# A quantum dynamic belief decision making model

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

###### Abstract

The sure thing principle and the law of total probability are basic laws in
classic probability theory. A disjunction fallacy leads to the violation of
these two classical probability laws. In this paper, a new quantum dynamic
belief decision making model based on quantum dynamic modelling and Dempster-
Shafer (D-S) evidence theory is proposed to address this issue and model the
real human decision-making process. Some mathematical techniques are borrowed
from quantum mathematics. Generally, belief and action are two parts in a
decision making process. The uncertainty in belief part is represented by a
superposition of certain states. The uncertainty in actions is represented as
an extra uncertainty state. The interference effect is produced due to the
entanglement between beliefs and actions. Basic probability assignment (BPA)
of decisions is generated by quantum dynamic modelling. Then BPA of the extra
uncertain state and an entanglement degree defined by an entropy function
named Deng entropy are used to measure the interference effect. Compared the
existing model, the number of free parameters is less in our model. Finally, a
classical categorization decision-making experiment is illustrated to show the
effectiveness of our model.

###### keywords:

Quantum dynamic model; Dempster-Shafer evidence theory; the sure thing
principle; disjunction fallacy; interference effect; categorization decision-
making experiment; Deng entropy

##  1 Introduction

The sure thing principle introduced by Jim Savage[1] is a fundamental
principle in economics and probability theory. It means that if one prefers
action $A$ over $B$ under state of the world $X$, while action $A$ is also
preferred under the opposite state of world $\neg X$, then it can be concluded
that one will still prefer action $A$ over $B$ under the state of the world is
unspecified. The law of total probability is a fundamental rule relating
marginal probabilities to conditional probabilities. It expresses the total
probability of an outcome which can be realized via several distinct events.
However, many experiments and studies have shown that the sure thing principle
and the law of total probability can be violated due to the disjunction
effect. The disjunction fallacy is an empirical finding in which the
proportion taking the target gamble under the unknown condition falls below
both of the proportions taking the target gamble under each of the known
conditions. The same person takes the target gamble under both known
conditions, but then rejects the target gamble under the unknown condition[2].

Generally, the sure thing principle and the law of total probability are basic
probability laws. However, the disjunction fallacy leads to the violation of
them. To explain it, many studies have been proposed. The original explanation
was a psychological idea based on the failure of consequential reasoning under
unknown condition[3]. More recently, to explain these paradoxical findings,
the theory of quantum probability has been introduced in the quantum cognition
and decision making process. Quantum information has a wide application, like
in user security[4], quantum communication[5] and so on. Quantum probability
is an effective approach to psychology[6, 7, 8, 9]. It has been widely applied
to psychology and decision making fields by explaining order effect[10, 11,
12], disjunction effect[13], the interference effect of categorization[14],
prisoner’s dilemma[15], conceptual combinations[16, 17], quantum game
theory[18, 19, 20] and so on. To explain the disjunction fallacy which leads
to the violation of the sure thing principle and the law of total probability,
many models have been proposed, such as a quantum dynamical model[21, 22],
quantum prospect decision theory[23, 24, 25, 26] and quantum-like Bayesian
networks[27] etc. Besides quantum-like approach toolbox was also proposed[28,
29, 30].

In this paper, a new quantum dynamic belief decision making model based on
quantum dynamic modelling and D-S evidence theory is proposed to explain the
disjunction fallacy. Dempster-Shafer evidence theory was proposed by Dempster
in 1967[31] and modified by Shafer in 1978[32]. Evidential reasoning is an
approach handing the evidence is D-S theory[33, 34]. And Deng entropy[35] is
an efficient tool to measure the information volume of evidence[36, 37]. Many
applications in realistic projects have shown the power of D-S evidence theory
handling uncertain information[38, 39, 40, 41]. Besides, it has been applied
in quantum information and quantum probability in many works[42, 43, 44, 45].
In our model, D-S theory is used to extend the action state space and
uncertainty during the decision-making process is the crucial factor of
measuring the interference effect.

Consider the prisoner dilemma (PD) paradigm[3], two players need to decide
independently whether to cooperate with opponent or to defect against
opponent. Beliefs and actions are two parts in a decision-making process. The
real situation in action part is that some participants can not make a precise
choice to cooperate or to defect for sure, but they are forced to make a final
decision in the experiment. In our model, the uncertainty in action is
represented in a specific uncertain state and the uncertainty in belief is
represented by a superposition of certain states. As the beliefs and actions
are entangled in some degree, the interference effect can be produced. Unlike
the previous models, the interference effect is measured by the distribution
of the uncertain state in action. The quantum dynamic model is applied to
generate the basic probability assignment (BPA), which measures the supporting
of a decision. Then the uncertain information will be distributed by using
Deng entropy. A classical categorization decision-making experiment is
illustrated in this paper to show the effectiveness of our model. Though an
extra uncertain state is introduced, the new model is more succinct as the
free parameters decrease compared with classical quantum dynamic models.
Because the entanglement degree is calculated by an entropy function rather
being set as a free parameter.

The rest of the paper is organized as follows. In Section 2, the preliminaries
of basic theories employed are briefly introduced. The new quantum dynamic
belief decision making model is proposed in Section 3. Then a categorization
decision-making experiment is illustrated and our new model is applied to it
in Section 4. Finally, Section 5 comes to the conclusion.

##  2 Preliminaries

###  2.1 Quantum dynamic model

The quantum dynamic model model first proposed by Busemeyer $etal.$ in
2006[46] is formulated as a random walk decision process. The evolution of
complex valued probability amplitudes over time is described. The interference
effect can be generated in a quantum model which is not possible in a
classical Markov model. The quantum dynamic model assumes that a participant
has some potential to be in every state in the beginning. Thus the person’s
state is a superposition of all possible $n$ states

$\left|\psi\right\rangle={\psi_{1}}\left|{{S_{1}}}\right\rangle+{\psi_{2}}\left|{{S_{2}}}\right\rangle+\ldots{\psi_{n}}\left|{{S_{n}}}\right\rangle$

and the initial state corresponds to an amplitude distribution
$\psi\left({\rm{0}}\right)$ represented by the $n\times 1$ matrix.

$\psi\left({\rm{0}}\right){\rm{=}}\left[{\begin{array}[]{*{20}{c}}{{\psi_{\rm{1}}}}\\\
{{\psi_{2}}}\\\ \vdots\\\ {{\psi_{n}}}\end{array}}\right].$

During the decision making process, the 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)

where $H$ is a Hamiltonian matrix: ${H^{\dagger}}=H$, $H$ has elements
${h_{i}j}$ in row $i$ column $j$ representing the instantaneous rate of change
to $\left|i\right\rangle$ from $\left|j\right\rangle$. Eq. (1) has a matrix
exponential solution:

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

where matrix $U={e^{-iHt}}$ is a unitary matrix: ${U^{\dagger}}U=I$. It
finally guarantees that $\psi\left(t\right)$ always has unit length.

For $t={t_{2}}-{t_{1}}$, the transition probabilities ${T_{ij}}$ of observing
state $i$ at time ${t_{2}}$ given that state $j$ was observed at time
${t_{1}}$ is determined as

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

where ${U_{ij}}$ is the line $i$, column $j$ element of the unitary matrix
$U$.

Based on the above definition, the amplitude distribution of person’s state
evolves to $\psi\left(t\right)$ from the initial $\psi\left(0\right)$ across
time $t$ as Eq. (4), which shows the dynamic in a decision making process.

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

###  2.2 Dempster-Shafer evidence theory

Let $F$ denote a finite set composed of all possible values of the random
variable $X$. The elements of set $F$ are mutually exclusive. $F$ is called
the frame of discernment. Let ${2^{F}}$ denote the power set of $F$ whose each
element corresponds to a subset of values of $X$. Basic probability assignment
(BPA) is a mapping from ${2^{F}}$ to $\left[{0,1}\right]$, defined as[31, 32]:

$m:{2^{F}}\to\left[{0,1}\right]$ (5)

satisfying

$\sum\limits_{A\in{2^{F}}}{m\left(A\right)}=1$ (6)

and

$m\left(\emptyset\right)=0{\kern 1.0pt}{\kern 1.0pt}$ (7)

The mass function $m$ represents supporting degree to proposition $A$. A mass
function corresponds to a belief ($Bel$) function and a plausibility($Pl$)
function respectively.

Given $m:{2^{U}}\to\left[{0,1}\right]$, $Bel\left(A\right)$ function
represents the whole belief degree to the proposition $A$, defined as

$Bel\left(A\right)=\sum\limits_{B\subseteq A}{m\left(B\right)}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}\forall A\subseteq{2^{F}}$ (8)

$Pl$ function represents the belief degree of not denying proposition $A$,
defined as

$Pl\left(A\right)=1-Bel\left({\bar{A}}\right)=\sum\limits_{B\cap
A\neq\emptyset}{m\left(B\right)}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}\forall A\subseteq{2^{F}}$ (9)

where $\bar{A}={2^{F}}-A$.

As $Pl\left(A\right)\geq Bel\left(A\right)$, $Pl$ function and $Bel$ function
are upper and lower limit of the supporting of $A$.

In the following, an game of picking ball will be used to show the D-S
theory’s ability of handling uncertainty. There are two boxes filled with some
balls as shown in Fig. 1. Left box is contended with red balls and right box
is contended with blue balls. The number of balls in each box is unknown. Now,
a ball is picked randomly from two boxes. The probability of picking from left
box $P1$ is known as 0.4 while picking from right box $P2$ is known as 0.6. It
is easy to obtain that the probability of picking a red ball is 0.4 while
picking a blue ball is 0.6 based on probability theory.

Figure 1: A game of picking ball which can be handled by probability theory

Now, the situation changes as shown in Fig. 2. The left box is contended with
right balls while the right box is contended with red and blue balls. The
exact number of the balls in each box is still unknown and the ratio of red
balls with blue balls is completely unknown. The probabilities of selecting
from two boxes keep the same, $P1=0.4$ and $P2=0.6$. The question is what the
probability that a red ball is picked is. Due to the lack of information, the
question can not be addressed in probability theory. However, D-S evidence
theory can effectively handle it. We can obtain a BPA that
$m\left(R\right)=0.4$ and $m\left({R,B}\right)=0.6$. Then the uncertainty is
well expressed in the frame of D-S theory.

Figure 2: A game of picking ball where probability theory is unable but D-S
evidence theory is able to handle

###  2.3 Pignistic probability transformation

The term ”pignistic” proposed by Smets[47] is originated from the word pignus,
meaning bet in Latin. Pignistic probability transformation (PPT) has a wide
application in decision making. Let $m$ be a BPA on a frame of discernment
$F$, then the PPT function is defined as

${Bet\left({A}\right)=\sum\nolimits_{A\subseteq
B}{\frac{{m\left({B}\right)}}{{\left|{B}\right|}}}}$ (10)

where ${\left|{B}\right|}$ denotes the number of elements in set $B$. This is
called as the cardinality of $B$.

