PITT-PACC-2321

# Optimizing Fictitious States for Bell Inequality Violation
in Bipartite Qubit Systems

Kun Cheng<EMAIL_ADDRESS>School of Physics, Peking University, Beijing,
100871, China  PITT PACC, Department of Physics and Astronomy,
University of Pittsburgh, 3941 O’Hara St., Pittsburgh, PA 15260, USA  Tao Han
<EMAIL_ADDRESS>PITT PACC, Department of Physics and Astronomy,
University of Pittsburgh, 3941 O’Hara St., Pittsburgh, PA 15260, USA  Matthew
Low<EMAIL_ADDRESS>PITT PACC, Department of Physics and Astronomy,
University of Pittsburgh, 3941 O’Hara St., Pittsburgh, PA 15260, USA

###### Abstract

There is a significant interest in testing quantum entanglement and Bell
inequality violation in high-energy experiments. Since the analyses in high-
energy experiments are performed with events statistically averaged over phase
space, the states used to determine observables depend on the choice of
coordinates through an event-dependent basis and are thus not genuine quantum
states, but rather “fictitious states.” We prove that if Bell inequality
violation is observed with a fictitious state, then it implies the same for a
quantum sub-state. We further show analytically that the basis which
diagonalizes the spin-spin correlations is optimal for constructing fictitious
states, and for maximizing the violation of Bell’s inequality.

Introduction

Quantum mechanics and the theory of relativity are the two pillars of modern
physics. Quantum entanglement and the violation of Bell’s inequality [1] are
the unambiguous tests of quantum principles, and have been verified in
numerous low-energy experiments. Recently, there has been a significant
interest in testing quantum entanglement and Bell inequality violation in
high-energy experiments [2]. While it is important to test the principles of
quantum mechanics deeply into the relativistic kinematic regime, certain
aspects of high-energy experiments make these test non-trivial and even
ambiguous.

Quantum states, represented by density matrices, are the fundamental objects
in quantum mechanics. In high-energy experiments, each event specified by a
multi-dimensional phase space point is a measurement of an underlying quantum
state. Collecting a large data sample with different phase space
configurations is equivalent to a phase space integration weighted by a matrix
element squared, which does not result in a quantum state when the average is
done in event-dependent frames. Such an object has been labelled a “fictitious
state” [3].

Fictitious states are a convex sum of quantum sub-states, but are not added in
a way that corresponds to a quantum state. Consequently, fictitious states
lose many properties that are held by quantum states. While observables
computed using quantum states are basis-independent, observables computed
using fictitious states are basis-dependent in general. For instance, the Bell
inequality violation of a fictitious state becomes basis-dependent.

In this work, we show that the fictitious states do maintain some properties
of quantum states. In particular, the Bell inequality violation of a
fictitious state implies the same for some quantum sub-state. We present the
optimal choice of basis for constructing fictitious states, and identify the
basis that maximizes the Bell inequality violation.

Fictitious States and their Basis Dependence

Consider a quantum state that is described by the density matrix $\rho$. With
$N$ measurements, we construct the quantum state $\overline{\rho}$

$\overline{\rho}=\frac{1}{N}\sum_{a=1}^{N}\rho_{a},$ (1)

where $\rho_{a}$ is a quantum sub-state from a single measurement $a$.

We first start with the simple example of a spin measurement on a single
qubit, where we can decompose a quantum sub-state as

$\rho_{a}=\frac{1}{2}\left(\mathbb{I}_{2}+\sum_{i=1}^{3}B_{a,i}\sigma_{i}\right).$
(2)

The reconstructed quantum state is an average over the $N$ measurements

$\displaystyle\overline{\rho}$
$\displaystyle=\frac{1}{2}\left(\mathbb{I}_{2}+\sum_{i=1}^{3}\overline{B}_{i}\sigma_{i}\right)\quad{\rm
with}\quad\overline{B}_{i}=\frac{1}{N}\sum_{a=1}^{N}B_{a,i}.$ (3)

The spin of the qubit is measured by $\langle\sigma_{i}\rangle$:

$\displaystyle\langle\sigma_{i}\rangle$
$\displaystyle=\frac{1}{N}\sum_{a=1}^{N}\operatorname{Tr}(\sigma_{i}\rho_{a})=\frac{1}{2N}\sum_{a=1}^{N}\operatorname{Tr}\left(\sigma_{i}(\sum_{j=1}^{3}B_{a,j}\sigma_{j})\right)$
$\displaystyle=\frac{1}{2}\operatorname{Tr}\left(\sigma_{i}(\sum_{j=1}^{3}\overline{B}_{j}\sigma_{j})\right)=\overline{B}_{i}=\operatorname{Tr}(\sigma_{i}\overline{\rho}).$
(4)

Next, consider reconstructing the state with a different measurement scheme
where rather than using the same reference axis for each measurement $a$:
$\sigma_{i}=\vec{\sigma}\cdot\hat{e}_{i}$, instead a different reference axis
is used for each measurement $a$:
$\sigma_{i,a}=\vec{\sigma}\cdot\hat{e}_{i,a}$. We call this scheme using an
event-dependent basis, and the previous event-independent scheme using a fixed
basis. They are related by an event-dependent rotation $R_{ij,a}$ such that
$\hat{e}_{i}=R_{ij,a}\hat{e}_{j,a}$.

Using an event-dependent basis, the basis for each measured quantum sub-state
is different

$\rho_{a}=\frac{1}{2}\left(\mathbb{I}_{2}+\sum_{i=1}^{3}B^{\prime}_{a,i}\sigma_{i,a}\right).$
(5)

The corresponding reconstructed averaged state is

$\overline{\rho}_{\rm
fic}=\frac{1}{2}\left(\mathbb{I}_{2}+\sum_{i=1}^{3}\overline{B}^{\prime}_{i}\sigma_{i}\right)\quad{\rm
with}\quad\overline{B}^{\prime}_{i}=\frac{1}{N}\sum_{a=1}^{N}B^{\prime}_{a,i}.$
(6)

Under the parametrization of Eq. (5), the measured value of the spin
$\langle\sigma_{i}\rangle$ compares to the calculated value
$\operatorname{Tr}(\sigma_{i}\overline{\rho}_{\rm fic})$ as

$\displaystyle\langle\sigma_{i}\rangle$
$\displaystyle=\frac{1}{N}\sum_{a=1}^{N}\operatorname{Tr}(\sigma_{i}\rho_{a})=\frac{1}{2N}\sum_{a=1}^{N}\operatorname{Tr}\left(\sigma_{i}(\sum_{j=1}^{3}B^{\prime}_{a,j}\sigma_{j,a})\right)$
$\displaystyle=\frac{1}{N}\sum_{a=1}^{N}\sum_{j=1}^{3}\left(R_{ij,a}B^{\prime}_{a,j}\right)\neq\overline{B}^{\prime}_{i}=\operatorname{Tr}(\sigma_{i}\overline{\rho}_{\rm
fic}),$ (7)

where we have used $\operatorname{Tr}(\sigma_{i}\sigma_{j,a})/2=R_{ij,a}$. As
such, the fictitious state reconstructed using the event-dependent basis does
not yield the same result as the genuine quantum state.

Fictitious states, given by $\overline{\rho}_{\rm fic}$, are still a convex
sum of quantum sub-states, but they are not added in a way that corresponds to
a quantum state. This drawback is more pronounced for other observables beyond
spin. For any space-time-dependent observable $\mathcal{O}$, the predicted
result $\operatorname{Tr}(\mathcal{O}\,\overline{\rho}_{\rm fic})$ would
depend on the phase space configuration and thus on the choice of the Lorentz
frame, as well as the coordinate axes.

A general remark is in order. Eq. (1) can always be rewritten as a sum of
weighted events $\overline{\rho}\propto\sum_{a}\omega_{a}\rho_{a}$ where the
weight $\omega_{a}$ denotes the production rate, or it can be changed from a
summation to a phase space integral $\overline{\rho}\propto\int\frac{\rm
d\sigma}{\rm d\Omega}\rho_{\Omega}{\rm d\Omega}$. These formulations are often
more suitable for the quantum states produced at collider experiments. All of
our discussion still holds as long as $\overline{\rho}$ is a convex sum of
quantum sub-states. Additionally, while the previous example is based on the
spin measurement of a spin-1/2 particle, we emphasize that the preceding
discussion applies to any fictitious state and its corresponding quantum
state.

For a generic qubit state, the density matrix can be expressed as a $2\times
2$ matrix $\rho=\rho_{\alpha\beta}\ket{\alpha}\bra{\beta}$. Hereafter, the
summation of repeated indices on density matrices is assumed. With respect to
a different basis choice, the qubit state in the primed basis corresponds to
an $SU(2)$ transformation
$\ket{\beta^{\prime}}=\ket{\alpha}U_{\alpha\beta^{\prime}}$, which acts on the
density matrix according to $\rho^{\prime}=U^{\dagger}\rho U$.

