# Bipartite quantum state discrimination and decomposable entanglement witness

Donghoon Ha  Department of Applied Mathematics and Institute of Natural
Sciences, Kyung Hee University, Yongin 17104, Republic of Korea  Jeong San Kim
<EMAIL_ADDRESS>Department of Applied Mathematics and Institute of Natural
Sciences, Kyung Hee University, Yongin 17104, Republic of Korea

###### Abstract

We consider bipartite quantum state discrimination using positive-partial-
transpose measurements and show that minimum-error discrimination by positive-
partial-transpose measurements is closely related to entanglement witness. By
using the concept of decomposable entanglement witness, we establish
conditions on minimum-error discrimination by positive-partial-transpose
measurements. We also provide conditions on the upper bound of the maximum
success probability over all possible positive-partial-transpose measurements.
Finally, we illustrate our results using examples of multidimensional
bipartite quantum states.

##  I Introduction

Quantum nonlocality is a fascinating phenomenon that occurs in multipartite
quantum systemshoro2009 ; chid2013 ; brun2014 . In discriminating multipartite
quantum states, quantum nonlocality occurs when optimal state discrimination,
which gives the maximum success probability overall possible measurements,
cannot be realized only by _local operations and classical
communication_(LOCC)chit20142 ; chef2000 ; berg2007 ; barn20091 ; bae2015 .
Orthogonal quantum states can always be perfectly discriminated using
appropriate global measurement. However, there exist some multipartite
orthogonal quantum states that cannot be perfectly discriminated only by LOCC
measurementsbenn19991 ; ghos2001 . Moreover, some multipartite nonorthogonal
quantum states cannot be optimally discriminated using only LOCC
measurementspere1991 ; chit2013 . Nonetheless, characterizing local
discrimination of quantum states is still a hard task due to the lack of good
mathematical structure for LOCC.
The first nonlocal phenomenon in quantum state discrimination was shown
through nine $3\otimes 3$ pure orthogonal product statesbenn19991 . On the
other hand, it was shown that nonlocality does not occur in discriminating any
two multipartite pure stateswalg2000 ; virm2001 . Since then, there have been
several studies focused on local indistinguishability or local
distinguishability of mutually orthogonal stateslu2010 ; akib2018 ; ha20222 ;
cohe2023 . In particular, the local distinguishability of two $2\otimes 2$
pure orthogonal entangled states has been experimentally demonstratedlu2010 .
It was also shown that the nonlocality of the $N$-fold quantum state ensemble
can disappear asymptoticallyakib2018 . Recently, an upper bound of the maximum
success probability overall separable measurements was established in the
optimization for local unambiguous discrimination of multipartite quantum
statesha20222 . Moreover, necessary conditions for perfect discrimination by
asymptotic LOCC were established in discriminating orthogonal pure
statescohe2023 .
The quantum nonlocal phenomenon also arises in the correlations of a
multipartite quantum system. Entanglement is a well-known quantum correlation
that cannot be realized using only LOCChoro2009 . The nonlocal property of
entanglement can be used as a useful resource in various quantum information
processing tasks such as quantum cryptography, teleportation, and local
discrimination of multipartite quantum stateseker1991 ; benn1993 ; chit2019 ;
band2021 . For this reason, much attention has been shown for characterizing
quantum entanglementamic2008 ; guhn2009 ; kett2020 .
An important task of characterizing quantum entanglement is to design ways to
detect the existence of entanglement. _Entanglement witness_(EW) is an
observable having non-negative mean value for any separable state, but
negative for some entangled stateshoro1996 ; terh2000 ; lewe2000 ; chru2014 .
As EW detects the existence of entanglement that is an important quantum
nonlocality, it is natural to ask whether EW can also be used to characterize
the limit on local discrimination of quantum states, another important quantum
nonlocality.
Here, we provide an answer to the question by establishing a specific relation
between the properties of EW and _positive-partial-transpose_(PPT)
measurements, a mathematically well-structured set of measurements having LOCC
measurements as special cases. By using the concept of decomposability of
operators, we show that the minimum-error discrimination of bipartite quantum
states using PPT measurements strongly depends on the existence of
_decomposable entanglement witness_(DEW). More precisely, we establish
conditions on minimum-error discrimination by PPT measurements in terms of
DEW. We also provide conditions on the maximum success probability over all
possible PPT measurements. Finally, we illustrate our results using examples
of multidimensional bipartite quantum states.
Because PPT measurements have LOCC measurements as special cases, our results
provide a useful method to detect the nonlocality arising in quantum state
discrimination. Moreover, our results can be applied to any ensemble of
bipartite states in an arbitrary dimension, whereas the previous resultslu2010
; akib2018 ; cohe2023  are only valid for some restricted cases with certain
conditions. We also note that our results provide a systematic way to
construct a bipartite quantum state ensemble showing nonlocality in quantum
state discrimination.
This paper is organized as follows. In Section II, we first recall the
definitions and some properties about PPT measurements and DEW. We also recall
the definition of minimum-error discrimination as well as some useful
properties of the optimal measurements. In Section III, we provide conditions
on the maximum success probability over all possible PPT measurements
(Theorems 1 and 2). In Section IV, we provide conditions on minimum-error
discrimination by PPT measurements in terms of DEW(Theorems 3 and 4). Finally,
we illustrate our results using examples of multidimensional bipartite quantum
states. In Section V, we summarize our results with possible applications and
future works. We also discuss a systematic way to construct a bipartite
quantum state ensemble showing nonlocality in quantum state discrimination.

##  II Preliminaries

For a bipartite Hilbert space
$\mathcal{H}=\mathbb{C}^{d_{1}}\otimes\mathbb{C}^{d_{2}}$, let $\mathbb{H}$ be
the set of all Hermitian operators acting on $\mathcal{H}$. A bipartite
quantum state is expressed by a density operator that is $\rho\in\mathbb{H}$
with positive semidefiniteness $\rho\succeq 0$ and unit trace
$\operatorname{Tr}\rho=1$. A measurement is represented by a positive
operator-valued measure that is $\\{M_{i}\\}_{i}\subseteq\mathbb{H}$
satisfying positive semidefiniteness $M_{i}\succeq 0$ for all $i$ and the
completeness relation $\sum_{i}M_{i}=\mathbbm{1}$, where $\mathbbm{1}$ is the
identity operator in $\mathbb{H}$. When a measurement $\\{M_{i}\\}_{i}$ is
performed on the quantum state $\rho$, the probability of obtaining the
measurement outcome corresponding to $M_{j}$ is $\operatorname{Tr}(\rho
M_{j})$.

######  Definition 1.

$E\in\mathbb{H}$ is called _PPT_ if $E^{\operatorname{PT}}\succeq 0$, where
the superscript $\operatorname{PT}$ is to indicate the partial transposition
pere1996 ; pptp . Similarly, we say that $\\{E_{i}\\}_{i}\subseteq\mathbb{H}$
is _PPT_ if $E_{i}$ is PPT for all $i$.

We denote the set of all positive-semidefinite PPT operators in $\mathbb{H}$
as

$\mathbb{PPT}_{+}=\\{E\in\mathbb{H}\,|\,E\succeq
0,~{}E^{\operatorname{PT}}\succeq 0\\},$ (1)

and its dual set as $\mathbb{PPT}_{+}^{*}$, that is,

$\mathbb{PPT}_{+}^{*}=\\{E\in\mathbb{H}\,|\,\operatorname{Tr}(EF)\geqslant
0~{}\forall F\in\mathbb{PPT}_{+}\\}.$ (2)

A measurement is called a _LOCC measurement_ if it can be implemented by LOCC,
and a measurement $\\{M_{i}\\}_{i}$ is called a _PPT measurement_ if $M_{i}$
is PPT for all $i$. We note that every LOCC measurement is a PPT measurement
chit20142 .

###  II.1 Decomposable entanglement witness

######  Definition 2.

$W\in\mathbb{H}$ is called _decomposable_ if it can be written as a sum of a
positive-semidefinite operator and a PPT operator in $\mathbb{H}$, that is,

$W=P+Q^{\operatorname{PT}}$ (3)

for some $P\succeq 0$ and $Q\succeq 0$.

The following proposition provides an equivalent condition of the
decomposability in Eq. (3) lewe2000 ; chru2014 .

######  Proposition 1.

$W\in\mathbb{H}$ is decomposable if and only if

$W\in\mathbb{PPT}_{+}^{*},$ (4)

where $\mathbb{PPT}_{+}^{*}$ is defined in Eq. (2).

A positive-semidefinite operator $E\in\mathbb{H}$ is called _separable_ if it
can be expressed as a sum of positive-semidefinite product operators, that is,

$E=\sum_{l}A_{l}\otimes B_{l},$ (5)

where $A_{l}$ and $B_{l}$ are positive-semidefinite operators on
$\mathbb{C}^{d_{1}}$ and $\mathbb{C}^{d_{2}}$ of $\mathcal{H}$, respectively.
We denote the set of all positive-semidefinite _separable_ operators in
$\mathbb{H}$ as

$\mathbb{SEP}=\\{E\in\mathbb{H}\,|\,E\succeq 0,~{}E:\mbox{separable}\\}.$ (6)

We also denote the dual set of $\mathbb{SEP}$ as

$\mathbb{SEP}^{*}=\\{E\in\mathbb{H}\,|\,\operatorname{Tr}(EF)\geqslant
0~{}\forall F\in\mathbb{SEP}\\}.$ (7)

######  Definition 3.

$W\in\mathbb{H}$ is called an _EW_ if it is in $\mathbb{SEP}^{*}$ but not
positive semidefinite, that is,

$W\in\mathbb{SEP}^{*},~{}W\not\succeq 0.$ (8)

In particular, an EW $W\in\mathbb{H}$ is called a _DEW_ if it is decomposable.

Figure 1: The relationship of the subsets of $\mathbb{H}$. $\mathbb{H}_{+}$ is
the set of all positive-semidefinite operators and $\mathbb{PPT}$ is the set
of all PPT operators. $\mathbb{PPT}_{+}$ is the intersection of
$\mathbb{H}_{+}$ and $\mathbb{PPT}$. The shaded area
$\\{W\in\mathbb{PPT}_{+}^{*}\,|\,W\notin\mathbb{H}_{+}\\}$ is the set of all
DEWs.

From Proposition 1, we can see that $W\in\mathbb{H}$ is a DEW if and only if

$W\in\mathbb{PPT}_{+}^{*},~{}W\not\succeq 0.$ (9)

We note that $\mathbb{PPT}_{+}^{*}\subseteq\mathbb{SEP}^{*}$ since
$\mathbb{SEP}\subseteq\mathbb{PPT}_{+}$. Figure 1 illustrates the relationship
of the subsets of $\mathbb{H}$.

###  II.2 Minimum-error discrimination of bipartite quantum states

For a _bipartite_ quantum state ensemble,

$\mathcal{E}=\\{\eta_{i},\rho_{i}\\}_{i=1}^{n},$ (10)

where the state $\rho_{i}$ is prepared with the probability
$\eta_{i}\in[0,1]$, let us consider the situation of discriminating the states
from $\mathcal{E}$ using a measurement $\\{M_{i}\\}_{i=1}^{n}$. Here, the
detection of $M_{i}$ means that we guess the prepared state as $\rho_{i}$.
The _minimum-error discrimination_ of $\mathcal{E}$ is to achieve the optimal
success probability,

$p_{\rm G}(\mathcal{E})=\max_{\rm
measurement}\sum_{i=1}^{n}\eta_{i}\operatorname{Tr}(\rho_{i}M_{i}),$ (11)

where the maximum is taken over all possible measurements hels1969 . We note
that a measurement $\\{M_{i}\\}_{i=1}^{n}$ gives the optimal success
probability $p_{\rm G}(\mathcal{E})$ if and only if it satisfies the following
condition hole1974 ; yuen1975 ; barn20092 ; bae2013 :

$\sum_{j=1}^{n}\eta_{j}\rho_{j}M_{j}-\eta_{i}\rho_{i}\succeq 0~{}\forall
i=1,\ldots,n.$ (12)

When the available measurements are limited to PPT measurements, we denote the
maximum success probability by

$p_{\operatorname{PPT}}(\mathcal{E})=\max_{\begin{subarray}{c}\rm PPT\\\ \rm
measurement\end{subarray}}\sum_{i=1}^{n}\eta_{i}\operatorname{Tr}(\rho_{i}M_{i}),$
(13)

where the maximum is taken over all possible PPT measurements. Similarly, we
denote

$p_{\rm L}(\mathcal{E})=\max_{\begin{subarray}{c}\rm LOCC\\\ \rm
measurement\end{subarray}}\sum_{i=1}^{n}\eta_{i}\operatorname{Tr}(\rho_{i}M_{i}),$
(14)

where the maximum is taken over all possible LOCC measurements.
Because $p_{\rm PPT}(\mathcal{E})$ is the maximum value over the set of all
PPT measurements, which is a proper subset of the set of all measurements,
$p_{\rm PPT}(\mathcal{E})$ is a lower bound of $p_{\rm G}(\mathcal{E})$.
Moreover, $p_{\rm L}(\mathcal{E})$ is a lower bound of $p_{\rm
PPT}(\mathcal{E})$ since every LOCC measurement is a PPT measurement chit20142
. Thus, we have

$p_{\rm L}(\mathcal{E})\leqslant p_{\rm PPT}(\mathcal{E})\leqslant p_{\rm
G}(\mathcal{E}).$ (15)

As $p_{\rm L}(\mathcal{E})$ and $p_{\operatorname{PPT}}(\mathcal{E})$ have the
same objective function to maximize, the first inequality in (15) becomes an
equality if and only if there exists a LOCC measurement realizing $p_{\rm
PPT}(\mathcal{E})$. Similarly, the second inequality in (15) becomes an
equality if and only if there exists a PPT measurement realizing $p_{\rm
G}(\mathcal{E})$.