###  2.4 Deng entropy

In order to measure the information volume of a BPA, Deng entropy has been
proposed[35], which is defined as follows

${E_{d}}=-\sum\limits_{i}{m\left({{X_{i}}}\right){{\log}_{2}}\frac{{m\left({{X_{i}}}\right)}}{{{2^{\left|{{X_{i}}}\right|}}-1}}}$
(11)

where ${{X_{i}}}$ is a proposition in BPA $m$ and ${\left|{{X_{i}}}\right|}$
is the cardinality of ${{X_{i}}}$ representing the number of elements in it.
For example, for BPA
${m_{1}}:{m_{1}}\left(a\right)={m_{1}}\left(b\right)={m_{1}}\left({ab}\right)=\frac{1}{3}$,
where proposition $ab$ means that both proposition $a$ and $b$ are possible,
namely it is uncertain whether belongs to $a$ or $b$.

${E_{d}}\left({{m_{1}}}\right)=-\frac{1}{3}\times{\log_{2}}{\textstyle{{{1\mathord{\left/{\vphantom{13}}\right.\kern-1.2pt}3}}\over{{2^{1}}-1}}}-\frac{1}{3}\times{\log_{2}}{\textstyle{{{1\mathord{\left/{\vphantom{13}}\right.\kern-1.2pt}3}}\over{{2^{1}}-1}}}-\frac{1}{3}\times{\log_{2}}{\textstyle{{{1\mathord{\left/{\vphantom{13}}\right.\kern-1.2pt}3}}\over{{2^{2}}-1}}}=2.11$

For BPA ${m_{2}}:{m_{2}}\left(a\right)={m_{2}}\left(b\right)=\frac{1}{2}$

${E_{d}}\left({{m_{2}}}\right)=-\frac{1}{2}\times{\log_{2}}{\textstyle{{{1\mathord{\left/{\vphantom{12}}\right.\kern-1.2pt}2}}\over{{2^{1}}-1}}}-\frac{1}{2}\times{\log_{2}}{\textstyle{{{1\mathord{\left/{\vphantom{12}}\right.\kern-1.2pt}2}}\over{{2^{1}}-1}}}=1$

Specially, when the BPA is consisted of singleton sets as ${m_{2}}$, namely
$\left|{{X_{i}}}\right|$ of all the proposition in BPA is 1. Deng entropy
degenerates to Shannon entropy.

${E_{d}}=-\sum\limits_{i}{m\left({{X_{i}}}\right){{\log}_{2}}\frac{{m\left({{X_{i}}}\right)}}{{{2^{1}}-1}}}=-\sum\limits_{i}{m\left({{X_{i}}}\right){{\log}_{2}}}m\left({{X_{i}}}\right)$
(12)

###  2.5 Quantum entanglement in D-S evidence theory

In this part, the correlation between Deng entropy and entanglement is
discussed.

Let us consider an example as follows. Suppose 32 students participated a
course examination and one of them won the first place. In order to know who
is the first one, we go to ask their course teacher. But the teacher does not
want to directly tell us. Instead, she just answers “Yes” or “No” to our
questions. The problem is how many times do we need ask at most in order to
know who is the first one? Assume the times is $t$, it is easy to answer the
problem through calculating the information volume by using information
entropy

$t={\log_{2}}32=5$

Now, let’s consider another situation. Assume we have been told that there are
students tied for first. In this case, how many times do we need ask at most
to know who are the first ones? In this case, obviously

$t\geq{\log_{2}}32$

According to Deng entropy, the information volume is as follows

$\begin{array}[]{l}{E_{d}}=-\frac{1}{{{2^{32}}-1}}\times{\log_{2}}\left({\frac{{{\textstyle{1\over{{2^{32}}-1}}}}}{{{2^{1}}-1}}}\right)-\frac{1}{{{2^{32}}-1}}\times{\log_{2}}\left({\frac{{{\textstyle{1\over{{2^{32}}-1}}}}}{{{2^{2}}-1}}}\right)\\\
{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}-\cdots-\frac{1}{{{2^{32}}-1}}\times{\log_{2}}\left({\frac{{{\textstyle{1\over{{2^{32}}-1}}}}}{{{2^{32}}-1}}}\right)\approx
48\end{array}$

In Deng and Deng (2014)[48], the conclusion is that we need 32 times to
determine the top 1 student(s). However, it is not correct since that,
according to the result above, we need 48 times to obtain the result. Deng and
Deng proposed that entanglement is the key to cause the difference of these
two values. Because the entanglement brings a larger information volume and
the extra information is measured by Deng entropy.

##  3 Proposed method

The quantum dynamic belief decision making model based on quantum dynamic
modelling and D-S evidence theory can model the decision making process.
Beliefs and actions are two main parts in a decision making process. The
uncertainty of beliefs is represented by a superposition of certain states
which collapses when the beliefs become certain. When the beliefs are
uncertain, the interference effect is produced due to the entanglement between
beliefs and actions. The concept of interference effect in quantum cognition
is that the states can interference with each other, which leads to the
violation of the law of total probability. The uncertainty of actions is
expressed as an extra uncertain state with D-S theory. It is rational to
assume that some people hesitate to make a precise action decision during a
decision making process. However, a final action decision has to be made as an
uncertain action is not an outcome. In classical theory, uncertain information
in actions can be distributed into basic probabilities using PPT[47]. However,
in some cases the beliefs and actions are entangled in some degree, which
means that beliefs and actions need to be coordinated. It has been verified
that participants feel the need to coordinate beliefs and actions.[49, 50]. In
a entangled system, uncertainty in actions should be handled differently to
measure the interference effect. To address it, an entanglement degree is
defined in our model using Deng entropy. The flow chart of our model is shown
in Fig. 3.

Figure 3: The flow chart of proposed model

In the following, an example of PD paradigm is used to illustrate the proposed
model. In a PD game, participants need to decide whether to cooperate (C) or
to defect (D) against the opponent independently. In the study, participants
are requested to make a decision under the condition that the opponent’s
decision is known and that the opponent’s decision is totally unknown. The
result is that the probability of defecting under known condition is larger
than the one under unknown condition, which violates the law of total
probability. Based on the flow chart, the proposed model can be listed step by
step as follows:

Step 1: Determine the frame of discrimination

When the opponent’s decision is known to defect (D) or to cooperate (C), the
outcome of the game will be either to defect (D) or to cooperate (C). Thus the
basic states ”DD”, ”CD”, ”DC” and ”DD” consist the frame of discrimination,
where, for example, ”CD” represents the participant decide to cooperate
knowing the opponent defects. Then extra state ”CDD”, ”CDC” and $\emptyset$
will be included to fill the power set of this discrimination. State ”CDD”
represents that the participant is uncertain to cooperate or defect when
knowing the opponent defects and it can be denoted as ”UD”. Same as it, state
”CDC” can be denoted as ”UC”. And the empty set $\emptyset$ is irrespective in
the model as it is meaningless.

When the opponent’s decision is totally unknown, the outcome of the game will
be either to defect (D) or to cooperate (C). Thus the basic states ”DU” and
”CU” consist the frame of discrimination, where, for example, ”DU” represents
the participant decide to defect unknowing the opponent’s decision. Then extra
state ”CDU” and $\emptyset$ will be included to fill the power set of this
discrimination. State ”CDU” presents that the participant is uncertain to
cooperate or defect when unknowing the opponent’s decision and it can be
denoted as ”UU”. And the empty set $\emptyset$ is irrespective in the model as
it is meaningless.

Step 2: Representation of beliefs and actions

As the states have been extended in Step 1, the initial state involves six
combination of beliefs and actions

$\left\\{{\left|{{B_{D}}{A_{D}}}\right\rangle,\left|{{B_{D}}{A_{U}}}\right\rangle,\left|{{B_{D}}{A_{C}}}\right\rangle,\left|{{B_{C}}{A_{D}}}\right\rangle,\left|{{B_{C}}{A_{U}}}\right\rangle,\left|{{B_{C}}{A_{C}}}\right\rangle}\right\\}$

where, for example, ${\left|{{B_{C}}{A_{D}}}\right\rangle}$ symbolizes the
event in which the player believes the opponent will cooperate but the player
intends to act by defecting. The model assumes that at the beginning of a
game, the person has some potential to be in every circle in Fig. 4 which
illustrates the possible transitions among the six states. So the person’s
state is a superposition of the six basis states

$\begin{array}[]{l}\left|\psi\right\rangle={\psi_{DD}}\cdot\left|{{B_{D}}{A_{D}}}\right\rangle+{\psi_{UD}}\cdot\left|{{B_{D}}{A_{U}}}\right\rangle+{\psi_{CD}}\cdot\left|{{B_{D}}{A_{C}}}\right\rangle+{\psi_{DC}}\cdot\left|{{B_{C}}{A_{D}}}\right\rangle\\\
{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}+{\psi_{UC}}\cdot\left|{{B_{C}}{A_{U}}}\right\rangle+{\psi_{CC}}\cdot\left|{{B_{C}}{A_{C}}}\right\rangle\end{array}$
(13)

Figure 4: Transition diagram in the PD game

and the initial state corresponds to an amplitude distribution represented by
the $6\times 1$ column matrix

$\psi\left({\rm{0}}\right){\rm{=}}\left[{\begin{array}[]{*{20}{c}}{{\psi_{DD}}}\\\
{{\psi_{UD}}}\\\ {{\psi_{CD}}}\\\ {{\psi_{DC}}}\\\ {{\psi_{UC}}}\\\
{{\psi_{CC}}}\end{array}}\right].$

where, for example, ${{{\left|{{\psi_{DC}}}\right|}^{2}}}$ is the probability
of observing state $\left|{{B_{C}}{A_{D}}}\right\rangle$ initially. The
squared length of $\phi$ must equal one: ${\psi^{\dagger}}\cdot\psi=1$.

Step 3: Inferences based on prior information

During the decision process, new information at time ${t_{1}}$ changes the
initial state ar time $t=0$ into a new state at time ${t_{1}}$. For example,
if the opponent is known to defect, the amplitude distribution across states
changes to

$\psi\left({{t_{1}}}\right)=\frac{1}{{\sqrt{{{\left|{{\psi_{DD}}}\right|}^{2}}+{{\left|{{\psi_{UD}}}\right|}^{2}}+{{\left|{{\psi_{CD}}}\right|}^{2}}}}}\left[{\begin{array}[]{*{20}{c}}{{\psi_{DD}}}\\\
{{\psi_{UD}}}\\\ {{\psi_{CD}}}\\\ 0\\\ 0\\\
0\end{array}}\right]=\left[{\begin{array}[]{*{20}{c}}{{\psi_{D}}}\\\
\textbf{0}\end{array}}\right].$ (14)

where
${{{\left|{{\psi_{DD}}}\right|}^{2}}+{{\left|{{\psi_{UD}}}\right|}^{2}}+{{\left|{{\psi_{CD}}}\right|}^{2}}}$
equals the initial probability that the opponent defects (before given any
information).

However, if the opponent’s information is totally unknown, the amplitude
distribution will remain the same as the initial $t=0$.

