# On the Capacity of Zero-Drift First Arrival Position Channels in Diffusive
Molecular Communication

Yen-Chi Lee  Hon Hai (Foxconn) Research Institute
Taipei, Taiwan
<EMAIL_ADDRESS>Jen-Ming Wu  Hon Hai (Foxconn) Research Institute
Taipei, Taiwan
<EMAIL_ADDRESS>Min-Hsiu Hsieh  Hon Hai (Foxconn) Research Institute
Taipei, Taiwan
<EMAIL_ADDRESS>
###### Abstract

Recent advancements in understanding the impulse response of the first arrival
position (FAP) channel in molecular communication (MC) have illuminated its
Shannon capacity. While Lee et al. shed light on FAP channel capacities with
vertical drifts, the zero-drift scenario remains a conundrum, primarily due to
the challenges associated with the heavy-tailed Cauchy distributions whose
first and second moments do not exist, rendering traditional mutual
information constraints ineffective. This paper unveils a novel
characterization of the zero-drift FAP channel capacity for both 2D and 3D.
Interestingly, our results reveal a 3D FAP channel capacity that is double its
2D counterpart, hinting at a capacity increase with spatial dimension growth.
Furthermore, our approach, which incorporates a modified logarithmic
constraint and an output signal constraint, offers a simplified and more
intuitive formula (similar to the well-known Gaussian case) for estimating FAP
channel capacity.

###### Index Terms:

diffusive molecular communication, absorbing receivers, first arrival position
(FAP), channel capacity, Cauchy distribution.

##  I Introduction

Molecular communication (MC) represents a novel communication paradigm rooted
in the transmission of information through molecular exchange [1, 2]. Owing to
its nano-scale applicability and biocompatibility, MC emerges as a potent
communication technique for nano-networks [3, 4]. Within these systems, tiny
message molecules (MM) act as the principal information carriers. A
propagation mechanism is necessary for transporting MMs to the receiver (Rx),
and this mechanism can be either diffusive (a.k.a. diffusion-based) [5], flow-
based [6], or due to an engineered transport system like molecular motors [7,
8]. Among these different propagation mechanisms, diffusive MC (sometimes in
combination with a drift field) has been the most prevalent approach for MC
theoretical research [9].

The molecular communication (MC) receiver’s reception mechanism can be
bifurcated into two primary categories, as delineated by [9]: i) passive
reception and ii) active reception. Within the realm of active reception, this
study considers the widely recognized fully-absorbing Rx [10]. It is
postulated that the Rx possesses the capability to precisely ascertain both
the time [11] and the position [12, 13, 14] at the initial contact of the MM
with the Rx. To investigate the Shannon capacity [15] for this novel category
of position channels, we adopt a straightforward geometric configuration for
the receiver, specifically, a spacious receiving plane [16]. It is worth
noting that this spacious plane model can serve as an approximation,
particularly when the transmission distance is significantly shorter than the
dimensions of the receiver, as discussed in prior works [12, 17]. Note that
MMs are removed upon their interaction with the absorptive receiving plane
[10, 13]. For a visual representation of this system model, please refer to
Fig. 1.

There are various physical attributes of MMs that can convey information. The
concept of the First Arrival Position (FAP) as an information-carrying
property in MC literature was first introduced in the paper by Lee et al. in
2016 [12]. Subsequently, this idea was further developed, particularly in the
context of two-dimensional (2D) scenarios, in a later study conducted by
Pandey et al. in 2018 [13]. While the majority of studies within the realm of
MC primarily concentrate on the First Arrival Time (FAT)111Readers interested
in gaining a more comprehensive understanding of FAT-type modulation and its
associated channel characteristics are encouraged to refer to [11, 18] for
detailed information. for absorbing receivers, it is noteworthy to acknowledge
at least two reasons for giving due consideration to FAP as a viable
alternative.

  * 1)

For every independent channel use—specifically, for each transmission of a
single MM—the FAT information is one-dimensional (1D). In contrast, the FAP-
type modulation can carry information in higher dimensions, up to $n-1$, in an
$n$-dimensional space. Consequently, the capacity of FAP channels, for each
individual channel use, can increase significantly in higher-dimensional
scenarios. This potentially allows FAP channels to surpass FAT channels in
terms of capacity. We will establish the capacity formulas for FAP channels in
Section IV and validate this assertion for 2D and 3D cases.

  * 2)

The second consideration pertains to time efficiency. In scenarios involving
multiple-MM transmission, MMs may arrive out of sequence due to the inherent
unpredictability of the diffusion process, leading to cross-over effects as
documented in [19, 9]. To mitigate cross-over effects in the MC system, the
transmission duration for each symbol must not be overly short.222A simple way
to conceptualize this is to imagine the necessity for a _guard interval_
between successive timing symbols. Given this, for applications where time
efficiency is paramount, exploring FAP-type modulation as an alternative
solution might be advisable.

In this paper, our primary attention centers on one-shot transmission [20],
referring specifically to transmission using a singular MM. To delve into the
channel capacity of either FAP or FAT channels, initiating with a quantitative
delineation of the channel impulse response is imperative. The one-shot FAT
channel can be succinctly characterized as a time-invariant additive channel
[11]: $t_{\text{out}}=t_{\text{in}}+t_{\text{n}},$ where $t_{\text{in}}$
represents the release time, $t_{\text{out}}$ denotes the arrival time, and
$t_{\text{n}}$ is the random time delay resulting from propagation dynamics.
As established in [11], the variable $t_{\text{n}}$ adheres to the inverse
Gaussian distribution. This channel is thus recognized as the Additive Inverse
Gaussian Noise (AIGN) channel within the MC field. Subsequent research in [11,
18] has presented bounds on the capacity of the AIGN channel and has further
characterized the capacity-achieving input time distribution.

For FAP channels operating within $n$-dimensional spaces, the one-shot channel
model is elegantly defined through an additive vector form, as detailed in
[21]: $\mathbf{x}_{\text{out}}=\mathbf{x}_{\text{in}}+\mathbf{x}_{\text{n}},$
where $\mathbf{x}_{\text{in}}$ signifies the releasing position,
$\mathbf{x}_{\text{out}}$ indicates the arriving position, and
$\mathbf{x}_{\text{n}}$ captures the random position deviation induced by
propagation mechanisms. It is crucial to recognize that the vectors
$\mathbf{x}_{\text{in}}$, $\mathbf{x}_{\text{out}}$, and
$\mathbf{x}_{\text{n}}$ reside within the Euclidean vector space
$\mathbb{R}^{n-1}$. This configuration arises since one dimension is earmarked
for physical movement, effectively excluding it from bearing information in
that particular dimension. Though past studies have articulated the
probability density function of $\mathbf{x}_{\text{n}}$ [12, 13, 21], the
quest to comprehend the channel capacity of this distinctive position channel
is ongoing. A recent exploration by Lee et al. [17] has elucidated the FAP
channel capacity in contexts characterized by a vertical drift toward Rx.
However, the channel capacity remains elusive in zero-drift scenarios. This
manuscript aims to address this lacuna, unveiling precise formulas for the
zero-drift FAP channel capacity for 2D and 3D dimensions, and introducing a
novel logarithmic constraint, as detailed in Sections III and IV.

The remainder of this paper is structured as follows. Section II provides a
succinct review of mutual information and the Shannon channel capacity
framework. Section III outlines the channel model under scrutiny and
introduces the $\alpha$-power constraint. Our core findings are presented in
Section IV. We draw our conclusions in Section V.

##  II Preliminary

This section revisits foundational concepts surrounding mutual information and
channel capacity. Consider $X$ as a random variable with corresponding
probability density function (pdf) $f$. The support of $f$ is represented as
$\mathcal{X}$. The differential entropy $h(X)$ can be expressed as
$h(X):=\mathbb{E}_{X}[-\ln f(X)]=-\int_{\mathcal{X}}[\ln f(x)]f(x)dx.$
Leveraging the above entropy definition, the mutual information between the
input and output of the additive channel $Y=X+N$ is:

$\displaystyle\begin{split}I(X;Y)&=h(Y)-h(Y|X)=h(Y)-h(X+N|X)\\\
&=h(Y)-h(X|X)-h(N|X)\\\ &=h(Y)-h(N).\end{split}$ (1)

Here, the assumption is that noise $N$ operates independently of signal $X$.
Remember that the (Shannon) channel capacity is captured as
$C=\sup_{f_{X}(x)}I(X;Y).$ The supremum is evaluated over all possible $X$
distributions under a selected constraint. For point-to-point communication or
single-input-single-output (SISO) scenarios, output constraints can be adopted
interchangeably, resulting in $C=\sup_{f_{Y}(x)}I(X;Y).$

Constraint selection is intimately tied to the noise distribution. To enhance
understanding, we’ll first contemplate a straightforward instance prior to
diving into the specifics of FAP channels. For the additive Gaussian channel,
where noise follows a Gaussian distribution, the prevailing power measure
embraces the second moment constraint:
$\mathbb{E}[X^{2}]=\int_{\mathcal{X}}x^{2}f(x)dx\leq P.$ The constant $P\geq
0$ denotes the upper power limit for all input signals. Based on this
constraint, the channel capacity can be expressed as: $C=\sup_{f_{X}(x):\
\mathbb{E}[X^{2}]\leq P}I(X;Y).$ Consulting a standard reference such as [22],
the supremum of $I(X;Y)$ is unveiled as
$\frac{1}{2}\ln\left(\frac{\sigma^{2}+P}{\sigma^{2}}\right),$ with the
capacity-achieving distribution being Gaussian with variance $P$, i.e.
$X\sim\mathcal{N}(0,P)$.

For subsequent comparison (presented in Section IV) purpose, we can set
$A^{2}:=\sigma^{2}+P$, yielding

$\displaystyle\begin{split}C_{\text{G}}=\frac{1}{2}\ln\left(\dfrac{A^{2}}{\sigma^{2}}\right)=\ln\left(\dfrac{A}{\sigma}\right),\end{split}$
(2)

where the subscript G hints the Gaussian channel.

##  III Channel Model

Figure 1:  This figure presents an abstract FAP channel model featuring
hyperplane-shaped transmitters (Tx) and receivers (Rx). In this model, the
transmitter is assumed to be transparent, permitting particles to traverse it
without encountering any forces. It is important to observe that the emission
point resides on the Tx hyperplane, and the Rx registers the arrival point
when the information particle initially hits the Rx. Note that we consider
_zero drift_ case for our main results.

We consider an additive vector channel: $\mathbf{Y}=\mathbf{X}+\mathbf{N}$
where the noise $\mathbf{N}$ is (multivariate) Cauchy distributed.333 We
postpone the reason why FAP channels reduce to Cauchy channels under zero-
drift assumption to Section IV-A.  Because the Cauchy distributions444 For a
detailed characterization of Cauchy density functions, one can refer to
Appendix A.  are heavy-tailed [23], traditional signal processing theory,
which is tailored for finite second moment signals, does not apply directly to
the Cauchy case.

Of the heavy-tailed probability models, the alpha-stable family [24] stands
out, having demonstrated its prowess in simulating multiple access
interference [25] and co-channel interference from a Poisson-distributed field
of interferers [26]. In alpha-stable channels, the second moment is not a
viable power measure due to its inherent infinity.

The task of discerning the channel capacity for alpha-stable additive
noise—specifically when $\alpha\geq 1$—under an $r$-th absolute moment input
constraint was adeptly addressed by [27] for symmetric alpha-stable noise
scenarios. Their exploration led to the identification of an optimal capacity-
achieving input that is both discrete and compactly defined. Meanwhile, a
distinct concept of power, termed as the “geometric power,” was ushered in by
[28] to cater to heavy-tailed distributions [23]. Within the context of this
manuscript, we introduce an alternative power characterization schema for the
$\alpha$-stable family, which is referred to as the $\alpha$-power, as
detailed in [29]. It’s pertinent to highlight that while the genesis of this
framework traces back to [29], our work repurposes its essence to adeptly
navigate the challenges posed by zero-drift FAP channels in the realm of
molecular communication.

Let us define a relative power measure $P_{\alpha}(\mathbf{X})$ satisfying the
following:

  * P1)

$P_{\alpha}(\mathbf{X})\geq 0$, with equality if and only if $\mathbf{X}=0$
almost surely.

  * P2)

$P_{\alpha}(k\mathbf{X})=|k|P_{\alpha}(\mathbf{X})$, for any $k\in\mathbb{R}$.

For $\alpha$-stable distributions, it is known (see [29]) that

$\begin{cases}\mathbb{E}_{\mathbf{X}}\big{[}\ln(1+\left\lVert\mathbf{X}\right\rVert^{2})\big{]}<\infty,\quad&\alpha<2;\\\
\mathbb{E}_{\mathbf{X}}\big{[}\left\lVert\mathbf{X}\right\rVert^{2}\big{]}<\infty,\quad&\alpha=2.\end{cases}$
(3)

We choose the power measure $P_{1}(\cdot)$ for multivariate Cauchy-like
distributions as follows:

$\mathbb{E}_{\mathbf{X}}\left[\ln\left(1+\left\lVert\frac{\mathbf{X}}{P_{1}(\mathbf{X})}\right\rVert^{2}\right)\right]=w_{2}\left(\frac{1+p}{2};\frac{p}{2}\right),$
(4)

where the constant evaluation function $w_{2}(t;\alpha)$ can be written
explicitly as the difference of two digamma functions:
$w_{2}(t;a)=\psi(t)-\psi(t-a),\text{\ for\ }t>a.$ Note that the values of
$w_{2}$ function at $p=1$ and $p=2$ are $2\ln(2)$ and $2\ln(e)$ respectively.