##  III PPT measurements and quantum state discrimination

In this section, we provide the first main result of our paper. We first
consider an upper bound of $p_{\operatorname{PPT}}(\mathcal{E})$ in Eq. (13),
and show that this upper bound is equal to
$p_{\operatorname{PPT}}(\mathcal{E})$. We further provide conditions on the
maximum success probability over all possible PPT measurements. The results
will be used to obtain necessary and/or sufficient conditions for minimum-
error discrimination by PPT measurements in the next section.
For a bipartite quantum state ensemble $\mathcal{E}$ in Eq. (10), we define
$\mathbb{H}_{\operatorname{PPT}}(\mathcal{E})$ as

$\mathbb{H}_{\operatorname{PPT}}(\mathcal{E})=\\{H\in\mathbb{H}\,|\,H-\eta_{i}\rho_{i}\in\mathbb{PPT}_{+}^{*}~{}\mbox{for
any}~{}i\in\\{1,\ldots,n\\}\\},$ (16)

where $\mathbb{PPT}_{+}^{*}$ is defined in Eq. (2). In other words,
$\mathbb{H}_{\operatorname{PPT}}(\mathcal{E})$ is the set of all $H$ such that
$H-\eta_{i}\rho_{i}$ is decomposable in $\mathbb{H}$, for all $i=1,\ldots,n$.
We further define

$\mathbb{H}_{\operatorname{DEW}}(\mathcal{E})=\\{H\in\mathbb{H}_{\operatorname{PPT}}(\mathcal{E})\,|\,H-\eta_{j}\rho_{j}~{}\mbox{is
a DEW}~{}\mbox{for some}~{}j\in\\{1,\ldots,n\\}\\},$ (17)

that is, a subset of $\mathbb{H}_{\operatorname{PPT}}(\mathcal{E})$ satisfying
$H-\eta_{j}\rho_{j}\not\succeq 0$ for some $j\in\\{1,\ldots,n\\}$. From the
argument after Definition 3, we can see that

$H\in\mathbb{H}_{\operatorname{PPT}}(\mathcal{E})\setminus\mathbb{H}_{\operatorname{DEW}}(\mathcal{E})$
(18)

if and only if

$H-\eta_{i}\rho_{i}\succeq 0~{}\forall i=1,\ldots,n.$ (19)

Now, let us consider the minimum quantity

$q_{\operatorname{PPT}}(\mathcal{E})=\min_{H\in\mathbb{H}_{\operatorname{PPT}}(\mathcal{E})}\operatorname{Tr}H,$
(20)

which is an upper bound of $p_{\rm PPT}(\mathcal{E})$cose2013 , that is,

$p_{\rm PPT}(\mathcal{E})\leqslant q_{\operatorname{PPT}}(\mathcal{E}).$ (21)

The following theorem shows that $p_{\rm PPT}(\mathcal{E})$ in Eq. (13) is
equal to $q_{\operatorname{PPT}}(\mathcal{E})$ in Eq. (20). The proof of
Theorem 1 is given in Appendix A.

######  Theorem 1.

For a bipartite quantum state ensemble
$\mathcal{E}=\\{\eta_{i},\rho_{i}\\}_{i=1}^{n}$,

$p_{\rm PPT}(\mathcal{E})=q_{\operatorname{PPT}}(\mathcal{E}).$ (22)

For a given ensemble $\mathcal{E}=\\{\eta_{i},\rho_{i}\\}_{i=1}^{n}$, the
following theorem provides a necessary and sufficient condition for a PPT
measurement $\\{M_{i}\\}_{i=1}^{n}$ and
$H\in\mathbb{H}_{\operatorname{PPT}}(\mathcal{E})$ to realize
$p_{\operatorname{PPT}}(\mathcal{E})$ and
$q_{\operatorname{PPT}}(\mathcal{E})$, respectively.

######  Theorem 2.

For a bipartite quantum state ensemble
$\mathcal{E}=\\{\eta_{i},\rho_{i}\\}_{i=1}^{n}$, a PPT measurement
$\\{M_{i}\\}_{i=1}^{n}$ and
$H\in\mathbb{H}_{\operatorname{PPT}}(\mathcal{E})$, $\\{M_{i}\\}_{i=1}^{n}$
realizes $p_{\operatorname{PPT}}(\mathcal{E})$ and $H$ provides
$q_{\operatorname{PPT}}(\mathcal{E})$ if and only if

$\operatorname{Tr}[M_{i}(H-\eta_{i}\rho_{i})]=0~{}~{}\forall i=1,\ldots,n.$
(23)

###### Proof.

Let us suppose that $\\{M\\}_{i=1}^{n}$ and $H$ give
$p_{\operatorname{PPT}}(\mathcal{E})$ and
$q_{\operatorname{PPT}}(\mathcal{E})$, respectively. From
$M_{i}\in\mathbb{PPT}_{+}$ and $H-\eta_{i}\rho_{i}\in\mathbb{PPT}_{+}^{*}$ for
all $i=1,\ldots,n$, we have

$\operatorname{Tr}[M_{i}(H-\eta_{i}\rho_{i})]\geqslant 0~{}\forall
i=1,\ldots,n.$ (24)

We note that

$\sum_{i=1}^{n}\operatorname{Tr}[M_{i}(H-\eta_{i}\rho_{i})]=\operatorname{Tr}H-\sum_{i=1}^{n}\eta_{i}\operatorname{Tr}(\rho_{i}M_{i})=q_{\operatorname{PPT}}(\mathcal{E})-p_{\operatorname{PPT}}(\mathcal{E})=0,$
(25)

where the first equality is from $\sum_{i=1}^{n}M_{i}=\mathbbm{1}$, the second
equality is due to the assumption of $H$ and $\\{M_{i}\\}_{i=1}^{n}$, and the
last equality is by Theorem 1. Inequality (24) and Eq. (25) lead us to
Condition (23).
Conversely let us assume that $\\{M_{i}\\}_{i=1}^{n}$ and $H$ satisfy
Condition (23). This assumption implies

$q_{\operatorname{PPT}}(\mathcal{E})=p_{\operatorname{PPT}}(\mathcal{E})\geqslant\sum_{i=1}^{n}\eta_{i}\mathrm{Tr}(\rho_{i}M_{i})=\sum_{i=1}^{n}\eta_{i}\mathrm{Tr}(\rho_{i}M_{i})+\sum_{i=1}^{n}\mathrm{Tr}[M_{i}(H-\eta_{i}\rho_{i})]=\mathrm{Tr}H\geqslant
q_{\operatorname{PPT}}(\mathcal{E}),$ (26)

where the first equality follows from Theorem 1, the second equality is from
Condition (23), the last equality is due to $\sum_{i=1}^{n}M_{i}=\mathbbm{1}$,
and the first and second inequalities are from the definitions of
$p_{\operatorname{PPT}}(\mathcal{E})$ and
$q_{\operatorname{PPT}}(\mathcal{E})$, respectively. Inequality (26) leads us
to

$\sum_{i=1}^{n}\eta_{i}\mathrm{Tr}(\rho_{i}M_{i})=p_{\operatorname{PPT}}(\mathcal{E}),~{}~{}\operatorname{Tr}H=q_{\operatorname{PPT}}(\mathcal{E}).$
(27)

Thus, $\\{M_{i}\\}_{i=1}^{n}$ and $H$ give
$p_{\operatorname{PPT}}(\mathcal{E})$ and
$q_{\operatorname{PPT}}(\mathcal{E})$, respectively. ∎

We note that $H\in\mathbb{H}_{\operatorname{PPT}}(\mathcal{E})$ giving
$q_{\operatorname{PPT}}(\mathcal{E})$ is generally not unique (see Example 2
in Section IV.2). However, the following corollary states the case that
$H\in\mathbb{H}_{\operatorname{PPT}}(\mathcal{E})$ providing
$q_{\operatorname{PPT}}(\mathcal{E})$ is unique.

######  Corollary 1.

For a bipartite quantum state ensemble
$\mathcal{E}=\\{\eta_{i},\rho_{i}\\}_{i=1}^{n}$, we have

$p_{\operatorname{PPT}}(\mathcal{E})=\eta_{1},$ (28)

if and only if

$\eta_{1}\rho_{1}-\eta_{i}\rho_{i}\in\mathbb{PPT}_{+}^{*}~{}~{}\forall
i=2,\ldots,n.$ (29)

In this case, $\eta_{1}\rho_{1}$ is the only element of
$\mathbb{H}_{\operatorname{PPT}}(\mathcal{E})$ providing
$q_{\operatorname{PPT}}(\mathcal{E})$.

###### Proof.

Let $\\{M_{i}\\}_{i=1}^{n}$ be the measurement that $M_{1}=\mathbbm{1}$ and
$M_{2},\ldots,M_{n}$ are the zero operator in $\mathbb{H}$. We first assume
Eq. (28) and consider $H\in\mathbb{H}_{\operatorname{PPT}}(\mathcal{E})$
providing $q_{\operatorname{PPT}}(\mathcal{E})$. Since $\\{M_{i}\\}_{i=1}^{n}$
is obviously a PPT measurement giving $p_{\operatorname{PPT}}(\mathcal{E})$,
it follows from Theorem 2 that $\operatorname{Tr}(H-\eta_{1}\rho_{1})=0$. From
$H-\eta_{1}\rho_{1}\in\mathbb{PPT}_{+}^{*}$ and Lemma 1 in Appendix A, we have
$H=\eta_{1}\rho_{1}$. Thus, $H\in\mathbb{H}_{\operatorname{PPT}}(\mathcal{E})$
together with the definition of $\mathbb{H}_{\operatorname{PPT}}(\mathcal{E})$
leads us to Condition (29).
Conversely, let us suppose Condition (29) and consider $H=\eta_{1}\rho_{1}$.
Condition (29) implies $H\in\mathbb{H}_{\operatorname{PPT}}(\mathcal{E})$. The
PPT measurement $\\{M_{i}\\}_{i=1}^{n}$ and
$H\in\mathbb{H}_{\operatorname{PPT}}(\mathcal{E})$ satisfy Condition (23).
Therefore, we have

$p_{\operatorname{PPT}}(\mathcal{E})=q_{\operatorname{PPT}}(\mathcal{E})=\operatorname{Tr}H=\eta_{1},$
(30)

where the first equality is from Theorem 1 and the second equality follows
from Theorem 2. ∎

When Eq. (28) of Corollary 1 holds, the maximum success probability
$p_{\operatorname{PPT}}(\mathcal{E})$ can be achieved without the help of
measurement, simply by guessing $\rho_{1}$ is prepared. As we can check in the
proof of Corollary 1, the choice of $\rho_{1}$ in Corollary 1 can be
arbitrary. That is, any of $\\{\rho_{i}\\}_{i=1}^{n}$ can be used to play the
role of $\rho_{1}$ in Corollary 1.

##  IV Minimum-error discrimination by PPT measurements

In this section, we provide another main result of our paper showing that the
minimum-error discrimination of bipartite quantum states using PPT
measurements strongly depends on the existence of DEW. More precisely, we
establish necessary and/or sufficient conditions for minimum-error
discrimination by PPT measurements in terms of DEW.
For a bipartite quantum state ensemble $\mathcal{E}$ in Eq. (10), the minimum-
error discrimination can be realized by PPT measurements if and only if

$p_{\operatorname{PPT}}(\mathcal{E})=p_{\rm G}(\mathcal{E}),$ (31)

where $p_{\rm G}(\mathcal{E})$ and $p_{\operatorname{PPT}}(\mathcal{E})$ are
defined in Eqs. (11) and (13), respectively. Here, we provide necessary and/or
sufficient conditions for Eq. (31) in terms of DEW.

