi(k)|^{2}dx,$ (51)

and we obtain the form for Shannon entropy $(S_{k})$ as

$S_{k}=-\int_{-\infty}^{\infty}\bigg{(}\frac{1}{\pi}\sqrt{\frac{2}{45}}\bigg{)}^{1/2}e^{\frac{-k^{2}}{\lambda}}\bigg{[}\frac{1.174}{\sqrt{\lambda}}-\frac{0.348}{\lambda^{3/2}}(\lambda-k^{2})\bigg{]}^{2}\\\
\Bigg{(}\frac{1}{2}\bigg{(}\frac{1}{\pi}\sqrt{\frac{2}{45}}\bigg{)}-\frac{k^{2}}{\lambda}+2\text{ln}\bigg{[}\frac{1.174}{\sqrt{\lambda}}-\frac{0.348}{\lambda^{3/2}}(\lambda-k^{2})\bigg{]}\Bigg{)}dk.$

The numerical result of Shannon entropy for constant mass is presented in the
following table:

$\lambda$ |  $S_{x}$ |  $S_{k}$ |  $S_{x}+S_{k}$ |  $1+\text{ln}\pi$
---|---|---|---|---
0.15 |  1.892 |  0.2565 |  2.149 |  2.1447
0.18 |  1.727 |  0.4434 |  2.170 |  2.1447
0.20 |  1.638 |  0.5353 |  2.174 |  2.1447
0.22 |  1.562 |  0.6903 |  2.252 |  2.1447
0.25 |  1.466 |  0.6960 |  2.162 |  2.1447
0.28 |  1.385 |  0.7619 |  2.147 |  2.1447
0.30 |  1.340 |  0.810 |  2.150 |  2.1447
Table 5:  Numerical results of the Shannon entropy

####  V.1.2 Fisher’s Information measure

For an observable $x$, Fisher information ($F_{x}$) is given by

$F_{x}=\int_{-\infty}^{\infty}|\psi(x)|^{2}\bigg{[}\frac{d}{dx}\text{ln}|\psi(x)|^{2}\bigg{]}^{2}dx>0,$
(52)

and we obtain the form for Fisher information $(F_{x})$ as

$F_{x}=\int_{-\infty}^{\infty}\Bigg{(}\bigg{(}\frac{1}{\pi}\sqrt{\frac{2}{45}}\bigg{)}^{1/2}e^{-\lambda
x^{2}}\bigg{[}1-0.087(4\lambda x^{2}-2)\bigg{]}^{2}\\\ \Bigg{[}-2\lambda
x+\frac{1.392\lambda x}{1-0.087(4\lambda x^{2}-2)}\Bigg{]}^{2}\Bigg{)}dx.$

In reciprocal space, Fisher information for an observable $k$, ($F_{k}$) is
given by

$F_{k}=\int_{-\infty}^{\infty}|\phi(k)|^{2}\bigg{[}\frac{d}{dk}\text{ln}|\phi(k)|^{2}\bigg{]}^{2}dk>0,$
(53)

and we obtain the form for Fisher information $(F_{k})$ as

$F_{k}=\int_{-\infty}^{\infty}\Bigg{(}\bigg{(}\frac{1}{\pi}\sqrt{\frac{2}{45}}\bigg{)}^{1/2}e^{\frac{-k^{2}}{\lambda}}\bigg{[}\frac{1.174}{\sqrt{\lambda}}-\frac{0.348}{\lambda^{3/2}}(\lambda-k^{2})\bigg{]}^{2}\\\
\Bigg{[}\frac{-2k}{\lambda}+\frac{1.392}{\lambda^{3/2}(\frac{1.174}{\sqrt{\lambda}}-\frac{0.348}{\lambda^{3/2}}(\lambda-k^{2}))}\Bigg{]}^{2}\Bigg{)}dk.$

We present the numerical result of Fisher’s information measure in the
following table:

$\lambda$ |  $F_{x}$ |  $F_{k}$ |  $\sigma^{2}_{x}$ |  $\sigma^{2}_{k}$ |  $F_{x}\cdot F_{k}$
---|---|---|---|---|---
0.01 |  0.1834 |  557.04 |  139.26 |  0.0458 |  102.16
0.02 |  0.2595 |  196.94 |  49.235 |  0.0648 |  51.105
0.05 |  0.4103 |  49.823 |  12.455 |  0.1026 |  20.442
0.08 |  0.5190 |  24.618 |  6.154 |  0.1298 |  12.777
0.10 |  0.5803 |  17.615 |  4.403 |  0.1450 |  10.222
0.15 |  0.7107 |  9.588 |  2.397 |  0.1777 |  6.814
0.20 |  0.8206 |  6.228 |  1.557 |  0.2051 |  5.110
0.25 |  0.9175 |  4.456 |  1.114 |  0.2293 |  4.088
Table 6:  Results for the uncertainty relation and Fisher information measure
Figure 8: Fisher’s information measure for soliton in position space (constant
mass) in symmetric well. Figure 9: Fisher’s information measure for soliton in
momentum space (constant mass) in symmetric well.

###  V.2 Mass distribution

For a mass distribution $m(x)=\frac{m_{0}}{1+x^{2}}$, Schrodinger equation is
modified as follows

$\frac{-\hbar^{2}}{2m}\frac{d^{2}\psi(x)}{dx^{2}}+\frac{-\hbar^{2}}{2}\frac{m^{\prime}(x)}{m(x)^{2}}\frac{d\psi(x)}{dx}+\Bigg{[}\frac{\hbar^{2}}{4}\frac{m^{\prime\prime}(x)}{m(x)^{2}}-\\\
\frac{3\hbar^{2}}{8}\frac{m^{\prime}(x)^{2}}{m(x)^{3}}+V_{0}\bigg{(}\frac{1-\lambda
xcot(\lambda x)}{(\lambda x)^{2}}\bigg{)}\Bigg{]}\psi(x)=E\psi(x).$ (54)

We adopt stationary perturbation theory and variation principle to obtain the
wave function [40]. Complete derivation of wave function is presented in
Appendix (VII.2). We get the wave function as follows

$\psi(x)\approx\frac{1}{\sqrt{\pi}}\frac{e^{\frac{-x^{2}}{2}}}{\sqrt{1+x^{2}}}\bigg{(}1-\frac{\lambda^{2}x^{2}}{10}+\frac{\lambda^{4}x^{4}}{126}-\frac{\lambda^{6}x^{6}}{2520}\bigg{)},$
(55)

In momentum space, the wave function is obtained through fourier transforming
the wave function in position space.