##  IV Main Results

In this section, we delve into the intricacies of the FAP communication
channel. We commence by demonstrating in Section IV-A that as the drift
velocity in the fluid medium approaches zero, the 2D and 3D FAP channels
simplify to univariate and bivariate Cauchy distributions, respectively.

Proceeding to Sections IV-B and IV-C, we integrate the concept of the
$\alpha$-power, a relative power measure documented in [29], into MC
discussions. We explicitly define the signal space requisite to depict the no-
drift FAP channels. Employing our introduced logarithmic constraint on the
output signal space, we ascertain the Shannon capacity of the FAP channel for
both 2D and 3D configurations in a closed-form manner. Key insights regarding
the FAP channel capacity are encapsulated in Theorem 1 and Theorem 21.

Notice that in the literature, while [30] provided a capacity formula for the
univariate Cauchy channel (not for MC purpose) under a logarithmic constraint
applied to the input signal space, directly extending their findings to
higher-dimensional scenarios is challenging due to intricate integration
procedures, as evident in [30, Appendix I]. One way to tackle this is
replacing the input constraints with output constraints, as demonstrated in
Sections IV-B and IV-C.

###  IV-A FAP Channel Reduces to Cauchy Channel Under Zero Drift Condition

####  IV-A1 In 2D Space

From [21] we know that the FAP density function in 2D space, with a drift
velocity $\mathbf{v}=(v_{1},v_{2})$, can be written as:

$\displaystyle\begin{split}f_{Y\mid X}(\mathbf{y}|\mathbf{x})=&\
\dfrac{|\mathbf{v}|\lambda}{\sigma^{2}\pi}\exp\left\\{\dfrac{-v_{2}\lambda}{\sigma^{2}}\right\\}\exp\left\\{\dfrac{-v_{1}(x_{1}-y_{1})}{\sigma^{2}}\right\\}\\\
&\cdot\dfrac{K_{1}\left(\dfrac{|\mathbf{v}|}{\sigma^{2}}\sqrt{(x_{1}-y_{1})^{2}+\lambda^{2}}\right)}{\sqrt{(x_{1}-y_{1})^{2}+\lambda^{2}}},\end{split}$
(5)

where $\mathbf{x}=(x_{1},\lambda)$, $\mathbf{y}=(y_{1},0)$, and $\lambda$ is
the transmission distance between transmitter (Tx) and Rx. Note that in Eq.
(5), $\sigma^{2}$ is the microscopic diffusion coefficient which can be
related to the macroscopic diffusion coefficient $D$ through the relation
$\sigma^{2}=2D$, see [4]. Note also that the special function $K_{1}(\cdot)$
in Eq. (5) is the modified Bessel function of the second kind (see [31]) with
order $\nu=1$.

Next we show that when drift velocity approaches zero, the 2D FAP density
reduces to a univariate Cauchy distribution. We use a limit property about the
special function $K_{1}(\cdot)$ from [32]:
$\lim_{x\xrightarrow{}0}xK_{1}(x)=1.$ Using this limit property, we can do the
following calculation:

$\displaystyle\begin{split}&\ f_{Y\mid
X;\mathbf{v}=\mathbf{0}}(\mathbf{y}|\mathbf{x})\\\ =&\
\lim_{|\mathbf{v}|\xrightarrow{}\mathbf{0}}\frac{|\mathbf{v}|\lambda}{\sigma^{2}\pi}\cdot\frac{K_{1}\left(\frac{|\mathbf{v}|}{\sigma^{2}}\sqrt{(x_{1}-y_{1})^{2}+\lambda^{2}}\right)}{\sqrt{(x_{1}-y_{1})^{2}+\lambda^{2}}}\\\
=&\
\dfrac{\lambda}{\pi}\lim_{|\mathbf{v}|\xrightarrow{}\mathbf{0}}\Bigg{[}\frac{|\mathbf{v}|}{\sigma^{2}}\sqrt{(x_{1}-y_{1})^{2}+\lambda^{2}}\\\
&\quad\quad\quad\quad\quad\cdot\frac{K_{1}\left(\frac{|\mathbf{v}|}{\sigma^{2}}\sqrt{(x_{1}-y_{1})^{2}+\lambda^{2}}\right)}{(x_{1}-y_{1})^{2}+\lambda^{2}}\Bigg{]}\\\
=&\ \dfrac{\lambda}{\pi}\frac{1}{(x_{1}-y_{1})^{2}+\lambda^{2}}.\end{split}$
(6)

This calculation result can be interpreted as follows. Let $x_{1}$ be the
input position and $y_{1}$ be the output position. The displacement of
position is an additive noise $n$, following the density function:
$\frac{\lambda}{\pi}\frac{1}{n^{2}+\lambda^{2}},$ which is Cauchy distributed.
Please refer to Appendix A for the form of density function of Cauchy random
variables.

####  IV-A2 In 3D Space

From [21, 12] we know that the FAP density function in 3D diffusion channel,
with a drift velocity $\mathbf{v}=(v_{1},v_{2},v_{3})$, can be written as:

$\displaystyle f_{Y\mid X}(\mathbf{y}|\mathbf{x})$ (7) $\displaystyle=$
$\displaystyle\frac{\lambda}{2\pi}\exp\left\\{-\dfrac{v_{3}\lambda}{\sigma^{2}}\right\\}\exp\left\\{\dfrac{v_{1}}{\sigma^{2}}\left(y_{1}-x_{1}\right)+\dfrac{v_{2}}{\sigma^{2}}\left(y_{2}-x_{2}\right)\right\\}$
$\displaystyle\cdot\exp\left\\{-\dfrac{|\mathbf{v}|}{\sigma^{2}}\left\lVert\mathbf{y}-\mathbf{x}\right\rVert\right\\}\left[\frac{1+\frac{|\mathbf{v}|}{\sigma^{2}}\left\lVert\mathbf{y}-\mathbf{x}\right\rVert}{\left\lVert\mathbf{y}-\mathbf{x}\right\rVert^{3}}\right],$

where $\sigma^{2}$ is the microscopic diffusion coefficient, the scale
parameter $\lambda$ is the transmission distance between Tx and Rx plane,
$v_{3}$ is the longitudinal component of drift velocity in the direction
parallel to the transmission direction, and $v_{1}$, $v_{2}$ are the
components perpendicular (or transverse) to the transmission direction. Here
we use the symbol $\left\lVert\cdot\right\rVert$ to represent the Euclidean
norm. Namely,
$\left\lVert\mathbf{y}-\mathbf{x}\right\rVert=\sqrt{(y_{1}-x_{1})^{2}+(y_{2}-x_{2})^{2}+\lambda^{2}},$
where $\mathbf{x}=(x_{1},x_{2},\lambda)$, $\mathbf{y}=(y_{1},y_{2},0)$ are
position vectors in $\mathbb{R}^{3}$.

Next we show that when drift velocity approaches zero, the 3D FAP density
reduces to a bivariate Cauchy distribution. Note that when
$\mathbf{v}=\mathbf{0}$, all the exponential terms in Eq. (7) become
$e^{0}=1$, so that we have:

$\displaystyle\begin{split}&\ f_{Y\mid
X;\mathbf{v}=\mathbf{0}}(\mathbf{y}|\mathbf{x})\\\ =&\
\frac{\lambda}{2\pi}\lim_{|\mathbf{v}|\xrightarrow{}\mathbf{0}}\frac{1+\frac{|\mathbf{v}|}{\sigma^{2}}\left\lVert\mathbf{y}-\mathbf{x}\right\rVert}{\left\lVert\mathbf{y}-\mathbf{x}\right\rVert^{3}}=\
\frac{\lambda}{2\pi}\frac{1}{\left\lVert\mathbf{y}-\mathbf{x}\right\rVert^{3}}\\\
=&\
\dfrac{\lambda}{2\pi}\dfrac{1}{\big{(}(y_{1}-x_{1})^{2}+(y_{2}-x_{2})^{2}+\lambda^{2}\big{)}^{3/2}}.\end{split}$
(8)

That is, if we regard the position channel as:
$\mathbf{y}=\mathbf{x}+\mathbf{n},$ then $\mathbf{n}$ follows a bivariate
Cauchy distribution, see Eq. (35) in Appendix A for its pdf.

###  IV-B Channel Capacity for 2D FAP Channel Under Logarithmic Constraint

As mentioned in Section IV-A, a 2D FAP channel reduces to a Cauchy channel
when the drift velocity approaches zero. Since we are considering a point-to-
point (or SISO) communication scenario, the input constraint is actually
equivalent to the output constraint. We choose to adopt output constraint to
simplify the calculation.

It is known that a Cauchy distribution $X\sim{\rm Cauchy}(0,k)$ maximizes the
entropy among all RVs555This is called the principle of maximum entropy (see
[33]) for Cauchy distribution. that satisfy

$\mathbb{E}_{X}\ln\left[1+\left(\frac{X}{k}\right)^{2}\right]=2\ln(2),$ (9)

or equivalently,

$\mathbb{E}_{X}\ln\left[1+X^{2}\right]=2\ln(1+k).$ (10)

The corresponding maximum entropy value is $\ln(4\pi k)$, see Appendix A. Note
that the parameter $k$ restricts the dispersion of all feasible random
variables $X$.

Consider a 2D FAP zero-drift channel (as shown in Section IV-A), the input-
output relation can be written as

$Y=X+N,$ (11)

where $N\sim{\rm Cauchy}(0,\lambda)$. For continuous-variable channel capacity
problem, we need to specify a family of distributions (i.e. feasible set) that
are under consideration in order to prevent the capacity value from being
infinite. Instead of writing down explicitly the constraint equality for $X$
or $Y$, we define

$\displaystyle\begin{split}\mathcal{D}(A)=\ &\Big{\\{}\text{distributions\ }Y\
\Big{|}\ \exists k\in[\lambda,A]\\\ &\quad\quad\quad\text{\ such that\
}\mathbb{E}_{Y}\ln\Big{[}1+\Big{(}\frac{Y}{k}\Big{)}^{2}\Big{]}=2\ln(2)\Big{\\}}.\end{split}$
(12)

The parameter $A$ appeared in Eq. (12) indicates the “largest allowed
dispersion” for the distributions in $\mathcal{D}(A)$. For later use, we also
define

$\displaystyle\begin{split}\mathcal{D}_{k}=\ &\Big{\\{}\text{distributions\
}Y\ \Big{|}\
\mathbb{E}_{Y}\ln\Big{[}1+\Big{(}\frac{Y}{k}\Big{)}^{2}\Big{]}=2\ln(2)\Big{\\}},\end{split}$
(13)

so that we have $\mathcal{D}(A)=\bigcup_{k\in[\lambda,A]}\mathcal{D}_{k}.$
With these notations, now we can state our main results in the following two
theorems.

######  Theorem 1.

The capacity of 2D FAP channel (11) is

$C_{\rm{2D,FAP}}=C(A,\lambda)=\ln\left(\dfrac{A}{\lambda}\right)$ (14)

under the output logarithmic constraint $Y\in\mathcal{D}(A)$ for some
prescribed dispersion level $A$, where $A\geq\lambda$. In addition, the
corresponding capacity achieving output distribution is $Y^{*}\sim{\rm
Cauchy}(0,A),$ or equivalently, $X^{*}\sim{\rm Cauchy}(0,A-\lambda).$

Given that the proof of Theorem 1 closely parallels that of Theorem 21, we
defer the proof details to Appendix B and proceed directly to the examination
of the 3D case.

###  IV-C Channel Capacity for 3D FAP Channel Under Logarithmic Constraint

It is known (see [33],[34]) that a bivariate Cauchy distribution
$\mathbf{X}\sim{\rm Cauchy}_{2}(0,\Sigma=\text{diag}(k^{2},k^{2}))$ maximizes
the entropy among all bivariate RVs that satisfy

$\mathbb{E}_{\mathbf{X}}\ln\left[1+\left\lVert\frac{\mathbf{X}}{k}\right\rVert^{2}\right]=2\ln\left(e\right),$
(15)

The corresponding maximum entropy value is $\ln(2\pi e^{3}k^{2})$, see
Appendix A. Note that the parameter $k$ restricts the dispersion of all
feasible random vectors $\mathbf{X}$.

Consider a 3D FAP zero-drift channel (as shown in Section IV-A), the input-
output relation can be written as

$\mathbf{Y}=\mathbf{X}+\mathbf{N},$ (16)

where $\mathbf{N}\sim{\rm
Cauchy}_{2}(\mathbf{0},\Sigma=\text{diag}(\lambda^{2},\lambda^{2}))$. We
define

$\displaystyle\begin{split}\mathcal{D}_{\text{bi}}(A)=\
&\Bigg{\\{}\text{distributions\ }\mathbf{Y}\ \Bigg{|}\ \exists
k\in[\lambda,A]\\\ &\quad\quad\quad\text{\ such that\
}\mathbb{E}_{\mathbf{Y}}\ln\Big{[}1+\left\lVert\frac{\mathbf{Y}}{k}\right\rVert^{2}\Big{]}=2\ln(e)\Bigg{\\}}.\end{split}$
(17)

The parameter $A$ appeared in Eq. (17) indicates the “largest allowed
dispersion” for the distributions in $\mathcal{D}_{\text{bi}}(A)$. For later
use, we also define

$\displaystyle\begin{split}\mathcal{D}_{k,\text{bi}}=\
&\Bigg{\\{}\text{distributions\ }\mathbf{Y}\ \Bigg{|}\
\mathbb{E}_{\mathbf{Y}}\ln\Big{[}1+\left\lVert\frac{\mathbf{Y}}{k}\right\rVert^{2}\Big{]}=2\ln(e)\Bigg{\\}},\end{split}$
(18)

so that we have

$\mathcal{D}_{\text{bi}}(A)=\bigcup_{k\in[\lambda,A]}\mathcal{D}_{k,\text{bi}}.$
(19)

The 3D FAP capacity formula is stated in the following theorem.