$\begin{array}[]{l}\psi\left({{t_{1}}}\right)=\psi\left(0\right)=\left[{\begin{array}[]{*{20}{c}}{\sqrt{{{\left|{{\psi_{DD}}}\right|}^{2}}+{{\left|{{\psi_{UD}}}\right|}^{2}}+{{\left|{{\psi_{CD}}}\right|}^{2}}}\cdot{\psi_{D}}}\\\
{\sqrt{{{\left|{{\psi_{DC}}}\right|}^{2}}+{{\left|{{\psi_{UC}}}\right|}^{2}}+{{\left|{{\psi_{CC}}}\right|}^{2}}}\cdot{\psi_{C}}}\end{array}}\right]\\\
=\sqrt{{{\left|{{\psi_{DD}}}\right|}^{2}}+{{\left|{{\psi_{UD}}}\right|}^{2}}+{{\left|{{\psi_{CD}}}\right|}^{2}}}\cdot\left[{\begin{array}[]{*{20}{c}}{{\psi_{D}}}\\\
\textbf{0}\end{array}}\right]+\sqrt{{{\left|{{\psi_{DC}}}\right|}^{2}}+{{\left|{{\psi_{UC}}}\right|}^{2}}+{{\left|{{\psi_{CC}}}\right|}^{2}}}\cdot\left[{\begin{array}[]{*{20}{c}}\textbf{0}\\\
{{\psi_{C}}}\end{array}}\right]\end{array}$

The equation shows that the initial state is a superposition formed by a
weighted sum of the amplitude distribution for the two known conditions.

Step 4: Obtain BPA based on payoffs

During the decision making process, the participants need to evaluate the
payoffs in order to select an appropriate action, which evolve the state at
time ${t_{1}}$ into a new state at time ${t_{2}}$. The evolution of the state
during this time period corresponds the thought process leading to a action
decision, defection, uncertainty or cooperation. Based on the preliminaries,
an unitary matrix $U={e^{-iHt}}$ is defined to satisfy the solution of
Schr${\ddot{o}}$dinger equation.

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

$\psi\left({{t_{2}}}\right)$ is the amplitude distribution across states after
evolution based on payoffs.

$H=\left[{\begin{array}[]{*{20}{c}}{{H_{D}}}&0\\\
0&{{H_{C}}}\end{array}}\right]$ (15)

with

${H_{D}}=\left({\begin{array}[]{*{20}{c}}{{h_{D}}}&0&1\\\ 0&1&0\\\
1&0&{-{h_{D}}}\end{array}}\right)$ (16)

and

${H_{C}}=\left({\begin{array}[]{*{20}{c}}{{h_{C}}}&0&1\\\ 0&1&0\\\
1&0&{-{h_{C}}}\end{array}}\right)$ (17)

The $3\times 3$ Hamiltonian matrix ${H_{D}}$ applies when the participant
believes the opponent will defect, and the other $3\times 3$ Hamiltonian
matrix ${H_{C}}$ applies when the participant believes the opponent will
cooperate. The parameter ${h_{D}}$ is a function of the difference between the
reward for defecting relative to cooperating given the opponent will defect,
and ${h_{C}}$ is a function of the difference between the reward for defecting
relative to cooperating given the opponent will cooperate. The Hamiltonian
matrix $H$ transforms the state probabilities to favor defection, cooperation
or uncertainty, depending on the reward function.

Each decision corresponds to a measurement of the state at time ${t_{2}}$. To
obtain the BPA of certain decision, we can use an according measurement matrix
$M$ to pick out the state. As the state $\psi$ is a $6\times 1$ column matrix,
the measurement matrix is defined as a $6\times 6$ one.

$M=\left({\begin{array}[]{*{20}{c}}{{M_{D}}}&\textbf{0}\\\
\textbf{0}&{{M_{C}}}\end{array}}\right),$ (18)

where all the elements is in the form of a $3\times 3$ matrix. ${{M_{D}}}$ is
used for measuring when believing the opponent will defect, and ${{M_{C}}}$ is
used for measuring when believing the opponent will cooperate. For example, to
obtain the BPA of defecting, the measurement matrix is defined as

${M_{D}}={M_{C}}=\left[{\begin{array}[]{*{20}{c}}1&0&0\\\ 0&0&0\\\
0&0&0\end{array}}\right]$

Then we can obtain the BPA of defecting in unknown condition using Eq.(19).

$m\left(DU\right)=M\cdot\psi\left({{t_{2}}}\right)$ (19)

The BPA of defecting in the condition that the opponent is known to defect can
be obtained by Eq.(20).

$m\left(DD\right)=({{{\left|{{\psi_{DD}}}\right|}^{2}}+{{\left|{{\psi_{UD}}}\right|}^{2}}+{{\left|{{\psi_{CD}}}\right|}^{2}}})\cdot
M\cdot\psi\left({{t_{2}}}\right)$ (20)

Step 5: Calculate entanglement degree

Now, the BPAs of decisions have been obtained. The BPA in known condition is

${m_{1}}=\left\\{{{m_{1}}\left({DD}\right),{m_{1}}\left({UD}\right),{m_{1}}(CD),{m_{1}}\left({DC}\right),{m_{1}}\left({UC}\right),{m_{1}}(CC)}\right\\}.$
(21)

As the participant’s beliefs are certain, no interference effect will be
produced. The BPA of uncertain state is distributed based on classical $PPT$.
The cardinality of $DD$, $CD$, $DC$ and $CC$ is 1. The cardinality of $UD$ and
$UC$ is 2 as the actions are uncertain.

The BPA in unknown condition is

${m_{2}}=\left\\{{{m_{2}}\left(DU\right),{m_{2}}\left(UU\right),{m_{2}}(CU)}\right\\}.$
(22)

It should be noticed that

${m_{2}}\left(DU\right)={m_{1}}\left({DD}\right)+{m_{1}}\left({DC}\right)$
${m_{2}}\left(UU\right)={m_{1}}\left({UD}\right)+{m_{1}}\left({UC}\right)$
${m_{2}}\left(CU\right)={m_{1}}\left({CD}\right)+{m_{1}}\left({CC}\right)$

As the opponent’s decision is totally unknown and entanglement exists between
beliefs and action, the BPA of uncertain state is distributed based on an
entanglement degree. Then the interference effect will be produced. The
cardinality of $D$ and $C$ is 2 as the beliefs are uncertain. The cardinality
of $UU$ is 3 as both the beliefs and actions are uncertain.

Then an entanglement degree is defined in our model using Deng entropy.

###### Definition 3.1

Let ${m_{1}}$ and ${m_{2}}$ be two BPAs which illustrate the supporting of
decisions under different frames of discernment. ${m_{1}}$ is under the known
condition and ${m_{2}}$ is under the unknown condition. ${E_{d1}}$ and
${E_{d2}}$ are information volume of ${m_{1}}$ and ${m_{2}}$ respectively
measured by Deng entropy. The entanglement degree $\gamma$ is defined as
follows:

$\gamma=\frac{{{E_{d2}}-{E_{d1}}}}{{{E_{d2}}}}$ (23)

Step 6: Obtain probability distribution

As the uncertain state is not an outcome of the game, the BPA of uncertain
state should be distributed into cooperation state and defection state. For
BPA ${m_{1}}$,

${P_{1}}\left(D\right)={m_{1}}\left({DD}\right)+\frac{1}{2}{m_{1}}\left({UD}\right)+{m_{1}}\left({DC}\right)+\frac{1}{2}{m_{1}}\left({UC}\right).$
(24)

For BPA ${m_{2}}$,

${P_{2}}\left(D\right)={m_{2}}\left(DU\right)+\left({\frac{1}{2}\pm\gamma}\right){m_{2}}\left(UU\right).$
(25)

where $\gamma$ is an entanglement degree, $\pm$ corresponds to produce
positive interference effect and negative interference effect respectively.

Obviously, ${P_{1}}\left(D\right)$ is different from ${P_{2}}\left(D\right)$.
The difference value is the interference effect $Int$.

$Int={P_{2}}\left(D\right)-{P_{1}}\left(D\right)=\pm\gamma\cdot{m_{2}}\left(UU\right)$
(26)

In sum, based on the above steps, the decision making process is modelled and
the interference effect can be effectively measured.

##  4 Categorization decision-making experiment

###  4.1 Experiment

Townsend $etal.$[51] proposed a categorization decision-making experiment to
study the interactions between categorization and decision making. In the
experiment, pictures of faces varying along face width and lip thickness are
shown to participants. Generally, the faces can be distributed into two
different kinds: one is a narrow (N) face with a narrow width and thick lips;
the other one is a wide (W) face with a wide width and thin lips (see Figure 6
for example). The participants are informed that N face had a 0.60 probability
to come from the ”bad” guy population while W face had a 0.60 probability to
come from the ”good” guy population.

Figure 5: Example faces used in a categorization decision-making experiment

The experiment can be classified into two parts. One part is in
categorization-decision making (C-D) condition: participants are asked to
categorize a face as belonging to either a ”good” (G) guy or ”bad” (B) guy
following make a decision whether to ”attack” (A) or to to ”withdraw” (W). The
other part is decision-making along (D-along) condition: participants are
asked to only make an action decision. The experiment included a total of 26
participants, but each participant provided 51 observations for the C-D
condition for a total of ${\rm{26}}\times{\rm{51=1326}}$ observations, while
each person produced 17 observations for the D condition for a total of
${\rm{17}}\times{\rm{26=442}}$ total observations.

The experiment results are shown in Table 1.

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

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:

$P\left(A\right)=P(G)\cdot P\left({A|G}\right)+P\left(B\right)\cdot
P\left({A|B}\right)$ (27)

Accordingly, the column label as $P\left({A}\right)$ represents the
probability of attacking in D-along condition. As shown in Table 1, some
deviation exist between ${P_{T}}$ and $P\left({A}\right)$ for both types of
face which is called the disjunction effect. However, the disjunction effect
is prominent for narrow type faces, while the disjunction effect is weak for
wide type faces.

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

Table 2: Results of other categorization decision-making experiments

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 $etal.$(2000)[51] | 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 $etal.$(2009)[22] | 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[14] | 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[14] | 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[14] | 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 $etal.$(2009)[22], the classical experiment is replicated.

  * 2

In Wang and Busemeyer(2016)[14], 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 only the result of C-D trial is used here. In experiment
3, the reward for attacking bad people is a bit less than the other two.

###  4.2 Application

In the following, our model will be applied to explain the disjunction effect
for narrow type faces.

Step 1: Determine the frame of discrimination

In C-D condition, the outcome will be either to A or W given categorizing the
face as G or B. Thus the basic states ”AG”,”WG”,”AB” and ”WB” consist the
frame of discernment, where, for example, ”AG” represents the participant
decide to attack when categorizing the face as a good guy. Then extra state
”AWG”, ”AWB” and $\emptyset$ will be included to fill the power set of this
discrimination. State ”AWG” represents that the participant is uncertain to
attack or withdraw given categorizing the face as a good guy and it can be
denoted as ”UG”. Same as it, state ”AWB” can be denoted as ”UB”. And the empty
set $\emptyset$ is irrespective in the model as it is meaningless.