###  IV.1 Necessary condition for $p_{\operatorname{PPT}}(\mathcal{E})=p_{\rm
G}(\mathcal{E})$

######  Theorem 3.

For a bipartite quantum state ensemble
$\mathcal{E}=\\{\eta_{i},\rho_{i}\\}_{i=1}^{n}$, if
$p_{\operatorname{PPT}}(\mathcal{E})=p_{\rm G}(\mathcal{E})$, then there does
not exist PPT measurement $\\{M_{i}\\}_{i=1}^{n}$ satisfying

$\sum_{i=1}^{n}\eta_{i}\rho_{i}M_{i}\in\mathbb{H}_{\operatorname{DEW}}(\mathcal{E}),$
(32)

where $\mathbb{H}_{\operatorname{DEW}}(\mathcal{E})$ is defined in Eq. (17).
Moreover, if $\\{M_{i}\\}_{i=1}^{n}$ is a PPT measurement satisfying Condition
(32), we have

$p_{\operatorname{PPT}}(\mathcal{E})=\sum_{i=1}^{n}\eta_{i}\operatorname{Tr}(\rho_{i}M_{i}).$
(33)

###### Proof.

Let us suppose $\\{M_{i}\\}_{i=1}^{n}$ is a PPT measurement satisfying
Condition (32), and consider

$H=\sum_{i=1}^{n}\eta_{i}\rho_{i}M_{i}.$ (34)

Equation (33) holds because

$\sum_{i=1}^{n}\eta_{i}\operatorname{Tr}(\rho_{i}M_{i})\leqslant
p_{\operatorname{PPT}}(\mathcal{E})=q_{\operatorname{PPT}}(\mathcal{E})\leqslant\operatorname{Tr}H=\sum_{i=1}^{n}\eta_{i}\operatorname{Tr}(\rho_{i}M_{i}),$
(35)

where the first and second inequalities follow from the definitions of
$p_{\operatorname{PPT}}(\mathcal{E})$ and
$q_{\operatorname{PPT}}(\mathcal{E})$, respectively, the first equality is due
to Theorem 1, and the second equality is from the definition of $H$. This
proves the second statement of our theorem.
Now, assume $p_{\operatorname{PPT}}(\mathcal{E})=p_{\rm G}(\mathcal{E})$. If
there exists a PPT measurement $\\{M_{i}^{\prime}\\}_{i=1}^{n}$ satisfying
Condition (32), the second statement of our theorem implies that
$\\{M_{i}^{\prime}\\}_{i=1}^{n}$ gives the optimal success probability $p_{\rm
G}(\mathcal{E})$ due to Eq. (33). From the optimality condition in Eq. (12),
we have

$\sum_{j=1}^{n}\eta_{j}\rho_{j}M_{j}^{\prime}-\eta_{i}\rho_{i}\succeq
0~{}~{}\forall i=1,\ldots,n,$ (36)

which contradicts Condition (32). Thus, there does not exist PPT measurement
satisfying Condition (32). This proves the first statement of our theorem. ∎

######  Example 1.

For any integer $d\geqslant 2$, let us consider the two-qudit state ensemble
$\mathcal{E}=\\{\eta_{i,j}^{(k,l)},\rho_{i,j}^{(k,l)}\\}_{i,j,k,l}$ consisting
of $2d(d-1)$ mixed states with equal prior probabilities,

$\displaystyle\eta_{i,j}^{(k,l)}=\frac{1}{2d(d-1)},~{}\rho_{i,j}^{(k,l)}=\frac{\lambda}{3}(\Psi_{i,j}^{(k)}+\Pi_{i,j}^{(l)})+(1-\lambda)\sigma,$
$\displaystyle
i,j\in\\{0,1,\ldots,d-1\\}~{}\mbox{with}~{}i<j,~{}~{}k,l\in\\{1,2\\},$ (37)

where $\sigma$ is an arbitrary two-qudit state, $0<\lambda\leqslant 1$, and

$\displaystyle\Psi_{i,j}^{(1)}=\frac{1}{2}(\ket{ii}+\ket{jj})(\bra{ii}+\bra{jj}),$
$\displaystyle~{}\Pi_{i,j}^{(1)}=\ket{ii}\\!\bra{ii}+\ket{jj}\\!\bra{jj},$
$\displaystyle\Psi_{i,j}^{(2)}=\frac{1}{2}(\ket{ii}-\ket{jj})(\bra{ii}-\bra{jj}),$
$\displaystyle~{}\Pi_{i,j}^{(2)}=\ket{ij}\\!\bra{ij}+\ket{ji}\\!\bra{ji}.$
(38)

For a PPT measurement $\\{M_{i,j}^{(k,l)}\\}_{i,j,k,l}$ with

$M_{i,j}^{(1,1)}=M_{i,j}^{(2,1)}=\frac{1}{2(d-1)}\Pi_{i,j}^{(1)},~{}M_{i,j}^{(1,2)}=M_{i,j}^{(2,2)}=\frac{1}{2}\Pi_{i,j}^{(2)},$
(39)

we show that Condition (32) holds with respect to the ensemble in Eq. (37) and
the measurement in Eq. (39). It is straightforward to verify that

$\displaystyle\Big{[}\sum_{i^{\prime},j^{\prime},k^{\prime},l^{\prime}}\eta_{i^{\prime},j^{\prime}}^{(k^{\prime},l^{\prime})}\rho_{i^{\prime},j^{\prime}}^{(k^{\prime},l^{\prime})}M_{i^{\prime},j^{\prime}}^{(k^{\prime},l^{\prime})}-\eta_{i,j}^{(k,1)}\rho_{i,j}^{(k,1)}\Big{]}^{\operatorname{PT}}$
$\displaystyle=\frac{\lambda}{4d(d-1)}\Big{[}\Psi_{i,j}^{(5-k)}+\frac{1}{3}\Psi_{i,j}^{(k+2)}+\hat{\Pi}_{i,j}^{(1)}+\frac{2}{3}\hat{\Pi}_{i,j}^{(2)}\Big{]}\succeq
0,$
$\displaystyle\sum_{i^{\prime},j^{\prime},k^{\prime},l^{\prime}}\eta_{i^{\prime},j^{\prime}}^{(k^{\prime},l^{\prime})}\rho_{i^{\prime},j^{\prime}}^{(k^{\prime},l^{\prime})}M_{i^{\prime},j^{\prime}}^{(k^{\prime},l^{\prime})}-\eta_{i,j}^{(k,2)}\rho_{i,j}^{(k,2)}$
$\displaystyle=\frac{\lambda}{4d(d-1)}\Big{[}\Psi_{i,j}^{(3-k)}+\frac{1}{3}\Psi_{i,j}^{(k)}+\hat{\Pi}_{i,j}^{(1)}+\frac{2}{3}\hat{\Pi}_{i,j}^{(2)}\Big{]}\succeq
0,$ $\displaystyle\forall
i,j\in\\{0,1,\ldots,d-1\\}~{}\mbox{with}~{}i<j,~{}~{}\forall k\in\\{1,2\\},$
(40)

where

$\displaystyle\Psi_{i,j}^{(3)}=\frac{1}{2}(\ket{ij}+\ket{ji})(\bra{ij}+\bra{ji}),~{}$
$\displaystyle\hat{\Pi}_{i,j}^{(1)}=\sum_{i^{\prime}=0}^{d-1}\ket{i^{\prime}i^{\prime}}\\!\bra{i^{\prime}i^{\prime}}-\Pi_{i,j}^{(1)},$
$\displaystyle\Psi_{i,j}^{(4)}=\frac{1}{2}(\ket{ij}-\ket{ji})(\bra{ij}-\bra{ji}),~{}$
$\displaystyle\hat{\Pi}_{i,j}^{(2)}=\sum_{\begin{subarray}{c}i^{\prime},j^{\prime}=0\\\
i^{\prime}\neq
j^{\prime}\end{subarray}}^{d-1}\ket{i^{\prime}j^{\prime}}\\!\bra{i^{\prime}j^{\prime}}-\Pi_{i,j}^{(2)}.$
(41)

From the positivity in (40), we have

$\displaystyle\sum_{i^{\prime},j^{\prime},k^{\prime},l^{\prime}}\eta_{i^{\prime},j^{\prime}}^{(k^{\prime},l^{\prime})}\rho_{i^{\prime},j^{\prime}}^{(k^{\prime},l^{\prime})}M_{i^{\prime},j^{\prime}}^{(k^{\prime},l^{\prime})}-\eta_{i,j}^{(k,l)}\rho_{i,j}^{(k,l)}\in\mathbb{PPT}_{+}^{*}$
$\displaystyle\forall
i,j\in\\{0,1,\ldots,d-1\\}~{}\mbox{with}~{}i<j,~{}~{}\forall k,l\in\\{1,2\\}.$
(42)

Furthermore, a straightforward calculation leads us to

$\displaystyle\operatorname{Tr}\Big{[}\Big{(}\sum_{i^{\prime},j^{\prime},k^{\prime},l^{\prime}}\eta_{i^{\prime},j^{\prime}}^{(k^{\prime},l^{\prime})}\rho_{i^{\prime},j^{\prime}}^{(k^{\prime},l^{\prime})}M_{i^{\prime},j^{\prime}}^{(k^{\prime},l^{\prime})}-\eta_{i,j}^{(k,1)}\rho_{i,j}^{(k,1)}\Big{)}\Psi_{i,j}^{(k)}\Big{]}=-\frac{\lambda}{12d(d-1)}$
$\displaystyle\forall
i,j\in\\{0,1,\ldots,d-1\\}~{}\mbox{with}~{}i<j,~{}~{}\forall k\in\\{1,2\\}.$
(43)

From Eq. (43), we have

$\displaystyle\sum_{i^{\prime},j^{\prime},k^{\prime},l^{\prime}}\eta_{i^{\prime},j^{\prime}}^{(k^{\prime},l^{\prime})}\rho_{i^{\prime},j^{\prime}}^{(k^{\prime},l^{\prime})}M_{i^{\prime},j^{\prime}}^{(k^{\prime},l^{\prime})}-\eta_{i,j}^{(k,1)}\rho_{i,j}^{(k,1)}\not\succeq
0$ $\displaystyle\forall
i,j\in\\{0,1,\ldots,d-1\\}~{}\mbox{with}~{}i<j,~{}~{}\forall k\in\\{1,2\\}.$
(44)

From Eqs. (42) and (44), the ensemble in Eq. (37) and the measurement in Eq.
(39) satisfy Condition (32).
The second statement of Theorem 3 implies that
$p_{\operatorname{PPT}}(\mathcal{E})$ is the expectation of the measurement in
Eq. (39) about the ensemble in Eq. (37):

$p_{\operatorname{PPT}}(\mathcal{E})=\frac{6+\lambda(2d-3)(d+2)}{12d(d-1)}.$
(45)

From the first statement of Theorem 3, we have

$p_{\operatorname{PPT}}(\mathcal{E})<p_{\rm G}(\mathcal{E}).$ (46)

We also note that the measurement in Eq. (39) is a LOCC measurement because it
can be implemented by performing the same local measurement
$\\{\ket{l}\\!\bra{l}\\}_{l=0}^{d-1}$ on each party. Thus, we have

$p_{\rm
L}(\mathcal{E})=p_{\operatorname{PPT}}(\mathcal{E})=\frac{6+\lambda(2d-3)(d+2)}{12d(d-1)}.$
(47)

###  IV.2 Necessary and sufficient condition for
$p_{\operatorname{PPT}}(\mathcal{E})=p_{\rm G}(\mathcal{E})$

######  Theorem 4.

For a bipartite quantum state ensemble
$\mathcal{E}=\\{\eta_{i},\rho_{i}\\}_{i=1}^{n}$,
$p_{\operatorname{PPT}}(\mathcal{E})=p_{\rm G}(\mathcal{E})$ if and only if
there exists $H\in\mathbb{H}_{\operatorname{PPT}}(\mathcal{E})$ such that it
provides $q_{\operatorname{PPT}}(\mathcal{E})$ but does not satisfy

$H\in\mathbb{H}_{\operatorname{DEW}}(\mathcal{E}),$ (48)

or equivalently, there is $H\in\mathbb{H}$ satisfying Condition (18) and
$\operatorname{Tr}H=q_{\operatorname{PPT}}(\mathcal{E})$.