$\phi(k)\approx\frac{1}{\sqrt{\pi}}\frac{e^{\frac{-k^{2}}{2}}}{\sqrt{1+\frac{k^{2}}{\lambda^{2}}}}\bigg{(}1+\frac{\lambda^{2}k^{2}}{10}+\frac{\lambda^{4}k^{4}}{126}+\frac{\lambda^{6}k^{6}}{2520}\bigg{)}.$
(56)

####  V.2.1 Shannon entropy

Probability of finding the particle in the state $\psi(x,t)$ between the
spatial interval $x$ and $x+dx$ describes as
$|\psi(x,t)|^{2}\approx|\psi(x)|^{2}$. Shannon’s entropy in position space
($S_{x}$) is given by

$S_{x}=-\int_{-\infty}^{\infty}|\psi(x)|^{2}\text{ln}|\psi(x)|^{2}dx,$ (57)

and we obtain the form for Shannon entropy $(S_{x})$ as

$S_{x}=-\int_{-\infty}^{\infty}\Bigg{(}\frac{1}{\pi}\frac{e^{-x^{2}}}{(1+x^{2})}\bigg{(}1-\frac{\lambda^{2}x^{2}}{10}+\frac{\lambda^{4}x^{4}}{126}-\frac{\lambda^{6}x^{6}}{2520}\bigg{)}^{2}\Bigg{)}\\\
\Bigg{(}-\text{ln}\pi-x^{2}-\text{ln}(1+x^{2})+2\text{ln}\bigg{(}1-\frac{\lambda^{2}x^{2}}{10}+\frac{\lambda^{4}x^{4}}{126}-\frac{\lambda^{6}x^{6}}{2520}\bigg{)}\Bigg{)}dx.$

In reciprocal space,

$S_{k}=-\int_{-\infty}^{\infty}|\phi(k)|^{2}\text{ln}|\phi(k)|^{2}dk,$ (58)

and we obtain the form for Shannon entropy $(S_{k})$ as

$S_{k}=-\int_{-\infty}^{\infty}\frac{1}{\pi}\frac{e^{-k^{2}}}{(1+\frac{k^{2}}{\lambda^{2}})}\bigg{(}1+\frac{\lambda^{2}k^{2}}{10}+\frac{\lambda^{4}k^{4}}{126}+\frac{\lambda^{6}k^{6}}{2520}\bigg{)}^{2}\\\
\Bigg{(}-\text{ln}\pi-k^{2}-\text{ln}(1+\frac{k^{2}}{\lambda^{2}})+2\text{ln}\bigg{(}1+\frac{\lambda^{2}k^{2}}{10}+\frac{\lambda^{4}k^{4}}{126}+\frac{\lambda^{6}k^{6}}{2520}\bigg{)}\Bigg{)}dk.$

Finding an analytical solution for this particular equation will be
complicated. So, we use some numerical techniques to find Shannon entropy for
this system. The numerical result of Shannon entropy is presented in the
following table:

$\lambda$ |  $S_{x}$ |  $S_{k}$ |  $S_{x}+S_{k}$ |  $1+\text{ln}\pi$
---|---|---|---|---
2.0 |  0.6033 |  1.6511 |  2.254 |  2.1447
2.1 |  0.6034 |  1.8394 |  2.443 |  2.1447
2.2 |  0.6107 |  2.0343 |  2.645 |  2.1447
2.3 |  0.6289 |  2.2036 |  2.832 |  2.1447
2.4 |  0.6630 |  2.2848 |  2.948 |  2.1447
2.5 |  0.7190 |  2.1661 |  2.885 |  2.1447
2.6 |  0.8035 |  1.6592 |  2.463 |  2.1447
Table 7:  Numerical results of the Shannon entropy

####  V.2.2 Fisher’s Information measure

For an observable $x$, Fisher information ($F_{x}$) is given by

$F_{x}=\int_{-\infty}^{\infty}|\psi(x)|^{2}\bigg{[}\frac{d}{dx}\text{ln}|\psi(x)|^{2}\bigg{]}^{2}dx>0,$
(59)

and we obtain the form for Fisher’s information $(F_{x})$ as

$F_{x}=\int_{-\infty}^{\infty}\Bigg{(}\frac{1}{\pi}\frac{e^{-x^{2}}}{(1+x^{2})}\bigg{(}1-\frac{\lambda^{2}x^{2}}{10}+\frac{\lambda^{4}x^{4}}{126}-\frac{\lambda^{6}x^{6}}{2520}\bigg{)}^{2}\Bigg{)}\\\
\Bigg{[}\frac{d}{dx}\Bigg{(}-\text{ln}\pi-x^{2}-\text{ln}(1+x^{2})+2\text{ln}\bigg{(}1-\frac{\lambda^{2}x^{2}}{10}+\frac{\lambda^{4}x^{4}}{126}-\frac{\lambda^{6}x^{6}}{2520}\bigg{)}\Bigg{)}\Bigg{]}^{2}dx.$

In reciprocal space, Fisher information for an observable $k$, ($F_{k}$) is
given by

$F_{k}=\int_{-\infty}^{\infty}|\phi(k)|^{2}\bigg{[}\frac{d}{dk}\text{ln}|\phi(k)|^{2}\bigg{]}^{2}dk>0.$
(60)

and we obtain the form for Fisher’s information $(F_{k})$ as

$F_{k}=\int_{-\infty}^{\infty}\frac{1}{\pi}\frac{e^{-k^{2}}}{(1+\frac{k^{2}}{\lambda^{2}})}\bigg{(}1+\frac{\lambda^{2}k^{2}}{10}+\frac{\lambda^{4}k^{4}}{126}+\frac{\lambda^{6}k^{6}}{2520}\bigg{)}^{2}\\\
\Bigg{[}\frac{d}{dk}\Bigg{(}-\text{ln}\pi-k^{2}-\text{ln}(1+\frac{k^{2}}{\lambda^{2}})+2\text{ln}\bigg{(}1+\frac{\lambda^{2}k^{2}}{10}+\frac{\lambda^{4}k^{4}}{126}+\\\
\frac{\lambda^{6}k^{6}}{2520}\bigg{)}\Bigg{)}\Bigg{]}^{2}dk.$

$\lambda$ |  $F_{x}$ |  $F_{k}$ |  $\sigma^{2}_{x}$ |  $\sigma^{2}_{k}$ |  $F_{x}\cdot F_{k}$
---|---|---|---|---|---
0.01 |  1.3600 |  50.009 |  12.502 |  0.34 |  68.012
0.02 |  1.3595 |  25.0406 |  6.260 |  0.3398 |  34.042
0.03 |  1.3596 |  16.7331 |  4.183 |  0.1942 |  22.750
0.04 |  1.3597 |  12.5902 |  3.1476 |  0.3399 |  17.119
0.05 |  1.3597 |  10.1128 |  2.5282 |  0.3399 |  13.750
0.08 |  1.36003 |  6.4257 |  1.6064 |  0.3400 |  8.739
0.1 |  1.3606 |  5.2142 |  1.3036 |  0.3402 |  7.094
Table 8:  Results for the uncertainty relation and Fisher information measure
Figure 10: Fisher’s information measure for soliton in position space (mass
distribution) in symmetric well. Figure 11: Fisher’s information measure for
soliton in momentum space (mass distribution) in symmetric well.