For spin-1/2 particles, the quantization basis choice is made by choosing
reference axes $\hat{e}_{i}\ (i=1,2,3)$ to measure their spin. Different
reference axes are related by an $SO(3)$ rotation
$\hat{e}_{i}^{\prime}=\sum_{j}R_{ij}\hat{e}_{j}$. Consequently, the $2\times
2$ density matrix of a qubit expressed in the basis using $\hat{e}_{i}$ and
the basis using $\hat{e}^{\prime}_{i}$ are related by

$\rho^{\hat{e}^{\prime}_{i}}_{\alpha^{\prime}\beta^{\prime}}=U^{\dagger}_{\alpha^{\prime}\alpha}\rho^{\hat{e}_{i}}_{\alpha\beta}U_{\beta\beta^{\prime}}.$
(8)

The $SO(3)$ rotation from $\hat{e}_{i}$ to $\hat{e}^{\prime}_{i}$ is given by
$R_{ij}=\operatorname{Tr}(U^{\dagger}\sigma_{i}U\sigma_{j})/2$, which is a
generalization from the earlier example.

Considering the state averaged over a large ensemble with a different basis
$\hat{e}_{i,a}$ for each event $a$,

$\overline{\rho}[{\hat{e}_{i,a}}]=\frac{1}{N}\sum_{a=1}^{N}\rho_{a}^{\hat{e}_{i,a}}=\frac{1}{N}\sum_{a=1}^{N}U^{\dagger}_{a}\rho_{a}^{\hat{e}_{i}}U_{a},$
(9)

where $\rho[{\hat{e}_{i,a}}]$ denotes the reconstruction of the state using an
event-dependent basis. This relation recasts our previous conclusion in a
general form. Although the genuine quantum state averaged in two different –
but fixed – bases are equal with respect to a unitary rotation, as shown in
Eq. (8), the density matrix $\overline{\rho}[\hat{e}_{i,a}]$ reconstructed
using an event-dependent basis is generally not equivalent to the genuine
quantum state reconstructed using a fixed basis:
$\overline{\rho}[\hat{e}_{i,a}]\neq
U^{\dagger}\overline{\rho}^{\hat{e}_{i}}U$. Therefore, the density matrix
averaged using an event-dependent basis is a fictitious state. Moreover, the
density matrix averaged using two different event-dependent basis schemes are
not equivalent, i.e., $\overline{\rho}[\hat{e}_{i,a}]\neq
U^{\dagger}\overline{\rho}[\hat{e}^{\prime}_{i,a}]U$. In other words,
fictitious states are basis-dependent, which leaves room for an optimal choice
of the basis for a given physical purpose.

This formalism applies equally to two qubits – the bipartite qubit system –
which is the simplest system that can exhibit entanglement and Bell inequality
violation. In collider experiments, this system is the most extensively
studied via the production of a top quark and an anti-top quark [2, 4, 3].

A bipartite qubit system is constructed by a direct product of the two qubits,
and its density matrix is a $4\times 4$ matrix
$\rho_{\alpha\bar{\alpha},\beta\bar{\beta}}$ expressed in the basis
$\ket{\alpha}\otimes\ket{\bar{\alpha}}$. Following the previous notation, the
$4\times 4$ density matrix of a bipartite qubit system expressed in different
bases are related by

$\displaystyle\rho^{\hat{e}^{\prime}_{i}}_{\beta^{\prime}\bar{\beta}^{\prime},\alpha^{\prime}\bar{\alpha}^{\prime}}$
$\displaystyle=U^{\dagger}_{\beta^{\prime}\bar{\beta}^{\prime},\beta\bar{\beta}}\rho^{\hat{e}_{i}}_{\beta\bar{\beta},\alpha\bar{\alpha}}U_{\alpha\bar{\alpha},\alpha^{\prime}\bar{\alpha}^{\prime}},$
(10) $\displaystyle
U_{\alpha\bar{\alpha},\alpha^{\prime}\bar{\alpha}^{\prime}}$
$\displaystyle\equiv U_{\alpha\alpha^{\prime}}\otimes
U_{\bar{\alpha}^{\prime}\bar{\alpha}^{\prime}}.$

In the following, we focus on optimizing the choice of fictitious state for
the purpose of testing Bell inequality violation.

The Bell Inequality Violation of Fictitious States

To study the Bell inequality violation of a bipartite qubit system, it is
convenient to parameterize the $4\times 4$ density matrix as

$\displaystyle\rho=\frac{1}{4}$
$\displaystyle\bigg{(}\mathbb{I}_{2}\otimes\mathbb{I}_{2}+\sum_{i=1}^{3}B_{i}^{+}\sigma_{i}\otimes\mathbb{I}_{2}+\sum_{i=1}^{3}B_{i}^{-}\mathbb{I}_{2}\otimes\sigma_{i}$
$\displaystyle+\sum_{i,j=1}^{3}C_{ij}\sigma_{i}\otimes\sigma_{j}\bigg{)}.$
(11)

The correlation matrix $\mathbf{C}$ – written above in component form as
$C_{ij}$ – encodes the correlations relevant for Bell’s inequality. The Bell’s
inequality for bipartite qubit systems is the Clauser-Horne-Shimony-Holt
inequality (CHSH) [5], written as

$\displaystyle\left|\vec{a}_{1}\cdot\mathbf{C}\cdot(\vec{b}_{1}-\vec{b}_{2})+\vec{a}_{2}\cdot\mathbf{C}\cdot(\vec{b}_{1}+\vec{b}_{2})\right|\leq
2$ (12)

where $\vec{a}_{1,2}\ (\vec{b}_{1,2})$ are normalized directions to measure
the spin of the first (second) qubit. Classical theories, including theories
with local hidden variables, satisfy this inequality. The violation of Bell’s
inequality is a consequence of genuine quantum mechanical behavior.

The optimal choice of the four spin measurements in Eq. (12) is known to be
[6]

$\displaystyle\mathscr{B}(\rho)$
$\displaystyle=\max_{\vec{a}_{1},\vec{a}_{2},\vec{b}_{1},\vec{b}_{2}}\left|\vec{a}_{1}\cdot\mathbf{C}\cdot(\vec{b}_{1}-\vec{b}_{2})+\vec{a}_{2}\cdot\mathbf{C}\cdot(\vec{b}_{1}+\vec{b}_{2})\right|$
$\displaystyle=2\max_{i\neq j}\sqrt{\mu_{i}^{2}+\mu_{j}^{2}},$ (13)

where $\mu_{i}^{2}$ are the eigenvalues of $\mathbf{C}^{T}\mathbf{C}$. In the
following, we assume that the correlation matrix $\mathbf{C}$ is symmetric, in
which case $\mu_{i}$ are eigenvalues of $\mathbf{C}$.

When we reparametrize the density matrix using another spin basis:
$\ket{\alpha}\to\ket{\beta}U_{\beta\alpha}$, the correlation matrix
$\mathbf{C}$ transforms as an $SO(3)$ tensor: $\mathbf{C}\to
R^{T}\mathbf{C}R$. Therefore, for a genuine quantum state, $\mathscr{B}(\rho)$
is basis-independent. For a fictitious state, however, its correlation matrix
is obtained by averaging the correlation matrices of quantum sub-states using
an event-dependent basis

$\displaystyle\overline{\mathbf{C}}[\hat{e}_{i,a}]=\frac{1}{N}\sum_{a=1}^{N}\mathbf{C}_{a}^{\hat{e}_{i,a}}=\frac{1}{N}\sum_{a=1}^{N}R^{T}_{a}\mathbf{C}_{a}^{\hat{e}_{i}}R_{a},$
(14)

where $\mathbf{C}_{a}^{\hat{e}_{i}}$ is the correlation matrix of the sub-
state $a$ expressed in a fixed basis and $R_{a}$ is an event-dependent
rotation from a fixed basis $\hat{e}_{i}$ to the event-dependent basis
$\hat{e}_{i,a}$. Generally, the correlation matrices of two fictitious states
averaged in different bases, $\overline{\mathbf{C}}[\hat{e}_{i,a}]$ and
$\overline{\mathbf{C}}[\hat{e}_{i,a}^{\prime}]$, yield different results and
different eigenvalues. Thus, the violation of Bell’s inequality is also basis-
dependent, i.e.,
$\mathscr{B}(\rho[\hat{e}_{i,a}])\neq\mathscr{B}(\rho[\hat{e}_{i,a}^{\prime}])$.

It is important to realize that establishing the Bell inequality violation of
a fictitious state $\overline{\rho}_{\rm fic}$ implies the same for some
quantum sub-state $\rho_{a}$. While this fits the naive expectation, we prove
this in the following.