###### Proof.

Let $\\{M_{i}\\}_{i=1}^{n}$ be a PPT measurement providing
$p_{\operatorname{PPT}}(\mathcal{E})$. We first suppose
$p_{\operatorname{PPT}}(\mathcal{E})=p_{\rm G}(\mathcal{E})$ and consider

$H=\sum_{i=1}^{n}\eta_{i}\rho_{i}M_{i}.$ (49)

Since the measurement $\\{M_{i}\\}_{i=1}^{n}$ gives the optimal success
probability $p_{\rm G}(\mathcal{E})$ due to the assumption of
$\\{M_{i}\\}_{i=1}^{n}$ and $p_{\operatorname{PPT}}(\mathcal{E})=p_{\rm
G}(\mathcal{E})$, we have from the optimality condition in Eq. (12) that

$\displaystyle
H-\eta_{i}\rho_{i}=\sum_{j=1}^{n}\eta_{j}\rho_{j}M_{j}-\eta_{i}\rho_{i}\succeq
0~{}~{}\forall i=1,\ldots,n.$ (50)

Therefore, $H$ satisfies Condition (18). Moreover, we have

$\operatorname{Tr}H=\sum_{i=1}^{n}\eta_{i}\operatorname{Tr}(\rho_{i}M_{i})=p_{\operatorname{PPT}}(\mathcal{E})=q_{\operatorname{PPT}}(\mathcal{E}),$
(51)

where the first equality is from Eq. (49), the second equality follows from
the assumption of $\\{M_{i}\\}_{i=1}^{n}$, and the last equality is due to
Theorem 1.
Conversely, let us assume $H$ is an element of $\mathbb{H}$ satisfying
Condition (18) and $\operatorname{Tr}H=q_{\operatorname{PPT}}(\mathcal{E})$.
Thus, the positivity in (19) is satisfied in terms of $H$. For each
$i=1,\ldots,n$, the positive-semidefinite operators $M_{i}$ and
$H-\eta_{i}\rho_{i}$ are orthogonal since they satisfy Condition (23) from
Theorem 2.
The optimality condition in Eq. (12) holds for the measurement
$\\{M_{i}\\}_{i=1}^{n}$ because

$\sum_{j=1}^{n}\eta_{j}\rho_{j}M_{j}-\eta_{i}\rho_{i}=\sum_{j=1}^{n}\eta_{j}\rho_{j}M_{j}+\sum_{k=1}^{n}(H-\eta_{k}\rho_{k})M_{k}-\eta_{i}\rho_{i}=H-\eta_{i}\rho_{i}\succeq
0~{}~{}\forall i=1,\ldots,n,$ (52)

where the first equality is from the orthogonality of $M_{i}$ and
$H-\eta_{i}\rho_{i}$ for each $i=1,\ldots,n$ and the second equality is from
$\sum_{i=1}^{n}M_{i}=\mathbbm{1}$. Thus, we have

$p_{\rm
G}(\mathcal{E})=\sum_{i=1}^{n}\eta_{i}\operatorname{Tr}(\rho_{i}M_{i})=p_{\operatorname{PPT}}(\mathcal{E}),$
(53)

where the second equality is due to the assumption of $\\{M_{i}\\}_{i=1}^{n}$.
∎

If $p_{\operatorname{PPT}}(\mathcal{E})=p_{\rm G}(\mathcal{E})$, Theorem 4
implies that there must exist
$H\in\mathbb{H}_{\operatorname{PPT}}(\mathcal{E})\setminus\mathbb{H}_{\operatorname{DEW}}(\mathcal{E})$
giving $q_{\operatorname{PPT}}(\mathcal{E})$. In this case, there possibly
exists another Hermitian operator $H^{\prime}$ satisfying
$H^{\prime}\in\mathbb{H}_{\operatorname{DEW}}(\mathcal{E})$ and
$\operatorname{Tr}H^{\prime}=q_{\operatorname{PPT}}(\mathcal{E})$, which is
illustrated in the following example.

######  Example 2.

Let us consider the two-qubit state ensemble
$\mathcal{E}=\\{\eta_{i},\rho_{i}\\}_{i=1}^{3}$ consisting of three pure
orthogonal states with equal prior probabilities,

$\displaystyle\eta_{1}=\frac{1}{3},~{}\rho_{1}=\ket{0}\\!\bra{0}\otimes\ket{0}\\!\bra{0},$
$\displaystyle\eta_{2}=\frac{1}{3},~{}\rho_{2}=\ket{1}\\!\bra{1}\otimes\ket{1}\\!\bra{1},$
$\displaystyle\eta_{3}=\frac{1}{3},~{}\rho_{3}=\Psi_{+},$ (54)

where

$\Psi_{\pm}=\ket{\Psi_{\pm}}\\!\bra{\Psi_{\pm}},~{}\ket{\Psi_{\pm}}=\frac{1}{\sqrt{2}}(\ket{01}\pm\ket{10}).$
(55)

Since the states $\rho_{1}$ and $\rho_{2}$ are product states, we have

$p_{\rm L}(\mathcal{E})=1.$ (56)

We note that three $2\otimes 2$ pure orthogonal states can be perfectly
discriminated by a finite-round LOCC if and only if at least two of them are
product stateswalg2002 . Since $p_{\rm G}(\mathcal{E})$ is bounded above by 1,
it follows from Inequality (15) and Eq. (56) that

$p_{\rm L}(\mathcal{E})=p_{\operatorname{PPT}}(\mathcal{E})=p_{\rm
G}(\mathcal{E})=1.$ (57)

Furthermore, we have

$q_{\operatorname{PPT}}(\mathcal{E})=p_{\operatorname{PPT}}(\mathcal{E})=1,$
(58)

where the first equality follows from Theorem 1 and the second equality is due
to Eq. (57).
Now, let us consider the Hermitian operator

$H=\frac{1}{3}(\Phi_{+}+\Phi_{-})+\frac{1+t}{6}\Psi_{+}+\frac{1-t}{6}\Psi_{-},$
(59)

where $\Psi_{\pm}$ is defined in Eq. (55), $0\leqslant t\leqslant 1$, and

$\Phi_{\pm}=\ket{\Phi_{\pm}}\\!\bra{\Phi_{\pm}},~{}\ket{\Phi_{\pm}}=\frac{1}{\sqrt{2}}(\ket{00}\pm\ket{11}).$
(60)

Equations (58) and (59) imply

$\operatorname{Tr}H=1=q_{\operatorname{PPT}}(\mathcal{E}).$ (61)

A straightforward calculation leads us to

$\displaystyle
H-\eta_{1}\rho_{1}=\frac{1}{3}\rho_{2}+\frac{1+t}{6}\Psi_{+}+\frac{1-t}{6}\Psi_{-}\succeq
0,$ $\displaystyle
H-\eta_{2}\rho_{2}=\frac{1}{3}\rho_{1}+\frac{1+t}{6}\Psi_{+}+\frac{1-t}{6}\Psi_{-}\succeq
0,$ (62) $\displaystyle
H-\eta_{3}\rho_{3}=\frac{1+t}{6}(\Phi_{+}+\Phi_{-})+\frac{1-t}{3}\Phi_{-}^{\operatorname{PT}}\in\mathbb{PPT}_{+}^{*}.$

Due to Eqs. (61) and (62), $H$ is an element of
$\mathbb{H}_{\operatorname{PPT}}(\mathcal{E})$ giving
$q_{\operatorname{PPT}}(\mathcal{E})$ regardless of $t\in[0,1]$. Moreover,
$H\in\mathbb{H}_{\operatorname{DEW}}(\mathcal{E})$ for all $t\in[0,1)$ because

$\operatorname{Tr}[(H-\eta_{3}\rho_{3})\Psi_{+}]=-\frac{1-t}{6}.$ (63)

For a bipartite quantum state ensemble
$\mathcal{E}=\\{\eta_{i},\rho_{i}\\}_{i=1}^{n}$ where $H$ is the only element
of $\mathbb{H}_{\operatorname{PPT}}(\mathcal{E})$ giving
$q_{\operatorname{PPT}}(\mathcal{E})$, Theorem 4 tells us that
$p_{\operatorname{PPT}}(\mathcal{E})=p_{\rm G}(\mathcal{E})$ if and only if
there is no DEW in $\\{H-\eta_{i}\rho_{i}\\}_{i=1}^{n}$. From Corollary 1,
$\eta_{1}\rho_{1}$ is the only element of
$\mathbb{H}_{\operatorname{PPT}}(\mathcal{E})$ providing
$q_{\operatorname{PPT}}(\mathcal{E})$ when Condition (29) holds. Thus, we have
the following corollary.

######  Corollary 2.

For a bipartite quantum state ensemble
$\mathcal{E}=\\{\eta_{i},\rho_{i}\\}_{i=1}^{n}$ with Condition (29),
$p_{\operatorname{PPT}}(\mathcal{E})=p_{\rm G}(\mathcal{E})$ if and only if
there is no DEW in $\\{\eta_{1}\rho_{1}-\eta_{i}\rho_{i}\\}_{i=2}^{n}$.

######  Example 3.

For any integer $d\geqslant 2$, let us consider the two-qu$d$it state ensemble
$\mathcal{E}=\\{\eta_{1},\rho_{1}\\}\cup\\{\eta_{i,j}^{(k)},\rho_{i,j}^{(k)}\\}_{i,j,k}$
consisting of $1+2d(d-1)$ states,

$\displaystyle\eta_{1}=\frac{d}{5d-4},~{}\rho_{1}=\frac{1}{d^{2}}\mathbbm{1},$
$\displaystyle\eta_{i,j}^{(k)}=\frac{2}{d(5d-4)},~{}\rho_{i,j}^{(k)}=(1-\lambda)\Psi_{i,j}^{(k)}+\frac{\lambda}{d^{2}}\mathbbm{1},$
$\displaystyle i,j\in\\{0,1,\ldots,d-1\\}~{}\mbox{with}~{}i<j,~{}k=1,2,3,4,$
(64)

where $0\leqslant\lambda<1$ and $\Psi_{i,j}^{(k)}$ is defined in Eqs. (38) and
(41).

From a straightforward calculation, we can verify that

$\displaystyle\eta_{1}\rho_{1}-\eta_{i,j}^{(k)}\rho_{i,j}^{(k)}$
$\displaystyle=$
$\displaystyle\frac{1}{d^{3}(5d-4)}\Big{[}(d^{2}-2\lambda)(\mathbbm{1}-\mathbbm{1}_{i,j})+(d^{2}-2\lambda)\mathbbm{1}_{i,j}-2d^{2}(1-\lambda)\Psi_{i,j}^{(k)}\Big{]}$
(65) $\displaystyle=$
$\displaystyle\frac{1}{d^{3}(5d-4)}\Big{[}(d^{2}-2\lambda)(\mathbbm{1}-\mathbbm{1}_{i,j})+\lambda(d^{2}-2)\mathbbm{1}_{i,j}+2d^{2}(1-\lambda)\Psi_{i,j}^{(5-k)\operatorname{PT}}\Big{]}\in\mathbb{PPT}_{+}^{*}$
$\displaystyle\forall
i,j\in\\{0,1,\ldots,d-1\\}~{}\mbox{with}~{}i<j,~{}~{}\forall
k\in\\{1,2,3,4\\}.$

where

$\mathbbm{1}_{i,j}=(\ket{i}\\!\bra{i}+\ket{j}\\!\bra{j})\otimes(\ket{i}\\!\bra{i}+\ket{j}\\!\bra{j}),$
(66)

and the second equality in Eq. (65) is due to

$\displaystyle\frac{1}{2}\mathbbm{1}_{i,j}-\Psi_{i,j}^{(k)}=\Psi_{i,j}^{(5-k)\operatorname{PT}}~{}\forall
i,j\in\\{0,1,\ldots,d-1\\}~{}\mbox{with}~{}i<j,~{}~{}\forall
k\in\\{1,2,3,4\\}.$ (67)

From Corollary 1 and Eq. (65), we have

$p_{\operatorname{PPT}}(\mathcal{E})=\eta_{1}=\frac{d}{5d-4}.$ (68)

For each $i,j,k$, we can easily see from the first equality in Eq. (65) that
$\eta_{1}\rho_{1}-\eta_{i,j}^{(k)}\rho_{i,j}^{(k)}\succeq 0$ if and only if

$\operatorname{Tr}\Big{[}\Big{(}\eta_{1}\rho_{1}-\eta_{i,j}^{(k)}\rho_{i,j}^{(k)}\Big{)}\Psi_{i,j}^{(k)}\Big{]}=\frac{2\lambda
d^{2}-2\lambda-d^{2}}{d^{3}(5d-4)}\geqslant 0.$ (69)

Therefore, it follows from Corollary 2 that
$p_{\operatorname{PPT}}(\mathcal{E})=p_{\rm G}(\mathcal{E})$ if and only if

$\lambda\geqslant\frac{d^{2}}{2(d^{2}-1)}.$ (70)

##  V Discussion

We have considered bipartite quantum state discrimination and shown that the
minimum-error discrimination of bipartite quantum states using PPT
measurements strongly depends on the existence of DEW. We have established
conditions on minimum-error discrimination by PPT measurements, that is,
$p_{\operatorname{PPT}}(\mathcal{E})=p_{\rm G}(\mathcal{E})$, in terms of
DEW(Theorems 3 and 4). We have also provided conditions on the maximum success
probability over all possible PPT measurements(Theorems 1 and 2). Our results
have been illustrated by examples of multidimensional bipartite quantum
states.
Quantum nonlocality is a genuine phenomenon of multipartite quantum systems
without any classical counterpart. Quantum nonlocality is a key ingredient
making quantum states outperform the classical ones in various quantum
information processing tasks such as quantum teleportation and quantum
cryptographyeker1991 ; benn1993 . It is also known that this quantum
nonlocality plays an important role in quantum algorithms which are more
powerful than any classical onesdeut1992 ; shor1994 . Quantum nonlocality also
occurs in discriminating multipartite quantum states; in discriminating states
from a bipartite quantum state ensemble $\mathcal{E}$, quantum nonlocality
occurs when optimal state discrimination cannot be realized only by LOCC
measurement, that is, $p_{\rm L}(\mathcal{E})<p_{\rm G}(\mathcal{E})$.
Our results here can be used to detect the occurrence of nonlocality in
quantum state discrimination. Violation of the condition in Theorem 3 or 4
means $p_{\operatorname{PPT}}(\mathcal{E})<p_{\rm G}(\mathcal{E})$, which
consequently implies $p_{\rm L}(\mathcal{E})<p_{\rm G}(\mathcal{E})$. Thus,
Condition (32) of Theorem 3 and Condition (48) of Theorem 4 can be used as
sufficient conditions on $p_{\rm L}(\mathcal{E})<p_{\rm G}(\mathcal{E})$.
Corollary 2 provides a useful method to make a bipartite quantum state
ensemble $\mathcal{E}=\\{\eta_{i},\rho_{i}\\}_{i=1}^{n}$ showing nonlocality,
that is, $p_{\rm L}(\mathcal{E})<p_{\rm G}(\mathcal{E})$, by means of DEW: For
a given DEW $W$, let us consider the bipartite quantum state ensemble
$\mathcal{E}=\\{\eta_{i},\rho_{i}\\}_{i=1}^{2}$ where