######  Theorem 2.

The capacity of 3D FAP channel (16) is

$C_{\rm{3D,FAP}}=C(A,\lambda)=2\ln\left(\dfrac{A}{\lambda}\right)$ (20)

under the output logarithmic constraint $\mathbf{Y}\in\mathcal{D}_{{\rm
bi}}(A)$ for some prescribed dispersion level $A$, where $A\geq\lambda$. In
addition, the corresponding capacity achieving output distribution is

$\mathbf{Y}^{*}\sim{\rm Cauchy}_{2}(0,{\rm diag}(A^{2},A^{2})).$ (21)

###### Proof.

For the additive channel we are considering, the mutual information can be
written as

$I(\mathbf{X};\mathbf{Y})=h(\mathbf{Y})-h(\mathbf{N}).$ (22)

The problem of finding the channel capacity turns out to be an optimization
problem:
$C(A,\lambda)=\sup_{\mathbf{Y}\in\mathcal{D}_{\text{bi}}(A)}I(\mathbf{X};\mathbf{Y}).$
We have

$\displaystyle C(A,\lambda)$
$\displaystyle=\sup_{\mathbf{Y}\in\mathcal{D}_{\text{bi}}(A)}\Big{\\{}h(\mathbf{Y})-h(\mathbf{N})\Big{\\}}$
(23)
$\displaystyle=\Big{\\{}\sup_{\mathbf{Y}\in\mathcal{D}_{\text{bi}}(A)}h(\mathbf{Y})\Big{\\}}-\ln(2\pi
e^{3}\lambda^{2}),$ (24)

where

$\displaystyle\sup_{\mathbf{Y}\in\mathcal{D}_{\text{bi}}(A)}h(\mathbf{Y})$
$\displaystyle=\sup_{k\in[\lambda,A]}\Big{\\{}\sup_{\mathbf{Y}\in\mathcal{D}_{k,\text{bi}}}h(\mathbf{Y})\Big{\\}}$
(25) $\displaystyle=\sup_{k\in[\lambda,A]}\ln(2\pi e^{3}k^{2})$ (26)
$\displaystyle=\ln(2\pi e^{3}A^{2}).$ (27)

Combining equations (24) and (27) yields

$\displaystyle\begin{split}C(A,\lambda)=&\ \ln(2\pi e^{3}A^{2})-\ln(2\pi
e^{3}\lambda^{2})\\\ =&\
\ln\left(\dfrac{A^{2}}{\lambda^{2}}\right)=2\ln\left(\dfrac{A}{\lambda}\right).\end{split}$
(28)

Notice that the equalities (25)-(27) occur if and only if $\mathbf{Y}$
distributes as ${\rm Cauchy}_{2}(0,\text{diag}(A^{2},A^{2}))$. Hence, we have
proved Theorem 21. ∎

We encapsulate the primary findings of this section in Table I. Note that
within this context, $A$ represents the output signal dispersion level
specific to FAP channels. Moreover, by defining $A^{2}$ as $\sigma^{2}+P$, we
establish a correspondence between the Gaussian case and the 2D FAP case. To
visually illustrate these capacity curves, we present a graphical
representation in Fig. 2.

Table I: 2D and 3D FAP channel capacity curves
with a comparison to Gaussian channels Channel Type | Gaussian | 2D FAP | 3D FAP
---|---|---|---
Constraint |  $L^{2}$ constr. | log constr. | log constr.
Capacity Formula | $\frac{1}{2}\ln\left(\dfrac{\sigma^{2}+P}{\sigma^{2}}\right)$ | $\ln\left(\dfrac{A}{\lambda}\right)$ | $2\ln\left(\dfrac{A}{\lambda}\right)$
Parameter Range | N/A | $A\geq\lambda$ | $A\geq\lambda$

Figure 2:  Illustration depicting the capacity curves in relation to the
output signal dispersion level $A$ for both 2D and 3D FAP channels. It is
worth noting that the Gaussian curve coincides with the 2D FAP curve when we
equate $A^{2}$ to $\sigma^{2}+P$ in the Gaussian case.

##  V Conclusion

This study examines the capacity problem for zero-drift first arrival position
(FAP) communication channel in diffusive molecular communication (MC) systems.
Our findings reveal that when the drift velocity approaches zero, both the 2D
and 3D FAP channels reduce to univariate and bivariate Cauchy distributions,
respectively.

Although a recent work [17] investigate the FAP channel capacity under the
case of positive _vertical drift_ , at the time of this writing, the
determination of the Shannon capacity for the FAP channels with _zero drift_
(i.e. the multidimensional Cauchy channels) remains an open problem. This
challenge stems from the heavy-tailed nature of the Cauchy distribution, which
belongs to a broader family of $\alpha$-stable distributions that often lack a
second moment. As a result, traditional energy constraints such as the
variance of random variables are inadequate for characterizing the signal
space.

To address this challenge, we introduce a relative power measure, the
$\alpha$-power, into the field of molecular communication. We explicitly
define the signal space required to characterize the zero-drift FAP channel
capacity and derive the Shannon capacity for the FAP channel in 2D and 3D
spaces in closed-form under a fresh logarithmic constraint. The capacity
formulations articulated in Theorem 1 and Theorem 21 underscore an intriguing
phenomenon: the 3D FAP channel’s capacity eclipses its 2D counterpart by a
factor of two when benchmarked against the identical dispersion parameter $A$.
This revelation amplifies the perspective that as spatial dimension $n$
expands, FAP channels inherently harness greater informational capacity.

## References

  * [1] T. Nakano, A. W. Eckford, and T. Haraguchi, _Molecular Communication_. Cambridge University Press, 2013.
  * [2] P.-C. Yeh, K.-C. Chen, Y.-C. Lee, L.-S. Meng, P.-J. Shih, P.-Y. Ko, W.-A. Lin, and C.-H. Lee, “A new frontier of wireless communication theory: diffusion-based molecular communications,” _IEEE Wireless Communications_ , vol. 19, no. 5, pp. 28–35, 2012.
  * [3] I. F. Akyildiz, F. Brunetti, and C. Blázquez, “Nanonetworks: A new communication paradigm,” _Computer Networks_ , vol. 52, no. 12, pp. 2260–2279, 2008.
  * [4] N. Farsad, H. B. Yilmaz, A. Eckford, C.-B. Chae, and W. Guo, “A comprehensive survey of recent advancements in molecular communication,” _IEEE Communications Surveys & Tutorials_, vol. 18, no. 3, pp. 1887–1919, 2016.
  * [5] M. Pierobon and I. F. Akyildiz, “Capacity of a diffusion-based molecular communication system with channel memory and molecular noise,” _IEEE Transactions on Information Theory_ , vol. 59, no. 2, pp. 942–954, 2012.
  * [6] S. Kadloor, R. S. Adve, and A. W. Eckford, “Molecular communication using brownian motion with drift,” _IEEE Transactions on NanoBioscience_ , vol. 11, no. 2, pp. 89–99, 2012.
  * [7] M. Moore, A. Enomoto, T. Nakano, R. Egashira, T. Suda, A. Kayasuga, H. Kojima, H. Sakakibara, and K. Oiwa, “A design of a molecular communication system for nanomachines using molecular motors,” in _Proc. 4th Annu. IEEE Int. Conf. Pervasive Comput. Commun. Workshops (PERCOMW)_ , Pisa, Italy, Mar. 2006, pp. 554–559.
  * [8] M. Gregori and I. F. Akyildiz, “A new nanonetwork architecture using flagellated bacteria and catalytic nanomotors,” _IEEE Journal on selected areas in communications_ , vol. 28, no. 4, pp. 612–619, 2010.
  * [9] V. Jamali, A. Ahmadzadeh, W. Wicke, A. Noel, and R. Schober, “Channel modeling for diffusive molecular communication—a tutorial review,” _Proceedings of the IEEE_ , vol. 107, no. 7, pp. 1256–1301, 2019.
  * [10] H. B. Yilmaz, A. C. Heren, T. Tugcu, and C.-B. Chae, “Three-dimensional channel characteristics for molecular communications with an absorbing receiver,” _IEEE Communications Letters_ , vol. 18, no. 6, pp. 929–932, 2014.
  * [11] K. V. Srinivas, A. W. Eckford, and R. S. Adve, “Molecular communication in fluid media: The additive inverse gaussian noise channel,” _IEEE transactions on information theory_ , vol. 58, no. 7, pp. 4678–4692, 2012.
  * [12] Y.-C. Lee, C.-C. Chen, P.-C. Yeh, and C.-H. Lee, “Distribution of first arrival position in molecular communication,” in _Proc. IEEE Int. Symp. Information Theory_ , Barcelona, Spain, July 2016, pp. 1033–1037.
  * [13] N. Pandey, R. K. Mallik, and B. Lall, “Molecular communication: The first arrival position channel,” _IEEE Wireless Communications Letters_ , vol. 8, no. 2, pp. 508–511, 2018.
  * [14] B. C. Akdeniz, N. A. Turgut, H. B. Yilmaz, C.-B. Chae, T. Tugcu, and A. E. Pusane, “Molecular signal modeling of a partially counting absorbing spherical receiver,” _IEEE Transactions on Communications_ , vol. 66, no. 12, pp. 6237–6246, 2018.
  * [15] C. E. Shannon, “A mathematical theory of communication,” _The Bell system technical journal_ , vol. 27, no. 3, pp. 379–423, 1948.
  * [16] S. S. Andrews, “Accurate particle-based simulation of adsorption, desorption and partial transmission,” _Physical biology_ , vol. 6, no. 4, p. 046015, 2009.
  * [17] Y.-C. Lee, Y.-F. Lo, J.-M. Wu, and M.-H. Hsieh, “Characterizing first arrival position channels: Noise distribution and capacity analysis,” _arXiv preprint arXiv:2306.08353v2_ , 2023.
  * [18] H. Li, S. M. Moser, and D. Guo, “Capacity of the memoryless additive inverse Gaussian noise channel,” _IEEE Journal on Selected Areas in Communications_ , vol. 32, no. 12, pp. 2315–2329, 2014.
  * [19] A. W. Eckford, “Molecular communication: Physically realistic models and achievable information rates,” _arXiv preprint arXiv:0812.1554_ , 2008.
  * [20] Y. Murin, N. Farsad, M. Chowdhury, and A. Goldsmith, “Optimal detection for one-shot transmission over diffusion-based molecular timing channels,” _IEEE Transactions on Molecular, Biological and Multi-Scale Communications_ , vol. 4, no. 2, pp. 43–60, 2018.
  * [21] Y.-C. Lee, J.-M. Wu, and M.-H. Hsieh, “A unified framework for calculating first arrival position density in molecular communication,” arXiv:2201.04476v3 [cs.IT], 2022.
  * [22] T. M. Cover, _Elements of Information Theory_. John Wiley & Sons, 1999.
  * [23] K. Sigman, “A primer on heavy-tailed distributions,” _Queueing systems_ , vol. 33, no. 1-3, p. 261, 1999.
  * [24] R. D. Pierce, “Application of the positive alpha-stable distribution,” in _Proceedings of the IEEE Signal Processing Workshop on Higher-Order Statistics_. IEEE, 1997, pp. 420–424.
  * [25] H. El Ghannudi, L. Clavier, N. Azzaoui, F. Septier, and P.-A. Rolland, “$\alpha$-stable interference modeling and cauchy receiver for an ir-uwb ad hoc network,” _IEEE Transactions on Communications_ , vol. 58, no. 6, pp. 1748–1757, 2010.
  * [26] K. Gulati, B. L. Evans, J. G. Andrews, and K. R. Tinsley, “Statistics of co-channel interference in a field of poisson and poisson-poisson clustered interferers,” _IEEE Transactions on Signal Processing_ , vol. 58, no. 12, pp. 6207–6222, 2010.
  * [27] J. Fahs and I. Abou-Faycal, “On the capacity of additive white alpha-stable noise channels,” in _2012 IEEE International Symposium on Information Theory Proceedings_. IEEE, 2012, pp. 294–298.
  * [28] J. G. Gonzalez, D. W. Griffith, and G. R. Arce, “Zero-order statistics: a signal processing framework for very impulsive processes,” in _Proceedings of the IEEE Signal Processing Workshop on Higher-Order Statistics_. IEEE, 1997, pp. 254–258.
  * [29] J. Fahs and I. Abou-Faycal, “Information measures, inequalities and performance bounds for parameter estimation in impulsive noise environments,” _IEEE Transactions on Information Theory_ , vol. 64, no. 3, pp. 1825–1844, 2017.
  * [30] ——, “A Cauchy input achieves the capacity of a Cauchy channel under a logarithmic constraint,” in _Proc. IEEE Int. Symp. Information Theory_ , Honolulu, HI, USA, July 2014, pp. 3077–3081.
  * [31] W. W. Bell, _Special Functions for Scientists and Engineers_. Courier Corporation, 2004.
  * [32] Z.-H. Yang and Y.-M. Chu, “On approximating the modified bessel function of the second kind,” _Journal of inequalities and applications_ , vol. 2017, no. 1, pp. 1–8, 2017.
  * [33] J. N. Kapur, _Maximum-entropy Models in Science and Engineering_. John Wiley & Sons, 1989.
  * [34] S. Kotz and S. Nadarajah, _Multivariate t-Distributions and Their Applications_. Cambridge University Press, 2004.
  * [35] M. Shao and C. L. Nikias, “Signal processing with fractional lower order moments: stable processes and their applications,” _Proceedings of the IEEE_ , vol. 81, no. 7, pp. 986–1010, 1993.