Assume that all quantum sub-states $\rho_{a}$ satisfy the CHSH inequality,
i.e., for any four spatial directions $\vec{a}_{1},\vec{a}_{2},\vec{b}_{1}$,
and $\vec{b}_{2}$,

$\displaystyle\vec{a}_{1}\cdot\mathbf{C}_{a}\cdot(\vec{b}_{1}-\vec{b}_{2})+\vec{a}_{2}\cdot\mathbf{C}_{a}\cdot(\vec{b}_{1}+\vec{b}_{2})\in[-2,2],$
(15)

where $\mathbf{C}_{a}$ is the correlation matrix of the $a^{\rm th}$ sub-state
expressed in an arbitrary basis. Here and after, we omit the basis choice
label for simplicity when referring to a general event-dependent basis such
that $\mathbf{C}_{a}^{\hat{e}_{i,a}}$ is written as $\mathbf{C}_{a}$ and
$\overline{\mathbf{C}}[\hat{e}_{i,a}]$ is written as $\overline{\mathbf{C}}$.
The sum of Eq. (15) is convex and yields

$\displaystyle\frac{1}{N}\sum_{a=1}^{N}\left[\vec{a}_{1}\cdot\mathbf{C}_{a}\cdot(\vec{b}_{1}-\vec{b}_{2})+\vec{a}_{2}\cdot\mathbf{C}_{a}\cdot(\vec{b}_{1}+\vec{b}_{2})\right]$
$\displaystyle=$
$\displaystyle~{}\vec{a}_{1}\cdot\overline{\mathbf{C}}\cdot(\vec{b}_{1}-\vec{b}_{2})+\vec{a}_{2}\cdot\overline{\mathbf{C}}\cdot(\vec{b}_{1}+\vec{b}_{2})\in[-2,2].$

This implies that a fictitious state constructed from quantum sub-states, all
of which satisfy Bell’s inequality, must also satisfy Bell’s inequality.
Conversely, establishing Bell inequality violation in a fictitious state shows
there must be a quantum sub-state that violates Bell’s inequality.

The Optimal Basis to Test Bell Inequality Violation

Since the construction of a fictitious state is basis-dependent, one may hope
to find an optimal basis for a specified physical application. We show that
the optimal basis to test Bell’s inequality is the one that diagonalizes the
spin correlation of each quantum sub-state, and we call it the diagonal basis.
The diagonal basis $\hat{e}^{\rm diag}_{i,a}$ is generally an event-dependent
basis, and the correlation matrix of each quantum sub-state in the diagonal
basis is

$\mathbf{C}^{\rm diag}_{a}=\begin{pmatrix}\mu_{1,a}&0&0\\\ 0&\mu_{2,a}&0\\\
0&0&\mu_{3,a}\\\ \end{pmatrix}.$ (16)

Without loss of generality, the eigenvalues are ordered as
$\mu_{1,a}\geq\mu_{2,a}\geq\mu_{3,a}$. The correlation matrix of the
fictitious state averaged in the diagonal basis,

$\overline{\mathbf{C}}^{\rm diag}=\frac{1}{N}\sum_{a=1}^{N}\mathbf{C}^{\rm
diag},$ (17)

is also diagonal, and three eigenvalues of $\overline{\mathbf{C}}^{\rm diag}$
are $\bar{\mu}_{i}\equiv\frac{1}{N}\sum_{a=1}^{N}\mu_{i,a}$.

Next, consider another arbitrary basis $\hat{e}_{i,a}$ that is related to the
diagonal basis by a possibly event-dependent rotation $R_{a}$. The correlation
matrix of the fictitious state in this basis is

$\overline{\mathbf{C}}=\frac{1}{N}\sum_{a=1}^{N}\mathbf{C}_{a}=\frac{1}{N}\sum_{a=1}^{N}R^{T}_{a}\mathbf{C}^{\rm
diag}_{a}R_{a}.$ (18)

The three eigenvalues of $\overline{\mathbf{C}}$, which we denote as
$\bar{c}_{i}$, satisfy the following relations:

$\displaystyle\bar{c}_{1}+\bar{c}_{2}+\bar{c}_{3}=\bar{\mu}_{1}+\bar{\mu}_{2}+\bar{\mu}_{3}=\operatorname{Tr}(\overline{\mathbf{C}}),$
(19)
$\displaystyle\bar{\mu}_{1}\geq\bar{c}_{i}\geq\bar{\mu}_{3}\quad(\text{for
any~{}}i=1,2,3).$ (20)

Eq. (19) is a direct consequence from the trace of Eqs. (17) and (18). To
prove Eq. (20), which states that $\bar{c}_{i}$ are bounded by
$\bar{\mu}_{i}$, we first denote the three eigenvectors of
$\overline{\mathbf{C}}$ as $\hat{v}_{i}$. The corresponding eigenvalue is then

$\displaystyle\bar{c}_{i}=\hat{v}_{i}\cdot\overline{\mathbf{C}}\cdot\hat{v}_{i}=\frac{1}{N}\sum_{a=1}^{N}\left(\hat{v}_{i}\cdot\mathbf{C}_{a}\cdot\hat{v}_{i}\right).$
(21)

Applying Eq. (16) and $\mathbf{C}_{a}=R^{T}_{a}\mathbf{C}^{\rm
diag}_{a}R_{a}$, we find that

$\hat{v}_{i}\cdot\mathbf{C}_{a}\cdot\hat{v}_{i}=\sum_{\ell=1}^{3}\left|(R_{a}\cdot\hat{v}_{i})_{\ell}\right|^{2}\mu_{\ell,a}$
(22)

is a convex sum of $\mu_{i,a}$. This leads to

$\mu_{1,a}\geq\hat{v}_{i}\cdot\mathbf{C}_{a}\cdot\hat{v}_{i}\geq\mu_{3,a}.$
(23)

Therefore, Eq. (20) holds as a convex sum of Eq. (23).

With Eqs. (19) and (20), we are ready to show that the diagonal basis
maximizes $\mathscr{B}(\overline{\rho}[\hat{e}_{i,a}])$, by proving that for
any $i\neq j$, there exist $k\neq\ell$ satisfying

$\bar{c}_{i}^{2}+\bar{c}_{j}^{2}~{}\leq~{}\bar{\mu}_{k}^{2}+\bar{\mu}_{\ell}^{2},$
(24)

where $\bar{\mu}_{i}$ and $\bar{c}_{i}$ are the eigenvalues of the correlation
matrix averaged in the diagonal basis and an arbitrary basis, respectively.

The relative signs of $\bar{\mu}_{1}$, $\bar{\mu}_{2}$ and $\bar{\mu}_{3}$ can
be divided into the following three cases, and we prove Eq. (24) case by case.

  1. (a)

$\bar{\mu}_{1}\geq\bar{\mu}_{2}\geq\bar{\mu}_{3}\geq 0$,

  2. (b)

$0\geq\bar{\mu}_{1}\geq\bar{\mu}_{2}\geq\bar{\mu}_{3}$,

  3. (c)

$\bar{\mu}_{1}\geq 0\geq\bar{\mu}_{3}$.

Case (a). In this case both $\bar{c}_{i}$ and $\bar{c}_{j}$ are positive.
Without loss of generality, we assume $\bar{c}_{i}\geq\bar{c}_{j}$.

Though $\bar{c}_{i}$ is bounded by $\bar{\mu}_{1}$ and $\bar{\mu}_{3}$, it can
be either larger or smaller than $\bar{\mu}_{2}$:

Case (a1) $0\leq\bar{c}_{i}\leq\bar{\mu}_{2}$. We have
$\bar{c}_{i}^{2}+\bar{c}_{j}^{2}\leq
2\bar{\mu}_{2}^{2}\leq\bar{\mu}_{1}^{2}+\bar{\mu}_{2}^{2}$.

Case(a2) $\bar{\mu}_{2}\leq\bar{c}_{i}\leq\bar{\mu}_{1}$. First,
$\bar{\mu}_{2}\leq\bar{c}_{i}\leq\bar{\mu}_{1}$ leads to

$\displaystyle-\frac{\bar{\mu}_{1}-\bar{\mu}_{2}}{2}\leq\frac{\bar{\mu}_{1}+\bar{\mu}_{2}}{2}-\bar{c}_{i}\leq$
$\displaystyle\frac{\bar{\mu}_{1}-\bar{\mu}_{2}}{2}.$ (25)

Second, combining Eq. (19) and Eq. (20), we have
$\bar{c}_{i}+\bar{c}_{j}\leq\bar{\mu}_{1}+\bar{\mu}_{2}$ for $i\neq j$.
Consequently,

$\bar{c}_{i}^{2}+\bar{c}_{j}^{2}\leq\bar{c}_{i}^{2}+\left(\bar{\mu}_{1}+\bar{\mu}_{2}-\bar{c}_{i}\right)^{2}.$
(26)

Next, we define a function

$\displaystyle f(\Delta)$
$\displaystyle=\left(\frac{\bar{\mu}_{1}+\bar{\mu}_{2}}{2}+\Delta\right)^{2}+\left(\frac{\bar{\mu}_{1}+\bar{\mu}_{2}}{2}-\Delta\right)^{2}$
$\displaystyle=\frac{(\bar{\mu}_{1}+\bar{\mu}_{2})^{2}}{2}+2\Delta^{2},$ (27)

which satisfies $f(\Delta_{1})\leq f(\Delta_{2})$ for
$|\Delta_{1}|\leq|\Delta_{2}|$. Then $\bar{\mu}_{1}^{2}+\bar{\mu}_{2}^{2}$ and
the r.h.s. of Eq. (26) and can be rewritten using $f(\Delta)$:

$\displaystyle\bar{c}_{i}^{2}+\left(\bar{\mu}_{1}+\bar{\mu}_{2}-\bar{c}_{i}\right)^{2}$
$\displaystyle=f\left(\frac{\bar{\mu}_{1}+\bar{\mu}_{2}}{2}-\bar{c}_{i}\right),$
(28) $\displaystyle\bar{\mu}_{1}^{2}+\bar{\mu}_{2}^{2}$
$\displaystyle=f\left(\frac{\bar{\mu}_{1}-\bar{\mu}_{2}}{2}\right).$ (29)

From Eq. (25), the r.h.s. of Eq. (28) is smaller than the r.h.s. of Eq. (29),
then
$\bar{c}_{i}^{2}+\left(\bar{\mu}_{1}+\bar{\mu}_{2}-\bar{c}_{i}\right)^{2}\leq\bar{\mu}_{1}^{2}+\bar{\mu}_{2}^{2}$.
In combination with Eq. (26), we reach our conclusion,
$\bar{c}_{i}^{2}+\bar{c}_{j}^{2}\leq\bar{\mu}_{1}^{2}+\bar{\mu}_{2}^{2}$.