Based on the findings, it is observed that there is a phenomenon of
”information propagation” in the solitonic mass distribution when it is
exposed to the confining potential $V(x)=V_{0}\bigg{(}\frac{1-\lambda
xcot(\lambda x)}{(\lambda x)^{2}}\bigg{)}$.

##  VI Summary and Remarks

This study focused on examining Shannon entropy and Fisher information
concerning a constant mass and solitonic mass distribution. The work explored
the solutions of the stationary Schrodinger equation, where the mass depends
on the position and is exposed to a quartic ($ax^{2}+bx^{4}$) and symmetric
potential ($V_{0}\bigg{(}\frac{1-\lambda xcot(\lambda x)}{(\lambda
x)^{2}}\bigg{)}$). After obtaining the analytical solutions of the model, the
investigation proceeded to study the Shannon entropy and Fisher’s information
associated with the ground state energy levels. The observations indicate that
the solitonic mass distribution under the quartic potential exhibits higher
Shannon entropy values compared to the case with a constant mass. Hence, the
combined value of Shannon entropy (i.e., the sum of $S_{x}$ and $S_{k}$) is
greater for the solitonic mass distribution than for a constant mass scenario.
On the other hand, the Fisher information is higher for the solitonic mass
distribution in comparison to the constant mass scenario. In the context of
information [41] and communication theories [42], based on our observations,
we can deduce that the solitonic system with mass distribution possesses
attributes that empower it to manage unpredictability, convey abundant
information, demonstrate heightened responsiveness, and function effectively
in various communication and data processing scenarios. Moreover, this system
also displays greater capacity, resilience, and versatility, leading to
enhanced performance and reliability in contemporary communication situations
than the system with a constant mass. Thus for effective communication, a
solitonic mass distribution can be preferred.

When considering solitons in symmetric potential, we notice distinct behavior
compared to that in quartic potential. The results show that the solitonic
mass distribution within the symmetric well has greater Shannon entropy values
compared to a constant mass scenario. As a consequence, the combined Shannon
entropy (the sum of $S_{x}$ and $S_{k}$) is greater for the solitonic mass
distribution than the constant mass scenario. On the contrary, the Fisher
information is higher for constant mass in contrast to mass distribution. In
information theory [41], a solitonic system with constant mass distribution
reflects a system with increased unpredictability, richer information content,
and heightened sensitivity to changes. This combination leads to a greater
capacity for data transmission, improved parameter estimation, and enhanced
performance in communication and data processing. The system can efficiently
handle diverse information and is more robust against noise and disturbances,
making it well-suited for various applications in information theory and
communication. However, managing the higher complexity and resource
requirements of such a system may be necessary. Nonetheless, in communication
systems [42], employing a solitonic system with constant mass leads to greater
capacity, improved noise resilience, and enhanced data transmission
capabilities, making them well-suited for modern communication scenarios and
emerging technologies. Therefore, based on our observations, we can infer that
an improved quantum communication can be attained using two types of solitonic
systems: one under a symmetric well with constant mass, and the other under a
quartic potential with a mass distribution.

In conclusion, the extension of the current work to investigate Shannon and
Fisher’s information measure in the context of Sine-Gordon solitons and
solitons in Bose-Einstein condensates (BEC) within the Gross-Pitaevskii
equation (GPE) framework will provide a deeper and more comprehensive
understanding of the underlying complexities of these intriguing systems. The
exploration of information entropy, both from a Shannon and Fisher
perspective, will unveil valuable insights into the intrinsic nature of
solitonic phenomena. By quantifying the uncertainty and information content
associated with these soliton structures, we gain a quantitative measure of
the intricate balance between coherence, localization, and the underlying
dynamics of the systems under consideration. Studying information entropy in
Sine-Gordon solitons will not only elucidate the richness of these
topologically nontrivial solutions but will also establish a connection
between information theory and field theory [43]. The quantification of
information entropy in this context will shed light on the interplay between
spatial distribution, energy concentration, and the stability of solitonic
configurations. Such insights not only will enhance our theoretical
understanding but also paves the way for potential applications in fields such
as data encoding, signal processing, and quantum information. Extending the
analysis to solitons in BEC gases governed by the GPE will introduce a quantum
mechanical dimension to the study. The intricate balance between quantum
coherence and nonlinear interactions inherent in BEC systems will add a layer
of complexity to the information entropy analysis. The quantification of
information content in these systems will offer a unique perspective on the
emergence and behavior of solitons in a quantum context, contributing to the
broader understanding of quantum coherence and correlations. The culmination
of this work will highlight the universality of information entropy as a
powerful tool for characterizing and comparing soliton dynamics across
different physical systems.

##  VII Appendix

###  VII.1 Solitons in Quartic potential - Wave function computation

####  VII.1.1 Constant mass

Our aim is to solve eq.(19). We employ the technique by assuming the solution
of this particular differential equation, viz, Gaussian wave function.
Previously, this assumption is used only to solve the system’s quadratic
potentials with time dependence. However, in [44], it is shown that the same
method also works in the case of quartic potential. The main difference is
that all the systems which are studied using this formulation work only for
time-dependent cases, but we work with solitonic systems which are time-
independent. This formulation (assumption) works even in our case. The
Gaussian function is assumed to have some parameters $r,s,q$ and a
normalization factor of $N$. We assume the solution $\psi(x)$ as

$\psi(x)=N\text{exp}\bigg{[}rx^{2}+sx+q\bigg{]}.$ (61)

Substituting this assumption in this differential equation eq. (19) we get

$\frac{d^{2}\psi}{dx^{2}}=(2rx+s)^{2}\psi(x)+2r\psi(x),$

Thus we get,

$(2rx+s)^{2}\psi(x)+2r=Cx^{4}+Bx^{2}-A,$

By comparing the co-coefficients, we get the wave function of this particular
system to be

$\psi(x)=N\text{exp}\bigg{[}\frac{-A}{2}x^{2}\bigg{]},$ (62)