##  Appendix A Density Function and Entropy of Multivariate Cauchy
Distribution

The Cauchy distribution, named after Augustin Cauchy, is a continuous
probability distribution often used in statistics as a canonical example of
distribution since both its expected value and its variance are undefined. It
is also known, especially among physicists, as the Lorentz distribution. In
mathematics, it is closely related to the fundamental solution for the Laplace
equation in the half-plane, and it is one of the few distributions that is
stable (see [24, 27]) and has a density function that can be expressed
analytically. (Other examples are normal distribution and Lévy distribution.)

The (univariate) Cauchy distribution has the probability density function
(PDF) which can be expressed as:

$f(x;x_{0},\gamma)=\dfrac{1}{\pi\gamma}\dfrac{\gamma^{2}}{(x-x_{0})^{2}+\gamma^{2}}=\dfrac{1}{\pi\gamma}\dfrac{1}{1+(\frac{x-x_{0}}{\gamma})^{2}},$
(29)

where $x_{0}\in\mathbb{R}$ is the location parameter, and $\gamma>0$ is the
scaling parameter (see [35]). For the purpose of exploring the channel
capacity, since the location parameter $x_{0}$ is irrelevant to the entropy,
we may assume without loss of generality that $x_{0}=0$, yielding the so
called symmetry Cauchy with PDF as:

$f(x;\gamma)=\dfrac{\gamma}{\pi}\dfrac{1}{x^{2}+\gamma^{2}}=\dfrac{1}{\pi\gamma}\dfrac{1}{1+(\frac{x}{\gamma})^{2}}.$
(30)

The entropy of Cauchy distribution can be evaluated by direct calculation.
From Eq. (29), we have

$h(X)=\ln(4\pi\gamma)$ (31)

whenever $X\sim\rm{Cauchy}(x_{0},\gamma)$, and $x_{0}$ can be chosen
arbitrarily.

As for multivariate Cauchy distribution, our notation system mainly follows
[34]. Consider a $p$-dimensional Euclidean random vector
$\mathbf{X}=(X_{1},\cdots,X_{p})^{\mathsf{\scriptstyle{T}}}$ which follows
multivariate Cauchy distribution. We use the notation $\mathbf{X}\sim{\rm
Cauchy}_{p}(\boldsymbol{\mu},\Sigma)$ to specify the parameters, where
$\boldsymbol{\mu}\in\mathbb{R}^{p\times 1}$ is the location vector, and
$\Sigma\in\mathbb{R}^{p\times p}$ is the scale matrix describing the shape of
the distribution. The PDF of multivariate Cauchy distribution is given by the
following formula:

$\displaystyle\begin{split}&f_{\mathbf{X}}^{(p)}(\mathbf{x};\boldsymbol{\mu},\Sigma)\\\
=&\
\dfrac{\Gamma(\frac{1+p}{2})}{\Gamma(\frac{1}{2})\pi^{\frac{p}{2}}|\Sigma|^{\frac{1}{2}}[1+(\mathbf{x}-\boldsymbol{\mu})^{\top}\Sigma^{-1}(\mathbf{x}-\boldsymbol{\mu})]^{\frac{1+p}{2}}}.\end{split}$
(32)

Note that $\Sigma$ is by nature a positive-definite square matrix. For our
later purpose, we mainly work in the case $p=2$. We provide the PDF of this
special case here for convenience:

$f_{\mathbf{X}}^{(2)}(\mathbf{x};\boldsymbol{\mu},\Sigma)=\dfrac{\Gamma(\frac{3}{2})}{\pi^{\frac{3}{2}}|\Sigma|^{\frac{1}{2}}[1+(\mathbf{x}-\boldsymbol{\mu})^{\top}\Sigma^{-1}(\mathbf{x}-\boldsymbol{\mu})]^{\frac{3}{2}}}.$
(33)

Eq. (33) is the so-called bivariate Cauchy distribution.

Similar to the univariate case, we can without loss of generality set
$\boldsymbol{\mu}=\mathbf{0}$ in equations (32) and (33) for the purpose of
entropy and channel capacity analysis. When $\boldsymbol{\mu}=\mathbf{0}$, the
multivariate Cauchy is called central. For the case that the Cauchy is
central, we abuse the notations ${\rm R}$ and $\Sigma$ to mean the same scale
matrix. We shall assume all Cauchy distributions considered in this paper to
be central from now on. In addition, for the special case that $\Sigma$ is a
diagonal matrix, namely

$\Sigma=\text{diag}(\gamma^{2},\gamma^{2})=\begin{bmatrix}\gamma^{2}&0\\\
0&\gamma^{2}\end{bmatrix},$ (34)

the central bivariate Cauchy density function can be simplified into a more
concise form:

$\dfrac{1}{2\pi}\dfrac{\gamma}{(n_{1}^{2}+n_{2}^{2}+\gamma^{2})^{3/2}},$ (35)

where $\mathbf{n}=(n_{1},n_{2})$ is the Cauchy random vector.

Finally, the (differential) entropy of $p$-variate central Cauchy distribution
can be found in [4]. We briefly state the results here for later use. Suppose
the scale matrix ${\rm R}$ of a $p$-variate Cauchy $\mathbf{X}$ is prescribed,
the entropy of $\mathbf{X}$ can be expressed as

$h(\mathbf{X};{\rm R})=\dfrac{1}{2}\ln|{\rm R}|+\Phi(p),$ (36)

where $|{\rm R}|$ represents the determinant of matrix ${\rm R}$; $\Phi(p)$ is
a constant depending only on dimension $p$, and it can be evaluated through

$\displaystyle\begin{split}\Phi(p)=&\
\ln\left[\dfrac{\pi^{\frac{p}{2}}}{\Gamma(\frac{p}{2})}B\left(\frac{p}{2},\dfrac{1}{2}\right)\right]\\\
&\quad+\dfrac{1+p}{2}\left[\psi\left(\frac{1+p}{2}\right)-\psi\left(\frac{1}{2}\right)\right].\end{split}$
(37)

In the above expression, $B(x,y)=\frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)}$ is
the so-called beta function, and

$\psi(t):=\dfrac{d}{dt}\big{[}\ln\Gamma(t)\big{]}$ (38)

is known as the digamma function.

##  Appendix B Proof of Theorem 1

In this appendix section, we prove Theorem 1 step by step. For the additive
channel

$Y=X+N$ (39)

we are considering, the mutual information can be written as
$I(X;Y)=h(Y)-h(N)$ according to Section II. The issue of finding the capacity
of channel (11) turns out to be an optimization problem:

$\displaystyle C(A,\lambda)$ $\displaystyle=\sup_{Y\in\mathcal{D}(A)}I(X;Y).$
(40)

The calculation is as follows. We have

$\displaystyle C(A,\lambda)$
$\displaystyle=\sup_{Y\in\mathcal{D}(A)}\Big{\\{}h(Y)-h(N)\Big{\\}}$ (41)
$\displaystyle=\Big{\\{}\sup_{Y\in\mathcal{D}(A)}h(Y)\Big{\\}}-\ln(4\pi\lambda),$
(42)

where

$\displaystyle\sup_{Y\in\mathcal{D}(A)}h(Y)$
$\displaystyle=\sup_{k\in[\lambda,A]}\Big{\\{}\sup_{Y\in\mathcal{D}_{k}}h(Y)\Big{\\}}$
(43) $\displaystyle=\sup_{k\in[\lambda,A]}\ln(4\pi k)$ (44)
$\displaystyle=\ln(4\pi A).$ (45)

Combining equations (42) and (45) yields

$C(A,\lambda)=\ln(4\pi
A)-\ln(4\pi\lambda)=\ln\left(\dfrac{A}{\lambda}\right).$ (46)

Notice that the equalities (43)-(45) hold if and only if $Y$ distributes as
${\rm Cauchy}(0,A)$. Hence, Theorem 1 is proved.

##  Appendix C Some Useful Facts about Cauchy Distribution

In this appendix, we delve into specific properties of Cauchy distributions,
building upon the foundational probability density function formula and
entropy evaluations presented in Appendix A.

The first two properties initially highlight the behavior of the independent
sum of Cauchy distributions. To elucidate, the univariate symmetric Cauchy
distribution remains invariant under independent addition, as articulated in
the subsequent lemma.

######  Lemma 1.

Given two independent Cauchy random variables, $U\sim{\rm Cauchy}(0,\sigma)$
and $V\sim{\rm Cauchy}(0,\tau)$, their sum is described as:

$Z=U+V\sim{\rm Cauchy}(0,\sigma+\tau).$ (47)

When considering the independent summation of bivariate Cauchy distributions,
the lemma below encapsulates the relationship.

######  Lemma 2.

With $\mathbf{U}\sim{\rm Cauchy}_{2}\Big{(}\mathbf{0},\Sigma_{1}={\rm
diag}(\sigma^{2},\sigma^{2})\Big{)}$ and $\mathbf{V}\sim{\rm
Cauchy}_{2}\Big{(}\mathbf{0},\Sigma_{2}={\rm diag}(\tau^{2},\tau^{2})\Big{)}$
representing two independent bivariate Cauchy random vectors, their summation
follows:

$\displaystyle\begin{split}\mathbf{Z}&=\mathbf{U}+\mathbf{V}\\\ &\sim{\rm
Cauchy}_{2}\Big{(}\mathbf{0},\Sigma_{3}={\rm
diag}\left((\sigma+\tau)^{2},(\sigma+\tau)^{2}\right)\Big{)}.\end{split}$ (48)

Note that in Lemma 2, we employ the notation

${\rm diag}(a,b)=\begin{bmatrix}a&0\\\ 0&b\end{bmatrix}$ (49)

to succinctly represent diagonal matrices.

The final property examines the linear combination of components of a Cauchy
vector, $\mathbf{X}$. Given an arbitrary constant vector
$\mathbf{v}\in\mathbb{R}^{p\times 1}$, the lemma below delineates the
relationship.

######  Lemma 3.

For a vector $\mathbf{X}\sim{\rm Cauchy}_{p}(\boldsymbol{\mu},\Sigma)$, the
relation is given by:

$\mathbf{v}^{\intercal}\mathbf{X}\sim{\rm
Cauchy}(\mathbf{v}^{\intercal}\boldsymbol{\mu},\mathbf{v}^{\intercal}\Sigma\mathbf{v}),$
(50)

where $\boldsymbol{\mu}$ signifies the location vector of $\mathbf{X}$ and
$\Sigma$ represents its scale matrix.


concave. By combining Theorem A.5 with Lemma 5.10, we have an even simpler
expression of the rank expansion lower bound.

######  Corollary A.6.

Let everything be defined as in Theorem A.4, if $\sigma_{A}(x)$,
$\sigma_{B}(x)$ are log-log concave, $r_{A}-\sigma_{A}(n_{A}-x)$ and
$r_{B}-\sigma_{B}(n_{B}-x)$ are log-log convex, on $(0,n_{A})$ and $(0,n_{B})$
respectively, e.g., polynomials and logarithm functions listed in Proposition
5.9, then

$R_{C}(k)=\min_{\begin{subarray}{c}k_{A}\in[d_{A},n_{A}],~{}k_{B}\in[d_{B},n_{B}],\\\
k_{A}\in\left\\{d_{A},n_{A}\right\\}\text{ or
}k_{B}\in\left\\{d_{B},n_{B}\right\\},\\\ k_{A}k_{B}\geqslant
k\end{subarray}}\sigma_{A}(k_{A})\cdot\sigma_{B}(k_{B})$

is a rank expansion lower bound for $\bm{C}$.


# A quantum dynamic belief model to explain the interference effects of
categorization on decision making

Zichang He  Wen Jiang  School of Electronics and Information, Northwestern
Polytechnical University, Xi’an, Shaanxi, 710072, China

###### Abstract

Categorization is necessary for many decision making tasks. However, the
categorization process may interfere the decision making result and the law of
total probability can be violated in some situations. To predict the
interference effect of categorization, some model based on quantum probability
has been proposed. In this paper, a new quantum dynamic belief(QDB) model is
proposed. Considering the precise decision may not be made during the process,
the concept of uncertainty is introduced in our model to simulate real human
thinking process. Then the interference effect categorization can be predicted
by handling the uncertain information. The proposed model is applied to a
categorization decision-making experiment to explain the interference effect
of categorization. Compared with other models, our model is relatively more
succinct and the result shows the correctness and effectiveness of our model.

###### keywords:

Quantum dynamic model; Uncertain information; Interference effect of
categorization; Law of total probability; categorization decision-making
experiment

##  1 Introduction

The fields of cognition and decision making are well-developed. As an
important and normal part of decision making process, categorization is widely
involved in decision making tasks in reality. Without a precise categorization
many decisions will not be able to make. For instance, doctors must classify
the tumor as a benign one or a malignant one before doing the surgery; judges
need to categorize the defendant as guilty or innocent before making a
judgement; unexpected aircrafts need to be categorized as an enemy or not
before commanders make decision. Categorization is regarded as a necessary
part during the decision making, however, the effect of categorization is
rarely studied separately.