Case (b). In this case both $\bar{c}_{i}$ and $\bar{c}_{j}$ are negative.
Without loss of generality, we assume $|\bar{c}_{i}|\geq|\bar{c}_{j}|$, i.e.,
$\bar{c}_{i}\leq\bar{c}_{j}$.

Case (b1) $0\geq\bar{c}_{i}\geq\bar{\mu}_{2}$. We have
$\bar{c}_{i}^{2}+\bar{c}_{j}^{2}\leq
2\bar{\mu}_{2}^{2}\leq\bar{\mu}_{2}^{2}+\bar{\mu}_{3}^{2}$.

Case (b2) $\bar{\mu}_{2}\geq\bar{c}_{i}\geq\bar{\mu}_{3}$. Firstly, from
$\bar{\mu}_{2}\geq\bar{c}_{i}\geq\bar{\mu}_{3}$, we have

$-\frac{\bar{\mu}_{2}-\bar{\mu}_{3}}{2}\leq\frac{\bar{\mu}_{2}+\bar{\mu}_{3}}{2}-\bar{c}_{i}\leq\frac{\bar{\mu}_{2}-\bar{\mu}_{3}}{2}.$
(30)

Then, combining Eq. (19) and Eq. (20), we have
$\bar{c}_{i}+\bar{c}_{j}\geq\bar{\mu}_{2}+\bar{\mu}_{3}$ (note that now both
$\bar{c}_{i}$ and $\bar{c}_{j}$ are negative), which leads to

$\bar{c}_{i}^{2}+\bar{c}_{j}^{2}\leq\bar{c}_{i}^{2}+\left(\bar{\mu}_{2}+\bar{\mu}_{3}-\bar{c}_{i}\right)^{2}.$
(31)

Similar to case (a), we define a function

$\displaystyle g(\Delta)$
$\displaystyle=\left(\frac{\bar{\mu}_{2}+\bar{\mu}_{3}}{2}+\Delta\right)^{2}+\left(\frac{\bar{\mu}_{2}+\bar{\mu}_{3}}{2}-\Delta\right)^{2}$
$\displaystyle=\frac{(\bar{\mu}_{2}+\bar{\mu}_{3})^{2}}{2}+2\Delta^{2},$ (32)

that satisfies $g(\Delta_{1})\leq g(\Delta_{2})$ for
$|\Delta_{1}|\leq|\Delta_{2}|$. We rewrite
$\bar{\mu}_{2}^{2}+\bar{\mu}_{3}^{2}$ and the r.h.s of Eq. (31) as

$\displaystyle\bar{c}_{i}^{2}+\left(\bar{\mu}_{2}+\bar{\mu}_{3}-\bar{c}_{i}\right)^{2}$
$\displaystyle=g\left(\frac{\bar{\mu}_{2}+\bar{\mu}_{3}}{2}-\bar{c}_{i}\right),$
(33) $\displaystyle\bar{\mu}_{2}^{2}+\bar{\mu}_{3}^{2}$
$\displaystyle=g\left(\frac{\bar{\mu}_{2}-\bar{\mu}_{3}}{2}\right).$ (34)

From Eq. (30), the r.h.s. of Eq. (33) is smaller than the r.h.s. of Eq. (34),
which implies that
$\bar{c}_{i}^{2}+\left(\bar{\mu}_{2}+\bar{\mu}_{3}-\bar{c}_{i}\right)^{2}\leq\bar{\mu}_{2}^{2}+\bar{\mu}_{3}^{2}$.
In combination with Eq. (31), we reach our conclusion
$\bar{c}_{i}^{2}+\bar{c}_{j}^{2}\leq\bar{\mu}_{2}^{2}+\bar{\mu}_{3}^{2}$.

Case (c). In this case we have no constraints on the sign of $\bar{c}_{i}$ and
$\bar{c}_{j}$, so we enumerate all the possibilities.

Case (c1)  both $\bar{c}_{i}$ and $\bar{c}_{j}$ are positive. The proof in
case (a) only relies on the fact that both $\bar{c}_{i}$ and $\bar{c}_{j}$ are
positive, therefore, we have
$\bar{c}_{i}^{2}+\bar{c}_{j}^{2}\leq\bar{\mu}_{1}^{2}+\bar{\mu}_{2}^{2}$.

Case (c2)  both $\bar{c}_{i}$ and $\bar{c}_{j}$ are negative. The proof in
case (b) only relies on the fact that both $\bar{c}_{i}$ and $\bar{c}_{j}$ are
negative, therefore, we have
$\bar{c}_{i}^{2}+\bar{c}_{j}^{2}\leq\bar{\mu}_{2}^{2}+\bar{\mu}_{3}^{2}$.

Case (c3)  $\bar{c}_{i}$ is positive while $\bar{c}_{j}$ is negative. Then
$|\bar{c}_{i}|<|\bar{\mu}_{1}|$ and $|\bar{c}_{j}|<|\bar{\mu}_{3}|$, and we
have $\bar{c}_{i}^{2}+\bar{c}_{j}^{2}\leq\mu_{1}^{2}+\mu_{3}^{2}$.

In summary, we have proven that the fictitious state averaged in the diagonal
basis gives the largest violation of Bell’s inequality,
$\mathscr{B}(\overline{\rho}^{\rm diag})\geq\mathscr{B}(\overline{\rho})$.

Conclusions

The density matrix of a quantum state is reconstructed from the expectation
value of observables, which are measured from an average over a large event
ensemble. When the measurements are performed in an event-dependent frame, one
reconstructs a fictitious state, rather than a genuine quantum state.

We showed that, although the fictitious state loses most properties of a
genuine quantum state and its construction is basis-dependent, an observation
of Bell inequality violation based on an analysis of a fictitious state would
still imply Bell inequality violation for a quantum sub-state. Moreover, the
basis-dependence of fictitious states leaves a freedom of a basis choice for
its construction. We further demonstrated that the optimal basis for a
bipartite qubit system that maximizes the violation of Bell’s inequality is
the one that diagonalizes the spin-spin correlations.

Our observations have important consequences for actual quantum systems. In
particular, quantum mechanical systems, constructed from the spins of final
state particles at colliders commonly use event-dependent bases. For a given
kinematic configuration, the axes of the diagonal basis can change which
impacts how the spins should be measured. For example, in the top and anti-top
system at a proton-proton collider the optimal basis is very close to using
fixed axes in the lab frame when the top has low transverse momentum, while
the optimal basis is very close to using axes aligned along the top quark
momentum direction at high transverse momentum [7].

Given the widespread applications of fictitious states at particle colliders,
there are many possible future directions exploring other properties of
fictitious states. Beyond Bell inequality violation, it can also be shown that
the diagonal basis yields the optimal fictitious state for entanglement [7].
The orientation of the diagonal basis depends on the specific process in
question and is potentially different in each application.

In closing, the use of event-dependent bases in quantum experiments leads to
the reconstruction of fictitious states rather than quantum states. We have
shown that fictitious states still have utility in demonstrating the presence
of Bell inequality violation in a quantum system and that the optimal basis,
which diagonalizes the spin correlation matrix, maximizes the violation of
Bell’s inequality. We hope these observations about fictitious states will
clarify their usage in high-energy experiments and establish the groundwork
for future developments.

###### Acknowledgements.

This work was supported in part by the U.S. Department of Energy under grant
No. DE-SC0007914 and in part by the Pitt PACC. TH would like to thank the
Aspen Center for Physics, where part of this work is complete, which is
supported by the National Science Foundation (NSF) grant PHY-1607611. ML is
also supported by the National Science Foundation under grant No. PHY-2112829.
KC is supported in part by the National Science Foundation of China under
grant No. 12235001.

## References

  * Bell [1964] J. S. Bell, Physics Physique Fizika 1, 195 (1964).
  * Afik and de Nova [2021] Y. Afik and J. R. M. n. de Nova, Eur. Phys. J. Plus 136, 907 (2021), arXiv:2003.02280 [quant-ph] .
  * Afik and de Nova [2022] Y. Afik and J. R. M. n. de Nova, Quantum 6, 820 (2022), arXiv:2203.05582 [quant-ph] .
  * Fabbrichesi  _et al._ [2021] M. Fabbrichesi, R. Floreanini, and G. Panizzo, Phys. Rev. Lett. 127, 161801 (2021), arXiv:2102.11883 [hep-ph] .
  * Clauser  _et al._ [1969] J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, Phys. Rev. Lett. 23, 880 (1969).
  * Horodecki  _et al._ [1995] R. Horodecki, P. Horodecki, and M. Horodecki, Physics Letters A 200, 340 (1995).
  * [7] K. Cheng, T. Han, and M. Low, to appear .