$\displaystyle\eta_{1}=\frac{\operatorname{Tr}(P+W)}{\operatorname{Tr}(2P+W)},~{}\rho_{1}=\frac{P+W}{\operatorname{Tr}(P+W)},$
$\displaystyle\eta_{2}=\frac{\operatorname{Tr}P}{\operatorname{Tr}(2P+W)},~{}\rho_{2}=\frac{P}{\operatorname{Tr}P},$
(71)

with any positive-semidefinite operator $P$ satisfying

$P+W\succeq 0.$ (72)

Since $\eta_{1}\rho_{1}-\eta_{2}\rho_{2}$ is proportional to the DEW $W$,
$p_{\operatorname{PPT}}(\mathcal{E})<p_{\rm G}(\mathcal{E})$ holds from
Corollary 2. Thus, Inequality (15) leads us to $p_{\rm L}(\mathcal{E})<p_{\rm
G}(\mathcal{E})$. In other words, Corollary 2 provides a systematic way to
construct a bipartite quantum state ensemble showing nonlocality in quantum
state discrimination.
Corollary 2 can also be used to construct a bipartite quantum state ensemble
$\mathcal{E}=\\{\eta_{i},\rho_{i}\\}_{i=1}^{n}$ with $n>2$ showing nonlocality
in quantum state discrimination. For a set of DEWs $\\{W_{i}\\}_{i=2}^{n}$,
let us consider the bipartite quantum state ensemble
$\mathcal{E}=\\{\eta_{i},\rho_{i}\\}_{i=1}^{n}$ where

$\displaystyle\eta_{1}=\frac{\operatorname{Tr}\mathbbm{1}}{\operatorname{Tr}(n\mathbbm{1}-\sum_{j=2}^{n}\lambda_{j}W_{j})},~{}\rho_{1}=\frac{\mathbbm{1}}{\operatorname{Tr}\mathbbm{1}},$
$\displaystyle\eta_{i}=\frac{\operatorname{Tr}(\mathbbm{1}-\lambda_{i}W_{i})}{\operatorname{Tr}(n\mathbbm{1}-\sum_{j=2}^{n}\lambda_{j}W_{j})},~{}\rho_{i}=\frac{\mathbbm{1}-\lambda_{i}W_{i}}{\operatorname{Tr}(\mathbbm{1}-\lambda_{i}W_{i})},~{}i=2,\ldots,n,$
(73)

with any set of positive real numbers $\\{\lambda_{i}\\}_{i=2}^{n}$ satisfying

$\mathbbm{1}-\lambda_{i}W_{i}\succeq 0~{}~{}\forall i=2,\ldots,n.$ (74)

Because $\eta_{1}\rho_{1}-\eta_{i}\rho_{i}$ is proportional to $W_{i}$ for any
$i\in\\{2,\ldots,n\\}$, $p_{\operatorname{PPT}}(\mathcal{E})<p_{\rm
G}(\mathcal{E})$ holds from Corollary 2. Thus, Inequality (15) leads us to
$p_{\rm L}(\mathcal{E})<p_{\rm G}(\mathcal{E})$.
As our results establish a specific relation between the properties of EW and
PPT measurements, it is natural to investigate the relationship between EW and
other measurements such as the set of all separable measurements. It is also
an interesting future work to investigate if EW can be used for the
optimization of other state discrimination strategies besides minimum-error
discrimination.

This work was supported by Basic Science Research Program(Grant No.
NRF-2020R1F1A1A010501270) and Quantum Computing Technology Development
Program(Grant No. NRF-2020M3E4A1080088) through a National Research Foundation
of Korea grant funded by the Korea government(Ministry of Science and
Information & Communications Technology(ICT)).

##  Appendix A Proof of Theorem 1

As we already have Inequality (21), it is enough to show that

$p_{\operatorname{PPT}}(\mathcal{E})\geqslant
q_{\operatorname{PPT}}(\mathcal{E}).$ (75)

######  Lemma 1.

If $E\in\mathbb{PPT}_{+}^{*}$ and $E\neq 0_{\mathbb{H}}$, then
$\operatorname{Tr}E>0$, where $0_{\mathbb{H}}$ is the zero operator in
$\mathbb{H}$.

###### Proof.

The proof is by contradiction. We first note that
$\operatorname{Tr}E=\operatorname{Tr}(\mathbbm{1}E)\geqslant 0$ because
$\mathbbm{1}\in\mathbb{PPT}_{+}$ and $E\in\mathbb{PPT}_{+}^{*}$. Thus, let us
suppose $\operatorname{Tr}E=0$.
For an arbitrary orthonormal product basis $\\{|e_{i}\rangle\\}_{i=1}^{D}$ of
the bipartite Hilbert space
$\mathcal{H}=\mathbb{C}^{d_{1}}\otimes\mathbb{C}^{d_{2}}$, we have

$\sum_{i=1}^{D}\operatorname{Tr}(E|e_{i}\rangle\\!\langle
e_{i}|)=\operatorname{Tr}(E\mathbbm{1})=\operatorname{Tr}E=0,$ (76)

where $D=d_{1}d_{2}$. From $E\in\mathbb{PPT}_{+}^{*}$ and
$|e_{i}\rangle\\!\langle e_{i}|\in\mathbb{PPT}_{+}$ for all $i=1,\ldots,D$, we
have

$\operatorname{Tr}(E|e_{i}\rangle\\!\langle e_{i}|)\geqslant 0~{}~{}\forall
i=1,\ldots,D.$ (77)

Equation (76) and Inequality (77) lead us to

$\operatorname{Tr}(E|e_{i}\rangle\\!\langle e_{i}|)=0~{}~{}\forall
i=1,\ldots,D.$ (78)

Since the choice of $\\{|e_{i}\rangle\\}_{i=1}^{D}$ can be arbitrary,

$\operatorname{Tr}(E|e\rangle\\!\langle e|)=0$ (79)

for any product vector $|e\rangle\in\mathcal{H}$, therefore

$\operatorname{Tr}(EF)=0~{}~{}\forall F\in\mathbb{SEP},$ (80)

where $\mathbb{SEP}$ is defined in Eq. (6).
We note that $\mathbb{SEP}$ spans $\mathbb{H}$. To see this, we first note
that the set of all positive-semidefinite operators on $\mathbb{C}^{d_{k}}$
spans the set of all Hermitian operators on $\mathbb{C}^{d_{k}}$ for each
$k=1,2$. Moreover, every $F\in\mathbb{H}$ can be represented as a summation of
product Hermitian operators,

$F=\sum_{l}A_{l}\otimes B_{l},$ (81)

where $A_{l}$ and $B_{l}$ are Hermitian operators on $\mathbb{C}^{d_{1}}$ and
$\mathbb{C}^{d_{2}}$, respectively. Therefore, Eq. (80) leads us to

$\operatorname{Tr}(EF)=0~{}~{}\forall F\in\mathbb{H}.$ (82)

Equation (82) means $E=0_{\mathbb{H}}$ which contradicts the assumption of the
lemma. Thus, $\operatorname{Tr}E>0$. ∎

Let us consider the set

$\displaystyle\mathcal{S}(\mathcal{E})=\Big{\\{}\big{(}\sum_{i=1}^{n}\eta_{i}\operatorname{Tr}(\rho_{i}M_{i})-p,\mathbbm{1}-\sum_{i=1}^{n}M_{i}\big{)}\in\mathbb{R}\times\mathbb{H}\,\Big{|}\,p>p_{\rm
PPT}(\mathcal{E}),~{}M_{i}\in\mathbb{PPT}_{+}~{}\forall
i=1,\ldots,n\Big{\\}},$ (83)

where $\mathbb{R}$ is the set of all real numbers. We note that
$\mathcal{S}(\mathcal{E})$ is a convex set due to the convexity of
$\mathbb{PPT}_{+}$ in Eq. (1). Moreover, $\mathcal{S}(\mathcal{E})$ does not
have the origin $(0,0_{\mathbb{H}})$ of $\mathbb{R}\times\mathbb{H}$,
otherwise there is a PPT measurement $\\{M_{i}\\}_{i=1}^{n}$ with

$\sum_{i=1}^{n}\eta_{i}\operatorname{Tr}(\rho_{i}M_{i})>p_{\operatorname{PPT}}(\mathcal{E}),$
(84)

and this contradicts the optimality of $p_{\operatorname{PPT}}(\mathcal{E})$
in Eq. (13). We also note that the Cartesian product
$\mathbb{R}\times\mathbb{H}$ can be considered as a real vector space with an
inner product defined as

$\langle(a,A),(b,B)\rangle=ab+\operatorname{Tr}(AB)$ (85)

for $(a,A),(b,B)\in\mathbb{R}\times\mathbb{H}$.
Since $\mathcal{S}(\mathcal{E})$ and the single-element set
$\\{(0,0_{\mathbb{H}})\\}$ are disjoint convex sets, it follows from the
separating hyperplane theoremboyd2004 ; sht  that there is
$(\gamma,\Gamma)\in\mathbb{R}\times\mathbb{H}$ satisfying

$\displaystyle(\gamma,\Gamma)\neq(0,0_{\mathbb{H}}),$ (86)
$\displaystyle\langle(\gamma,\Gamma),(r,G)\rangle\leqslant
0~{}~{}\forall(r,G)\in\mathcal{S}(\mathcal{E}).$ (87)

Suppose

$\displaystyle\operatorname{Tr}\Gamma\leqslant\gamma
p_{\operatorname{PPT}}(\mathcal{E}),$ (88)
$\displaystyle\Gamma-\gamma\eta_{i}\rho_{i}\in\mathbb{PPT}_{+}^{*}~{}\forall
i=1,\ldots,n,$ (89) $\displaystyle\gamma>0.$ (90)

From Conditions (89) and (90), the Hermitian operator $H=\Gamma/\gamma$ is an
element of $\mathbb{H}_{\operatorname{PPT}}(\mathcal{E})$ in Eq. (16). Thus,
the definition of $q_{\operatorname{PPT}}(\mathcal{E})$ in Eq. (20) leads us
to

$q_{\operatorname{PPT}}(\mathcal{E})\leqslant\operatorname{Tr}H.$ (91)

Moreover, Condition (88) implies

$\operatorname{Tr}H\leqslant p_{\operatorname{PPT}}(\mathcal{E}).$ (92)

Inequalities (91) and (92) complete the proof of Inequality (75).
The rest of this section is to prove Conditions (88), (89) and (90).