Upon normalising we get the final wave equation to be

$\psi(x)=\sqrt{\frac{A}{\pi}}\text{exp}\bigg{[}\frac{-A}{2}x^{2}\bigg{]}.$
(63)

####  VII.1.2 Mass distribution

The Schrodinger equation for the given mass distribution is given by,

$\frac{-\hbar^{2}}{2m_{0}}(1+x^{2})\frac{d^{2}\psi}{dx^{2}}+x^{2}\psi+x^{4}\psi=E\psi,$
(64)
$\frac{d^{2}\psi}{dx^{2}}-\frac{2m_{0}}{\hbar^{2}}x^{2}\psi=\frac{-2m_{0}E}{\hbar^{2}}\frac{1}{1+x^{2}}\psi,$
$\frac{d^{2}\psi}{dx^{2}}+\frac{2m_{0}}{\hbar^{2}}\bigg{(}\frac{E}{1+x^{2}}-x^{2}\bigg{)}\psi=0.$

We find the solution analytically by Perturbation theory, total Hamiltonian is
given by $H=H_{0}+H_{1}$. The corresponding potential for $H_{0}$ is $V_{0}$
which is equal to $x^{2}$ and $H_{1}=\lambda x^{4}$ (perturbation term), where
$\lambda$ is small. The wave function corresponding to $H_{0}$ (zeroth order
perturbation theory) is

$\psi_{0}(x)=A\text{exp}\bigg{[}\frac{-x^{2}}{2(1+x^{2})}\bigg{]}H_{n}(x),$
(65)

where, $H_{n}$ is the Hermite polynomial for a given state $n$, $A$ is the
normalization constant. However, we are interested in computing the wave
function only for the ground state (i.e. $n=0$). Therefore, we get the Hermite
polynomial to be, $H_{0}=1$. Therefore, the wave function for the system in
the ground state is given by

$\psi_{0}(x)=A\text{exp}\bigg{[}\frac{-x^{2}}{2(1+x^{2})}\bigg{]},$ (66)

here, $\lambda x^{4}$ term is looked as a perturbation to the $x^{2}$ term.
Thus the wave function perturbed to the first order is given by

$\psi_{1}(x)=\sum_{n}\frac{\bra{n}H_{1}\ket{0}}{E_{0}-E_{n}}\psi_{0}(x),$ (67)

For ground state, we get the perturbed wave function ($\psi_{1}(x)$) as

$\psi(x)=\frac{A}{(1+x^{2})^{1/4}}\text{exp}\bigg{[}\frac{-x^{2}}{2(1+x^{2})}\bigg{]}.$
(68)

###  VII.2 Solitons in Symmetric well - Wave function computation

####  VII.2.1 Constant mass

For a constant mass, the Schrodinger equation for symmetric potential is given
by

$\frac{d^{2}\psi(x)}{dx^{2}}+\frac{2m_{0}}{\hbar^{2}}\Bigg{[}E-V_{0}\bigg{(}\frac{1-\lambda
xcot(\lambda x)}{(\lambda x)^{2}}\bigg{)}\Bigg{]}\psi(x)=0,$ (69)

We find the solution analytically by Perturbation theory, total Hamiltonian is
given by $H=H_{0}+H_{1}$. The corresponding potential for $H_{0}$ is $V_{0}$
which is equal to the potential of the harmonic oscillator. We use stationary
perturbation theory to obtain the wave function and energy levels for this
particular system. This potential (up to $\mathcal{O}(x^{6})$ term) is
expressed as

$V(x)\approx V_{0}\bigg{(}\frac{1}{3}+\frac{1}{45}(\lambda
x)^{2}+\frac{2}{945}(\lambda x)^{4}+\frac{1}{4725}(\lambda x)^{6}\bigg{)}.$

The unperturbed hamiltonian is $H_{0}=\frac{V_{0}}{45}(\lambda x)^{2}$ and the
perturbed hamiltonian ($H_{1}$) is given by,

$H_{1}(x)=V_{0}\bigg{(}\frac{1}{3}+\frac{2}{945}(kx)^{4}+\frac{1}{4725}(kx)^{6}\bigg{)}$
(70)

The wave function of solitons in first-order perturbation theory is given by

$\psi^{(0)}_{n}(u)=\bigg{(}\frac{\lambda}{\pi}\bigg{)}^{1/4}\frac{1}{\sqrt{2^{n}}n!}H_{n}(u)e^{\frac{-u^{2}}{2}},$
(71)

where, $u=\sqrt{\lambda}x$ and $H_{n}$ is the Hermite polynomial and the
values are as follows:

$H_{0}=1$ $H_{1}=2u$ $H_{2}=4u^{2}-2$ $H_{3}=8u^{3}-12u$
$H_{4}=16u^{4}-48u^{2}+12$

For ground states ($n=0$), we get the wave function $(\psi^{(0)}_{0})$ to be

$\psi^{(0)}_{0}(x)=\bigg{(}\frac{\lambda}{\pi}\sqrt{\frac{2}{45}}\bigg{)}^{1/4}\text{exp}\bigg{[}-\lambda\frac{x^{2}}{2}\bigg{]}.$
(72)

Eigen value for unperturbed system reads $E^{(0)}_{0}=\frac{1}{2}$.

$\psi^{(0)}_{1}(x)=\psi^{(0)}_{0}+\frac{\bra{\psi^{(0)}_{2}}H_{1}(x^{4})\ket{\psi^{(0)}_{0}}}{E^{(0)}_{0}-E^{(2)}_{0}}\psi_{2}^{(0)}+\frac{\bra{\psi^{(0)}_{4}}H_{1}(x^{4})\ket{\psi^{(0)}_{0}}}{E^{(0)}_{0}-E^{(0)}_{4}}\psi_{4}^{(0)}.$
(73)
$\psi^{(0)}_{1}(x)=\psi^{(0)}_{0}+\frac{\bra{2}H_{1}(x^{4})\ket{0}}{E^{(0)}_{0}-E^{(2)}_{0}}\psi_{2}^{(0)}+\frac{\bra{4}H_{1}(x^{2})\ket{0}}{E^{(0)}_{0}-E^{(0)}_{4}}\psi_{4}^{(0)}$

We only look into unperturbed and the second order state. Thus we get the wave
function as follows

$\psi(x)=\psi^{(0)}_{0}+\frac{\bra{2}H_{1}\ket{0}}{E^{(0)}_{0}-E^{(2)}_{0}}.$
(74)