# Label Noise-Resistant Mean Teaching for Weakly Supervised Fake News
Detection

Jingyi Xie†, Jiawei Liu†, Zheng-Jun Zha* University of Science and Technology
of China, China
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Fake news spreads at an unprecedented speed, reaches global audiences and
poses huge risks to users and communities. Most existing fake news detection
algorithms focus on building supervised training models on a large amount of
manually labeled data, which is expensive to acquire or often unavailable. In
this work, we propose a novel label noise-resistant mean teaching approach
(LNMT) for weakly supervised fake news detection. LNMT leverages unlabeled
news and feedback comments of users to enlarge the amount of training data and
facilitates model training by generating refined labels as weak supervision.
Specifically, LNMT automatically assigns initial weak labels to unlabeled
samples based on semantic correlation and emotional association between news
content and the comments. Moreover, in order to suppress the noises in weak
labels, LNMT establishes a mean teacher framework equipped with label
propagation and label reliability estimation. The framework measures a weak
label similarity matrix between the teacher and student networks, and
propagates different valuable weak label information to refine the weak
labels. Meanwhile, it exploits the consistency between the output class
likelihood vectors of the two networks to evaluate the reliability of the weak
labels and incorporates the reliability into model optimization to alleviate
the negative effect of noisy weak labels. Extensive experiments show the
superior performance of LNMT.

##  1 Introduction

††footnotetext: † Equal contribution.††footnotetext: * Corresponding author.

In recent years, the boom of social media has created a straight path from
content producers to content consumers, changing the way that people acquire
information, express feelings and exchange opinions Dou et al. (2021).
However, such a convenient platform also promotes the wide dissemination of
fake news (false or misleading information dissimulated in news articles to
mislead people) as it enables people to create and spread any information at
any time with a low cost. The dissemination of fake news may cause many
negative effects, including social panics and financial losses. For example,
fake news about the COVID-19 pandemic can prevent healthy behaviors and
increase the spread of the virus Lyu et al. (2021). Thus, automatic fake news
detection has become an urgent problem attracting much research effort.

Automatic fake news detection aims to design a classifier model to identify a
given news as real or fake Tschiatschek et al. (2018). Early studies mainly
focus on traditional machine learning models based on feature engineering Jin
et al. (2017b). After the emergence of deep learning technique, many deep
learning based methods Bian et al. (2020); Khoo et al. (2020); Jin et al.
(2017a) have been proposed and greatly improved the performance, due to the
powerful ability of automatically learning informative representations from
samples. Nevertheless, these deep learning based methods rely on large amount
of labeled samples to train supervised models. Creating such large training
data is extremely expensive and time-consuming, requiring the annotators to
have sufficient knowledge about the events. To fully utilize the ability of
deep learning models in fake news detection, it is essential to tackle the
problem of labeling fake news.

A promising solution is to exploit the feedback reports of users (i.e.,
comments) who read the news article. Most popular social medial platforms
offer a way for users to report their comments about the news article, which
contain significant information that are highly relevant to fake news
detection Shu et al. (2019); Qian et al. (2018), in the form of opinions,
stances, and emotion, etc. The reports of users can be served as weak sources
of labels, which bear implicit judgments of the users about the news, and are
more easily available. The large amount of users’ reports help alleviate the
annotation shortage problem for fake news detection. Shu et al. Shu et al.
(2020) proposes to exploit multiple weak supervision signals from different
sources from user engagements with the news content to generate weak labels
for unlabeled samples, and jointly leverage the limited labeled samples along
with these weakly labeled samples to train a fake news detector in a meta-
learning framework. However, the generated weak labels are unavoidably noisy
due to inaccurate prediction model and the meaningless or poor-quality
comments, which hinder model learning and performance improvement. Therefore,
how to obtain refined weakly labeled samples is a major issue. Wang et al.
Wang et al. (2020) attempts to address this issue, and proposes a preliminary
reinforced weakly-supervised framework (WeFEND) that uses reinforcement
learning technique to directly select high-quality samples from the estimated
weak labels to train the fake news detector. However, in WeFEND, the annotator
assigns weak labels simply based on user reports and is trained in an off-line
manner. It is independent from the training process of the reinforced selector
and the fake news detector, therefore the reliability of the estimated weak
labels cannot be improved by themselves and a large number of inaccurate
weakly labeled samples are directly discarded from the subsequent training
process of fake news detector. Moreover, WeFEND neglects the emotion
association between news content and the feedback reports, which contain
complementary and useful clues for fake news detection Giachanou et al.
(2019); Zhang et al. (2021).

In this work, we propose a novel label noise-resistant mean teaching approach
(LNMT) for weakly supervised fake news detection, which leverages unlabeled
news and feedback comments of users to enlarge the amount of training data and
facilitates model training by generating refined labels as weak supervision.
Specifically, as illustrated in Figure 1, LNMT establishes a mean teacher
framework with different learning ability and information interaction. The
framework consists of a teacher network and a student network that are based
on the same baseline model. The training process of the mean teacher framework
has two stages. In the first training stage, we pre-train a baseline model on
the limited manually labeled samples, which is then used as the initialization
of the teacher and student networks, and automatically assigns initial weak
labels for unlabeled samples. The baseline model simultaneously explores the
semantic correlation and the emotional association between news content and
the feedback reports for enhancing the quality of the initial weak labels. In
the second training stage, in order to depress noises in the weak labels, the
mean teacher framework measures weak label similarity matrix of the unlabeled
samples between the two networks, and propagates different valuable weak label
information encoded in the two networks to refine the weak labels. Meanwhile,
the framework exploits the consistency between the output class likelihood
vectors of the two networks to evaluate the reliability of weak labels, and
incorporates the reliability into model optimizing for alleviating the
negative effect on noisy weak labels. By jointly performing the label
propagation (LP) and label reliability estimation (LR) strategies, the
framework can effectively leverage the limited manually labeled samples and
vast refined weakly labeled samples to train a satisfactory fake news
detector. Extensive experiments on a benchmark dataset show the effectiveness
of the proposed approach.

The main contributions of this paper are as follows: (1) Considering the
shortage of high-quality labels, we propose a novel label noise-resistant mean
teaching (LNMT) method for weakly supervised fake news detection. (2) LNMT
simultaneously explores semantic correlation and emotional association in news
and user reports to assign initial weak labels with reasonable quality. (3)
With the proposed label propagation and label reliability estimation
strategies, LNMT has the ability to refine weak labels and alleviate the
negative effect of noisy weak labels.

Figure 1: The overall architecture of the proposed LNMT. (a) The detailed
structure of the baseline model. (b) The detailed structure of the mean
teacher framework equipped with label propagation and label reliability
estimation strategies.

##  2 Related Work

Supervised Fake News Detection. Most existing methods mainly focus on
extracting rumor features from the text contents, user profiles and
propagation structures, and training supervised classifiers on labeled samples
Kwon et al. (2013); Zhao et al. (2015). Early traditional works mainly design
a complementary set of hand-crafted features. For example, Kwon et al. Kwon et
al. (2013) studied the rumor spreading pattern on Twitter by exploring three
sets of features: temporal, structural and linguistic. Recently, researchers
exploit deep learning models that automatically mine high-level representation
to identify fake news Khoo et al. (2020); Yu et al. (2017). For example, Wang
et al. Wang et al. (2021) proposed a detection framework named MetaFEND, which
can learn quickly with a few verified posts using the proposed meta neural
process networks for effective detection on newly arrived events.

Weakly Supervised Fake News Detection. Although supervised methods have made a
remarkable breakthrough on performance, they rely on large amount of labeled
samples, which are extremely expensive to obtain. To handle this challenge,
learning with weak supervision provides a promising solution. For eaxmple, Shu
et al. Shu et al. (2020) leveraged a small amount of labeled data and multi-
source weak signals from social engagements with their importance weights, to
train a fake news detector in a meta-learning framework. Wang et al. Wang et
al. (2020) proposed a reinforced weakly-supervised fake news detection
framework, which consists of an annotator, a reinforced selector and a fake
news detector. The annotator assigned weak labels for unlabeled samples based
on user reports. The reinforced selector employed reinforcement learning
technique to select high-quality samples from weakly labeled samples. The fake
news detector identified fake news based on the news content.