###### Proof of (88).

From Eq. (85), Inequality (87) can be rewritten as

$\operatorname{Tr}\Gamma-\sum_{i=1}^{n}\operatorname{Tr}[M_{i}(\Gamma-\gamma\eta_{i}\rho_{i})]\leqslant\gamma
p$ (93)

for all $p>p_{\operatorname{PPT}}(\mathcal{E})$ and all
$\\{M_{i}\\}_{i=1}^{n}\subseteq\mathbb{PPT}_{+}$. If $M_{i}=0_{\mathbb{H}}$
for all $i=1,\ldots,n$, Inequality (93) becomes Inequality (88) by taking the
limit of $p$ to $p_{\operatorname{PPT}}(\mathcal{E})$. ∎

###### Proof of (89).

For each $j\in\\{1,\ldots,n\\}$, let us consider an arbitrary
$M_{j}\in\mathbb{PPT}_{+}$ and $M_{i}=0_{\mathbb{H}}$ for all $i=1,\ldots,n$
with $i\neq j$. In this case, $\\{M_{i}\\}_{i=1}^{n}$ is clearly a subset of
$\mathbb{PPT}_{+}$, and Inequality (93) becomes

$\operatorname{Tr}\Gamma-\operatorname{Tr}[M_{j}(\Gamma-\gamma\eta_{j}\rho_{j})]\leqslant\gamma
p_{\operatorname{PPT}}(\mathcal{E})$ (94)

by taking the limit of $p$ to $p_{\operatorname{PPT}}(\mathcal{E})$.
Suppose $\Gamma-\gamma\eta_{j}\rho_{j}\notin\mathbb{PPT}_{+}^{*}$, then there
is $M\in\mathbb{PPT}_{+}$ with
$\operatorname{Tr}[M(\Gamma-\gamma\eta_{j}\rho_{j})]<0$. We note that
$M\in\mathbb{PPT}_{+}$ implies $tM\in\mathbb{PPT}_{+}$ for any $t>0$. Thus,
$\\{M_{i}\\}_{i=1}^{n}$ consisting of $M_{j}=tM$ for $t>0$ and $M_{i}=0$ for
all $i=1,\ldots,n$ with $i\neq j$ is also a subset of $\mathbb{PPT}_{+}$.
Now, Inequality (94) can be rewritten as

$\operatorname{Tr}\Gamma-\operatorname{Tr}[tM(\Gamma-\gamma\eta_{j}\rho_{j})]\leqslant\gamma
p_{\operatorname{PPT}}(\mathcal{E}).$ (95)

Since Inequality (95) is true for arbitrary large $t>0$, $\gamma
p_{\operatorname{PPT}}(\mathcal{E})$ can also be arbitrarily large. However,
this contradicts that both $\gamma$ and $p_{\operatorname{PPT}}(\mathcal{E})$
are finite. Thus, $\Gamma-\gamma\eta_{j}\rho_{j}\in\mathbb{PPT}_{+}^{*}$,
which completes the proof of (89). ∎

###### Proof of (90).

To show $\gamma\geqslant 0$, we assume $\gamma<0$. Let us consider
$\\{M_{i}\\}_{i=1}^{n}$ with $M_{i}=0_{\mathbb{H}}$ for all $i=1,\ldots,n$.
Since $\\{M_{i}\\}_{i=1}^{n}\subseteq\mathbb{PPT}_{+}$, Inequality (93)
becomes

$\operatorname{Tr}\Gamma\leqslant-\infty$ (96)

by taking the limit of $p$ to $\infty$. This contradicts that $\Gamma$ is
bounded. Thus $\gamma\geqslant 0$.
Now, let us suppose $\gamma=0$. In this case, Conditions (88) and (89) become

$\operatorname{Tr}\Gamma\leqslant 0,~{}~{}\Gamma\in\mathbb{PPT}_{+}^{*}.$ (97)

From Lemma 1 together with Condition (97), we have

$\Gamma=0_{\mathbb{H}},$ (98)

which contradicts Condition (86). Thus, $\gamma>0$. ∎

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# The Future of Traditional Fuel Vehicles (TFV) and New Energy Vehicles (NEV):
Creative Destruction or Co-existence?

Zhaojia Huang  Liang Zhang  Tianhao Zhi  Guangdong Provincial Key Laboratory
of Interdisciplinary Research and Application for Data Science, BNU-HKBU
United International College, Zhuhai 519087, China  Gies College of Business,
University of Illinois Urbana-Champaign, USA

###### Abstract

There is a rapid development and commercialization of new Energy Vehicles
(NEV) in recent years. Although traditional fuel vehicles (TFV) still occupy a
majority share of the market, it is generally believed that NEV is more
efficient, more environmental friendly, and has a greater potential of a
Schumpeterian “creative destruction” that may lead to a paradigm shift in auto
production and consumption. However, less is discussed regarding the potential
environmental impact of NEV production and future uncertainty in R&D
bottleneck of NEV technology and innovation. This paper aims to propose a
modelling framework based on Lux (1995) that investigates the long-term
dynamics of TFV and NEV, along with their associated environmental
externality. We argue that environmental and technological policies will play
a critical role in determining its future development. It is of vital
importance to constantly monitor the potential environmental impact of both
sectors and support the R&D of critical NEV technology, as well as curbing its
negative externality in a preemptive manner.

###### keywords:

New Energy Vehicles, Schumpeterian Growth, Externality
_JEL classification_ : G4, Q27, Q40

##  1 Introduction

New Energy Vehicles, or NEVs, have become increasingly popular globally in
recent years. According to WIND database, the quantity of NEVs had grown from
initially about 6,000 to more than 181,000 from 2017 to 2019 in China. The
number of NEVs in the US have almost doubled over the same period, as shown in
Fig. 1. Although traditional fuel vehicles (TFV) still occupy a majority of
the market, it is generally believed that NEV is more efficient, more
environmental friendly, and has a greater potential of a Schumpeterian
“creative destruction”111See Schumpeter (1942). that may lead to a paradigm
shift in auto production and consumption.

Figure 1: _Source:_ Wind Database

There are numerous research on various factors in NEV consumption and
production. Rezvani et al. (2015) and Gallagher and Muehlegger (2015)
investigate the relationship between consumer preference of NEV in relation to
taxation policy and oil price. Oliver and Rosen (2010) and Wang et al. (2020)
found that customers’ environmental awareness contributes to NEV consumption.
Masiero et al. (2016) study the role of government subsidy policy in promoting
NEV production.

Despite a general optimism in NEV industry at its infant stage, there are at
least two sources of uncertainty that may hinder the future of NEV innovation
and growth. _First_ , it is unclear whether NEV production technology will be
constrained by the current technological bottleneck in R&D in the longer term.
_Second_ , the production of NEV may also lead to another unforeseen
environmental challenge: The production of NEV, especially the permanent
magnetic engine demands a high consumption of rare earth elements (REEs) and
other pollutants such as lithium. 222Frank LIU, “EU Says “No” to Lithium, and
the Underlying Causes Are Worth Attention”, SMM, Retrieved from
https://news.metal.com/newscontent/101856562/EU-Says-“No”-to-Lithium-and-the-
Underlying-Causes-Are-Worth-Attention/
Mark Burton. “Battery Makers Warn EU That Lithium Proposal May Hurt EV
Sector”, Bloomberg, Retrieved from
https://www.bloomberg.com/news/articles/2022-07-04/battery-makers-warn-eu-
that-lithium-proposal-may-hurt-ev-sector  Hawkins et al. (2013) argues that
NEVs have higher environmental burden than ICEVs (Internal Combustion Engine
Vehicle) in production. Adriana et al. (2019) found that producing permanent
magnets will cause environmental externality. Once reaching a large scale, it
may lead to irreversible soil and water pollution that may have severe long-
term environmental consequences (Riba et al., 2016). Despite its importance,
the potential environmental adversity of NEV production is rarely discussed in
current economic literature, which may have a significant potential in
providing well-informed economic, as well as environmental policy guidance
(Jonathan and Florentin, ; Adriana et al., 2019).

The aim of this paper is twofold: _first_ , we propose a theoretical model in
studying the long-term dynamics of TFV and NEV production & consumption, as
well as its environmental externality. We adopt Lux (1995) in modelling the
switching and adoption of TFV and NEV from consumer’s perspective. _Second_ ,
we discuss the potential policy impact on the long-term dynamics based on our
model from three aspects: (i) _laissez-faire_ , (ii) one-sided regulation,
(iii) comprehensive regulation.

The rest of this paper is organized as follows: Section 2 sets up the model.
Section 3 provides some analytical results of the 2D and 3D model. Section 4
discusses the potential impacts of different policy regimes based on numerical
simulations of the 4D model. Finally, Section 5 concludes.

##  2 Model Set-up

###  2.1 Dynamics of transition probability $x$

Lux (1995) proposes a framework of modelling the heterogeneous interactions of
a large quantity of stochastic agents in terms of mean-field interaction. It
is assumed that the number of agents is large enough, so that a master
equation can be derived that captures the mean-field variable. In economics
context, this variable generally implies the average opinion of the economic
agents. There are numerous applications that adopt this framework in studying
opinion contagion and asset price bubbles (Hommes, 2005; Franke, 2012;
Chiarella et al., 2020).

We adopt Lux (1995) to model the switching mechanism of consumers.
Specifically, we assume that there are $2N$ number of consumers that
constantly switch between TFV and NEV based on the transitional probability: a
high transitional probability implies that the consumers are more likely to
switch from one state to another. The transitional probability is determined
by various factors that affect agent’s decision-making. Here we denote
${N_{E}}$ as the number of consumers who prefer NEVs while ${N_{F}}$ who
prefer TFVs. To simplify, we define $n={N_{E}}-{N_{F}}$ and ${x_{t}}=n/2N$.
Here $x$ has two implications. _First_ , $x$ defines the composition of TFV
and NEV in the market. $x=1$ implies that all consumers choose NEV, while
$x=-1$ implies the opposite. $x=0$ implies that there are equal number of TFV
and NEV consumers. Also, as a general case:

$\displaystyle N_{F}$ $\displaystyle=$ $\displaystyle(1-x)N,$ (1)
$\displaystyle\frac{dN_{F}}{dt}$ $\displaystyle=$
$\displaystyle(1-x)\frac{dN}{dt}-\frac{dx}{dt}N,$ (2) $\displaystyle N_{E}$
$\displaystyle=$ $\displaystyle(1+x)N$ (3) $\displaystyle\frac{dN_{E}}{dt}$
$\displaystyle=$ $\displaystyle(1+x)\frac{dN}{dt}+\frac{dx}{dt}N.$ (4)

_Second_ , $x$ captures the average opinion and herd behaviour of consumers.
According to Lux (1995), the consumers are more likely to switch to one
particular state if this is the choice of majority in the hypothetical model.

Here we let $p_{t}^{-+}$ be the probability per unit of time that a person
changes from TFVs to NEVs , and $p_{t}^{+-}$ be the probability change on an
opposite size. According to Lux (1995), the dynamics of $x$ can be derived as:

$\displaystyle\frac{dx}{dt}=p_{t}^{-+}(1-x_{t})-p_{t}^{+-}(1+x_{t}).$ (5)

Lux (1995) points out that the transitional probability possesses three
properties: (i) it has to be positive; (ii) the probability should reflect the
prevailing opinion (contagion), and (iii) the change of transition probability
is in proportion with $x$ ($dp/p=ax$). Hence it can be formulated as:

$\displaystyle p_{t}^{-+}$ $\displaystyle=$ $\displaystyle vexp(s),$ (6)
$\displaystyle p_{t}^{+-}$ $\displaystyle=$ $\displaystyle vexp(-s),$ (7)
$\displaystyle s$ $\displaystyle=$ $\displaystyle
a_{0}+a_{1}x+a_{2}\pi_{F}+a_{3}\pi_{E},$ (8)

where $s$ is the opinion formation index: a higher $s$ implies that the
consumers are more likely to switch from TFV to NEV, as _vice versa_.
$\pi_{F}$ and $\pi_{E}$ are the two environmental externality index caused by
TFV and NEV, which will be discussed in details in the following section.

It is important to note that the parameters in Eq. 8 will be our main focus of
policy analysis. The constant $a_{0}$ captures the consumer preference
regardless of environmental externality: a positive $a_{0}$ implies that the
consumers prefer NEV than TFV. This could be due to the technological &
economic advantage of NEV than TFV, as well as from the government subsidy
that renders NEV economically more attractive. $a_{1}$ is the herding
parameter: a larger $a_{1}$ implies that consumers are more prone to follow
decision of the majority: if a majority of the consumers choose NEV, then it
is more likely that the remaining TFV consumers will switch to NEV, and _vice
versa_. The parameter $a_{2}$ and $a_{3}$ are our regulatory levers. As the
government tries to manage carbon emission in various types of policies such
as carbon tax, TFV becomes less attractive, prompting more consumers to switch
to NEV. Therefore, $a_{2}>0$. Similarly, if the government strives to regulate
NEV pollution, it will make NEV less attractive, thus $a_{3}<0$.

Substituting Eq. 6-7 into Eq. 5, we obtain:

$\displaystyle\frac{dx}{dt}$ $\displaystyle=$ $\displaystyle
v[(1-x_{t})exp(s)-(1+x_{t})exp(-s)],$ (9) $\displaystyle s$ $\displaystyle=$
$\displaystyle a_{0}+a_{1}x+a_{2}\pi_{F}+a_{3}\pi_{E}.$ (10)

###  2.2 Dynamics of Environmental Externality $\pi$

We define two types of environmental externality: _(i)_ the instant
externality from production process (_Type I_), and _(ii)_ the ongoing
externality that stems from usage (_Type II_). Both TFVs and NEVs would
potentially produce both types of externality from production and usage. Yet
for the sake of analytical simplicity, we assume in this paper that NEV is the
primary contributor of _Type I_ externality (soil and water pollution due to
REE), and TFV is the primary contributor of _Type II_ externality (carbon
emission)333In reality however, NEV can also contribute significantly to _type
II_ externality. In the context of electric car for example, it depends on how
electricity is generated. Simiarly, TFV production also contributes to _Type
I_ externality. The assumption here is simply for the sake of analytical
simplicity.. Suppose $N_{x}$ is the number of vehicles, then _Type I_
externality ought to be in proportion with the increase of $N_{x}$ ($dN_{x}$),
and _Type II_ externality in proportion with $N_{x}$ itself. We assign two
parameters $\theta_{x}$ and $\gamma_{x}$ as these two types of externality
parameters respectively. Furthermore, the ecological system exhibits self-
purification, implying that in the long-term, the environmental pollution will
reduce from its own. We assign the two parameters $\alpha_{1}$ and
$\alpha_{2}$ captures the self-purification adjustment rate.