Therefore we get the final wave function as,

$\psi(u)=\bigg{(}\frac{1}{\pi}\sqrt{\frac{2}{45}}\bigg{)}^{1/4}e^{\frac{-u^{2}}{2}}\bigg{[}1-0.087(4u^{2}-2)\bigg{]},$
(75)

where, $u=\sqrt{\lambda}x$. We write the final expression as follows

$\psi(x)=\bigg{(}\frac{1}{\pi}\sqrt{\frac{2}{45}}\bigg{)}^{1/4}e^{\frac{-\lambda
x^{2}}{2}}\bigg{[}1-0.087(4\lambda x^{2}-2)\bigg{]}.$ (76)

####  VII.2.2 Mass distribution

The potential of symmetric well is given by

$V(x)=V_{0}\bigg{(}\frac{1-\lambda xcot(\lambda x)}{(\lambda x)^{2}}\bigg{)},$
(77)

For the sake of simplicity, we consider the value of $V_{0}$ to be $1$. This
potential can be expressed as,

$V(x)=-4\sum^{\infty}_{n=1}\frac{(-1)^{n}(2\lambda
x)^{2n-2}}{(2n!)}B_{2n},\hskip 15.00002pt\forall\lambda x<\pi.$ (78)

The first few terms are given by,

$\sum^{\infty}_{n=1}\frac{(-1)^{n}(2\lambda
x)^{2n-2}}{(2n!)}B_{2n}=\frac{-B_{2}}{2}+\frac{B_{4}}{6}(\lambda
x)^{2}-\cdots$ (79)

We use stationary perturbation theory to obtain the wave function and energy
levels for this particular system. This potential (up to $\mathcal{O}(x^{6})$
term) is expressed as

$V(x)=\frac{1}{3}+\frac{1}{45}(\lambda x)^{2}+\frac{2}{945}(\lambda
x)^{4}+\frac{1}{4725}(\lambda x)^{6}.$ (80)

The Schrodinger equation for the given mass distribution,
$m(x)=\frac{1}{1+x^{2}}$ (assuming the value of $m_{0}$ to be $1$), we get

$\frac{d^{2}\psi(x)}{dx^{2}}+\frac{2}{\hbar^{2}}\Bigg{[}E-V_{0}\bigg{(}\frac{1-\lambda
xcot(\lambda x)}{(\lambda
x)^{2}}\bigg{)}\Bigg{]}\bigg{(}\frac{1}{1+x^{2}}\bigg{)}\psi(x)=0,$ (81)

We work with perturbation theory for small values of $\lambda$ and the usual
convention $\hbar=1=V_{0}$. Thus we get

$\frac{d^{2}\psi(x)}{dx^{2}}+2\Bigg{[}E-\bigg{(}\frac{1}{3}+\frac{1}{45}(\lambda
x)^{2}+\frac{2}{945}(\lambda x)^{4}+\frac{1}{4725}(\lambda
x)^{6}\bigg{)}\Bigg{]}\\\ \bigg{(}\frac{1}{1+x^{2}}\bigg{)}\psi(x)=0,$ (82)

The first-order correction to the wave function $\psi(x)$ can be calculated
using the first-order perturbation theory formula

$\psi_{1}(x)=\sum_{n\neq
0}\frac{\bra{\psi^{(0)}_{n}}H_{1}\ket{\psi^{(0)}_{0}}}{E^{0}_{0}-E^{0}_{n}}\psi_{0}(x),$
(83)

where, $H_{1}$ is the perturbation term. In our case, it’s our usual
potential. Now, the first-order perturbation correction to the final ground
state wave function is,

$\psi(x)\approx\frac{1}{\sqrt{\pi}}\frac{e^{\frac{-x^{2}}{2}}}{\sqrt{1+x^{2}}}\bigg{(}1-\frac{\lambda^{2}x^{2}}{10}+\frac{\lambda^{4}x^{4}}{126}-\frac{\lambda^{6}x^{6}}{2520}\bigg{)}.$
(84)

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# Reading tea leaves? Polygenic scores and differences in traits among groups.

Graham Coop

###### Abstract

In the past decade, Genome-Wide Association Studies (GWAS) have delivered an
increasingly broad view of the genetic basis of human phenotypic variation.
One of the major developments from GWAS is polygenic scores, a genetic
predictor of an individual’s genetic predisposition towards a trait
constructed from GWAS. The success of GWAS and polygenic scores seems to
suggest that we will soon be able to settle debates about whether phenotypic
differences among groups are driven in part by genetics. However, answering
these questions is more complicated than it seems at first glance and touches
on many old issues about the interpretation of human genetic variation. In
this perspective piece I outline the ways in which issues of causality,
stratification, gene-by-environment interactions, and divergence among groups
all complicate the interpretation of among-population polygenic score
differences.

Debates about the contribution of genetics to differences among groups have a
long and contentious history. We have known for a long time that nearly all
traits are partially heritable, meaning that variation in the traits is due
both to genetic differences as well as environmental ones. The heritability of
traits within groups has led some observers—both scientists and non-
scientists—to conclude that between-group differences in average traits can
also be explained in part by genetics, approximately in proportion to their
heritability. Yet if a trait is highly heritable within a population, it does
not follow that differences between groups are due to genetics—instead
environmental and cultural differences could be the primary driver of between-
population differences (Feldman and Lewontin, 1975).

Recently, the field of genetics has made progress in identifying regions of
the genome (mostly single nucleotide polymorphisms, SNPs) that are associated
with differences among individuals within a population, using genome-wide
association studies (GWAS). GWAS have found SNPs associated with a dizzying
array of traits. Concurrently, sophisticated methods have emerged to estimate
the combined phenotypic effect of many genetic loci
citepvinkhuyzen2013estimation. The success of GWAS seems to suggest that we
will soon be able to settle debates about whether average phenotypic
differences among populations are driven in part by genetic differences.
Answering this question is harder than it seems at first glance, however. In
this perspective, I will explain some of the complications, including how
stratification, gene-by-environment interactions, and correlations among SNPs
can make it difficult or in some cases effectively impossible to use GWAS to
understand differences among populations.

Some of these complications are perhaps best illustrated with a toy example.
Say we perform a GWAS of the amount of tea that individuals in the UK drink
(e.g. in the UK Biobank, Taylor et al., 2018; Cole et al., 2019). On the basis
of this tea-drinking GWAS, a person might argue that we could learn about
France-UK differences in tea consumption by counting up the average number of
alleles for tea preference that individuals in the UK and France carry. If the
British, overall, are more likely to have alleles that are associated with
increased tea consumption than the French, then he might claim that we have
demonstrated that the difference between French and UK peoples’ preference for
tea is in part genetic. Our protagonist would assure us that these alleles are
polymorphic in both countries and that both environment and culture plays an
important role. He would further reassure us that there will be an overlapping
distribution of numbers of tea-drinking alleles in both countries, so he’s not
saying that all British people drink more tea than all French people for
genetic reasons. He’ll tell us he’s simply interested in showing that the
average difference in tea consumption is partly due to genetic differences
between groups.