##  3 Method

###  3.1 Architecture Overview

Fake news detection task can be defined as a binary classification problem,
i.e., each news article can be real ($\boldsymbol{y}=0$) or fake
($\boldsymbol{y}=1$). Let
$\\{\boldsymbol{X}^{l},\boldsymbol{Y}^{l}\\}=\\{(\boldsymbol{x}_{i}^{l},\boldsymbol{y}_{i}^{l})\\}_{i=1}^{N_{l}}$
denotes $N_{l}$ manually annotated news article $\boldsymbol{x}_{i}^{l}$ with
the associated label $\boldsymbol{y}_{i}^{l}$, and
$\boldsymbol{X}^{u}=\\{\boldsymbol{x}_{i}^{u}\\}_{i=1}^{N_{u}}$ denotes
$N_{u}$ unlabeled news article $\boldsymbol{x}_{i}^{u}$. In addition, each
news article $\boldsymbol{x}_{i}$ contains $M_{i}$ feedback reports of users
$\boldsymbol{R}_{i}=\\{\boldsymbol{r}_{i,1},...,\boldsymbol{r}_{i,M_{i}}\\}$.
Given the limited amount of labeled samples and large amount of unlabeled
samples, the objective is to effectively utilize the unlabeled data as weak
supervision to improve the performance. To this end, we propose a novel label
noise-resistant mean teaching approach (LNMT) for weakly supervised fake news
detection. The overall architecture is shown in Figure 1. It establishes a
mean teacher framework equipped with label propagation and label reliability
estimation strategies. The framework consists of a teacher network and a
student network, both of which are based on a baseline model. The training
process of the framework have two stages, which jointly leverages the limited
labeled samples and vast refined weakly labeled samples to train an effective
fake news detector.

###  3.2 The Baseline Model

The baseline model is the core component of the overall framework of LNMT,
which aims at automatically assigning initial weak labels to the unlabeled
samples, and enlarging the training set. It contains a transformer encoder and
an emotion-aware encoder. We employ the feedback reports provided by users as
weak supervision signals to train the baseline model in the first training
stage.

Transformer Encoder. Transformer Vaswani et al. (2017) has a strong ability to
extract semantic features by modeling long-term dependency and capturing the
sequence information through positional encoding. We thus design a transformer
encoder as the core of the baseline model to explore the semantic interaction
between news content and the reports, and learn the semantic feature
$\boldsymbol{f}_{sem}$. The building block of the transformer encoder is the
multi-head attention (MHA) layer Vaswani et al. (2017). An MHA layer consists
of a multi-head self-attention module and a fully connected feed-forward
network.

Given a sample consisting of a news article $\boldsymbol{x}$ and $M$ reports
$\boldsymbol{R}=\\{\boldsymbol{r}_{i}\\}_{i=1}^{M}$, the sample can be denoted
as
$\mathcal{W}=[\boldsymbol{w}_{1},\boldsymbol{w}_{2},...,\boldsymbol{w}_{M+1}]$,
where $\boldsymbol{w}_{i}$ denotes $i$-th sequence text (corresponding to the
news article or each report). Each sequence text
$\boldsymbol{w}_{i}=[\boldsymbol{w}_{i,1},\boldsymbol{w}_{i,2},...,\boldsymbol{w}_{i,n_{i}}]$
have $n_{i}$ words. The words in the news article and the reports are firstly
projected into continuous word embeddings. The sequences of word embeddings in
the news article and the reports are passed through $L_{w}$ layers of token-
level MHA to produce the complex sequence representations
$\\{\boldsymbol{z}_{i,j}\\}_{i=1}^{M+1}$. These layers allow information
interaction between tokens in a news article or a report. The sequence
features $\\{\boldsymbol{z}_{i,j}\\}_{i=1}^{M+1}$ are then aggregated by
attention mechanism to generate the semantic sentence representations
$\\{\boldsymbol{f}_{i}^{s}\\}_{i=1}^{M+1}$ for the news article and the
reports. $\boldsymbol{f}_{i}^{s}$ is calculated as following:

$\displaystyle\\{\boldsymbol{z}_{i,j}\\}_{j=1}^{n_{i}}$
$\displaystyle=\text{MHA}_{L_{w}}(\\{\boldsymbol{w}_{i,j}\\}_{j=1}^{n_{i}})$
$\displaystyle g_{i,j}$
$\displaystyle=\text{Softmax}(\boldsymbol{\gamma}_{w}^{T}\boldsymbol{z}_{i,j})$
$\displaystyle\boldsymbol{f}_{i}^{s}$
$\displaystyle=\sum_{j=1}^{n_{i}}g_{ij}\boldsymbol{z}_{i,j}$ (1)

where $\boldsymbol{w}_{i,j}$ denotes $j$-th word tokens of $i$-th sequence
text, $n_{i}$ is the number of word tokens for $i$-th sequence text.
$\boldsymbol{\gamma}_{w}\in\mathbb{R}^{d_{model}}$ denotes the learnable
training parameter of the linear transformation layer and $g_{ij}$ denotes the
importance scores of the word tokens. Furthermore, the obtained sentence
representations $\\{\boldsymbol{f}_{i}^{s}\\}_{i=1}^{M+1}$ are fed into extra
$L_{r}$ layers of report-level MHA, which model the information interactions
between the news article and the reports at report-level. Lastly, the
attention mechanism is used to interpolate the news article and the reports to
obtain the final semantic representation $\boldsymbol{f}_{sem}$ of the sample,
which is calculated as following:

$\displaystyle\\{\boldsymbol{z}_{i}^{s}\\}_{i=1}^{M+1}$
$\displaystyle=\text{MHA}_{L_{r}}(\\{\boldsymbol{f}_{i}^{s}\\}_{i=1}^{M+1})$
$\displaystyle\pi_{i}$
$\displaystyle=\text{Softmax}(\boldsymbol{\gamma}_{s}^{T}\boldsymbol{z}_{i}^{s})$
$\displaystyle\boldsymbol{f}_{sem}$
$\displaystyle=\sum_{i=1}^{M+1}\pi_{i}\boldsymbol{z}_{i}^{s}$ (2)

where $\boldsymbol{f}_{i}^{s}\in\mathbb{R}^{d}$, and
$\boldsymbol{\gamma}_{s}\in\mathbb{R}^{d_{model}}$ denotes the learnable
training parameter of the linear transformation layer.

Emotion-aware Encoder. Prior works Giachanou et al. (2019); Ajao et al. (2019)
have shown that there exists obvious relationship between news veracity and
the emotion information in the reports of users. Following this idea, we
design an emotion-aware encoder, which mines the emotion information contained
in the news article and the user reports, and explores the emotional
association between them to further enhance feature representation.
Specifically, the emotion-aware encoder firstly extracts emotion vectors
$\\{\boldsymbol{h}_{i}\\}\in\mathbb{R}^{d_{emo}}$ from the news article and
the reports. It then projects $\boldsymbol{h_{i}}$ into three different
representation subspaces $\boldsymbol{h}_{i}^{(q)}$,
$\boldsymbol{h}_{i}^{(k)}$ and $\boldsymbol{h}_{i}^{(v)}$, and applies an
emotion-aware attention module (at the report-level) with
$\boldsymbol{h}^{(q)}_{i}=\boldsymbol{h}^{(q)}_{i}+\boldsymbol{z}_{i}^{s}$ as
queries, $\boldsymbol{h}_{i}^{(k)}$ as keys, and $\boldsymbol{h}_{i}^{(v)}$ as
values, to generate the final emotion feature $\boldsymbol{f}_{emo}$ for the
sample. The feature $\boldsymbol{f}_{emo}$ is calculated as following:

$\displaystyle\boldsymbol{h}_{i}^{(q)}$
$\displaystyle=\boldsymbol{W}_{hq}^{T}\boldsymbol{h}_{i}+\boldsymbol{z}_{i}^{s},\;\boldsymbol{h}_{i}^{(k)}=\boldsymbol{W}_{hk}^{T}\boldsymbol{h}_{i},\;\boldsymbol{h}_{i}^{(v)}=\boldsymbol{W}_{hv}^{T}\boldsymbol{h}_{i}$
$\displaystyle\beta_{ji}$
$\displaystyle=\text{Softmax}(\frac{\boldsymbol{h}^{(q)}_{i}\boldsymbol{h}^{(k)T}_{j}}{\sqrt{d_{model}}})$
$\displaystyle\boldsymbol{f}_{emo}$
$\displaystyle=\frac{1}{M+1}\sum_{j=1}^{M+1}\sum_{i=1}^{M+1}\beta_{ji}\boldsymbol{h}_{i}^{(v)}$
(3)

where
$\boldsymbol{W}_{hq},\boldsymbol{W}_{hk},\boldsymbol{W}_{hv}\in\mathbb{R}^{d_{emo}\times
d_{model}}$ are the weight matrices of linear projections. $\beta_{ji}$
denotes emotion-aware attention, which measures the pairwise compatibility of
the emotion features. We concatenate the complementary two features
$\boldsymbol{f}_{c}=\boldsymbol{f}_{sem}\oplus\boldsymbol{f}_{emo}$ to produce
the reinforced semantic feature $\boldsymbol{f}_{c}$. The feature
$\boldsymbol{f}_{c}$ is then fed into a classifier layer for prediction. In
the first training stage, we employ a binary cross entropy loss to train the
baseline model with the labeled samples under supervised setting.

###  3.3 Mean Teacher Framework

Given the unlabeled set $\boldsymbol{X}^{u}$ with the corresponding reports,
the pre-trained baseline model can predict their weak labels, denoted as
$\hat{\boldsymbol{Y}}^{u}$. Nevertheless, the automatically annotated weak
labels are unavoidably noisy and directly utilizing these samples
$\\{\boldsymbol{X}^{u},\hat{\boldsymbol{Y}}^{u}\\}$ to train the model could
mislead feature learning and degrade the performance. The mean teacher
framework is designed to refine the weak labels by utilizing label
propagation, and evaluate the reliability of them for alleviating the negative
effect on noisy weakly labeled samples during modeling optimization.