In D-along condition, the outcome of the game will be either to A or W without
categorizing the face. Thus the basic states ”AU” and ”WU” consist the frame
of discrimination. Then extra state ”AWU” and $\emptyset$ will be included to
fill the power set of this discrimination. State ”AWU” presents that the
participant is uncertain to attack or withdraw without categorization and it
can be denoted as ”UU”. And the empty set $\emptyset$ is irrespective in the
model as it is meaningless.

Step 2: Representation of beliefs and actions

The initial state involves six combination of beliefs and actions

$\left\\{{\left|{{B_{G}}{A_{A}}}\right\rangle,\left|{{B_{G}}{A_{U}}}\right\rangle,\left|{{B_{G}}{A_{W}}}\right\rangle,\left|{{B_{B}}{A_{A}}}\right\rangle,\left|{{B_{B}}{A_{U}}}\right\rangle,\left|{{B_{C}}{A_{W}}}\right\rangle}\right\\}$

. Participants’ state is a superposition of the six basis states

$\begin{array}[]{l}\left|\psi\right\rangle={\psi_{DD}}\cdot\left|{{B_{D}}{A_{D}}}\right\rangle+{\psi_{UD}}\cdot\left|{{B_{D}}{A_{U}}}\right\rangle+{\psi_{CD}}\cdot\left|{{B_{D}}{A_{C}}}\right\rangle+{\psi_{DC}}\cdot\left|{{B_{C}}{A_{D}}}\right\rangle\\\
{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}+{\psi_{UC}}\cdot\left|{{B_{C}}{A_{U}}}\right\rangle+{\psi_{CC}}\cdot\left|{{B_{C}}{A_{D}}}\right\rangle\end{array}$
(28)

and the initial state corresponds to an amplitude distribution represented by
the $6\times{\rm{1}}$ column matrix

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

Step 3: Inferences based on prior information

In C-D condition, if the participant categorizes the face as ”good”, the state
changes to

$\psi\left({{t_{1}}}\right)=\frac{1}{{\sqrt{{{\left|{{\psi_{AG}}}\right|}^{2}}+{{\left|{{\psi_{UG}}}\right|}^{2}}+{{\left|{{\psi_{WG}}}\right|}^{2}}}}}\left[{\begin{array}[]{*{20}{c}}{{\psi_{AG}}}\\\
{{\psi_{UG}}}\\\ {{\psi_{WG}}}\\\ 0\\\ 0\\\
0\end{array}}\right]=\left[{\begin{array}[]{*{20}{c}}{{\psi_{G}}}\\\
\textbf{0}\end{array}}\right].$ (29)

If the participant categorizes the face as ”bad”, the state changes to

$\psi\left({{t_{1}}}\right)=\frac{1}{{\sqrt{{{\left|{{\psi_{AB}}}\right|}^{2}}+{{\left|{{\psi_{UB}}}\right|}^{2}}+{{\left|{{\psi_{WB}}}\right|}^{2}}}}}\left[{\begin{array}[]{*{20}{c}}0\\\
0\\\ 0\\\ {{\psi_{AB}}}\\\ {{\psi_{UB}}}\\\
{{\psi_{WB}}}\end{array}}\right]=\left[{\begin{array}[]{*{20}{c}}\textbf{0}\\\
{{\psi_{B}}}\end{array}}\right].$ (30)

In D-along condition, the state is a superposition formed by a weighted sum of
the amplitude distribution for two known conditions in C-D condition.

$\begin{array}[]{l}\psi\left({{t_{1}}}\right)=\sqrt{{{\left|{{\psi_{AG}}}\right|}^{2}}+{{\left|{{\psi_{UG}}}\right|}^{2}}+{{\left|{{\psi_{WG}}}\right|}^{2}}}\left[{\begin{array}[]{*{20}{c}}{{\psi_{G}}}\\\
\textbf{0}\end{array}}\right]\\\ {\kern
29.0pt}+\sqrt{{{\left|{{\psi_{AB}}}\right|}^{2}}+{{\left|{{\psi_{UB}}}\right|}^{2}}+{{\left|{{\psi_{WB}}}\right|}^{2}}}\left[{\begin{array}[]{*{20}{c}}\textbf{0}\\\
{{\psi_{B}}}\end{array}}\right].\end{array}$ (31)

Step 4: Obtain BPA based on payoffs

The evolution of the state obeys a Schr${\ddot{o}}$dinger equation (Eq. (1)).
The solution is

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

In this problem, $t$ is set to $\frac{\pi}{2}$ as Busemeyer $etal.$[22]. The
Hamiltonian matrix $H$ is

$H=\left[{\begin{array}[]{*{20}{c}}{{H_{G}}}&0\\\
0&{{H_{B}}}\end{array}}\right]$ (32)

with

${H_{G}}=\left({\begin{array}[]{*{20}{c}}{{h_{G}}}&0&1\\\ 0&1&0\\\
1&0&{-{h_{G}}}\end{array}}\right){\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}and{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{H_{B}}=\left({\begin{array}[]{*{20}{c}}{{h_{B}}}&0&1\\\ 0&1&0\\\
1&0&{-{h_{B}}}\end{array}}\right)$

In C-D condition, if the face is categorized as ”good”, the state changes to

$\psi\left({{t_{2}}}\right)={e^{-i\cdot H\cdot
t}}\cdot\psi({t_{\rm{1}}}){\rm{=}}\left[{\begin{array}[]{*{20}{c}}{{e^{-i\cdot{H_{G}}\cdot
t}}}&0\\\ 0&{{e^{-i\cdot{H_{B}}\cdot
t}}}\end{array}}\right]\cdot\left[{\begin{array}[]{*{20}{c}}{{\psi_{G}}}\\\
\textbf{0}\end{array}}\right]={e^{-i\cdot{H_{G}}\cdot t}}\cdot{\psi_{G}}$ (33)

If the face is categorized as ”bad”, the state changes to

$\psi\left({{t_{2}}}\right)={e^{-i\cdot H\cdot
t}}\cdot\psi({t_{\rm{1}}}){\rm{=}}\left[{\begin{array}[]{*{20}{c}}{{e^{-i\cdot{H_{G}}\cdot
t}}}&0\\\ 0&{{e^{-i\cdot{H_{B}}\cdot
t}}}\end{array}}\right]\cdot\left[{\begin{array}[]{*{20}{c}}\textbf{0}\\\
{{\psi_{B}}}\end{array}}\right]={e^{-i\cdot{H_{B}}\cdot t}}\cdot{\psi_{B}}$
(34)

In D-along condition, the state changes to

$\footnotesize\begin{array}[]{*{20}{l}}{\psi\left({{t_{2}}}\right)={e^{-i\cdot
H\cdot
t}}\cdot\psi(0){\rm{=}}\left[{\begin{array}[]{*{20}{c}}{{e^{-i\cdot{H_{G}}\cdot
t}}}&0\\\ 0&{{e^{-i\cdot{H_{B}}\cdot
t}}}\end{array}}\right]\cdot\left[{\begin{array}[]{*{20}{c}}{\sqrt{{{\left|{{\psi_{AG}}}\right|}^{2}}+{{\left|{{\psi_{UG}}}\right|}^{2}}+{{\left|{{\psi_{WG}}}\right|}^{2}}}\cdot{\psi_{G}}}\\\
{\sqrt{{{\left|{{\psi_{AB}}}\right|}^{2}}+{{\left|{{\psi_{UB}}}\right|}^{2}}+{{\left|{{\psi_{WB}}}\right|}^{2}}}\cdot{\psi_{B}}}\end{array}}\right]}\\\
\begin{array}[]{l}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern
1.0pt}=\sqrt{{{\left|{{\psi_{AG}}}\right|}^{2}}+{{\left|{{\psi_{UG}}}\right|}^{2}}+{{\left|{{\psi_{WG}}}\right|}^{2}}\cdot}{e^{-i\cdot{H_{G}}\cdot
t}}\cdot{\psi_{G}}{\rm{+}}\\\ {\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\kern
1.0pt}\sqrt{{{\left|{{\psi_{AB}}}\right|}^{2}}+{{\left|{{\psi_{UB}}}\right|}^{2}}+{{\left|{{\psi_{WB}}}\right|}^{2}}}\cdot{e^{-i\cdot{H_{B}}\cdot
t}}\cdot{\psi_{B}}\end{array}\end{array}$ (35)

The measure matrix is

$M=\left({\begin{array}[]{*{20}{c}}{{M_{1}}}&0\\\
0&{{M_{2}}}\end{array}}\right)$

where ${M_{1}}$ and ${M_{2}}$ is set as

${M_{1}}={M_{2}}=\left[{\begin{array}[]{*{20}{c}}1&0&0\\\ 0&0&0\\\
0&0&0\end{array}}\right],\left[{\begin{array}[]{*{20}{c}}0&0&0\\\ 0&1&0\\\
0&0&0\end{array}}\right]or\left[{\begin{array}[]{*{20}{c}}0&0&0\\\ 0&0&0\\\
0&0&1\end{array}}\right]$

respectively to pick out the state of attacking, uncertainty or defecting.

In C-D condition, the BPA of states ”AG”, ”UG” and ”WG” can be obtained by Eq.
(36).

$m=P\left(G\right)\cdot{\left\|{M\cdot{e^{-itH}}\cdot\psi\left({{t_{1}}}\right)}\right\|^{2}}$
(36)

where $P\left(G\right)$ is the probability that a face is categorized as ”G”,
equalling to
${\sqrt{{{\left|{{\psi_{AG}}}\right|}^{2}}+{{\left|{{\psi_{UG}}}\right|}^{2}}+{{\left|{{\psi_{WG}}}\right|}^{2}}}}$.

The BPA of states ”AB”, ”UB” and ”WB” can be obtained by Eq. (37).

$m=P\left(B\right)\cdot{\left\|{M\cdot{e^{-itH}}\cdot\psi\left({{t_{1}}}\right)}\right\|^{2}}$
(37)

where $P\left(B\right)$ is the probability that a face is categorized as ”B”,
equalling to
${\sqrt{{{\left|{{\psi_{AB}}}\right|}^{2}}+{{\left|{{\psi_{UB}}}\right|}^{2}}+{{\left|{{\psi_{WB}}}\right|}^{2}}}}$.

Thus the BPA in C-D condition is

$\begin{array}[]{l}{m_{1}}=\left({m\left({AG}\right),m\left({UG}\right),m\left({WG}\right),m\left({AB}\right),m\left({UB}\right),m\left({WG}\right)}\right)\\\
=\left({0.0414,0.0567,0.0720,0.3846,0.2767,0.1688}\right).\end{array}$

In D-along condition, the BPA of states ”AU”, ”UU” and ”WU” can be obtained by
Eq. (38).