$\displaystyle\frac{d\pi_{F}}{dt}$ $\displaystyle=$
$\displaystyle\gamma_{F}N_{F}-\alpha_{1}\pi_{F},$ (11) $\displaystyle=$
$\displaystyle\gamma_{F}N(1-x)-\alpha_{1}\pi_{F},$ (12)
$\displaystyle\frac{d\pi_{E}}{dt}$ $\displaystyle=$
$\displaystyle\theta_{E}\frac{dN_{E}}{dt}-\alpha_{2}\pi_{E},$ (13)
$\displaystyle=$
$\displaystyle\theta_{E}[\frac{dN}{dt}(1+x)+\frac{dx}{dt}N]-\alpha_{2}\pi_{E}.$
(14)

###  2.3 3D System of $\pi_{F}$, $\pi_{E}$ and $x$

Based on previous set-up, we propose the 3D system of $x$, $\pi_{F}$, and
$\pi_{E}$:

$\displaystyle\frac{dx}{dt}$ $\displaystyle=$ $\displaystyle
v[(1-x)exp(s)-(1+x)exp(-s)],$ (15) $\displaystyle\frac{d\pi_{F}}{dt}$
$\displaystyle=$ $\displaystyle\gamma_{F}N(1-x)-\alpha_{1}\pi_{F},$ (16)
$\displaystyle\frac{d\pi_{E}}{dt}$ $\displaystyle=$
$\displaystyle\theta_{E}[\frac{dN}{dt}(1+x)+\frac{dx}{dt}N]-\alpha_{2}\pi_{E},$
(17)

where:

$\displaystyle s$ $\displaystyle=$ $\displaystyle
a_{0}+a_{1}x+a_{2}\pi_{F}+a_{3}\pi_{E}.$ (18)

If the total number of vehicle $2N$ is fixed, then $\frac{dN}{dt}=0$. Hence:

$\displaystyle\frac{dx}{dt}$ $\displaystyle=$ $\displaystyle
v[(1-x)exp(s)-(1+x)exp(-s)],$ (19) $\displaystyle\frac{d\pi_{F}}{dt}$
$\displaystyle=$ $\displaystyle\gamma_{F}N(1-x)-\alpha_{1}\pi_{F},$ (20)
$\displaystyle\frac{d\pi_{E}}{dt}$ $\displaystyle=$
$\displaystyle\theta_{E}\frac{dx}{dt}N-\alpha_{2}\pi_{E},$ (21)

where:

$\displaystyle s$ $\displaystyle=$ $\displaystyle
a_{0}+a_{1}x+a_{2}\pi_{F}+a_{3}\pi_{E}.$ (22)

This 3D system becomes our baseline model of analysis. It is possible to
manipulate the policy as well as environmental parameters to analyse different
policy regimes. This will be discussed further in the following sections.

###  2.4 The 4D Model: The long-term growth rate of N ($g_{N}$)

The baseline model discussed in previous section does not allow growth in
total number of cars. In reality, $N$ may grow exponentially over time. Here
we incorporate the exponential growth of $N$ into the system, which yields:

$\displaystyle\frac{dx}{dt}$ $\displaystyle=$ $\displaystyle
v[(1-x)exp(s)-(1+x)exp(-s)],$ (23) $\displaystyle\frac{d\pi_{F}}{dt}$
$\displaystyle=$ $\displaystyle\gamma_{F}N(1-x)-\alpha_{1}\pi_{F},$ (24)
$\displaystyle\frac{d\pi_{E}}{dt}$ $\displaystyle=$
$\displaystyle\theta_{E}[\frac{dN}{dt}(1+x)+\frac{dx}{dt}N]-\alpha_{2}\pi_{E},$
(25) $\displaystyle\frac{dN}{dt}$ $\displaystyle=$ $\displaystyle g_{N}N,$
(26)

where:

$\displaystyle s$ $\displaystyle=$ $\displaystyle
a_{0}+a_{1}x+a_{2}\pi_{F}+a_{3}\pi_{E}.$ (27)

The next two sections will discuss the analytical side of the model, as well
as numerical analysis & policy experiments. We use the 3D model for analytical
discussion for the sake of mathematical simplicity, and the 4D model on
numerical simulation.

##  3 Local Stability Analysis

###  3.1 Analysis of the two-dimensional system

This section analyse the 3D model in terms of its fixed points and local
stability. We first reduce the 3D model into 2D model by assuming no rare
earth pollution has existed yet, which implies that $\theta_{E}=0$ and
$\pi_{E}=0$:

$\displaystyle\frac{dx}{dt}$ $\displaystyle=$ $\displaystyle
v[(1-x)exp(s)-(1+x)exp(-s)],$ (28) $\displaystyle=$ $\displaystyle
2v[tanh(s)-x]cosh(s),$ (29) $\displaystyle\frac{d\pi_{F}}{dt}$
$\displaystyle=$ $\displaystyle\gamma_{F}N(1-x)-\alpha_{1}\pi_{F},$ (30)

where:

$\displaystyle s$ $\displaystyle=$ $\displaystyle a_{0}+a_{1}x+a_{2}\pi_{F}.$
(31)

We set $LHS=0$ on both equations and derive the following isoclines:

$\displaystyle 0$ $\displaystyle=$ $\displaystyle 2v[tanh(s)-x]cosh(s),$ (32)
$\displaystyle 0$ $\displaystyle=$
$\displaystyle\gamma_{F}N(1-x)-\alpha_{1}\pi_{F}.$ (33)

We first analyse the steady state an local stability of the above 2D system.
The Jacobian is derived as:

$\displaystyle J$ $\displaystyle=$
$\displaystyle\begin{bmatrix}F_{x}&F_{\pi_{F}}\\\ G_{x}&G_{\pi_{F}}\\\
\end{bmatrix}$ (34) $\displaystyle=$
$\displaystyle\begin{bmatrix}2v(a_{1}cosh(s)-cosh(s)-a_{1}xsinh(s))&2va_{2}(cosh(s)-xsinh(s))\\\
-\gamma_{F}N&-\alpha_{1}\end{bmatrix}.$ (35)

Since it is difficult to find the exact solution of the fixed point in this
equation. As a special case, we set $a_{0}=-\frac{\gamma_{F}N}{\alpha_{2}}$
and $a_{2}=1$, and we can easily obtain an equilibrium ($x^{\star}=0$,
$\pi_{F}=-\frac{\gamma_{F}N}{\alpha_{2}}$). Furthermore, there is potentially
an emergence of other equilibria, depending on the value of economic factor
$a_{0}$ and $CO_{2}$ emission psychological account constant $a_{2}$ if and
only if $a_{0}+a_{2}\pi_{F}=0$. Therefore, the trace and determinant of the
Jacobian at a equilibrium (x = 0) is calculated as:

$\displaystyle Det(J)$ $\displaystyle=$
$\displaystyle-2v\alpha_{1}(a_{1}-1)+2va_{2}\gamma_{F}N,$ (36) $\displaystyle
Tr(J)$ $\displaystyle=$ $\displaystyle-\alpha_{1}+2v(a_{1}-1).$ (37)

The necessary and sufficient condition for the local stability for this
equilibrium is that $Det(J)>0$ and $Tr(J)<0$. This condition can be rewritten
as: $\alpha_{1}(a_{1}-1)>a_{2}\gamma_{F}N$, $\alpha_{1}>2v(a_{1}-1)$. This
condition implies that the self-purification parameter $\alpha_{1}$ plays a
crucial role in maintaining the stability of this special fixed point.

###  3.2 Analysis of the three-dimensional system

In this section, we introduce the production pollution $\pi_{E}$ in our model.
We obtain the system as follows:

$\displaystyle\frac{dx}{dt}$ $\displaystyle=$ $\displaystyle
v[(1-x)exp(s)-(1+x)exp(-s)],$ (38) $\displaystyle=$ $\displaystyle
2v[tanh(s)-x]cosh(s),$ (39) $\displaystyle\frac{d\pi_{F}}{dt}$
$\displaystyle=$ $\displaystyle\gamma_{F}N(1-x)-\alpha_{1}\pi_{F},$ (40)
$\displaystyle\frac{d\pi_{E}}{dt}$ $\displaystyle=$
$\displaystyle\theta_{E}\frac{dx}{dt}N-\alpha_{2}\pi_{E},$ (41)

where $\theta_{E}>0$ and $\alpha_{2}>0$ are constants and

$\displaystyle s$ $\displaystyle=$ $\displaystyle
a_{0}+a_{1}x+a_{2}\pi_{F}+a_{3}\pi_{E}.$ (42)

The Jacobian of this 3D system is

$\displaystyle J=\begin{bmatrix}F_{xx}&F_{\pi_{F}x}&F_{\pi_{E}x}\\\
F_{\pi_{F}x}&F_{\pi_{F}\pi_{F}}&F_{\pi_{F}\pi_{E}}\\\
F_{\pi_{E}x}&F_{\pi_{E}\pi_{F}}&F_{\pi_{E}\pi_{E}}\end{bmatrix}.$ (43)

where

$\displaystyle F_{xx}$ $\displaystyle=$ $\displaystyle
2v(a_{1}cosh(s)-cosh(s)-a_{1}xsinh(s)),$ (44) $\displaystyle F_{\pi_{F}x}$
$\displaystyle=$ $\displaystyle 2va_{2}(cosh(s)-xsinh(s)),$ (45)
$\displaystyle F_{\pi_{E}x}$ $\displaystyle=$ $\displaystyle
2va_{3}(cosh(s)-xsinh(s)),$ (46) $\displaystyle F_{\pi_{F}x}$ $\displaystyle=$
$\displaystyle-\gamma_{F}N,$ (47) $\displaystyle F_{\pi_{F}\pi_{F}}$
$\displaystyle=$ $\displaystyle-\alpha_{1},$ (48) $\displaystyle
F_{\pi_{F}\pi_{E}}$ $\displaystyle=$ $\displaystyle 0,$ (49) $\displaystyle
F_{\pi_{E}x}$ $\displaystyle=$ $\displaystyle
2\theta_{E}Nv(a_{1}cosh(s)-cosh(s)-a_{1}xsinh(s)),$ (50) $\displaystyle
F_{\pi_{E}\pi_{F}}$ $\displaystyle=$
$\displaystyle\theta_{E}N(2va_{2}(cosh(s)-xsinh(s))),$ (51) $\displaystyle
F_{\pi_{E}\pi_{E}}$ $\displaystyle=$
$\displaystyle-\alpha_{2}+\theta_{E}N(2va_{3}(cosh(s)-xsin(s))).$ (52)

We set

$\displaystyle 0$ $\displaystyle=$ $\displaystyle 2v[tanh(s)-x]cosh(s),$ (53)
$\displaystyle 0$ $\displaystyle=$
$\displaystyle\gamma_{F}N(1-x)-\alpha_{1}\pi_{F},$ (54) $\displaystyle 0$
$\displaystyle=$ $\displaystyle\theta_{E}\frac{dx}{dt}N-\alpha_{2}\pi_{E}.$
(55)

Since the close-form solution for the fixed point is hard to obtain, we
discuss a special case with $a_{0}=-\frac{\gamma_{F}N}{\alpha_{1}}$,$a_{2}=1$
and $a_{3}=-1$, so that we have a mid-point equilibrium for $x=0$. In this
special case, the fixed point is derived as: $x*=0$,
$\pi_{F}=\frac{\gamma_{F}N}{\alpha_{1}}$, $\pi_{E}=0$. The Jacobian is given
by:

$\displaystyle J$ $\displaystyle=$
$\displaystyle\begin{bmatrix}2v(a_{1}-1)&2v&-2v\\\
-\gamma_{F}N&-\alpha_{1}&0\\\
2v\theta_{E}N(a_{1}-1)&2vN\theta_{E}&-\alpha_{2}-2vN\theta_{E}\end{bmatrix}.$
(56)

which has the sign structure

$\displaystyle\begin{bmatrix}?&+&-\\\ -&-&0\\\ ?&+&-\end{bmatrix}.$ (57)

In a 3D dynamical system, the Routh-Hurwitz necessary and sufficient condition
for local stability is:

$\displaystyle Det(J)$ $\displaystyle<$ $\displaystyle 0,$ (58) $\displaystyle
Tr(J)$ $\displaystyle<$ $\displaystyle 0,$ (59) $\displaystyle
J_{1}+J_{2}+J_{3}$ $\displaystyle>$ $\displaystyle 0,$ (60) $\displaystyle-
Tr(J)(J_{1}+J_{2}+J_{3})+|J|$ $\displaystyle>$ $\displaystyle 0,$ (61)

where

$\displaystyle J_{1}$ $\displaystyle=$
$\displaystyle\begin{vmatrix}-\alpha_{1}&0\\\
2vN\theta_{E}&-\alpha_{2}-2vN\theta_{E}\end{vmatrix},$ (62) $\displaystyle
J_{2}$ $\displaystyle=$ $\displaystyle\begin{vmatrix}2v(a_{1}-1)&-2v\\\
2v\theta_{E}N(a_{1}-1)&-\alpha_{2}-2vN\theta_{E}\end{vmatrix},$ (63)
$\displaystyle J_{3}$ $\displaystyle=$
$\displaystyle\begin{vmatrix}2v(a_{1}-1)&2v\\\
-\gamma_{F}N&-\alpha_{1}\end{vmatrix}.$ (64)

The trace and determinant of the Jacobian at this fixed point is calculated as
:

$\displaystyle Det(J)$ $\displaystyle=$
$\displaystyle(-\alpha_{2}-2vN\theta_{E})(-2v\alpha_{1}(a_{1}-1)+2\gamma_{F}Nv)$
(66)
$\displaystyle-2v(-2\gamma_{F}N^{2}\theta_{E}v\theta_{E}+\alpha_{1}\theta_{E}(2vN(a_{1}-1))),$
$\displaystyle Tr(J)$ $\displaystyle=$
$\displaystyle-2v\alpha_{1}(a_{1}-1)+2v\gamma_{F}N-\alpha_{2}-2vN\theta_{E}.$
(67)

The necessary and sufficient condition of local stability for this fixed point
follows Routh-Hourwitz theorem.