The mean teacher framework consists of a teacher network $F_{t}$ and a student
network $F_{s}$, both of which are initialized by the pre-trained baseline
model in the first training stage. The teacher network $F_{t}$ acts as a
teacher that periodically produces soft weak labels for unlabeled samples,
which are employed as weak supervision signals for the student network
$F_{s}$. In the second training stage, the parameters of $F_{s}$ are updated
through back-propagation, while $F_{t}$ are updated through exponential moving
average policy on the parameters of $F_{s}$. This policy guarantees the two
networks have different learning ability and label evolving is stable
Tarvainen and Valpola (2017). The second training stage has $T$ training
generations (epochs). Before each generation, we feed all unlabeled samples
into $F_{t}$, and obtain the soft predictions
$\\{\boldsymbol{y}_{i}^{u}=\boldsymbol{p}_{i}^{u,t}\\}_{i=1}^{N_{u}}$,
$\boldsymbol{y}_{i}^{u}\in[0,1]$ will serve as the soft weak label for sample
$\boldsymbol{x}_{i}^{u}$ in the next generation. During each training
generation, the loss function $\mathcal{L}_{s}$ for the student network
$F_{s}$ is formulated as following:

$\displaystyle\mathcal{L}_{l}$
$\displaystyle=\frac{1}{B}\sum_{i=1}^{B}-[\boldsymbol{y}_{i}^{l}\log\boldsymbol{p}_{i}^{l}+(1-\boldsymbol{y}_{i}^{l})\log(1-\boldsymbol{p}_{i}^{l})]$
$\displaystyle\mathcal{L}_{u}$
$\displaystyle=\frac{1}{B}\sum_{i=1}^{B}-[\boldsymbol{y}_{i}^{u}\log\boldsymbol{p}_{i}^{u,s}+(1-\boldsymbol{y}_{i}^{u})\log(1-\boldsymbol{p}_{i}^{u,s})]$
$\displaystyle\mathcal{L}_{s}$ $\displaystyle=\mathcal{L}_{l}+\mathcal{L}_{u}$
(4)

where $\mathcal{L}_{l},\mathcal{L}_{u}$ are the losses for labeled and
unlabeled samples respectively. $\boldsymbol{p}_{i}^{l}$,
$\boldsymbol{p}_{i}^{u,s}$ are the soft prediction scores of the labeled and
unlabeled samples from the student network. After the parameters of $F_{s}$
updated, $F_{t}$ is updated as following:

$\boldsymbol{\theta}_{t}^{j}\leftarrow\alpha\boldsymbol{\theta}_{t}^{j}+(1-\alpha)\boldsymbol{\theta}_{s}^{j}$
(5)

where $\\{\boldsymbol{\theta}_{t}^{j}\\}$ and
$\\{\boldsymbol{\theta}_{s}^{j}\\}$ are the parameters of the teacher and
student networks, respectively. $\alpha$ is a momentum coefficient.

Label Propagation. Although weak labels can be gradually improved with the
proceeding of iteratively optimizing for the teacher network, there exists
some unavoidable label errors that cannot be refined and would be overfitted
as the training proceeds. As the teacher and student networks have different
learning ability, they are able to provide different valuable supervision
information to further assist in refining weak label errors. Thus, a label
propagation strategy is proposed to exploit information interaction between
the two networks. Specifically, before each training generation, we employ
both the teacher and student networks to obtain the hard predictions (hard
weak labels) of the unlabeled set $\boldsymbol{X}^{u}$, denoted as
$\\{\hat{\boldsymbol{y}}_{i}^{u,t}\\},\\{\hat{\boldsymbol{y}}_{i}^{u,s}\\}\in\\{0,1\\}$.
A label similarity matrix $\boldsymbol{C}\in\mathbb{R}^{2\times 2}$ is then
calculated via Intersection over Union (IoU) criterion between the predicting
results of unlabeled sets from the two networks:

$\boldsymbol{C}_{m,n}=\frac{|\\{\boldsymbol{X}^{u}:\hat{\boldsymbol{y}}_{i}^{u,s}=m\cap\hat{\boldsymbol{y}}_{i}^{u,t}=n\\}|}{|\\{\boldsymbol{X}^{u}:\hat{\boldsymbol{y}}_{i}^{u,s}=m\cup\hat{\boldsymbol{y}}_{i}^{u,t}=n\\}|}$
(6)

where $m,n\in\\{0,1\\}$, and $|\cdot|$ counts the number of the unlabeled
samples. $\boldsymbol{C}_{m,n}$ measures the consensus between the label class
$m$ in the student network and the label class $n$ in the teacher network.
$\boldsymbol{C}$ is then normalized such that each row sums to $1$. After the
estimated weak label similarity matrix obtained, the complementary weak
supervision information from the student network can be propagated to the
teacher network to refine the weak labels. The soft predictions
$\\{\boldsymbol{y}_{i}^{u}=\boldsymbol{p}_{i}^{u,t}\\}_{i=1}^{N_{u}}$ from the
teacher network are purified:

$[1-\boldsymbol{y}_{i}^{u},\boldsymbol{y}_{i}^{u}]\leftarrow(1-\beta)[1-\boldsymbol{y}_{i}^{u,s},\boldsymbol{y}_{i}^{u,s}]\boldsymbol{C}+\beta[1-\boldsymbol{y}_{i}^{u},\boldsymbol{y}_{i}^{u}]$
(7)

where $\boldsymbol{y}_{i}^{u,s}=\boldsymbol{p}_{i}^{u,s}\in[0,1]$ denotes the
soft prediction of the unlabeled sample from the student network. The refined
weak label $\boldsymbol{y}_{i}^{u}$ is employed in Eq. 4 and Eq. 5 to optimize
the mean-teacher framework.

Label Reliability Estimation. Apart from utilizing label propagation to refine
noisy weak labels, we propose to evaluate the credibility of the weak labels
and suppress the contribution of samples with error-prone weak labels in the
loss function for alleviating the negative effect. Based on the uncertainty
theory Zheng and Yang (2021); Kendall and Gal (2017), a network has
uncertainty on its predicted result of an input sample. The more uncertain
result a network predicts, the less reliable its output is, and thus is more
likely to generate incorrect weak labels. The uncertainty can be utilized to
softly assess the credibility of the weak labels. We evaluate the credibility
(uncertainty) of weak labels based on the output consistency (inconsistency)
between the teacher and student networks. A straightforward strategy is to
calculate the distance between the reinforced semantic features
$\boldsymbol{f}_{c}^{u,t}$ and $\boldsymbol{f}_{c}^{u,s}$ from the two
networks as the consistency. Nevertheless, this strategy does not consider the
global distribution information in the whole unlabeled set. We thus exploit
the class likelihood vectors $\boldsymbol{q}_{i}^{u,s}$,
$\boldsymbol{q}_{i}^{u,t}$ to evaluate the consistency. For each unlabeled
sample $\boldsymbol{x}_{i}^{u}$ and its reinforced semantic feature
$\boldsymbol{f}_{c}^{u,s}$ from the student network, we calculate its
similarity to 2-class center vector $\bar{\boldsymbol{f}}_{c}^{u,s}$ to obtain
the class likelihood vector $\boldsymbol{q}_{i}^{u,s}$ from the student
network as follows:

$\boldsymbol{q}_{i}^{u,s}=\text{Softmax}(\bar{\boldsymbol{f}}_{c}^{u,s}\cdot\boldsymbol{f}_{c,i}^{u,s})$
(8)

where $\bar{\boldsymbol{f}}_{c}^{u,s}\in\mathbb{R}^{2\times d}$ is the center
vector of 2 classes (fake or real) on the unlabeled set, which is calculated
by averaging all reinforced semantic features of the unlabeled samples
predicted fake or real on the unlabeled set. Similarly, the class likelihood
vector $\boldsymbol{q}_{i}^{u,t}$ from the teacher network is obtained.
Afterwards, we employ the Maximum Mean Discrepancy (MMD) with Gaussian kernel
Gretton et al. (2012) to measure the distance between the two class likelihood
vectors from the two networks as the uncertainty $u_{i}$ of the unlabeled
sample $\boldsymbol{x}_{i}^{u}$:

$u_{i}=\text{MMD}(\boldsymbol{q}_{i}^{u,t},\boldsymbol{q}_{i}^{u,s})$ (9)

The credibility of corresponding estimated weak label is calculated as
$\omega_{i}=\exp(-u_{i})$ for enabling stable optimizing. For an unlabeled
sample $x_{i}^{u}$ with high uncertainty $u_{i}$, a smaller weight
$\omega_{i}$ is employed to reduce its negative contribution in the loss
function. Thus, we incorporate $\omega_{i}$ into the binary cross entropy loss
in Eq. 4, and define the credibility-aware cross entropy loss as follows:

$\displaystyle\mathcal{L}_{u}=\frac{1}{B}\sum_{i=1}^{B}-\omega_{i}[\boldsymbol{y}_{i}^{u}\log\boldsymbol{p}_{i}^{u,s}+(1-\boldsymbol{y}_{i}^{u})\log(1-\boldsymbol{p}_{i}^{u,s})]$
(10)

The credibility-aware cross entropy loss is finally used to optimize the
framework on the unlabeled samples.