$m={\left\|{M\cdot{e^{-itH}}\cdot\psi\left({{t_{1}}}\right)}\right\|^{2}}$
(38)

Thus the BPA in D-along condition is

${m_{2}}=\left({m\left(AU\right),m\left(UU\right),m\left(WU\right)}\right)=\left({0.4259,0.3333,0.2407}\right).$

There are four free parameters existing in our model shown as Table 3.
${h_{G}}$ and ${h_{B}}$ are reward functions set by users. $P(G)$ and $P(B)$
are obtained from the experiment result.

Table 3: The value of free parameters Parameters | ${h_{G}}$ | ${h_{B}}$ | $P(G)$ | $P(B)$
---|---|---|---|---
Value | -0.1376 | 0.2033 | 0.83 | 0.17

Step 5: Calculate entanglement degree

The information volume of ${m_{1}}$ and ${m_{2}}$ is measured as following:

$\begin{array}[]{l}{E_{d1}}=-0.0414{\log_{2}}\left({\frac{{0.0414}}{{{2^{1}}-1}}}\right)-0.0567{\log_{2}}\left({\frac{{0.0567}}{{{2^{2}}-1}}}\right)-0.072{\log_{2}}\left({\frac{{0.072}}{{{2^{1}}-1}}}\right)\\\
-0.3846{\log_{2}}\left({\frac{{0.3846}}{{{2^{1}}-1}}}\right)-0.2767{\log_{2}}\left({\frac{{0.2767}}{{{2^{2}}-1}}}\right)-0.1688{\log_{2}}\left({\frac{{0.1688}}{{{2^{1}}-1}}}\right)\\\
{\kern 18.0pt}=2.7026\end{array}$ (39)
$\begin{array}[]{l}{E_{d2}}=-0.4259{\log_{2}}\left({\frac{{0.4259}}{{{2^{2}}-1}}}\right)-0.3333{\log_{2}}\left({\frac{{0.3333}}{{{2^{3}}-1}}}\right)-0.2407{\log_{2}}\left({\frac{{0.2407}}{{{2^{2}}-1}}}\right)\\\
{\kern 18.0pt}=3.5398\end{array}$ (40)

Then then entanglement degree $\gamma$ is calculated as

$\gamma=\frac{{{E_{d2}}-{E_{d1}}}}{{{E_{d2}}}}=0.2365.$ (41)

Step 6: Obtain probability distribution

In C-D condition, the probability of attack ${P_{1}}$ is

${P_{1}}\left(A\right)={m_{1}}\left({AG}\right)+\frac{1}{2}{m_{1}}\left({UG}\right)+{m_{1}}\left({AB}\right)+\frac{1}{2}{m_{1}}\left({UB}\right)=0.5927$

which is exactly the same as the experiment result.

In D-along condition, as a positive interference effect is produced, the
probability of attack ${P_{2}}$ is

${P_{2}}\left(A\right)={m_{2}}(AU)+\left({\frac{1}{2}+\gamma}\right){m_{2}}\left({UU}\right)=0.6715$

which is close to the experiment result.

The interference effect produced by categorization is measured as

$Int={P_{2}}\left(A\right)-{P_{1}}\left(A\right)=\gamma\cdot{m_{2}}\left(UU\right)=0.0788$

###  4.3 Comparison

Compare the obtained model results with the observed experiment results (for N
type face), the model results are close to the practical situation, which
verifies the correctness and effectiveness of our model. As Table 4 shows, the
interference effect is predicted and the average error rate is less than 1%.
The decision making process is modelled and the disjunction effect is well
explained in 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 $etal.$(2000) | Obs | 0.17 | 0.41 | 0.83 | 0.63 | 0.59 | 0.69
QDB | 0.17 | 0.4100 | 0.83 | 0.6301 | 0.5926 | 0.6715
Busemeyer $etal.$(2009) | Obs | 0.20 | 0.45 | 0.80 | 0.64 | 0.60 | 0.69
QDB | 0.20 | 0.4470 | 0.80 | 0.6340 | 0.5965 | 0.6689
Wang and Busemeyer(2016) Experiment 1 | Obs | 0.21 | 0.41 | 0.79 | 0.58 | 0.54 | 0.59
QDB | 0.21 | 0.4148 | 0.79 | 0.5916 | 0.5544 | 0.6241
Wang and Busemeyer(2016) Experiment 2 | Obs | 0.24 | 0.37 | 0.76 | 0.61 | 0.55 | 0.60
QDB | 0.24 | 0.3720 | 0.76 | 0.6162 | 0.5575 | 0.6247
Wang and Busemeyer(2016) Experiment 3 | Obs | 0.24 | 0.33 | 0.76 | 0.66 | 0.58 | 0.62
QDB | 0.24 | 0.3381 | 0.76 | 0.6454 | 0.5716 | 0.6417
Average | Obs | 0.21 | 0.39 | 0.79 | 0.62 | 0.57 | 0.64
QDB | 0.21 | 0.3964 | 0.79 | 0.6235 | 0.5758 | 0.6462

  * 1

Obs represents the observed experiment result.

  * 2

QDB represents the results of quantum dynamic belief model.

In the following, the comparison among our quantum dynamic belief model,
Markov belief-action (BA) model and quantum belief-action entanglement (BAE)
model will be made.

Markov BA model was proposed by Townsend $etal.$(2000)[51] to do the 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 6: The decision-making process of Markov BA model

In C-D condition, the probability of attacking equals to
$\phi(G)\cdot\phi(A|G)$ given the face is categorized as G and equals to
$\phi(B)\cdot\phi(A|B)$ given the face is categorized as B. In D-along
condition, the probability of attacking equals to the probability of reaching
the 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
that disjunction effect cannot be explained.

Quantum BAE model was initially proposed by Pothos and Busemeyer (2009)[21]
and it was improved by Wang and Busemeyer (2016)[14]. The model bases on
quantum dynamic modelling and the crucial factor to produce interference
effect is that a free entanglement parameter $c$ is defined. The unitary
matrix in Eq. (2) is ${e^{-i({H_{1}}+{H_{2}})t}}$, where ${H_{1}}$ is the same
as $H$ in our model. ${H_{2}}$ rotates inferences for G face to match W action
and rotates inferences for B face to match A action, which produces the
interference effect.

${H_{2}}=\frac{c}{{\sqrt{2}}}\left[{\begin{array}[]{*{20}{c}}{-1}&0&1&0\\\
0&1&0&1\\\ 1&0&1&0\\\ 0&1&0&{-1}\end{array}}\right]$

In order to compare the ability of predict the interference effect, we apply
three models to obtain the probability of attacking in D-along condition. The
comparison among the three models is shown in Fig. 7.

Figure 7: The comparison of probability of attacking in D-along condition

Both the quantum BAE model and our quantum dynamic belief model can predict
the interference effect while the Markov BA model could not. However, the
prediction result of our quantum dynamic belief model is more accurate. It
should be noticed that although an extra uncertain state is introduced in our
model, the number of free parameters decreases on the contrary. The
entanglement degree is measured by an entropy function in our model rather a
free parameter in the quantum BAE model. Based on the above, it is reasonable
to conclude that the QDB model is correct and efficient.

##  5 Conclusion

To explain the disjunction fallacy, a new quantum dynamic belief decision
making model is proposed in this paper. The model combines Dempster-Shafer
evidence theory with the quantum dynamic model. The uncertainty of belief is
represented by a superposition of certain states. And the uncertainty in
actions is represented as an extra uncertain state with D-S theory. The
entanglement between beliefs and actions can produce the interference effect
when the beliefs are uncertain. The interference effect is measured by the
uncertain state in actions and an entanglement degree defined by Deng entropy.
A classical categorization decision-making experiment is illustrated and the
new model modelling a real human decision-making process can well explain the
disjunction effect. In the end, the comparison with other models is made,
which shows the correctness and effectiveness of our model.

##  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)

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# EOD: The IEEE GRSS Earth Observation Database

###### Abstract

In the era of deep learning, annotated datasets have become a crucial asset to
the remote sensing community. In the last decade, a plethora of different
datasets was published, each designed for a specific data type and with a
specific task or application in mind. In the jungle of remote sensing
datasets, it can be hard to keep track of what is available already. With this
paper, we introduce EOD - the IEEE GRSS Earth Observation Database (EOD) - an
interactive online platform for cataloguing different types of datasets
leveraging remote sensing imagery.

Index Terms—  Deep Learning, Machine Learning, Datasets, Remote Sensing

##  1 Introduction

Annotated datasets for the purpose of training and evaluating machine learning
models have become a vital requirement for many branches of cutting-edge
research in remote sensing and Earth observation. In contrast to general
computer vision, whose target is the extraction of information from every-day
images containing every-day objects, such as furniture, animals, or road
signs, there is a much greater variety of image modalities and tasks in remote
sensing. In this field, people may want to work with classical optical
imagery, make use of multi- or hyperspectral sensing capabilities, or resort
to even more special sensor technologies such as laser scanning or synthetic
aperture radar. In addition to this there is a difference in scale with
spatial resolutions ranging from centimeters to kilometers and temporal
resolutions ranging from minutes (e.g. with drone-based imagery) to several
weeks. Moreover, there is a multitude of applications including object
detection and recognition (e.g. ship and oil spill detection), semantic
segmentation (e.g. land cover mapping), instance segmentation (e.g. building
footprint extraction), but also regression tasks such as the estimation of
bio-/geo-physical parameters (e.g. building height, soil moisture, biomass,
etc.). Given this diversity it becomes clear that we will not see a single
extremely large all-purpose image database, such as ImageNet111As a prime
example for an annotated computer vision dataset, ImageNet contains more than
14 million images depicting objects from more than 20,000 categories. in the
near future. Thus, as e.g. discussed by [1], we are overwhelmed by newly
published remote sensing datasets, all of which are annotated with a specific
combination of sensor modality, application, and geographic location. In order
to structure the available datasets and to provide an overview, the Image
Analysis and Data Fusion (IADF) Technical Committee of IEEE GRSS has created
the Earth Observation Database (EOD), which is described in this paper.

Fig. 1: The landing page of the EOD website.

##  2 Related Work

To the best of the authors’ knowledge there is not yet one central database
with the goal to provide a comprehensive catalogue of existing remote sensing
datasets. Data itself can be accessed via web portals of the corresponding
institutions that collected the data, e.g. the Copernicus data hub [2], or
open data programs of companies (e.g. [3, 4]). While this might include data
such as semantic maps that can serve as reference for training and evaluation
(e.g. the CORINE land cover [5]), these databases are limited to very specific
sensors and not intended to serve as a general purpose catalogue for Earth
observation datasets. Benchmark datasets are often only listed on the web
pages of individual research labs or institutions, which summarize all
datasets they have published, e.g. [6], [7] or [8]. Competition sites such as
Kaggle222https://www.kaggle.com/datasets or
CodaLab333https://codalab.lisn.upsaclay.fr/ as well as data storage sites such
as IEEE DataPort444https://ieee-dataport.org/ contain many datasets of
relevance but have a very different focus. They contain data from all kinds of
scientific disciplines, the provided information is not standardized, and the
search capabilities are not tailored towards factors of importance in Earth
observation and remote sensing. Evaluation servers such as
DASE555http://dase.grss-ieee.org/ provide access to public leaderboards and
allow a fair and unbiased evaluation based on standardized protocols. However,
the number of datasets is usually limited. Cloud services such as Amazon Web
Services666https://registry.opendata.aws/, Microsoft’s Planetary
Computer777https://planetarycomputer.microsoft.com/catalog, Radiant Earth’s
MLHub888https://www.mlhub.earth/, etc. provide access to selected datasets.
While they do aim to have a comprehensive list of datasets available, they are
limited to those that have been added into their databases. What all of these
commendable efforts have in common is that they focus on specific aspects such
as data hosting, evaluation, easy access for processing in the cloud, etc.,
that are different from simply cataloguing existing datasets. This leads to
databases that either contain a huge set of very diverse datasets (potentially
including but not limited to Earth observation data) in a rather unstructured
way, or consist of only a few carefully selected datasets.