Actually, a lot of living examples and experiments have shown that the
categorization will bring about interference effects on decision making. The
law of total probability will be broken due to the interference effect like
categorization decision-making experiment[1]. In order to explain the
phenomenon and predict the interference effect, some models have been proposed
like the quantum Belief-action entanglement(BAE) model proposed by Zheng Wang
and Busemeyer[2], an exemplar model account of feature inference proposed by
Nosofsky R.M[3] and so on. Quantum information has a wide application, like
user security[4, 5], quantum communication[6, 7] and so on. Quantum theory has
been widely applied in the field of cognition and decision[8, 9, 10, 11].
Quantum probability theory can solve many paradoxes like the violation of sure
thing principle[12, 13], the additive law of probability[14], the Ellsberg
paradox paradox[15] and many judgement errors[16]. The quantum model can also
be adopted to explain many phenomena which are irregular in classical theories
like order effect[17, 18, 19], disjunction effect[20], interference effect[21,
22] and so on. Among them, disjunction effect and interference effect are
widely studied. Many models have been proposed, such as quantum dynamical
model[20, 23], quantum-like approach[24, 21], quantum prospect decision
theory[10, 25] and quantum-like Bayesian networks[26] etc. Besides, quantum
theory can explain some classical phenomena like prisoner’s dilemma[27, 28].
And it is also widely applied in game theory[29, 30, 31].

In this paper, a quantum dynamic belief(QDB) model which introduces an
uncertain part to the decision making is proposed to predict the interference
effects of categorization on decision making. Quantum dynamic model first
proposed by Busemeyer $et.al$ in 2006[32] is formulated as random walk
decision process. The evolution of complex valued probability amplitudes over
time can be described in the dynamic model[23, 20]. Decision making and
optimization under uncertain environment is normal in reality and is heavily
studied[33, 34, 35]. Uncertainty information modeling and processing is still
an open issue[36, 37, 38]. It is reasonable to assume that uncertainty exists
in the decision making process. For example, in the test you are required to
make a choice between plan A or plan B, you may be confused to make a precise
choice, however the final decision is demanded. Considering it, the concept of
uncertainty is introduced in our model to simulate the real human thinking
process. A categorization decision-making experiment will be used to verify
the correctness and effectiveness of the new model. The uncertain information
will be handled differently in decision making along(D along) condition and
categorization-decision making(C-D) condition, which will predict the
interference effect well. Additionally, the new model is more succinct than
classical quantum belief-action entanglement(BAE) model.

The rest of the paper is organized as follows. In Section 2, the preliminaries
of the basic theory employed will be briefly introduced. And a categorization
decision-making experiment which violates the law of total probability is
illustrated in Section 3. Then our QDB model is proposed to predict the
interference effects and explain the results of the experiment in Section 4.
The model result and comparison is shown in Section 5. Finally, Section 6
comes to the conclusion.

##  2 Preliminaries

###  2.1 quantum dynamic model

The quantum dynamic model assumes that at the beginning of a game, a
participant is possible to be in every state, thus the initial belief state is
a superposition of all possible states in the form of a vector. The term
$\psi$ represents the quantum probability amplitude of the according state.
The term ${{{\left|{{\psi_{\rm{n}}}}\right|}^{\rm{2}}}}$ is the probability of
observing state ${{\psi_{\rm{n}}}}$ initially.

$\psi\left({\rm{0}}\right){\rm{=}}\frac{{\rm{1}}}{{\sqrt{{{\left|{{\psi_{\rm{1}}}}\right|}^{\rm{2}}}{\rm{+}}{{\left|{{\psi_{\rm{2}}}}\right|}^{\rm{2}}}{\rm{+}}\cdots{\rm{+}}{{\left|{{\psi_{\rm{n}}}}\right|}^{\rm{2}}}}}}\left[{{\psi_{\rm{1}}},{\psi_{2}},\ldots,{\psi_{n}}}\right]$

During the decision making process, the belief state will evolve across time
obeying a Schr${\ddot{o}}$dinger equation.

$\frac{d}{{dt}}\psi\left(t\right)=-i\cdot H\cdot\psi\left(t\right)$ (1)

which has a matrix exponential solution

$\psi\left({{t_{2}}}\right)={e^{-iHt}}\cdot\psi\left({{t_{1}}}\right)$ (2)

where $H$ is a Hamiltonian matrix $t$ to ensure a unitary matrix
$U={e^{-iHt}}$ which ensures the sum of probabilities of all states always
equals to 1.

For $t={t_{2}}-{t_{1}}$, the transition probabilities ${T_{ij}}$ is the square
of the unitary matrix ${U_{ij}}$.

${T_{ij}}\left(t\right)={\left|{{U_{ij}}\left(t\right)}\right|^{2}}$ (3)

which means the probability of observing state $i$ at time ${t_{2}}$ given
that state $j$ was observed at time ${t_{1}}$.

Based on the above definition, Eq.(2) could be rewritten as

$\psi\left(t\right)=U\cdot\psi\left(0\right)$ (4)

The person’s belief state evolves to $\psi\left(t\right)$ from the initial
$\psi\left(0\right)$ across time $t$, which shows the dynamic process of the
decision making. This model can produce interference effect which is
impossible in classical Markov model[23].

###  2.2 Pignistic probability transformation

Pignistic probability is widely applied to the field of decision making. The
term ”pignistic” proposed by Smets is originated from the word pignus, meaning
’bet’ in Latin. Principle of insufficient reason is used to assign the basic
probability of multiple-element set to singleton set. In other word, a belief
interval is distributed into the crisp one determined as[39]:

${bet\left({{A_{i}}}\right)=\sum\nolimits_{{A_{i}}\subseteq{A_{k}}}{\frac{{m\left({{A_{k}}}\right)}}{{\left|{{A_{k}}}\right|}}}}$
(5)

where ${\left|{{A_{k}}}\right|}$ denotes the number of elements in the set
called the cardinality. Eq.(5) is also called as Pignistic Probability
Transformation(PPT).

##  3 Categorization decision-making experiment

###  3.1 Experiment introduction

Townsend $et.al$(2000)[1] proposed a paradigm to study the interactions
between categorization and decision making. The paradigm was designed to test
Markov model initially. Subsequently, this paradigm was extended for quantum
models as well by Busemeyer $et.al$(2009)[23]. The paradigm consists of two
kinds of conditions, one is categorization decision-making(C-D) condition and
the other is decision along (D along) condition. In C-D condition, in each
trial, participants are shown pictures of faces, which vary along two
dimensions(face width and lip thickness) like Figure 5.

Figure 1: Example faces used in a categorization-decision experiment

Participants are asked to categorize the face as a ”good” guy or a ”bad” guy
and then make a decision to ”attack” or to ”withdraw”. The faces can be
roughly divided into two kinds: one kind is ”narrow” faces with narrow width
and thick lips; the other kind is ”wide” faces with wide width and thin lips.
The participates are informed that ”narrow” faces have a 0.60 probability of
belonging to the ”bad guy” population and ”wide” faces had a 0.60 probability
of belonging to the ”good guy” population. And participants are rewarded for
attacking ”bad guy” and withdrawing from ”good guy”. In D along condition, the
participants are asked to make a decision to attack or to withdraw directly
without categorizing. The faces shown to them are the same as those in C-D
condition. The whole experiment included a total of 26 participants, but each
participant provides 51 observations for C-D condition for a total of
${\rm{26}}\times{\rm{51=1326}}$ observations, while each person produces 17
observations for D along condition for a total of
${\rm{17}}\times{\rm{26=442}}$ observations.

###  3.2 Experiment results

The experiment results are shown in Table 1.

Table 1: The results of C-D condition and D along condition Face type | $P\left({G}\right)$ | $P\left({A|G}\right)$ | $P\left({B}\right)$ | $P\left({A|B}\right)$ | ${P_{T}}$ | $P\left({A}\right)$ | t
---|---|---|---|---|---|---|---
Wide | 0.84 | 0.35 | 0.16 | 0.52 | 0.37 | 0.39 | 0.5733
Narrow | 0.17 | 0.41 | 0.83 | 0.63 | 0.59 | 0.69 | 2.54

The column labeled $P\left({G}\right)$ represents the probability of
categorizing the face as a ”good buy”, the column labeled
$P\left({A|G}\right)$ represents the probability of attacking given
categorizing the face as a ”good guy”. The column labeled $P\left({B}\right)$
represents the probability of categorizing the face as a ”bad buy”. The column
labeled $P\left({A|B}\right)$ represents the probability of attacking given
categorizing the face as a ”bad guy”. And the column labeled ${P_{T}}$
represents the final probability of attacking in C-D condition which is
computed as follows.

$P\left(A\right)=P(G)\cdot P\left({A|G}\right)+P\left(B\right)\cdot
P\left({A|B}\right)$ (6)

Accordingly, the column label as $P\left({A}\right)$ represents the
probability of attacking in D along condition. As shown in Table 1, there
exist some deviation between ${P_{T}}$ and $P\left({A}\right)$ for both types
of face. However, the prominent deviation occurs as the narrow type faces were
shown, which produces positive interference effect, while the interference
effect is weak for the wide type faces. Using a paired $t$-test to test the
significance of the difference between ${P_{T}}$ and $P\left({A}\right)$
estimated from the 26 participants, the results shown in the $t$ column
indicated that the interference effect is statistically significant for the
narrow faces, but not for the wide faces.

The classical paradigm has been discussed in many works, the literatures of
studying the categorization decision-making experiment and their results are
shown below in Table 2.

Table 2: Works of literature which study the categorization decision-making
experiment

Literature | Type | $P\left({G}\right)$ | $P\left({A|G}\right)$ | $P\left({B}\right)$ | $P\left({A|B}\right)$ | ${P_{T}}$ | $P\left({A}\right)$
---|---|---|---|---|---|---|---
Townsend $et.al$(2000)[1] | W | 0.84 | 0.35 | 0.16 | 0.52 | 0.37 | 0.39
N | 0.17 | 0.41 | 0.83 | 0.63 | 0.59 | 0.69
Busemeyer $et.al$(2009)[23] | W | 0.80 | 0.37 | 0.20 | 0.53 | 0.40 | 0.39
N | 0.20 | 0.45 | 0.80 | 0.64 | 0.60 | 0.69
Wang and Busemeyer(2016) Experiment 1[2] | W | 0.78 | 0.39 | 0.22 | 0.52 | 0.42 | 0.42
N | 0.21 | 0.41 | 0.79 | 0.58 | 0.54 | 0.59
Wang and Busemeyer(2016) Experiment 2[2] | W | 0.78 | 0.33 | 0.22 | 0.53 | 0.37 | 0.37
N | 0.24 | 0.37 | 0.76 | 0.61 | 0.55 | 0.60
Wang and Busemeyer(2016) Experiment 3[2] | W | 0.77 | 0.34 | 0.23 | 0.58 | 0.40 | 0.39
N | 0.24 | 0.33 | 0.76 | 0.66 | 0.58 | 0.62
Average | W | 0.79 | 0.36 | 0.21 | 0.54 | 0.39 | 0.39
N | 0.21 | 0.39 | 0.79 | 0.62 | 0.57 | 0.64

  * 1

In Busemeyer $et.al$(2009)[23], the classical experiment is replicated.

  * 2

In Wang and Busemeyer(2016)[2], Experiment 1 uses a larger data set to
replicate the classical experiment. Experiment 2 introduce a new X-D trial
verse C-D trial and this table only use the result of C-D trial. In experiment
3, the reward for attacking bad people is a bit less than the other two.

According to the law of total probability, the probability of attacking should
be the same in two conditions. Hence, the categorization brings about
interference effect which makes a difference in decision making and breaks the
law of total probability.

##  4 Proposed method

In this section, our QDB model will be applied to explain and predict the
experiment results shown in last section. To begin with, the integral flow
chart of the model is shown in Figure 2.

Figure 2: The model flow chart

Step 1. Represent the initial state

In C-D condition, faces shown to participants are categorized as ”good”(G)
ones and ”bad”(B) ones. Accordingly, the basic action options for participants
is to withdraw(W) and to attack(A). However, in reality, it may be uncertain
for some participants to make a precise choice. Considering it, a new
temporary possible action labeled as uncertain(U) is brought into our model.
Hence, the experiment involves a set of six mutual and exhaustive events
$\left\\{{AG,UG,WG,AB,UB,WB}\right\\}$, where $AG$ symbolized the event that
the participants decided to attack given the face was categorized as a good
guy. Our model assumes that these six events correspond to six mutual and
exhaustive states of the participant
$\left\\{{\left|{AG}\right\rangle,\left|{UG}\right\rangle,\left|{WG}\right\rangle,\left|{AB}\right\rangle,\left|{UB}\right\rangle,\left|{WB}\right\rangle}\right\\}$
It should be noticed that the states ${\left|{UG}\right\rangle}$ and
${\left|{UB}\right\rangle}$ are intermediate states whose probabilities will
be divided into the other four states in the final decision making stage.

It is assumed that at the beginning of the experiment, a participant holds the
potential to be in every possible state and the states can transit with each
other like shown in Figure 3, namely the initial state is totally random.