At face value, this argument seems scientifically plausible; if there are
alleles for tea preference, then to determine whether the British peoples’
love of a good cuppa tea is genetic, we could simply count these alleles up
and compare them to the average allele counts in France. Adding up these tea
preference alleles for individuals is one way of calculating an individual’s
”polygenic score”. Polygenic scores can be thought of as an attempt at the
prediction of people’s trait values computed from genotype data. There are
several ways of calculating polygenic scores, and they have a range of
potential uses. For example, people have done GWAS for risk of coronary artery
disease and constructed polygenic scores in the hope of enabling early
detection and preventive care (Khera et al., 2018). Currently, these polygenic
scores tend not to explain much of the variation in a given trait, but the
size of studies is increasing, and predictions based on polygenic scores will
likely become more accurate within populations (but see Mostafavi et al.,
2019).

Polygenic scores constructed using GWAS information from a single population
are expected to differ among populations. The allele frequency at every locus
will vary among populations because of genetic drift— the compounding of
chance variation in which alleles are transmitted from generation to
generation—leads allele frequencies among populations to diverge over time. If
natural selection acts on the loci differently in the two populations, it will
also cause allele frequencies to differ, but genetic drift is sufficient to
cause allele frequency differences and will occur regardless of whether there
is selection on the trait. Since a polygenic score is just a weighted sum of
allele counts, it will also vary among populations in ways that are reasonably
well understood in the case of genetic drift (Berg and Coop, 2014).

Importantly, the existence of genetic differences among populations does not
imply that observed differences in phenotype among populations are the result
of genetics. It could be that French people tend to have higher polygenic
scores for tea-consumption than the British, but that this genetic
predisposition is hidden or counteracted by cultural influences. For example,
perhaps British people on average find bitter (tannin) tastes slightly less
palatable than French people, but this influence is overridden by the culture
of tea drinking in the UK. It is a simple point that genetics and environment
can act in opposite influences on a phenotypic difference between populations,
yet people often assume that differences in PGS should somewhat line up with
observed phenotypic differences as if that were not the case.

Beyond the fact that environment and culture can counteract or overwhelm the
influence of genetics, there’s another, more subtle problem: polygenic scores
are not strong statements about differences in the contribution of genetics to
phenotypic variation among groups. The issue is that GWAS studies do not point
to specific alleles _for_ tea preferences, only to alleles that are associated
with tea preference in the set of environments experienced by people enrolled
as participants in the GWAS in the UK Biobank. Similarly, as geneticists, we
often talk about height alleles and loci obtained from GWAS. But these are not
alleles _for_ height, but simply alleles that are associated with differences
in height within a population that was studied in a GWAS. There’s no guarantee
that alleles mapped within populations will affect the trait in the same way
in other populations and environments, nor (even if they do) that they will
explain differences between populations.

Complex traits are just that—complex. Most traits are incredibly polygenic,
likely involving tens of thousands of loci. These loci will act via many
pathways (Boyle et al., 2017), mediated by interactions with many
environmental and cultural factors. Some of our tea-GWAS SNPs may well be
enriched near olfactory receptors and genes expressed in relevant parts of
brain, and some may overlap with SNPs associated with caffeine sensitivity
(Taylor et al., 2018). But the majority may not—many may fall near genes with
no simple connection to tea drinking. For example, how thirsty you get may
play into their beverage choice, thus genetic variants in many physiological
pathways may be weakly associated with tea drinking. The rare cases where we
can confidently make a specific causal connection to a gene and through a
causal pathway all the way to phenotype may explain so little of the variance
that, while they may provide important clues to biology, they often won’t
allow us to state a general causal mechanism that underlies our genetic
predictions. This should not be taken as an anti-GWAS position. We have
learned a lot of new biology from GWAS (Visscher et al., 2017), and doubtless
will learn a lot more over the coming years. But GWAS are not a complete
solution to understanding the causes of variation, especially variation among
populations and across environments (Rosenberg et al., 2018). Let’s see some
reasons why.

## Population stratification

One long-standing concern in GWAS is the potential confounding of
environmental and genetic variation due to population stratification. To co-
opt a classic illustrative example of the problem (Lander and Schork,
1994),consider performing a GWAS of the phenotype ”drinks strong tea” in
Paris. Any allele at slightly higher frequency in English immigrants in Paris
than in other Parisians could be spuriously correlated with tea drinking.
These stratification issues will arise whether the cause of British preference
for tea drinking is environmental or partially genetic (Vilhjálmsson and
Nordborg, 2012). Such stratification issues may replicate across seemingly
independent datasets. There is no such thing as a truly independent sample of
humans–just samples with varying degrees of relatedness thus stratification
issues may replicate even across seemingly quite different samples.

Concerns about stratification drove human GWAS to be implemented primarily in
what were characterized as relatively genetically homogeneous populations,
with the hope that genotypes would be reasonably randomized across
environmental and genetic backgrounds within that population. To mitigate
against any remaining confounding by stratification, a broad range of
statistical techniques were developed Price et al. (2010); Bulik-Sullivan et
al. (2015). These advances have likely greatly reduced some of the issues of
stratification in modern GWAS, notably for the strongest associations.
However, even a small amount of residual stratification can potentially cause
serious problems for the interpretation of polygenic scores, as this subtle
bias is compounded across the loci used to construct a polygenic score. This
problem does not boil down to false-positive genetic associations, because
even true positives can have effect size estimates that are subtly biased by
stratification.

One vivid example is offered by recent issues with height polygenic scores.
Starting in 2012 (Turchin et al., 2012), a number of studies identified a
seemingly strong difference in polygenic scores for height between Northern
and Southern Europeans Berg and Coop (2014); Berg et al. (2017); Mathieson et
al. (2015); Robinson et al. (2015); Zoledziewska et al. (2015); Racimo et al.
(2018); Guo et al. (2018). To this end, they relied on scores that were
constructed from effect sizes estimated by the GIANT GWAS, a meta-analysis of
various European ancestries samples, and seemingly replicated their results in
various ways. However, when a number of studies tried to replicate the
European height results using GWAS effect sizes from the UK Biobank Bycroft et
al. (2018)—a large, independent sample—they found that the previously observed
patterns did not replicate (Berg et al., 2019; Sohail et al., 2019) and upon
further inspection, found problems with population stratification in the
original GIANT height GWAS and replication efforts. Many of the top hits did
replicate between the GIANT and UK Biobank studies, so arguably the GIANT
study fulfilled its main aim. However, GIANT was not a reliable basis for
building PGS, which compounds many subtly biased effect size estimates. Other
papers have begun to highlight potentially related issues Kerminen et al.
(2018); Haworth et al. (2019), demonstrating that stratification may be a
serious confounding issue for polygenic scores at even smaller geographic
scales.