What is missing so far is a central catalogue that provides an exhaustive list
of available datasets with their basic information, can be accessed and
extended by the community, and queried in a structured and interactive manner.
One of the most extensive and best-curated lists is the _Awesome Satellite
Imagery Datasets_ list [9]. While it is an open github project welcoming
contributions by anybody interested, it’s not a database with query
functionality. EOD aims to close this gap by providing an interactive
catalogue, curated by GRSS IADF, extendable and accessible by everybody, as an
easy access point to search for Earth observation datasets.

##  3 The Earth Observation Database

The EOD web page is reachable via https://eod-grss-ieee.com/. Its landing page
is depicted in Fig. 1. It displays both a query form for immediate search on
available datasets and advertises those datasets that were either most
recently added to the database or most popular. From this starting point,
three basic options are possible:

  * •

Starting an immediate search for a datasets following specified requirements
using the landing page query form.

  * •

Navigating to the full data catalogue by clicking on the _Datasets_ link in
the top frame of the landing page.

  * •

Submitting a new dataset by clicking on the _Submit_ link in the top frame of
the landing page.

Those three options are described in detail in the following.

###  3.1 Dataset Query

The simple query interface including the current selection options is shown in
Fig. 2. Generally, there are three features that can be used to identify
relevant datasets:

  * •

The underlying sensor technology.

  * •

The desired task/application.

  * •

The location of the dataset.

As an example, the query shown in Fig. 2 would search the database for
datasets containing both SAR and optical data annotated for semantic
segmentation purposes. Note that the selection of multiple options is
possible. A checkbox controls whether they are combined by a logical _AND_ or
_OR_. The location field is still left open, though. Note that either data
from a specific location or datasets covering multiple locations can be
requested.

Fig. 2: The EOD query interface. Fig. 3: The map view of the EOD data
catalogue.

###  3.2 Data Catalogue

The core of the EOD is the database of user-submitted datasets. It can be
viewed either in an illustrated list view, or in an interactive map view
indicating those datasets that cover only a specific location with a marker
(see Fig. 3).

Clicking on one of the markers in the map view will limit the list to datasets
at this specific location. Clicking on a dataset in the list will open a new
window displaying detailed information about this dataset which includes

  * •

Geographic location

  * •

Sensor modality

  * •

Task / Application

  * •

Size

  * •

URL to access the data

  * •

Number of views

  * •

A graphical representation

  * •

A brief description

A helpful function is the _compare_ option, which allows a direct side-to-side
comparison of this information from two or more datasets in a newly opened
window. As of the time of writing of this paper, there were 14 datasets in the
catalogue. Designed as a community platform, EOD requires people creating or
using remote sensing datasets to contribute actively by submitting new
datasets to the database.

###  3.3 Dataset Submission

As mentioned in the previous section, EOD is designed to be a collaborative
community platform that can be used by every interested group or person to
catalogue or advertise remote sensing datasets of any kind. New entries to the
database can be realized via the submission form, which is depicted in Fig. 4.

Fig. 4: The EOD submission form.

As can be seen, several pieces of information are required for every dataset:

  * •

_Submitter name and email address:_
This is just for reference for the database back end and administrators and
will not be published. The email address will only be used, if, for example,
important information about the dataset were missing or entered in an
erroneous way.

  * •

_Dataset name:_
Submitters should make sure to enter the correct name of the dataset, i.e. as
the dataset authors would want the dataset to be known in the community.

  * •

_MM/DD/YYY:_
This field shall indicate the date of publication of the dataset, _not_ the
date of submission. This is important for use cases where possible changes
might negatively impact the use of annotated data.

  * •

_Dataset location:_
Here, the location (i.e. address) can be specified. Alternatively, if a
dataset covers multiple locations, this can be specified as well.

  * •

_Dataset sensors and tasks:_
In these fields, the sensor types that were used to produce the observations
in the dataset as well as the tasks the data can be used for can be selected.

  * •

_Dataset size:_
The size of the dataset can be provided either in terms of MB or GB. It should
cover the size of the data to be downloaded.

  * •

_Link to dataset:_
This is one of the most crucial pieces of information: Where can the user
download the dataset for further usage.

  * •

_Dataset teaser image:_
The submitter should always provide a small teaser image that grasps the
nature of the dataset in a visual manner.

  * •

_Short description:_
Here, the dataset should be described in just a few concise sentences.

While it is very important that these information are entered with high care,
datasets will not immediately appear in the public catalogue – before
approval, they will be checked by IADF members for correctness and
consistency.

##  4 Call for Action

As mentioned before, EOD is intended as a community platform and will thus
thrive on contributions by the Earth observation and/or machine learning
communities. Thus, this paper shall be concluded by a call to action: If you
have created a dataset, or if you love to work with a dataset created by
somebody else, please submit the dataset to EOD if it’s not represented in the
database yet! It should be our common goal to make EOD the most comprehensive
database for EO-related datasets available in the world wide web.

##  5 ACKNOWLEDGMENTS

The authors would like to thank the IADF co-chair Gemine Vivone and the IADF
Working Group co-leads Seyed Ali Ahmadi, Francescopaolo Sica, and Xian Sun for
helping to enter dataset information for the initial set before launching the
webpage.

## References

  * [1] Michael Schmitt, Seyed Ali Ahmadi, and Ronny Hänsch,  “There is no data like more data - current status of machine learning datasets in remote sensing,”  in Proc. IGARSS, 2021, pp. 1206–1209.
  * [2] ESA,  “Copernicus open access hub,” https://scihub.copernicus.eu/, 2022\.
  * [3] Maxar,  “Maxar open data program,” https://www.maxar.com/open-data, 2022\.
  * [4] Capella Space,  “Capella space sar open data program,” https://www.capellaspace.com/community/capella-open-data, 2022.
  * [5] ESA,  “Corine land cover,” https://land.copernicus.eu/pan-european/corine-land-cover, 2022.
  * [6] Data Science in Earth Observation, TU Munich,  “Public datasets,” https://www.asg.ed.tum.de/sipeo/downloads/, 2022.
  * [7] Computational and Photogrammetric Vision Team, Wuhan University,  “Datasets,” http://www.captain-whu.com/repository_En.html, 2022\.
  * [8] ISPRS,  “ISPRS Test Project on Urban Classification, 3D Building Reconstruction and Semantic Labeling,” https://www2.isprs.org/commissions/comm2/wg4/benchmark/, 2022.
  * [9] Christoph Rieke,  “Awesome satellite imagery datasets,” https://github.com/chrieke/awesome-satellite-imagery-datasets, 2022.


# Phase retrieval by projections in $\mathbb{R}^{n}$ requires
${\large\bf{2n-2}}$ projections
(This paper has a gap)

Peter G. Casazza and Dorsa Ghoreishi  Department of Mathematics, University of
Missouri, Columbia, MO 65211-4100<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

We will answer the most significant open problem in real phase retrieval by
projections by showing it requires at least $2n-2$ projections to do phase
retrieval in $\mathbb{R}^{n}$.

###### 1991 Mathematics Subject Classification:

42C15

The authors were supported by NSF DMS 1609760 and 1906725

##  1\. Introduction

Phase retrieval is one of the most active and applied subjects in mathematics
today, with applications to pure mathematics, applied mathematics,
engineering, medicine, computer science, and more. Frame theory is fundamental
to the digitalizing of information for processing and storing. Phase retrieval
has applications to X-ray crystallography, electron microscopy, astronomical
imaging, optics, quantum physics, and much more. Phase retrieval will even be
needed to align the mirrors in the new James Webb Space Telescope.

######  Definition 1.1.

A family of vectors $\\{\phi_{i}\\}_{i=1}^{m}$ in $\mathbb{R}^{n}$ or
$\mathbb{C}^{n}$ does phase retrieval if for any non-zero
$x,y\in\mathbb{R}^{n}$ satisfying $|\langle x,\phi_{i}\rangle|=|\langle
y,\phi_{i}\rangle|$ for all $i\in[m]$ this will imply $x=cy$ with $|c|=1$.

A fundamental idea here is:

######  Definition 1.2.

A family of vectors $\\{\phi_{i}\\}_{i=1}^{m}$ in $\mathbb{R}^{n}$ has the
complement property if for all subsets $I\subset[m]$, either
$\textup{span}\\{\phi_{i}\\}_{i\in I}=\mathbb{R}^{n}$ or
$\textup{span}\\{\phi_{i}\\}_{i\in I^{c}}=\mathbb{R}^{n}$.

The most important classification of phase retrieval in $\mathbb{R}^{n}$
appeared in [1].

######  Theorem 1.3.

A family of vectors $\\{\phi_{i}\\}_{i=1}^{m}$ does phase retrieval in
$\mathbb{R}^{n}$ if and only if it has the complement property.

######  Corollary 1.4.

If a family of vectors $\\{\phi_{i}\\}_{i=1}^{m}$ satisfies phase retrieval in
$\mathbb{R}^{n}$, then $m\geq 2n-1$.

In some applications a signal must be reconstructed from the norms of higher
dimensional components. In X-ray crystallography such a problem arises in
crystal twinning [7]. In this scenario, there exists a similar phase retrieval
problem.

######  Definition 1.5.

A family of subspaces $\\{W_{i}\\}_{i=1}^{m}$ in $\mathbb{R}^{n}$or
$\mathbb{C}^{n}$ with respective orthogonal projections
$\\{P_{i}\\}_{i=1}^{m}$ satisfy phase retrieval if for all
$x,y\in\mathbb{R}^{n}$, $\|P_{i}x\|=\|P_{i}y\|$ for all $i\in[m]$, implies
$x=cy$ with $|c|=1$.

A deep study of phase retrieval by projections appears in [4]. This includes
the result:

######  Theorem 1.6.

For any $0<r_{i}<n$, $i\in[2n-1]$ there are subspaces
$\displaystyle\\{W_{i}\\}_{i=1}^{2n-1}$ in $\mathbb{R}^{n}$ doing phase
retrieval with dim $W_{i}=r_{i}$.

A fundamental result in this area from [4] is:

######  Theorem 1.7.

Let $\\{W_{i}\\}_{i=1}^{m}$ be subspaces of $\mathbb{R}^{n}$. The following
are equivalent:

  1. (1)

$\\{W_{i}\\}_{i=1}^{m}$ does phase retrieval.