Figure 3: Transition among states

Hence, the participant’s state is a superposition of the six basis states as
follows.

$\left|\psi\right\rangle{\rm{=}}{\psi_{AG}}\cdot\left|{AG}\right\rangle{\rm{+}}{\psi_{UG}}\cdot\left|{UG}\right\rangle{\rm{+}}{\psi_{WG}}\cdot\left|{WG}\right\rangle{\rm{+}}{\psi_{AB}}\cdot\left|{AB}\right\rangle{\rm{+}}{\psi_{UB}}\cdot\left|{UB}\right\rangle{\rm{+}}{\psi_{WB}}\cdot\left|{WB}\right\rangle$
(7)

In the meantime, the initial state corresponds to an amplitude distribution
represented by the ${\rm{6}}\times{\rm{1}}$ matrix.

$\psi\left(0\right)=\left[{\begin{array}[]{*{20}{c}}{{\psi_{AG}}}\\\
{{\psi_{UG}}}\\\ {{\psi_{WG}}}\\\ {{\psi_{AB}}}\\\ {{\psi_{UB}}}\\\
{{\psi_{WB}}}\end{array}}\right]$ (8)

where, ${\left|{{\psi_{AG}}}\right|^{\rm{2}}}$ is the probability of observing
state ${\left|{AG}\right\rangle}$ initially and the same as the others. It
should be noticed that the sum of all states’ probability must equal to one.

${\left|\psi\right|^{\rm{2}}}{\rm{=}}\psi^{\prime}\cdot\psi{\rm{=1}}$ (9)

Step 2. Update the state

As the experiment goes, more information is known to the participants. In C-D
condition, the categorization of the faces is informed to the participants.
Assume that new information at time ${{\rm{t}}_{1}}$ changes the initial state
$\psi\left(0\right)$ which is at time $t$=0 into a new state
$\psi\left({{{\rm{t}}_{1}}}\right)$.

In C-D condition, if the face is categorized as ”good”, the amplitude
distribution across states will change to

$\psi\left({{t_{1}}}\right)=\frac{1}{{\sqrt{{{\left|{{\psi_{AG}}}\right|}^{2}}+{{\left|{{\psi_{UG}}}\right|}^{2}}+{{\left|{{\psi_{WG}}}\right|}^{2}}}}}\left[{\begin{array}[]{*{20}{c}}{{\psi_{AG}}}\\\
{{\psi_{UG}}}\\\ {{\psi_{WG}}}\\\ 0\\\ 0\\\
0\end{array}}\right]=\left[{\begin{array}[]{*{20}{c}}{{\psi_{G}}}\\\
0\end{array}}\right]$ (10)

It means that the probabilities of being the states ${{\psi_{AB}}}$,
${{\psi_{UB}}}$ and ${{\psi_{WB}}}$ change into zero and the probabilities of
being the sates ${{\psi_{AG}}}$, ${{\psi_{UG}}}$ and ${{\psi_{WG}}}$ are
enlarged with the same proportion
$\frac{1}{{\sqrt{{{\left|{{\psi_{AG}}}\right|}^{2}}+{{\left|{{\psi_{UG}}}\right|}^{2}}+{{\left|{{\psi_{WG}}}\right|}^{2}}}}}$
to meet Eq. 9, where
${\sqrt{{{\left|{{\psi_{AG}}}\right|}^{2}}+{{\left|{{\psi_{UG}}}\right|}^{2}}+{{\left|{{\psi_{WG}}}\right|}^{2}}}}$
is the initial probability of categorizing the face as ”good”. And the
$2\times 1$ matrix $\left[{\begin{array}[]{*{20}{c}}{{\psi_{G}}}\\\
0\end{array}}\right]$ is the brief form of the state given categorizing the
face as ”good”.

If the face is categorized as ”bad”, the amplitude distribution across states
will change to

$\psi\left({{t_{1}}}\right)=\frac{1}{{\sqrt{{{\left|{{\psi_{AB}}}\right|}^{2}}+{{\left|{{\psi_{UB}}}\right|}^{2}}+{{\left|{{\psi_{WB}}}\right|}^{2}}}}}\left[{\begin{array}[]{*{20}{c}}0\\\
0\\\ 0\\\ {{\psi_{AB}}}\\\ {{\psi_{UB}}}\\\
{{\psi_{WB}}}\end{array}}\right]=\left[{\begin{array}[]{*{20}{c}}{{\psi_{G}}}\\\
0\end{array}}\right]$ (11)

It means that the probabilities of being the states ${{\psi_{AG}}}$,
${{\psi_{UG}}}$ and ${{\psi_{WG}}}$ change into zero and the probabilities of
being the sates ${{\psi_{AB}}}$, ${{\psi_{UB}}}$ and ${{\psi_{WB}}}$ are
enlarged with the same proportion
$\frac{1}{{\sqrt{{{\left|{{\psi_{AB}}}\right|}^{2}}+{{\left|{{\psi_{UB}}}\right|}^{2}}+{{\left|{{\psi_{WB}}}\right|}^{2}}}}}$
to meet Eq.9, where
${\sqrt{{{\left|{{\psi_{AB}}}\right|}^{2}}+{{\left|{{\psi_{UB}}}\right|}^{2}}+{{\left|{{\psi_{WB}}}\right|}^{2}}}}$
is the initial probability of categorizing the face as ”bad”. And the $2\times
1$ matrix $\left[{\begin{array}[]{*{20}{c}}0\\\
{{\psi_{B}}}\end{array}}\right]$ is the brief form of the state given
categorizing the face as ”bad”.

In D along condition, there is no new information to bring the change. So the
amplitude distribution across states will keep the same as the initial one.

$\footnotesize\begin{array}[]{l}\psi\left({{t_{1}}}\right)=\psi\left(0\right)=\left[{\begin{array}[]{*{20}{c}}{\sqrt{{{\left|{{\psi_{AG}}}\right|}^{2}}+{{\left|{{\psi_{UG}}}\right|}^{2}}+{{\left|{{\psi_{WG}}}\right|}^{2}}}\cdot{\psi_{G}}}\\\
{\sqrt{{{\left|{{\psi_{AB}}}\right|}^{2}}+{{\left|{{\psi_{UB}}}\right|}^{2}}+{{\left|{{\psi_{WB}}}\right|}^{2}}}\cdot{\psi_{B}}}\end{array}}\right]\\\
{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern
1.0pt}=\sqrt{{{\left|{{\psi_{AG}}}\right|}^{2}}+{{\left|{{\psi_{UG}}}\right|}^{2}}+{{\left|{{\psi_{WG}}}\right|}^{2}}}\left[{\begin{array}[]{*{20}{c}}{{\psi_{G}}}\\\
0\end{array}}\right]+\sqrt{{{\left|{{\psi_{AB}}}\right|}^{2}}+{{\left|{{\psi_{UB}}}\right|}^{2}}+{{\left|{{\psi_{WB}}}\right|}^{2}}}\left[{\begin{array}[]{*{20}{c}}0\\\
{{\psi_{B}}}\end{array}}\right]\end{array}$ (12)

The equation shows that the initial state for the participants under unknown
condition is formed by a weighted sum of the amplitude distributions for the
two known conditions.

Step 3. Strategies based on reward

The actions taken are always based on the according reward. Hence, in the
model, the decision maker must evaluate the rewards in order to select an
appropriate action, which changes the previous state at time ${{t_{1}}}$ into
a new state ${\psi\left({{t_{2}}}\right)}$ at time ${{t_{2}}}$. The evolution
of the state during the time period corresponds to the thinking process which
leads to a decision.

Based on the quantum dynamic model, the state evolution obeys a
Schr${\ddot{o}}$dinger equation driven by a $6\times 6$ Hamiltonian matrix
$H$:

$\frac{d}{{dt}}\psi\left(t\right)=-i\cdot H\cdot\psi\left(t\right)$ (13)

which has a matrix exponential solution for $t={t_{1}}-{t_{2}}$.

$\psi\left({{t_{2}}}\right)={e^{-i\cdot H\cdot t}}\cdot\psi(t)$ (14)

The unitary matrix is defined by

$U\left(t\right)={e^{-i\cdot H\cdot t}}$ (15)

which determines the transition probabilities according to
${T_{ij}}\left(t\right)={\left|{{U_{ij}}\left(t\right)}\right|^{2}}$. Let the
component of ${T_{ij}}$ denote as ${t_{ij}}$, which represents the probability
of observing state $i$ at time ${t_{2}}$ given that state $j$ was observed at
time ${t_{1}}$. To create a choice probability that reaches its maximum, the
process time parameter $t$ is set as ${\pi\mathord{\left/{\vphantom{\pi
2}}\right.\kern-1.2pt}2}$ in the model.

The Hamiltonian matrix $H$ is a Hermitian matrix, which satisfies ${H^{*}}=H$,
so that $U$ is a unitary matrix, which satisfies

${U^{*}}\cdot U=I$ (16)

The property is the key to guarantee that the state $\psi\left(t\right)$ keeps
a unit length. Assume

$H=\left[{\begin{array}[]{*{20}{c}}{{H_{G}}}&0\\\
0&{{H_{B}}}\end{array}}\right]$ (17)

with

${H_{G}}=\left({\begin{array}[]{*{20}{c}}{{h_{G}}}&0&1\\\ 0&1&0\\\
1&0&{-{h_{G}}}\end{array}}\right)$ (18)

and

${H_{B}}=\left({\begin{array}[]{*{20}{c}}{{h_{B}}}&0&1\\\ 0&1&0\\\
1&0&{-{h_{B}}}\end{array}}\right)$ (19)

The $3\times 3$ Hamiltonian matrix ${H_{G}}$ applies when the participant
categorizes the face as ”good”, and the other $3\times 3$ Hamiltonian matrix
${H_{B}}$ applies when the participant categorizes the face as ”bad”. The
parameter ${h_{G}}$ is a function of the difference between the reward for
attacking relative to withdrawing given categorizing the face as ”good” and
the parameter ${h_{B}}$ is a function of the difference between the reward for
attacking relative to withdrawing given categorizing the face as ”bad”.
According to the payoffs, the Hamiltonian matrix transforms the state
probabilities to attacking, uncertainty or withdrawing.

Based on the above, we can obtain the participant’s state at time ${t_{2}}$ as
follows: In C-D condition, if the face is categorized as ”good”, the state
will be

$\psi\left({{t_{2}}}\right)={e^{-i\cdot H\cdot
t}}\cdot\psi({t_{\rm{1}}}){\rm{=}}\left[{\begin{array}[]{*{20}{c}}{{e^{-i\cdot{H_{G}}\cdot
t}}}&0\\\ 0&{{e^{-i\cdot{H_{B}}\cdot
t}}}\end{array}}\right]\cdot\left[{\begin{array}[]{*{20}{c}}{{\psi_{G}}}\\\
0\end{array}}\right]={e^{-i\cdot{H_{G}}\cdot t}}\cdot{\psi_{G}}$ (20)

If the face is categorized as ”bad”, the state will be

$\psi\left({{t_{2}}}\right)={e^{-i\cdot H\cdot
t}}\cdot\psi({t_{\rm{1}}}){\rm{=}}\left[{\begin{array}[]{*{20}{c}}{{e^{-i\cdot{H_{G}}\cdot
t}}}&0\\\ 0&{{e^{-i\cdot{H_{B}}\cdot
t}}}\end{array}}\right]\cdot\left[{\begin{array}[]{*{20}{c}}0\\\
{{\psi_{B}}}\end{array}}\right]={e^{-i\cdot{H_{B}}\cdot t}}\cdot{\psi_{B}}$
(21)

In D along condition, the state does not change at time ${t_{1}}$ and it will
change into

$\footnotesize\begin{array}[]{*{20}{l}}{\psi\left({{t_{2}}}\right)={e^{-i\cdot
H\cdot
t}}\cdot\psi(0){\rm{=}}\left[{\begin{array}[]{*{20}{c}}{{e^{-i\cdot{H_{G}}\cdot
t}}}&0\\\ 0&{{e^{-i\cdot{H_{B}}\cdot
t}}}\end{array}}\right]\cdot\left[{\begin{array}[]{*{20}{c}}{\sqrt{{{\left|{{\psi_{AG}}}\right|}^{2}}+{{\left|{{\psi_{UG}}}\right|}^{2}}+{{\left|{{\psi_{WG}}}\right|}^{2}}}\cdot{\psi_{G}}}\\\
{\sqrt{{{\left|{{\psi_{AB}}}\right|}^{2}}+{{\left|{{\psi_{UB}}}\right|}^{2}}+{{\left|{{\psi_{WB}}}\right|}^{2}}}\cdot{\psi_{B}}}\end{array}}\right]}\\\
\begin{array}[]{l}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern
1.0pt}=\sqrt{{{\left|{{\psi_{AG}}}\right|}^{2}}+{{\left|{{\psi_{UG}}}\right|}^{2}}+{{\left|{{\psi_{WG}}}\right|}^{2}}\cdot}{e^{-i\cdot{H_{G}}\cdot
t}}\cdot{\psi_{G}}{\rm{+}}\\\ {\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern
1.0pt}\sqrt{{{\left|{{\psi_{AB}}}\right|}^{2}}+{{\left|{{\psi_{UB}}}\right|}^{2}}+{{\left|{{\psi_{WB}}}\right|}^{2}}}\cdot{e^{-i\cdot{H_{B}}\cdot
t}}\cdot{\psi_{B}}\end{array}\end{array}$ (22)

at time ${t_{2}}$. As we can see, the ${t_{2}}$ state for the participants
under unknown condition is still formed by a weighted sum of the amplitude
distributions for the two known conditions.