##  4 Simulation & Policy Analysis

This section provides some numerical simulation of the 3D model, which aims to
bring some insights of how different policy measures would affect the long-
term dynamics of TFVs and NEVs, as well as its associated environmental
externality of both types.

###  4.1 Scenario 1: Laissez-Faire

We first simulate an initial situation where the government does not
interfere. Also, we assume that there is no difference between TFV and NEV in
terms of constant consumer preference ($a_{0}=0$) as an initial condition. In
other words, the consumer will is indifferent between TFV and NEV in the
absence of government intervention. Neither side of the environmental
externality ($\pi_{n}$ and $\pi_{f}$) is regulated ($a_{2}=a_{3}=0$). The
growth rate of total number of vehicles are assumed to be zero ($g_{n}=0$) as
an initial setting. We first look at the corresponding Jacobian of scenario 1
before running the simulation:

$\displaystyle
J^{(1)}=\begin{bmatrix}2v[a_{1}cosh(a_{1}x)-cosh(a_{1}x)-a_{1}xsinh(a_{1}x)]&0&0\\\
-\gamma_{F}N&-\alpha_{1}&0\\\
0.2N(2v[a_{1}cosh(a_{1}x)-cosh(a_{1}x)-a_{1}xsinh(a_{1}x)])&0&-\alpha_{2}\end{bmatrix}.$
(68)

The trace and determinant of the Jacobian under this scenario is calculated
as:

$\displaystyle Det(J)$ $\displaystyle=$ $\displaystyle
2\alpha_{1}\alpha_{2}v[a_{1}cosh(a_{1}x)-cosh(a_{1}x)-a_{1}xsinh(a_{1}x)],$
(69) $\displaystyle Tr(J)$ $\displaystyle=$ $\displaystyle
2v[a_{1}cosh(a_{1}x)-cosh(a_{1}x)-a_{1}xsinh(a_{1}x)]-\alpha_{1}-\alpha_{2}.$
(70)

Besides,
$J^{(1)}_{1}=\alpha_{1}\alpha_{2},J^{(1)}_{2}=-2\alpha_{2}v[a_{1}cosh(a_{1}x)-cosh(a_{1}x)-a_{1}xsinh(a_{1}x)],J^{(1)}_{3}=-2\alpha_{1}v[a_{1}cosh(a_{1}x)-cosh(a_{1}x)-a_{1}xsinh(a_{1}x)]$.

In this scenario, the herding parameter plays a crucial role, since the main
driving force behind the model is the Lux (1995) dynamics. We also look at two
sub-scenarios where the herding behaviour is relatively weak ($a_{1}=0.5$) and
strong ($a_{1}=1.5$). In this situation, the initial condition of x will play
a crucial role in determining the dynamics of the system. Other parameters are
set as follows: $N=10$, $\gamma_{f}=0.9$, $\theta_{n}=0.2$, $v=0.6$,
$\alpha_{1}=0.03$, $\alpha_{2}=0.07$. The simulation results are provided
below:

As shown in Fig. 2, with a strong herding parameter ($a_{1}=1.5$) and an
initial advantage of TFV ($x_{0}=-0.1$), the system eventually converges to a
situation where a majority of the consumers opt for TFVs, and both $\pi_{f}$
and $\pi_{e}$ rose steadily over the period. On the other hand, when the
herding parameter is relatively weak ($a_{1}=0.5$), the number of TFV and NEV
eventually becomes equal, _ceteris paribus_. This is shown in Fig. 3

It is also worth to mention that in this scenario, the parameter $a_{0}$ plays
a crucial role in determining whether the system will eventually converge
toward TFV or NEV equilibrium in the absence of government interference
($a_{0}$ is solely determined by economic utility). The previous simulation
assumes that $a_{0}=0$, implying that NEV and TFV have the same economic
utility to consumers. Given a strong herding parameter and an initial state of
TFV majority, the system converges toward an TFV equilibrium, as the
simulation shows. However, if NEV is economically more advantageous
($a_{0}>0$), final equilibrium will favour NEV, and _vice versa_.

###  4.2 Scenario 2: One-sided Regulation

The second scenario looks at a situation when the government imposes one-sided
regulation, in which the government subsidizes NEV and punishes TFV,
overlooking the potential environmental externality caused by NEV production.
The support not only makes NEV more economically attractive through tax
benefit and other means ($a_{0}=1$), but also ignores the potential
environmental externality caused by NEV production at its initial phase
($a_{3}=0$). The rest of the parameters are set the same as in scenario 1
discussed previously, with $a_{1}=1.5$ implying a relative strong herd
incentive in consumers in a realistic sense. Also, we assume that $N$ grows
exponentially at $10\%$ _per annum_ over time ($g_{N}=0.1$).

As shown in Fig. 4, in such scenario, most consumers almost instantly choose
NEV due to the unparallel advantage from the one-sided policy, and the carbon-
related externality from TFV diminishes over time. However, there is a drastic
increase of $\pi_{e}$ arising from the close-to-exponential growth of NEVs.
The long-term environmental impact of NEV production can not be ignored, once
the NEV production reaches a large scale.

###  4.3 Scenario 3: Comprehensive Regulation

The third scenario looks at the case when both sectors are regulated. We set
$a_{2}=0.5$ and $a_{3}=-0.5$, which implies that externality from both sectors
will be regulated and punished. In this case, the government may still
subsidize the NEV sector at an initial stage ($a_{0}>0$), yet the government
will also impose corrective tax over NEV-related _Type I_ pollution, as soon
as $\pi_{e}$ grows and becomes larger ($a_{3}=-0.5$). In this scenario, the
parameter of initial subsidy will be crucial in determining whether consumers
will converge toward TFV or NEV at the end. A low $a_{0}$ implies that the
subsidy is not significant enough, and consumers may eventually choose TFV.
Yet a larger $a_{0}$ may lead to NEV equilibrium in the end. We simulate three
cases where $a_{0}=0.5$, $a_{0}=2.5$, $a_{0}=4.5$ respectively, as shown in
Fig. 5-7:

####  4.3.1 Macro Regulation: Manipulation of $g_{N}$

The previous section discusses the long-term dynamics of TFV and NEV when both
sectors are monitored and regulated. From the modelling perspective, it is
achieved by manipulating $a_{2}$ and $a_{3}$. Another aspect of comprehensive
regulation may be achieved by imposing the quota on total number of vehicles
regardless of TFVs and NEVs. The quota is set by the growth rate of total
number of vehicles ($g_{N}$), which is determined by aggregate externality
index $\Pi$:

$\displaystyle\Pi$ $\displaystyle=$ $\displaystyle k_{1}\pi_{F}+k_{2}\pi_{E}$
(71) $\displaystyle g_{N}$ $\displaystyle=$
$\displaystyle\bar{g}_{N}\exp(-\Pi)$ (72)

Eq. 71 defines the aggregate externality index determined by both $\pi_{F}$
and $\pi_{E}$. This will in turn determine the cap on $g_{N}$. If there is no
externality, then $g_{N}=\bar{g}_{N}$. The presence of externality implies
that $g_{N}$ needs to be lowered, as determined by Eq. 72.

The simulation results are shown in Fig. 8-9. According to this set of
simulation, we can see that regulation on $g_{N}$ can significantly reduce
both the number of vehicles ($N$), as well as the associated externality. The
real-world implication is that effective regulation requires not only on the
micro level in each sector, but also on the macro level on the growth of total
number of vehicles. As the government increases spending on public
transportation, $g_{N}$ can be effectively reduced to a desirable level, thus
reducing the overall level of environmental externality captured by $\Pi$.

####  4.3.2 Other possible scenarios

The previous sections have discussed the possible long-term dynamics of TFV &
NEV under different policy regimes with an emphasis on parameters in Eq. 23.
However, there are other parameters in Eq. 24-25 that may deserve further
discussion, since they are also subject to policy control. For example, the _R
& D_ on fuel efficiency may lower the parameter $\gamma_{F}$. Similarly, more
efficient use of rare-earth elements (REEs) and other related materials such
as lithium during NEV production would lower $\theta_{E}$. Also, new
development in circular economy may lead to more efficient recycling of
retired TFVs and NEVs, thus increasing the reversing parameters $\alpha_{1}$
and $\alpha_{2}$. These non-economic aspects are more difficult to determine
and measure, and requires careful cost & benefit analysis. These aspects are
beyond the scope of this paper, which will be investigated further in future
research.

##  5 Conclusion

The ever-worsening global warming and other fuel-related environmental issues
have led to an ongoing debate of whether new energy vehicles (NEVs) will
eventually “creatively destroy” traditional fuel vehicles (TFVs). It is
generally believed that NEVs are more efficient, produce zero emission, and
will bring a better future to our planet. However, there is a danger that NEV
production may also bring unintended environmental consequence due to the use
of rare earth elements (REEs) and other pollutants such as lithium, once
reaching a large scale of production. Despite its importance, the potential
environmental externality arising from NEV production is scantly discussed in
current literature.

This paper aims to fill this gap by proposing a simple, preliminary framework
that models the long-term dynamics of TFV and NEV growth, as well as their
associated environmental externality. We have defined two types of externality
in this paper: _Type I_ externality that comes from production and _Type II_
externality that comes from ongoing usage. We further conduct policy
experiments in three aspects: _laissez-faire_ , one-sided regulation, and
comprehensive regulation. We argue that neither _laissez-faire_ nor one-sided
regulation would bring us an optimal outcome in the long run, and it is
crucial to constantly monitor and regulate the _Type I_ externality as well
from NEV production at this infant stage of NEV development, so that water and
soil pollution is under control. Also, comprehensive regulation implies that
TFV and NEV may co-exist for a sustained period of time, so the total
externality will not stem from a single source that may bring an irreversible
damage to the ecological system. We also suggest that the regulation on the
growth of total number of vehicles is more desirable, since it directly
reduces the quantity of both TFVs and NEVs, thus significantly lowers total
externality. This can be achieved by a long-term commitment of public policies
that aims for better infrastructure building and better provision of public
transport system, so that the overall environmental impact is kept at its
minimum.

There are numerous paths that deserve further investigation. _First_ , the
model can potentially be calibrated with the use of both economic and
environmental data so that we can produce a more realistic long-term
projection. _Second_ , it is possible to derive a long-term “golden-rule”
policy recommendation that aims to maximize long-term economic benefit, while
minimizing total environmental externality. _Third_ , we may look at more
realistic consumer behaviour beyond the simple Lux (1995) assumptions. These
aspects will be discussed more thoroughly in future research.

## Acknowledgement

Our work was supported in part by the Guangdong Provincial Key Laboratory of
Interdisciplinary Research and Application for Data Science, BNU-HKBU United
International College (2022B1212010006) and in part by Guangdong Higher
Education Upgrading Plan (2021-2025) (UIC R0400024-21).

## Author Contributions

The authors are solely listed in alphabetical order, and all authors have made
equal contribution toward this research. Zhaojia Huang derived the analytical
results of the 2D and 3D model. Liang Zhang conducted the thorough literature
review. Tianhao Zhi ignited the initial idea, constructed the 2D & 3D model,
and worked on the numerical simulations & policy discussions. All remaining
errors are on our own.

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## Appendix: Simulation Results

Figure 2: Initial Run with $a_{1}=1.5$, _ceteris paribus_
Figure 3: Initial Run with $a_{1}=0.5$, _ceteris paribus_
Figure 4: Scenario 2 (_“Creative Destruction”_)
Figure 5: Scenario 3: $a_{0}=0.5$ (_Co-existence_)
Figure 6: Scenario 3: $a_{0}=2.5$ (_Co-existence_)
Figure 7: Scenario 3: $a_{0}=4.5$ (_Co-existence_)
Figure 8: $g_{N}$ grows at an exogenous rate ($\bar{g}_{N})$
Figure 9: $g_{N}$ under regulation according to $\Pi$