  2. (2)

Whenever we choose orthonormal bases $\\{u_{ij}\\}_{j=1}^{I_{i}}$ for $W_{i}$,
the family of vectors$\\{u_{ij}\\}_{i=1,j=1}^{\ m\ \ I_{i}}$ does phase
retrieval.

Edidin [8] proved a fundamental result concerning phase retrieval by
projections (See [5] for an elementary proof of this resut.)

######  Theorem 1.8.

A family of projections $\\{P_{i}\\}_{i=1}^{m}$ does phase retrieval in
$\mathbb{R}^{n}$ if and only if for every $0\not=x\in\mathbb{R}^{n}$,
$\textup{span}\\{P_{i}x\\}_{i=1}^{m}=\mathbb{R}^{n}$.

The following theorem in [11] shows a particular case of $\mathbb{R}^{4}$ in
which phase retrieval is possible with $2n-2=2(4)-2=6$ subspaces.

######  Theorem 1.9.

There are 6 two-dimensional subspaces of $\mathbb{R}^{4}$ satisfying phase
retrieval.

In [3] it was shown that there are 6 hyperplanes in $\mathbb{R}^{4}$ doing
phase retrieval.

The following result also appears in [3]. Since this result is central to this
paper, we include its short proof.

######  Theorem 1.10.

It takes at least $2n-2$ hyperplanes in $\mathbb{R}^{n}$ to do phase
retrieval.

###### Proof.

Let $\\{W_{i}\\}_{i=1}^{m}$ be a family of hyperplanes in $\mathbb{R}^{n}$
doing phase retrieval with respective projections $\\{P_{i}\\}_{i=1}^{m}$.
By way of contradiction, assume $m\leq 2n-3$. Choose a vector

$0\not=x\in\bigcap_{i=1}^{n-1}W_{i}$

so $P_{i}x=x$ for $i\in[n-1]$. Therefore we can write,
$\\{P_{i}x\\}_{i=1}^{m}=\\{x\\}\cup\\{P_{i}x\\}_{i=n}^{m}$. Notice that
$2n-3-(n-1)+1=n-1$ which shows $\\{P_{i}x\\}_{i=1}^{m}$ has at most $n-1$ non-
zero vectors and cannot span the space. This is contradicting Theorem 1.8.

∎

The main open question in this area has been about finding the least number of
subspaces satisfying phase retrieval. We will show in this paper that it takes
at least $2n-2$ subspaces to satisfy phase retrieval in $\mathbb{R}^{n}$.

Another open problem in this area is motivated by this theorem of Edidin [8]:

######  Theorem 1.11.

If $n=2^{k}-1$, for some $k$, then it takes $2n-1$ subspaces of
$\mathbb{R}^{n}$ to do phase retrieval.

So now the main open problem in this area is:

######  Problem 1.12.

Can we do phase retrieval in $\mathbb{R}^{n}$ with $2n-2$ subspaces whenever
$n\not=2^{k}-1$ for any $k$.

The complex case is significantly more complicated. It is known [2] that phase
retrieval can be done in $\mathbb{C}^{n}$ with $4n-4$ vectors. It was believed
this result was best possible. But Vinzant [9] showed that phase retrieval can
be done in $\mathbb{C}^{4}$ with 11 vectors (in other words $4n-5$ vectors).
So the major open problem here is:

######  Problem 1.13.

What is the minimum number of vectors required to do phase retrieval in
$\mathbb{C}^{n}$?

Phase retrieval by projections in $\mathbb{C}^{n}$ is still unknown.

######  Problem 1.14.

What is the minimum number of projections required to do phase retrieval in
$\mathbb{C}^{n}$

Notice that Theorem 1.8 (Edidin’s theorem) fails in $\mathbb{C}^{n}$. A
complex version of this theorem appears in [11], Theorem 2.1 (3). A completely
different approach to a complex version of Edidin’s theorem was proved in [6].
In this paper they associate vectors in $\mathbb{C}^{n}$ with rank 2
projections in $\mathbb{R}^{2n}$.

##  2\. Main Result

The following is immediate by Theorem 1.8.

######  Lemma 2.1.

If a family of projections $\\{P_{i}\\}_{i=1}^{m}$ does phase retrieval in
$\mathbb{R}^{n}$, then for every $0\not=x\in\mathbb{R}^{n}$, either
$\textup{span}\\{P_{i}x\\}_{i=2}^{m}=\mathbb{R}^{n}$ or
$span\\{P_{i}x\\}_{i=2}^{m}$ spans a hyperplane.
Moreover, if this is a hyperplane,
$P_{1}x\notin\textup{span}\\{P_{i}x\\}_{i=2}^{m}$.

The next result is fundamental to our proof.

######  Theorem 2.2.

Let a family of projections $\\{P_{i}\\}_{i=1}^{m}$ do phase retrieval on
$\mathbb{R}^{n}$ and let

$F=\\{x:\|x\|=1\mbox{ and
}H_{x}=\textup{span}\\{P_{i}x\\}_{i=2}^{m}\not=\mathbb{R}^{n}\\}.$

If $x_{1},x_{2}\in F$ and $H_{x_{1}}=H_{x_{2}}$, then $x_{1}=\pm x_{2}$.

###### Proof.

Since $H_{x_{1}}=H_{x_{2}}$ is a hyperplane, there is a vector $\|z\|=1$ so
that $z\perp P_{i}x_{1}$ and $z\perp P_{i}x_{2}$ for all $i=2,3,\ldots,m$.
Hence

$0=\langle z,P_{i}x_{1}\rangle=\langle P_{i}z,x_{1}\rangle=\langle
z,P_{i}x_{2}\rangle=\langle P_{i}z,x_{2}\rangle,\mbox{ for all
}i=2,3,\ldots,m.$

Since this family spans a hyperplane, $x_{1},x_{2}$ are both orthogonal to
this hyperplane and so $x_{1}=\pm x_{2}$. ∎

The main result will be based on the following theorem:

######  Theorem 2.3.

Let a family of subspaces $\\{W_{i}\\}_{i=1}^{m}$ do phase retrieval in
$\mathbb{R}^{n}$. If $dim\ W_{1}^{\perp}\geq 2$, there is a $z\in
W_{1}^{\perp}$ with $\|z\|=1$ so that $\\{W_{1}\cup
z\\}\cup\\{W_{2},\ldots,W_{m}\\}$ does phase retrieval.

###### Proof.

We will do this in steps.

(1) Choose orthonormal bases $\\{u_{ij}\\}_{j=1}^{I_{i}}$ for $W_{i}$ when
$i\in[m]$ so that these vectors satisfy phase retrieval. For $z\in
W_{1}^{\perp}$ let $W_{z}=\textup{span}\\{W_{1},z\\}$ and let $P_{z}$ be the
projection onto $W_{z}$.
We need to show there exists a $z\in W_{1}^{\perp}$ so that
$\textup{span}\\{P_{z},P_{2},\ldots,P_{n}\\}$ does phase retrieval in
$\mathbb{R}^{n}$. (2) Choose $x\in\mathbb{R}^{n}$ with $\|x\|=1$. If
$\textup{span}\\{P_{i}x\\}_{i=2}^{m}=\mathbb{R}^{n}$, this shows that if we
add a vector $z$ which is in $W_{1}^{\perp}$, the subspaces still satisfy
phase retrieval and we are done for this $x$.

Now we need to consider the cases when
$\textup{span}\\{P_{i}x\\}_{i=2}^{m}\neq\mathbb{R}^{n}$, (3) By Theorem 2.2,
if $\textup{span}\\{P_{i}x\\}_{i=2}^{m}$ =
$\textup{span}\\{P_{i}y\\}_{i=2}^{m}\not=\mathbb{R}^{n}$ and $\|x\|=\|y\|$,
then $x=\pm y$. (4) Note

$\textup{span}\\{P_{i}x\\}_{i=2}^{m}=\textup{span}\\{u_{kj}:\langle
P_{i}x,u_{kj}\rangle\not=0\mbox{ for some }i=1,2,\ldots,m\\}.$

It follows from (3) that there are at most a finite number of $x^{\prime}s$
for which $\textup{span}\\{P_{i}x\\}_{i=2}^{m}\not=\mathbb{R}^{n}$.Call these
$\\{x_{j}\\}_{j=1}^{k}$, and let
$V_{x_{j}}=\textup{span}\\{P_{i}x_{j}\\}_{i=2}^{m}$ and choose $\|y_{j}\|=1$
with $y_{j}\perp V_{x_{j}}$. (5)Now we need to find a $z\in W_{1}^{\perp}$
with $\|z\|=1$ and $P_{z}x_{j}\notin V_{x_{j}}$ for all $j\in[k]$.
Since there are a finite number of $V_{x_{j}}$, it suffices to show that for
all $j\in[k]$ if

(1) $A=\\{z\in W_{1}^{\perp}:\|z\|=1,\ P_{z}x_{j}\in
V_{x_{j}}=y_{j}^{\perp},\\}$

the family of lines in the unit ball of A going through the origin is a set of
measure zero relative to Lebesgue measure on $W_{1}^{\perp}$.

(6) We have $A\cap y_{j}^{\perp}=A\cap x_{j}^{\perp}=\emptyset$.

  * •

If $z\in A\cap x_{j}^{\perp}$ then:

$\displaystyle P_{z}x_{j}$ $\displaystyle=P_{1}x_{j}+\langle x_{j},z\rangle z$
$\displaystyle=P_{1}x_{j}\in y_{j}^{\perp}.$

It follows that
$\textup{span}\\{P_{i}x_{j}\\}_{i=1}^{m}=y_{j}^{\perp}\not=\mathbb{R}^{n}$. So
by Theorem 1.8, $\\{P_{i}\\}_{i=1}^{m}$ fails phase retrieval - a
contradiction.

  * •

If $z\in A\cap y_{j}^{\perp}$ then

$P_{z}x_{j}=P_{1}x_{j}+\langle x_{j},z\rangle z\in y_{j}^{\perp}.$

Since $y_{j}^{\perp}$ is a subspace and $z\in y_{j}^{\perp}$ it follows again
that $P_{1}x_{j}\in y_{j}{\color[rgb]{0,0,1}{{}^{\perp}}}$ and so
$\\{P_{i}\\}_{i=1}^{m}$ fails phase retrieval - again a contradiction.

(7) Claim: There is a constant $0<c\leq 1$ so that for every $z\in A$ we have

$\langle x_{j},z\rangle\langle y_{j},z\rangle=c.$

Proof of the claim: Let $z_{1},z_{2}\in A$. Then if
$P_{z_{1}}x_{j},P_{z_{2}}x_{j}\in y_{j}^{\perp}$, since $y_{j}^{\perp}$ is a
subspace,

$P_{z_{1}}x_{j}-P_{z_{2}}x_{j}=(P_{1}x_{j}+\langle x_{j},z_{1}\rangle
z_{1})-(P_{1}x_{j}+\langle x_{j},z_{2}\rangle z_{2})\in y_{j}^{\perp}.$

Hence

$\langle x_{j},z_{1}\rangle\langle y_{j},z_{1}\rangle-\langle
x_{j},z_{2}\rangle\langle y_{j},z_{2}\rangle=0.$

Notice that $c\not=0$, since if $c=0$, then for every $z\in A$ either $z\perp
x_{j}$ or $z\perp y_{j}$ which is not possible by (6).