Step 4. Make the final decision

After the previous three steps, the participants’s state has been obtained.
The last step is to calculate the probabilities according to the state. To
address it, a measure matrix $M$ is introduced to pick out the needed state.

$M=\left({\begin{array}[]{*{20}{c}}{{M_{G}}}&0\\\
0&{{M_{B}}}\end{array}}\right)$ (23)

where ${M_{G}}$ is the matrix used to pick out state given categorizing the
face as ”good” and ${M_{B}}$ is the matrix used to pick out state given
categorizing the face as ”bad”.

As the uncertain state is an assumed intermediate state, it should be divided
into attacking state and withdrawing state as follows based on Eq. (5):

$P\left(A\right)=P\left(W\right)=0.5\cdot P\left(N\right)$ (24)

In C-D condition, to pick out the state of attacking, we set

${M_{A}}=\left({\begin{array}[]{*{20}{c}}{{M_{GA}}}&0\\\
0&{{M_{BA}}}\end{array}}\right)$ (25)

with

${M_{GA}}={M_{BA}}=\left[{\begin{array}[]{*{20}{c}}1&0&0\\\ 0&0&0\\\
0&0&0\end{array}}\right]$ (26)

and then the probability of attacking given categorizing the face as ”good”
equals to

$\footnotesize\begin{array}[]{l}P\left({A|G}\right){\rm{=}}{\left\|{{M_{A}}\cdot{e^{-i\cdot
t\cdot
H}}\cdot\psi\left({{t_{1}}}\right)}\right\|^{\rm{2}}}{\rm{=}}{\left\|{\left[{\begin{array}[]{*{20}{c}}{{M_{GA}}}&{\rm{0}}\\\
{\rm{0}}&{{M_{BA}}}\end{array}}\right]\cdot\left[{\begin{array}[]{*{20}{c}}{{e^{-i\cdot
t\cdot{H_{G}}}}}&{\rm{0}}\\\ {\rm{0}}&{{e^{-i\cdot
t\cdot{H_{B}}}}}\end{array}}\right]\cdot\left[{\begin{array}[]{*{20}{c}}{{\psi_{G}}}\\\
0\end{array}}\right]}\right\|^{\rm{2}}}\\\ {\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\rm{=}}{\left\|{{M_{GA}}\cdot{e^{-i\cdot
t\cdot{H_{G}}}}\cdot{\psi_{G}}}\right\|^{\rm{2}}}\end{array}$ (27)

The probability of attacking given categorizing the face as ”bad” equals to

$\footnotesize\begin{array}[]{l}P\left({A|B}\right){\rm{=}}{\left\|{{M_{A}}\cdot{e^{-i\cdot
t\cdot
H}}\cdot\psi\left({{t_{1}}}\right)}\right\|^{\rm{2}}}{\rm{=}}{\left\|{\left[{\begin{array}[]{*{20}{c}}{{M_{GA}}}&{\rm{0}}\\\
{\rm{0}}&{{M_{BA}}}\end{array}}\right]\cdot\left[{\begin{array}[]{*{20}{c}}{{e^{-i\cdot
t\cdot{H_{G}}}}}&{\rm{0}}\\\ {\rm{0}}&{{e^{-i\cdot
t\cdot{H_{B}}}}}\end{array}}\right]\cdot\left[{\begin{array}[]{*{20}{c}}0\\\
{{\psi_{B}}}\end{array}}\right]}\right\|^{\rm{2}}}\\\ {\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\rm{=}}{\left\|{{M_{BA}}\cdot{e^{-i\cdot
t\cdot{H_{G}}}}\cdot{\psi_{B}}}\right\|^{\rm{2}}}\end{array}$ (28)

To pick out the state of uncertainty, we set

${M_{U}}=\left({\begin{array}[]{*{20}{c}}{{M_{GU}}}&0\\\
0&{{M_{BU}}}\end{array}}\right)$ (29)

with

${M_{GU}}={M_{BU}}=\left[{\begin{array}[]{*{20}{c}}0&0&0\\\ 0&1&0\\\
0&0&0\end{array}}\right]$ (30)

and then the probability of uncertain given categorizing the face as ”good”
equals to

$\footnotesize\begin{array}[]{l}P\left({U|G}\right){\rm{=}}{\left\|{{M_{U}}\cdot{e^{-i\cdot
t\cdot
H}}\cdot\psi\left({{t_{1}}}\right)}\right\|^{\rm{2}}}{\rm{=}}{\left\|{\left[{\begin{array}[]{*{20}{c}}{{M_{GU}}}&{\rm{0}}\\\
{\rm{0}}&{{M_{BU}}}\end{array}}\right]\cdot\left[{\begin{array}[]{*{20}{c}}{{e^{-i\cdot
t\cdot{H_{G}}}}}&{\rm{0}}\\\ {\rm{0}}&{{e^{-i\cdot
t\cdot{H_{B}}}}}\end{array}}\right]\cdot\left[{\begin{array}[]{*{20}{c}}{{\psi_{G}}}\\\
0\end{array}}\right]}\right\|^{\rm{2}}}\\\ {\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\rm{=}}{\left\|{{M_{GU}}\cdot{e^{-i\cdot
t\cdot{H_{G}}}}\cdot{\psi_{G}}}\right\|^{\rm{2}}}\end{array}$ (31)

Then the probability of uncertainty given categorizing the face as ”bad”
equals to

$\footnotesize\begin{array}[]{l}P\left({U|B}\right){\rm{=}}{\left\|{{M_{U}}\cdot{e^{-i\cdot
t\cdot
H}}\cdot\psi\left({{t_{1}}}\right)}\right\|^{\rm{2}}}{\rm{=}}{\left\|{\left[{\begin{array}[]{*{20}{c}}{{M_{GU}}}&{\rm{0}}\\\
{\rm{0}}&{{M_{BU}}}\end{array}}\right]\cdot\left[{\begin{array}[]{*{20}{c}}{{e^{-i\cdot
t\cdot{H_{G}}}}}&{\rm{0}}\\\ {\rm{0}}&{{e^{-i\cdot
t\cdot{H_{B}}}}}\end{array}}\right]\cdot\left[{\begin{array}[]{*{20}{c}}{\rm{0}}\\\
{{\psi_{B}}}\end{array}}\right]}\right\|^{\rm{2}}}\\\ {\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\rm{=}}{\left\|{{M_{BU}}\cdot{e^{-i\cdot
t\cdot{H_{B}}}}\cdot{\psi_{B}}}\right\|^{\rm{2}}}\end{array}$ (32)

Hence, the probability of finally deciding to attack equals to

$P\left(A\right)=P\left(G\right)P\left({A|G}\right)+P\left(B\right)P\left({A|B}\right){\rm{+}}0.5\left({P\left({U|G}\right)+P\left({U|B}\right)}\right)$
(33)

Take the obtained probabilities into Eq. (33), we can get

$\begin{array}[]{l}P\left(A\right)=\left({{{\left|{{\psi_{AG}}}\right|}^{2}}+{{\left|{{\psi_{UG}}}\right|}^{2}}+{{\left|{{\psi_{WG}}}\right|}^{2}}}\right)\cdot{\left\|{\left({{M_{GA}}+0.5\cdot{M_{GU}}}\right)\cdot{e^{-i\cdot
t\cdot{H_{G}}}}\cdot{\psi_{G}}}\right\|^{\rm{2}}}\\\ {\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}+\left({{{\left|{{\psi_{AB}}}\right|}^{2}}+{{\left|{{\psi_{UB}}}\right|}^{2}}+{{\left|{{\psi_{WB}}}\right|}^{2}}}\right)\cdot{\left\|{\left({{M_{BA}}+0.5\cdot{M_{BU}}}\right)\cdot{e^{-i\cdot
t\cdot{H_{B}}}}\cdot{\psi_{B}}}\right\|^{\rm{2}}}\end{array}$ (34)

In D along condition, without a precise categorization, it is reasonable to
assume that ${P\left({U|G}\right)}$ and ${P\left({U|B}\right)}$ will not be
produced. It means that the uncertainty state will be picked out and divided
in the same time rather than being produced at first and then being divided as
C-D condition. To address it, we set the measure matrix as Eq. (23) with

${M_{G}}={M_{B}}=\left[{\begin{array}[]{*{20}{c}}1&0&0\\\
0&{{\textstyle{1\over{\sqrt{2}}}}}&0\\\ 0&0&0\end{array}}\right]$ (35)

To meet the sum of $P\left(A\right)$ and $P\left(G\right)$ equals to one, the
measure coefficient of the uncertain state is
${{\textstyle{1\over{\sqrt{2}}}}}$. Then the probability of attacking in D
along condition equals to

$\footnotesize\begin{array}[]{l}P\left(A\right){\rm{=}}{\left\|{M\cdot{e^{-i\cdot
t\cdot H}}\cdot\psi\left(0\right)}\right\|^{\rm{2}}}\\\ {\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\rm{=}}{\left\|{\left[{\begin{array}[]{*{20}{c}}{{M_{G}}}&{\rm{0}}\\\
{\rm{0}}&{{M_{B}}}\end{array}}\right]\cdot\left[{\begin{array}[]{*{20}{c}}{{e^{-i\cdot
t\cdot{H_{G}}}}}&{\rm{0}}\\\ {\rm{0}}&{{e^{-i\cdot
t\cdot{H_{B}}}}}\end{array}}\right]\cdot\left[{\begin{array}[]{*{20}{c}}{\sqrt{{{\left|{{\psi_{AG}}}\right|}^{2}}+{{\left|{{\psi_{UG}}}\right|}^{2}}+{{\left|{{\psi_{WG}}}\right|}^{2}}}\cdot{\psi_{G}}}\\\
{\sqrt{{{\left|{{\psi_{AB}}}\right|}^{2}}+{{\left|{{\psi_{UB}}}\right|}^{2}}+{{\left|{{\psi_{WB}}}\right|}^{2}}}\cdot{\psi_{B}}}\end{array}}\right]}\right\|^{\rm{2}}}\\\
{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\rm{=}}\left({{{\left|{{\psi_{AG}}}\right|}^{2}}+{{\left|{{\psi_{UG}}}\right|}^{2}}+{{\left|{{\psi_{WG}}}\right|}^{2}}}\right)\cdot{\left\|{{M_{G}}\cdot{e^{-i\cdot
t\cdot{H_{G}}}}\cdot{\psi_{G}}}\right\|^{\rm{2}}}\\\ {\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}+\left({{{\left|{{\psi_{AB}}}\right|}^{2}}+{{\left|{{\psi_{UB}}}\right|}^{2}}+{{\left|{{\psi_{WB}}}\right|}^{2}}}\right)\cdot{\left\|{{M_{B}}\cdot{e^{-i\cdot
t\cdot{H_{G}}}}\cdot{\psi_{B}}}\right\|^{\rm{2}}}\end{array}$ (36)

Now, the probabilities of attacking in both C-D condition and D along
condition have been obtained. Compare Eq. 34 with Eq. 36, we can find that the
two probabilities are unequal. The crucial difference of two equations is the
measure matrix, which is also the key to predict the interference effect of
categorization on decision making. In C-D condition, the probability of
uncertain state undergoes the process of producing and dividing separately
while it is produced and divided in the same time in D along condition. The
inference effect of categorization acts on the uncertain state and then lead
to the violation of the law of total probability.

The parameters of the model is shown in Table 3 as follows.

Table 3: the model parameters

Parameters | ${h_{G}}$ | ${h_{B}}$ |  $P\left(A\right)$1 |  $P\left(A\right)$2
---|---|---|---|---
Type | Free | Free | Known | Known

  * 1

$P\left(A\right)$ equals to
${{{\left|{{\psi_{AG}}}\right|}^{2}}+{{\left|{{\psi_{UG}}}\right|}^{2}}+{{\left|{{\psi_{WG}}}\right|}^{2}}}$.

  * 2

$P\left(A\right)$ equals to
${{{\left|{{\psi_{AB}}}\right|}^{2}}+{{\left|{{\psi_{UB}}}\right|}^{2}}+{{\left|{{\psi_{WB}}}\right|}^{2}}}$.

###  4.1 Result and comparison

Applying our QDB model to the categorization decision-making experiment, we
can adjust the values of two free coefficients ${h_{G}}$ and ${h_{B}}$ and
obtain a series of probabilities. Compared the obtained model results with the
observed experiment results (narrow type face) in Table 2, the model result
shown in Table 4 is close to practical situation, which verifies the
correctness and effectiveness of our model.