For many traits, there may be no bright line between population stratification
and ‘real’ biological effects Lawson et al. (2019); Haworth et al. (2019);
Belsky et al. (2019). People in better health are more likely to move long
distances within the UK (Brimblecombe et al., 2000). If people tend to move to
areas with less tea drinking per capita, then alleles associated with overall
health wd then become alleles associated with decreased tea drinking.
Adjusting for genome-wide relatedness in the GWAS would likely not make this
effect go away—the induced correlations are beyond those expected due to
stratification due to genome-wide relatedness. Yet this too is a form of
stratification, one that may not replicate among groups.

## Gene-by-environment interactions (G x E)

The effect of an allele on any given phenotype is always measured in the
context of a particular set of environments. This issue is not new: debates
over the meaning of heritability and the genetics in the context of
environmental variation stretch back to the dawn of quantitative genetics and
debates between Hogben and Fisher (Tabery, 2008). These issues are
particularly difficult in humans, as we cannot raise humans in laboratory
environments or conduct randomized environments. Our behavioural, cultural and
societal practices will influence the ways in which genetic variants impact
phenotypic variation.

For example, there are cultural differences between the UK and France in
whether milk is taken with tea, in the types and quality of tea drunk, and in
the prominence of coffee. Do polymorphisms associated with bitter taste
sensitivity lead to different beverage outcomes in the two countries? What
role do parents, siblings, and peers play in shaping one’s choice of hot
drinks, and how do these interactions differ between countries? The genotypes
of other people can indirectly shape one’s phenotype by shaping one’s
environment (Kong et al., 2018). Presumably, all of these differences, and
many others, could mean that the genetic basis of variation in tea drinking
will differ between France and the UK. Therefore, the loci that influence tea
drinking in the UK could be somewhat different from those underlying
differences in tea drinking in France.

Even if we perform tea-drinking GWAS in multiple countries, we may not be able
to circumvent the issues of GxE. Suppose after our GWAS for tea drinking in
the UK and France, we find found that the genetic basis of the trait within
both countries to be correlated. What would be a high enough correlation to
constitute evidence of a genetic difference in phenotypic preferences between
countries? Observing a high genetic correlations in GWAS results amongst
groups does not mean that polygenic predictions are necessarily portable
between groups (even within groups with very similar ancestries Mostafavi et
al., 2019). Moreover, even if the polygenic score explained a lot of the
variance within each country, it may not explain much of the difference
between the countries. As one example: suppose, hypothetically, that people
who care greatly about their weight are more likely to drink tea (e.g., as
compared with soft drinks); then alleles that are correlated with body mass
index (BMI) in the UK Biobank will be alleles predicted to predispose one to
tea drinking. These loci may be reliably associated with BMI and tea drinking
in both the UK and France. Yet a difference in the frequency of loci
associated with BMI between the UK and France would not imply that differences
in tea drinking preferences among countries result from genetics. Suppose for
example that an individual’s preference for tea is not influenced by their
absolute BMI, but rather by their relative BMI within a country. In this
scenario, a polygenic score could be predictive of individual phenotypes
within multiple countries but have little predictive power in explaining
differences among those countries.

Without a thorough understanding of the causal biological and cultural
mechanisms by which GWAS SNPs interact with the range of environments
encountered by individuals, it may be hard to rule out GxE as a serious
confounder of inferences of polygenic scores across populations. These
problems do not just exist among populations. There’s systematic variation
across the UK in opinions about how milky tea should be, and whether milk
should be added before or after the tea (a matter of intense debate Fisher,
1935; Orwell, 1946; Salsburg, 2001). Tastes in tea and use of sugar vary
across socio-economic groups with UK (North, 2015). What phenotype, and which
set of environments, are we learning about when we examine polygenic scores?

## We don’t have the functional genetic markers

A third major hurdle that we face in understanding polygenic scores is that we
do not know the loci that are functionally important for trait variation, only
loci that are statistical proxies for them—sometimes called tag SNPs—that will
be nearby in the genome. (Technically the SNPs used to construct polygenic
scores are in linkage disequilibrium with the functional loci—meaning that
genotypes at the tag SNP are correlated with genotypes at the functional
locus—but unlikely to be the functional loci themselves.) To understand this
point, look at the example below (Figure 1). On the left is a cartoon of
people from the UK. Both of the filled circle alleles appear to be associated
with tea drinking. ( However, only one of them is the functional SNP
predisposing people toward tea drinking; the other SNP just happens to be
associated because the mutation there arose at a similar point in history on
the same genetic background. If we guess that the blue allele is the
functional one, we would predict that French people have a slightly weaker
preference to tea on the basis of this allele. But if we guess the red allele
is the functional one, we would predict that the UK and France have very
similar tea drinking habits on the basis of this locus.

What’s happened here is that the correlation between the alleles at the two
loci have changed due to different histories of recombination and genetic
drift. Now, such a strong change in the correlation of loci is unlikely
between two countries, such as Britain and France, that share so much of their
genetic history. However, it is a serious problem when comparing populations
that have been more distant from each other for a longer period of time. The
fact that the correlation between any two SNPs changes over evolutionary time
is a major candidate explanation for why polygenic scores lose predictive
ability as we move to populations that have been isolated for more of their
history from the population in which the GWAS was conducted (Carlson et al.,
2013; Marigorta and Navarro, 2013; Martin et al., 2019). For highly polygenic
traits, the associations we find may partially reflect collections of loosely
linked SNPs, whose relationship may be subject to change even between closely
related populations.

Figure 1: Each person has two chromosomes (horizontal black lines) and in this
small stretch of the genome there are two loci (red and blue SNPs), the
alleles of which are indicated by the presence/absence of a filled circle.
Whether an individual drinks a lot of tea is indicated by the tea cup next to
the individual (public domain tea cup). Obviously, this sample size is too
small, but it serves for illustration.