(8) After a rotation, we may assume
$X=\textup{span}\\{x_{j},y_{j}\\}=\mathbb{R}^{2}\subset\mathbb{R}^{n}$ with
canonical basis $\\{e_{1},e_{2}\\}$ and $x_{j}=e_{1}$.
Let $y_{j}=(e,f)\in\mathbb{R}^{2}$ and let $P$ be the projection onto $X$. We
want to find the points $v\in X$ so that $\langle x_{j},v\rangle\langle
y_{j},v\rangle=c$.

The points on the line $\ell_{1}$ orthogonal to $x_{j}=e_{1}$ and passing
through $(x,0)$ represents all the points in $\mathbb{R}^{2}$ whose inner
product with $x_{j}$ is $x$. Let $m$ be the slope of the line orthogonal to
$y_{j}$. The points on the line $\ell_{2}$ orthogonal to $y_{j}$ and passing
through $\frac{c}{x}(e,f)$ are the points in $\mathbb{R}^{2}$ whose inner
product with $y_{j}$ is $\frac{c}{x}$ since $e^{2}+f^{2}=1$. This line has
equation:

$y-\frac{c}{x}f=m(x-\frac{c}{x}e).$

That is:

$y=mx+\frac{c}{x}(f-em).$

So if $v\in\ell_{1}\cap\ell_{2}$ for some $c\leq x\leq 1$ then

$v=(x,mx+\frac{c}{x}(f-em)).$

Therefore, the points $v\in X$ so that $\langle x_{j},v\rangle\langle
y_{j},v\rangle=c$ are:

$\\{(x,mx+\frac{c}{x}(f-em)):c\leq x\leq 1\\}\hskip 14.45377pt(*)$

(9) We will answer the case where $dim\ W_{1}^{\perp}=2$. We will examine 3
cases.

Case 1: P maps $W_{1}^{\perp}$ onto $X$. Since $P$ is onto, there is a $v\in
W_{1}^{\perp}$ with $\|v\|=1$ and $Pv=re_{1}$. Let $v^{\perp}$ be the unit
vector in $W_{1}^{\perp}$ orthogonal to $v$ and
$y_{j}=(e,f)\in\mathbb{R}^{2}$. Let $Pv^{\perp}=(g,h)$. If $z\in A$,
$z=av^{\perp}+bv$ and $a^{2}+b^{2}=1$. By (8) and the argument above,
$Pz=(x,mx+\frac{c}{x}(f-em))$. Also,

$\displaystyle Pz$ $\displaystyle=aPv^{\perp}+bPv$
$\displaystyle=(ag,ah)+(br,0)$ $\displaystyle=(ag+br,ah).$

So in order for $\langle x,Pz\rangle\langle y,Pz\rangle=c$ we must have for
some $x$:

$ag+\sqrt{1-a^{2}}r=x\mbox{ and }ah=mx+\frac{c}{x}(f-em).$

Substituting the first equation into the second equation for $x$, since all
constants except $a$ are fixed, we have a quadratic equation in $a$. So A is
of cardinality at most two and the lines through A are a set of measure zero
in $W_{1}^{\perp}$.

Case 2: $PW_{1}^{\perp}=\emptyset$. In this case, if $z\in A$ then $Pz=0$. So
$\langle z,x_{j}\rangle=\langle Pz,x_{j}\rangle=0$. By (6), this is a
contradiction unless $A=\emptyset$.

Case 3: $PW_{1}^{\perp}$ is dimension one in $X$.

In this case, $x_{j}^{\perp}\cap W_{1}^{\perp}=y_{j}^{\perp}\cap
W_{1}^{\perp}$. So $x_{j}^{\perp}\cap W_{1}^{\perp}$ is a hyperplane in
$W_{1}^{\perp}$. Let $y_{j}=(e,f)\in\mathbb{R}^{2}$ and choose $\|v\|=1$ in
$W_{1}^{\perp}$ with $Pv^{\perp}=0$. Let $Pv=(g,h)$.

Then, if $z\in A$ with $z=av+bv^{\perp}$ by (8) and (*),

$Pz=(ag,ah)=(x,mx+\frac{c}{x}(f-em))$

Hence,

$ag=x\mbox{ and }ah=mx+\frac{c}{x}(f-em).$

Substituting the first equation in the second equation for $x$, all constants
except $a$ are fixed so this is a quadratic equation for $a$ and hence has at
most two solutions. That is, $A$ consists of at most two lines and so is a set
of measure zero in $W_{1}^{\perp}$.

(10) We check the case $dim\ W_{1}^{\perp}\geq 3$. Since $x_{j}^{\perp}\cap
y_{j}^{\perp}$ is a subset of $\mathbb{R}^{n}$ of codimension 1 or 2, we will
examine these two cases. Case 1: $(x_{j}^{\perp}\cap y_{j}^{\perp})\cap
W_{1}^{\perp}$ is a set of codimension 2 in $W_{1}^{\perp}$. Choose a two
dimensional subspace $V$ of $W_{1}^{\perp}$ so that
$W_{1}^{\perp}=V\oplus(x_{j}^{\perp}\cap y_{j}^{\perp})$.

Claim: $PV=X$.

Proof of claim: If not, then there is a $v\in V$ with $Pv=0$. That is,

$\langle v,x_{j}\rangle=\langle Pv,x_{j}\rangle=0=\langle v,y_{j}\rangle.$

But $v\perp(x_{j}^{\perp}\cap y_{j}^{\perp}$), a contradiction.

Choose $v\in V$, $\|v\|=1$ and $Pv=re_{1}$. Let $v^{\perp}$ be in V and
orthogonal to $v$ and $Pv^{\perp}=(g,h)$. It follows that every $z\in
W_{1}^{\perp}$ is of the form

$z=bv+av^{\perp}+u$

when $a^{2}+b^{2}+\|u\|^{2}=1$, and $P(u)=0$. By (8) and (*),

$Pz=(x,mx+\frac{c}{x}(f-em))$

But

$\displaystyle Pz$ $\displaystyle=bPv+aPv^{\perp}$
$\displaystyle=(br,0)+(ag,ah)$ $\displaystyle=(ag+br,ah).$

So we must have for some $x$:

$ag+br=x\mbox{ and }ah=mx+\frac{c}{x}(f-em).$

Now, from the second equation

$ah=m(ag+bj)+\frac{c}{ag+br}(f-em),$

All the constants here are fixed except a,b. This equation, represents a
quadratic equation in a,b. Hence,

$\\{z=av^{\perp}+bv:\langle x_{j},Pz\rangle\langle y_{j},Pz\rangle=c\\}$

is the union of two continuous quadratic graphs $G_{1},G_{2}$ in $V$. For
$z\in A$, $z=av+bv^{\perp}+u$, $\|z\|=1$. So

$A=\\{x+u:x\in G_{1},\ \|u\|^{2}=1-\|x\|^{2}\\}$

or

$A=\\{x+u:x\in G_{2},\ \|u\|^{2}=1-\|x\|^{2}\\}.$

In each case, since the unit ball of $(x_{j}^{\perp}\cap y_{j}^{\perp})\cap
W_{1}^{\perp}$ is a set of measure zero in $W_{1}^{\perp}$ and A is a curve on
the sphere of $W_{1}^{\perp}$ and since $dim\ W_{1}^{\perp}\geq 3$, the lines
through A are a set of measure zero in $W_{1}^{\perp}$.

Case 2: $(x_{j}^{\perp}\cap y_{j}^{\perp})\cap W_{1}^{\perp}$ is a set of
codimension 1 in $W_{1}^{\perp}$. Choose $v\in W_{1}^{\perp}$ with $v\perp
x_{j}$. Then every $z\in W_{1}^{\perp}$, $\|z\|=1$, is of the form $z=av+u$
and $u\in x_{j}^{\perp}$. Let $Pv=(g,h)$. Now, $Pz=Pav$ so

$Pz=aPv=(ag,ah).$

Hence, by (8) and (*),

$ag=x\mbox{ and }ah=mx+\frac{c}{x}(f-em).$

This yields

$ah=mag+\frac{c}{ag}(f-em).$

This quadratic equation in $a$ has at most two solutions $a_{1},a_{2}$. So $A$
is of the form

$\\{a_{i}v+u:\|u\|^{2}=1-a_{i}^{2}\\}.$

But these sets are two translates of a circle in $(x_{j}^{\perp}\cap
y_{j}^{\perp})\cap W_{1}^{\perp}$ and the lines through these sets form a set
of measure zero in $W_{1}^{\perp}$.

(11) Now, for any $x_{j}$ the set of all $z\in W_{1}^{\perp}$ with
$P_{z}x_{j}\in y_{1}^{\perp}$ is a set of measure zero in $W_{1}^{\perp}$. So
choose $z\in W_{1}^{\perp}$ which is in none of these $k$ subsets of
$W_{1}^{\perp}$ of measure zero. Now, for any $0\not=x\in\mathbb{R}^{n}$, if
$x\not=x_{j}$ for any $j\in[k]$, then $\\{P_{i}x\\}_{i=2}^{m}$ spans
$\mathbb{R}^{n}$. And if $x=x_{j}$ for some $j\in[k]$ then $P_{z}x_{j}\notin
y_{j}^{\perp}$ and so $\\{P_{z}x_{j},P_{i}x_{j}\\}_{i=2}^{m}$ spans
$\mathbb{R}^{n}$. That is, $\\{P_{z},P_{i}\\}_{i=2}^{m}$ does phase retrieval
in $\mathbb{R}^{n}$.

This completes the proof of the theorem. ∎

As a consequence, applying the theorem over and over, we have

######  Corollary 2.4.

Assume a family of subspaces $\\{W_{i}\\}_{i=1}^{m}$ with dim $W_{i}=k_{i}$,
$1\leq k_{i}\leq n-1$, for all $i=1,2,\ldots,$ does phase retrieval in
$\mathbb{R}^{n}$. The following hold:

  1. (1)

For any $1\leq k_{i}\leq r_{i}\leq n-1$, there are subspaces $W_{i}^{\prime}$
of $\mathbb{R}^{n}$ so that $W_{i}\subset W_{i}^{\prime}$, dim
$W_{i}^{\prime}=r_{i}$ and $\\{W_{i}^{\prime}\\}_{i=1}^{m}$ does phase
retrieval.

  2. (2)

There are hyperplanes $\\{W_{i}^{\prime}\\}_{i=1}^{m}$ with $W_{i}\subset
W_{i}^{\prime}$ and $\\{W_{i}^{\prime}\\}_{i=1}^{m}$ does phase retrieval. So
by Theorem 1.10, $m\geq 2n-2$.

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