Table 4: The result of QDB model

Literature |  | $P\left(G\right)$ |  $P\left({A|G}\right)$1 | $P\left(B\right)$ |  $P\left({A|B}\right)$2 | ${{P}_{T}}$ | $P\left(A\right)$
---|---|---|---|---|---|---|---
Townsend $et.al$(2000) | Obs | 0.17 | 0.41 | 0.83 | 0.63 | 0.59 | 0.69
QDB | 0.17 | 0.41 | 0.83 | 0.6296 | 0.5923 | 0.6756
Busemeyer $et.al$(2009) | Obs | 0.20 | 0.45 | 0.80 | 0.64 | 0.60 | 0.69
QDB | 0.2 | 0.4499 | 0.80 | 0.6409 | 0.6027 | 0.6860
Wang and Busemeyer(2016) Experiment 1 | Obs | 0.21 | 0.41 | 0.79 | 0.58 | 0.54 | 0.59
QDB | 0.21 | 0.41 | 0.79 | 0.5802 | 0.5444 | 0.6278
Wang and Busemeyer(2016) Experiment 2 | Obs | 0.24 | 0.37 | 0.76 | 0.61 | 0.55 | 0.60
QDB | 0.24 | 0.3702 | 0.76 | 0.6104 | 0.5528 | 0.6361
Wang and Busemeyer(2016) Experiment 3 | Obs | 0.24 | 0.33 | 0.76 | 0.66 | 0.58 | 0.62
QDB | 0.24 | 0.3296 | 0.76 | 0.6604 | 0.5810 | 0.6644
Average | Obs | 0.21 | 0.39 | 0.79 | 0.62 | 0.57 | 0.64
QDB | 0.21 | 0.39 | 0.79 | 0.6205 | 0.5721 | 0.6554

  * 1

$P\left({A|G}\right)$ in QDB model is calculated by
$P\left({A|G}\right)=P^{\prime}\left({A|G}\right)+0.5\cdot P(N|G)$.

  * 2

$P\left({A|B}\right)$ in QDB model is calculated by
$P\left({A|B}\right)=P^{\prime}\left({A|B}\right)+0.5\cdot P(N|B)$.

Figure 4 compares the observed probability of attacking with the model
predicted one. As the figure shows, the error between them is small and the
average error rate is just 2.4%. Hence, the interference effect of
categorization can be well predicted.

Figure 4: Comparison between observed result and predicted result

In the following, the comparison among our QDB model, Markov BA model and
quantum BAE model will be made.

Townsend et al[1] proposed the Markov model to do category-decision task. The
model assumes that the categorization and decision -making are two parts in
the chain, namely the categorization depends only on the face while the action
depends only on the categorization.

Figure 5: The decision-making process of Markov BA model

In C-D condition, the probability of attacking equals to
$\phi(G)\cdot\phi(A|G)$ if categorizing the face as a good one and equals
to$\phi(B)\cdot\phi(A|B)$ if categorizing the face as a bad one. In D along
condition, the probability of attacking equals the probability of reaching a
final state $A$ by two different paths
$\phi(A)=\phi(G)\cdot\phi(A|G)+\phi(B)\cdot\phi(A|B)$. Hence, the Markov BA
model follows the law of total probability, which means no interference effect
will be produced.

Quantum BAE model is initially proposed by Pothos and Busemeyer(2009)[20] and
it has further developed in [2] based on quantum dynamic model. The crucial
factor to produce interference effect is that a new entanglement parameter
$\gamma$ is defined. As the state G and W, State B and A is assumed to be
entangled in some degree, the interference effect can be produced.

The comparison result among the three models is shown in Table 5.

Table 5: Comparison among different models Literature | Model | Pt | P(A)
---|---|---|---
Townsend $et.al$(2000) | Obs | 0.59 | 0.69
QDB | 0.5923 | 0.6756
Quantum BAE | 0.56 | 0.63
Markov BA | 0.576 | 0.576
Busemeyer $et.al$(2009) | Obs | 0.60 | 0.69
QDB | 0.6027 | 0.6860
Quantum BAE | 0.56 | 0.63
Markov BA | 0.621 | 0.621
Wang and Busemeyer(2016) Experiment 1 | Obs | 0.54 | 0.59
QDB | 0.5444 | 0.6278
Quantum BAE | 0.5634 | 0.6214
Markov BA | 0.532 | 0.532
Wang and Busemeyer(2016) Experiment 2 | Obs | 0.55 | 0.60
QDB | 0.5528 | 0.6361
Quantum BAE | 0.6065 | 0.6315
Markov BA | 0.5979 | 0.5979
Wang and Busemeyer(2016) Experiment 3 | Obs | 0.58 | 0.62
QDB | 0.5810 | 0.6644
Quantum BAE | 0.6123 | 0.6323
Markob BA | 0.5316 | 0.5316
Average | Obs | 0.57 | 0.64
QDB | 0.5721 | 0.6554
Quantum BAE | 0.580 | 0.629
Makov BA | 0.572 | 0.572

Both the quantum BAE model and our QDB model can predict the interference
effect while the Markov BA model could not. However, the results of QDB model
are more close to the observed experiment result. In addition, our QDB model
is more succinct than the quantum BAE model as it has less parameters. Based
on the above, it is reasonable to conclude that the QDB model is efficient to
predict the interference effect of categorization.

##  5 Conclusion

In this paper, a quantum dynamic belief model is proposed to predict the
interference effects of categorizing on decision making. Quantum probability
theory is an effective tool to approach the psychology. Based on the quantum
dynamic model, the conception of uncertainty is introduced to the QDB model to
simulate the real process of human thinking. The model is applied to explain
the categorization decision-making experiment which verifies our model’s
correctness and effectiveness. And the comparison with other models is also
made. Admittedly, the model proposed can only predict the positive
interference effect. For future study like when the interference effect will
be produced still remains to be studied.

##  6 Acknowledgement

The work is partially supported by National Natural Science Foundation of
China (Grant No. 61671384), Natural Science Basic Research Plan in Shaanxi
Province of China (Program No. 2016JM6018), Aviation Science Foundation
(Program No. 20165553036), the Fund of SAST (Program No. SAST2016083), the
Seed Foundation of Innovation and Creation for Graduate Students in
Northwestern Polytechnical University (Program No. Z2016122).

## References

  * Townsend et al. [2000] J. T. Townsend, K. M. Silva, J. Spencer-Smith, M. J. Wenger,  Exploring the relations between categorization and decision making with regard to realistic face stimuli,  Pragmatics & Cognition 8 (2000) 83–105.
  * Wang and Busemeyer [2016] Z. Wang, J. R. Busemeyer,  Interference effects of categorization on decision making,  Cognition 150 (2016) 133–149.
  * Nosofsky [2015] R. M. Nosofsky,  An exemplar-model account of feature inference from uncertain categorizations.,  Journal of Experimental Psychology Learning Memory & Cognition 41 (2015).
  * Yu et al. [2015] F. Yu, D. Qiu, H. Situ, X. Wang, S. Long,  Enhancing user privacy in SARG04-based private database query protocols,  Quantum Information Processing 14 (2015) 4201–4210.
  * Sharma et al. [2016] V. Sharma, K. Thapliyal, A. Pathak, S. Banerjee,  A comparative study of protocols for secure quantum communication under noisy environment: single-qubit-based protocols versus entangled-state-based protocols,  Quantum Information Processing 15 (2016) 1–30.
  * Song et al. [2015] T.-T. Song, S.-J. Qin, Q.-Y. Wen, Y.-K. Wang, H.-Y. Jia,  Finite-key security analyses on passive decoy-state QKD protocols with different unstable sources,  Scientific Reports 5 (2015) 15276.
  * Gisin and Thew [2015] N. Gisin, R. Thew,  Quantum communication,  Quantum Information 57 (2015) 441–462.
  * Khrennikov [2004] A. Khrennikov,  Information dynamics in cognitive, psychological, social and anomalous phenomena,  Fundamental Theories of Physics 138 (2004) xvi,235.
  * Aerts and de Bianchi [2015] D. Aerts, M. S. de Bianchi,  The unreasonable success of quantum probability i: Quantum measurements as uniform fluctuations,  Journal of Mathematical Psychology 67 (2015) 51–75.
  * Yukalov and Sornette [2015] V. I. Yukalov, D. Sornette,  Quantum decision theory as quantum theory of measurement,  Physics Letters A 372 (2015) 6867–6871.
  * Blutner et al. [2013] R. Blutner, E. M. Pothos, P. Bruza,  A quantum probability perspective on borderline vagueness,  Topics in Cognitive Science 5 (2013) 711–736.
  * Khrennikov and Haven [2009] A. Y. Khrennikov, E. Haven,  Quantum mechanics and violations of the sure-thing principle: the use of probability interference and other concepts,  Journal of Mathematical Psychology 53 (2009) 378–388.
  * Cheon and Takahashi [2010] T. Cheon, T. Takahashi,  Interference and inequality in quantum decision theory,  Physics Letters A 375 (2010) 100–104.
  * Brainerd et al. [2015] C. Brainerd, Z. Wang, V. F. Reyna, K. Nakamura,  Episodic memory does not add up: Verbatim–gist superposition predicts violations of the additive law of probability,  Journal of memory and language 84 (2015) 224–245.
  * Alnowaihi and Dhami [2016] A. Alnowaihi, S. Dhami,  The ellsberg paradox: A challenge to quantum decision theory?,  Discussion Papers in Economics (2016).
  * Busemeyer et al. [2011] J. R. Busemeyer, E. M. Pothos, R. Franco, J. S. Trueblood,  A quantum theoretical explanation for probability judgment errors.,  Psychological review 118 (2011) 193.
  * Trueblood and Busemeyer [2011] J. S. Trueblood, J. R. Busemeyer,  A quantum probability account of order effects in inference,  Cognitive science 35 (2011) 1518–1552.
  * Wang and Busemeyer [2013] Z. Wang, J. R. Busemeyer,  A quantum question order model supported by empirical tests of an a priori and precise prediction,  Topics in Cognitive Science 5 (2013) 689–710.
  * Wang et al. [2014] Z. Wang, T. Solloway, R. M. Shiffrin, J. R. Busemeyer,  Context effects produced by question orders reveal quantum nature of human judgments,  Proceedings of the National Academy of Sciences 111 (2014) 9431–9436.
  * Pothos and Busemeyer [2009] E. M. Pothos, J. R. Busemeyer,  A quantum probability explanation for violations of ’rational’ decision theory.,  Proceedings of the Royal Society B Biological Sciences 276 (2009) 2171–8.
  * Nyman and Basieva [2011] P. Nyman, I. Basieva,  Quantum-like representation algorithm for trichotomous observables,  International Journal of Theoretical Physics 50 (2011) 3864–3881.
  * Busemeyer and Wang [2007] J. R. Busemeyer, Z. Wang,  Quantum information processing explanation for interactions between inferences and decisions.,  in: AAAI Spring Symposium: Quantum Interaction, pp. 91–97.
  * Busemeyer et al. [2009] J. R. Busemeyer, Z. Wang, A. Lambert-Mogiliansky,  Empirical comparison of markov and quantum models of decision making,  Journal of Mathematical Psychology 53 (2009) 423–433.
  * Nyman [2011] P. Nyman,  On the consistency of the quantum-like representation algorithm for hyperbolic interference,  Advances in Applied Clifford Algebras 21 (2011) 799–811.
  * Yukalov and Sornette [2011] V. I. Yukalov, D. Sornette,  Decision theory with prospect interference and entanglement,  Theory and Decision 70 (2011) 283–328.
  * Moreira and Wichert [2016] C. Moreira, A. Wichert,  Quantum-like bayesian networks for modeling decision making.,  Frontiers in Psychology 7 (2016).
  * Busemeyer and Pothos [2012] J. R. Busemeyer, E. M. Pothos,  Social projection and a quantum approach for behavior in prisoner’s dilemma,  Psychological Inquiry 23 (2012) 28–34.
  * Chen et al. [2003] L. Chen, H. Ang, D. Kiang, L. Kwek, C. Lo,  Quantum prisoner dilemma under decoherence,  Physics Letters A 316 (2003) 317–323.
  * Makowski [2009] M. Makowski,  Transitivity vs. intransitivity in decision making process c an example in quantum game theory,  Physics Letters A 373 (2009) 2125–2130.
  * Alonso-Sanz et al. [2015] R. Alonso-Sanz, M. Carvalho, H. Situ,  A quantum relativistic prisoner s dilemma cellular automaton,  International Journal of Theoretical Physics 470 (2015) 1–14.
  * Alonso-Sanz and Situ [2016] R. Alonso-Sanz, H. Situ,  On the effect of quantum noise in a quantum-relativistic prisoner s dilemma cellular automaton,  International Journal of Theoretical Physics 55 (2016) 5265–5279.
  * Busemeyer et al. [2006] J. R. Busemeyer, Z. Wang, J. T. Townsend,  Quantum dynamics of human decision-making,  Journal of Mathematical Psychology 50 (2006) 220–241.
  * Deng [2015] Y. Deng,  Generalized evidence theory,  Applied Intelligence 43 (2015) 530–543.
  * Du et al. [2015] W. Du, Y. Gao, C. Liu, Z. Zheng, Z. Wang,  Adequate is better: particle swarm optimization with limited-information,  Applied Mathematics and Computation 268 (2015) 832 – 838.
  * Akyar [2016] H. Akyar,  Fuzzy Risk Analysis for a Production System Based on the Nagel Point of a Triangle,  MATHEMATICAL PROBLEMS IN ENGINEERING 3080679 (2016) DOI:10.1155/2016/3080679.
  * He et al. [2012] Y. He, L. Hu, X. Guan, D. Han, Y. Deng,  New conflict representation model in generalized power space,  Journal of Systems Engineering and Electronics 23 (2012) 1–9.
  * Deng [2016] Y. Deng,  Deng entropy,  Chaos, Solitons & Fractals 91 (2016) 549–553.
  * Vourdas [2014] A. Vourdas,  Quantum probabilities as dempster-shafer probabilities in the lattice of subspaces,  Journal of Mathematical Physics 55 (2014) 823.
  * Smets and Kennes [1994] P. Smets, R. Kennes,  The transferable belief model,  Artificial Intelligence 66 (1994) 191–234.