A second, more subtle force can decrease the predictive validity of polygenic
scores. Assortative mating among individuals can drive rapid changes in the
SNPs associated with a trait. For example, if people who drink more tea tend
to have children with taller people, this pattern of assortative mating can
cause greater height and tea drinking to become associated (i.e., assortative
mating can generate genetic correlations Gianola, 1982). In other words,
height-increasing alleles will be associated with tea drinking because the
offspring of tea-drinking/tall couples will have alleles associated with both
tea drinking and height. Even after assortative mating has stopped, these
effects can persist for a few generations as they are slowly broken down by
Mendelian segregation and recombination, making them potentially hard to rule
out. Such associations need not hold in other populations, however, if they
have not experienced similar patterns of assortative mating. Therefore, sets
of loci that contribute to trait variation via genetic correlations may change
rapidly across environments or populations due to changes in patterns of
assortment.

## We will not map all of the alleles influencing trait differences among
populations

Other things equal, the statistical power of GWAS is higher for alleles that
are at intermediate frequency in the GWAS population than for alleles that are
at low or high frequency. The functional variants contributing to a trait will
differ in frequency among populations due to genetic drift and selection.
Therefore, GWAS conducted in one population will miss many of the loci
contributing to phenotypic variation in other populations. This may not be
much of a problem for comparing the UK and French population, as allele
frequencies are very similar in the two countries. However, it is potentially
a much bigger problem in comparing more distant populations.

A case in point is the genetics of skin pigmentation. The variants that were
mapped within European populations, though important in Europe, explain little
of the variation in skin pigmentation worldwide. Even variants that explain
lighter European skin pigmentation do not explain lighter skin pigmentation in
East Asians (e.g. see Norton et al., 2006; Edwards et al., 2010; Adhikari et
al., 2019). Loci important for explaining skin-pigmentation variation
worldwide were missed by studies focused on non-African populations and only
found as variation within Africa began to be explored (Crawford et al., 2017;
Martin et al., 2017). Furthermore, our understanding of the evolutionary
history of skin pigmentation in Europe is undergoing major revision in the
light of ancient DNA (Olalde et al., 2014). This history of major shifts in
our understanding of the genetics and evolutionary history of skin
pigmentation suggests that bold claims about other traits, based on incomplete
evidence, may not stand the test of time.

In the coming decade, we will likely uncover a surprising amount of
heterogeneity in the alleles controlling trait variation worldwide. Based on
genetic drift alone, we should expect as much: the alleles that explain most
variance in populations of European ancestry and those which explain the most
variance in those of East Asian ancestry will not be the same, because allele
frequencies drift over time. Also as a result of allele frequency change at
many loci, across populations, epistatic relationships among loci may also
change in unpredictable ways, confounding cross-population predictions. These
problems of different alleles contributing to traits in different populations
will be compounded for traits subject to natural selection (in addition to
genetic drift). Whether traits are subject to stabilizing selection or
directional selection (shared or divergent), selection will drive more rapid
turnover in the loci contributing to trait variation among populations
(Durvasula and Lohmueller, 2019).

Again, one can hope to address these issues by performing GWAS in multiple
worldwide populations, but the reality is that we will have a European-biased
view of genetic variation for some time to come, simply because of the size of
the studies in these populations dwarfs those that can currently be done
elsewhere.

## Conclusion

Undoubtedly the coming decades of human genomics will see advances in the
identification of functional loci, the size of GWAS performed worldwide, and
in the statistical methodologies used to understand trait variation. The
conceptual issues and pitfalls discussed here are fascinating problems in
their own right. Working to understand the interplay of GxE along with direct
and indirect effects of genetics promises to be a rich seam of work for many
years to come. Interpreting differences in mean polygenic scores poses many
challenges, but some promises. Understanding cross-population differences in
genetics may allow us to better understand the contribution of genetics to
differences in disease prevalence among populations (although many differences
likely reflect healthcare and environmental disparities Williams and Jackson,
2005; Williams and Mohammed, 2009). While the ability to use GWAS to learn
about whether natural selection has driven differences in the genetic
architecture of human traits among populations promises to open up new
intersections between anthropology, genetics, and archaeology; for example in
understanding the role of environmental change and genetics in shaping changes
in height and body proportions during the Neolithic (Cox et al., 2019). There
is also no doubt that we will come to understand much more about human
variation. However, our ability to perform GWAS to identify loci underlying
variation in traits among individuals vastly outstrips our ability to
understand the causal mechanisms underlying these differences. In many cases,
genetic contributions may not be separable from environmental and cultural
differences. Certainly making a case for the relative importance of genetics
in explaining differences among populations will involve a lot more work than
simply counting up the number of tea preference alleles in populations and
seeing how the averages differ.

Despite these complications, we are poised to see many partial, incomplete
(and in some cases initially downright incorrect) stories about the genetics
of population differences in traits. Already, we thought we knew something
about the evolution of polygenic height scores among European populations,
only to find that much of this result was driven by stratification– and that’s
for height, an easily measured and well-studied trait. Applied to other and
more fraught traits, this patchy understanding of the contribution of genetics
to phenotypic differences will be fertile ground for misleading claims.

At the core of this issue is an even more fundamental disconnect between talk
of polygenic scores and what some people seem to think they might learn from
this kind of research. Even if we could attribute some proportion of the
phenotypic difference among populations to differences in polygenic score,
such a result won’t address the question that many have in mind in asking
whether a difference is “genetic.” Saying a phenotypic difference among
individuals is genetic is often taken to imply that it is immutable or
unavoidable. Yet a difference in genetic predispositions are neither immutable
or “natural”, as many people have highlighted (Penrose, 1955). Many phenotypes
where genetics plays a role are modifiable, and presumably many ”genetic”
trait differences among people are environmentally mediated. Without at least
some working knowledge of causal mechanisms underlying the action of the
genetic variation contributing to a trait, we may often not know how
environment and culture shape the actions of these variants, nor how changes
in these factors may modify any role played by genetics. Even if our tea
polygenic scores were strongly predictive within and among populations,
cultural changes, e.g., a Europe-wide health food craze for drinking tea with
dinner, could presumably stand these results on their head. Will taking tea
with a meal moderate the role of caffeine-sensitivity SNPs; will exercise-
conscious people now drink more tea? Will we know enough about the interaction
of culture and genetics to predict this? If we do not, the statement that a
difference in polygenic scores plays a role in explaining a difference in
phenotypes among populations will shed little light on how we as individuals
or societies should view that difference. But will these critical subtleties
be lost in the public’s understanding of results based on polygenic scores?
Will such results be wrongly taken as supporting genetic determinism about
human variation?

Acknowledgements. I thank Jeremy Berg, Vince Buffalo, Nancy Chen, Doc Edge,
Emily Josephs, Molly Przeworski, and Jenny Tung for helpful discussions and/or
feedback on earlier drafts. This work was supported in part by the National
Institute of General Medical Sciences of the National Institutes of Health
(NIH R01 GM108779).

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