##  4 Experiments

Table 1: Performance comparison of the state-of-the-art methods on our Wechat dataset. Settings | Method | Accuracy | AUC-ROC | Fake News | Real News
---|---|---|---|---|---
Precision | Recall | F1 | Precision | Recall | F1
Supervised | LIWC-SVM | 58.0 | 60.9 | 57.9 | 53.6 | 55.6 | 56.8 | 62.4 | 59.5
LSTM | 74.1 | 80.5 | 87.2 | 56.1 | 68.3 | 67.5 | 92.1 | 77.9
CNN | 74.3 | 82.8 | 86.2 | 58.9 | 70.0 | 69.2 | 89.7 | 77.5
EANN | 76.1 | 79.4 | 86.5 | 62.9 | 72.8 | 71.4 | 89.3 | 79.3
DualEmotion | 77.4 | 81.7 | 84.9 | 66.4 | 74.5 | 73.8 | 88.4 | 80.4
| LNMT(S) | $78.7$ | $82.9$ | $85.2$ | $67.7$ | $75.4$ | $73.2$ | $89.7$ | $80.6$
Semi-Supervised | $\text{LSTM}+\text{Semi}$ | $75.6$ | $83.7$ | $84.9$ | $61.7$ | $71.4$ | $70.4$ | $89.6$ | $78.9$
$\text{CNN}+\text{Semi}$ | $75.4$ | $84.2$ | $85.1$ | $62.8$ | $72.3$ | $71.1$ | $88.0$ | $78.7$
LNMT(SS) | $79.6$ | $84.5$ | $82.8$ | $70.2$ | $75.9$ | $74.8$ | $89.0$ | $81.2$
Weakly-Supervised | $\text{LSTM}+\text{LP}+\text{LR}$ | 77.5 | 84.6 | 86.5 | 66.2 | $75.1$ | 74.3 | 88.8 | 81.0
$\text{CNN}+\text{LP}+\text{LR}$ | 78.1 | 84.9 | 86.2 | 68.0 | $76.0$ | 75.7 | 88.2 | 81.5
WeFEND | 79.9 | 84.3 | 85.9 | 72.3 | 78.5 | 75.9 | 87.3 | 81.2
| LNMT | 82.6 | 87.1 | 88.3 | 74.8 | 81.0 | 77.5 | 90.4 | 83.5

###  4.1 Dataset and Evaluation Metrics

The Wechat dataset Wang et al. (2020) is collected from WeChat’s Official
Accounts. In order to promote fake news detection, the WeChat’s Official
Account encourages users to report suspicious articles, and write feedback
messages to explain the reasons. Note that the publicly released version of
the Wechat dataset does not match the used version in the method WeFEND Wang
et al. (2020). WeFEND does not provide the information about the used version
of the Wechat dataset. In our experiments, we thus randomly select samples
from the publicly released version of the Wechat dataset to create 5 sub-
datasets (Following the same setting of the used version of the Wechat
dataset) for evaluation, each of which contains 2,440 news articles for
training, 1,000 news articles for validation, 1,740 news articles for testing
and 22,981 articles for the unlabeled set. Training, validation and test sets
contain an equal amount of fake and real news articles, respectively. Detailed
statistics of our dataset can be found in Supplementary Material. We average
the performance on the five sub-datasets as the final results. Following
previous methods, we adopt Accuracy, AUC-ROC and class-wise Precision, Recall
and $F_{1}$-Score as the evaluation metrics.

###  4.2 Implementation Details

We adopt Chinese Word Vectors to initialize the word embeddings in the
baseline model, which contains vectors of length 300 Li et al. (2018). We set
the input and output dimensions of linear projections in all feed-forward
networks and MHA layers ($\mathbb{R}^{d}$, $\mathbb{R}^{d_{model}}$) to 300.
Following the work Zhang et al. (2021), several emotion sources are adopted to
extract robust emotion vector ($\mathbb{R}^{d_{emo}}$) of length 47, which is
composed of emotion lexicon and intensity features, sentiment score and other
auxilary features. We adopt the Affective Lexicon Ontology to extract emotion
lexicon and intensity features with length 29, employ the dictionary HowNet to
calculate sentiment score with length 1 and extract other auxiliary features
with length 17. The number of MHA layers are set as $L_{w}=2$ and $L_{r}=4$.
We use a maximum sequence length of 40 and 100 for token-level and report-
level MHA layers in the baseline model, respectively. In the first training
stage, the input batch size of the baseline model is set to 48, which is
trained for 200 epochs by using Adam optimizer. We adopt a multi-stage
learning rate schedule, with the learning rate linearly increasing from 0 to
3e-5 in the first 8000 iterations, and decreasing linearly to 3e-7 in the
remaining iterations. In the second training stage, we set the momentum
coefficient $\alpha=0.999$ in Eq. 5, label propagating rate $\beta=0.9$ in Eq.
7. We adopt Adam optimizer to train the mean teacher framework with a fixed
learning rate 2e-6. The framework is optimized with batch size of 64, for 15
epochs.

###  4.3 Comparison to State-of-the-Art Methods

Table 2: Effectiveness of the emotion-aware encoder within the baseline model. Model | Accuracy | AUC-ROC
---|---|---
w/o emotion | 77.4 | 81.7
emotion | 78.7 | 82.9
Table 3: Effectiveness of label propagation and label reliability estimation within the mean teacher framework. Model | LP | LR | Accuracy | AUC-ROC
---|---|---|---|---
Stage 1 | - | - | 78.7 | 82.9
Stage 1 + Stage 2 | $\times$ | $\times$ | 79.5 | 84.0
✓ | $\times$ | 80.8 | 85.4
✓ | ✓ | 82.6 | 87.1

We compare LNMT with several fake news detection methods under supervised,
semi-supervised and weakly-supervised settings on our Wechat dataset in Table
1, including: LIWC-SVM Pennebaker et al. (2015), LSTM Ruchansky et al. (2017),
CNN Wang et al. (2018), EANN Wang et al. (2018), DualEmotionZhang et al.
(2021), WeFENDWang et al. (2020). The details of these compared methods and
LNMT under the three settings can be found in Supplementary Material. It can
be observed that LNMT achieves the best performance in terms of Accuracy, AUC-
ROC, Precision, Recall and $F_{1}$ metrics. In the supervised setting, all the
methods directly utilize the labeled samples to train the models and predict
the results on the testing set. Among the supervised learning approaches, deep
learning based ones such as LNMT(S), DualEmotion are superior to the
traditional method LIWC-SVM, which implies that neural network is more capable
in capturing representative features. Particularly, LNMT(S) employs the
transformer structure to capture semantic dependencies at the token-level and
report-level, and incorporates emotion information to improve the capacity.
This design derives more robust features and achieves the best result in
supervised setting. In the semi-supervised setting, all methods utilize
external unlabeled data and entropy minimization loss Grandvalet et al. (2005)
for training. Semi-supervised learning greatly increases the amount of
training samples, and brings a performance boost in LSTM+Semi, CNN+Semi and
LNMT(SS). Furthermore, LNMT(SS) improves the second best compared method
CNN+Semi by 4.2% in Accuracy and 0.3% AUC-ROC, respectively. In the weakly-
supervised setting, LSTM+LP+LR, CNN+LP+LR and our LNMT not only utilize
unlabeled samples as supplement, but also tackle noises in the weak labels by
using label propagation and label reliability estimation strategies. Such
strategies ensure more reliable training samples with refined weak labels
during model optimization. The experiment results show greater improvements in
performance than these models with semi-supervised learning scheme, achieving
a boosting of 3%-4% in Accuracy and AUC-ROC. The improvement verifies the
effectiveness of the proposed two strategies. Moreover, LNMT improves the
second best weakly-supervised method WeFEND by 2.7% in Accuracy and 2.8% in
AUC-ROC, which further demonstrates the superiority of LNMT.

###  4.4 Ablation Study

Analysis of the baseline model. In Table 2, we investigate the influence of
the emotion-aware encoder in the baseline model. “w/o emotion” denotes the
baseline model without using emotion-aware encoder, and “emotion” denotes the
whole the baseline model. It can be observed that without using emotion-aware
encoder leads to 1.3% and 1.2% drop in Accuracy and AUC-ROC respectively,
which confirms that exploring emotional association between news article and
reports can enhance the capacity of the learned features.

Analysis of LP and LR. We conduct the experiments to analyze the influence of
label propagation and label reliability estimation in the mean teacher
framework. As shown in Table 3, compared to the baseline model, the mean
teacher framework without using LP and LR achieves 79.5% and 84.0% in Accuracy
and AUC-ROC, obtaining 0.8% and 1.1% improvement in Accuracy and AUC-ROC,
respectively. This improvement indicates the teacher and student networks have
different learning ability, which help initially correct weak label errors. By
applying LP to the mean teacher framework, the model performance is improved
by 1.3% Accuracy and 1.4% AUC-ROC, which verifies the effectiveness of the
label propagation strategy to exploit information interaction between the two
networks for depressing noises in weak labels. By further applying LR to the
mean teacher framework, the model performance is improved by 1.8% Accuracy and
1.7% AUC-ROC, indicating the effectiveness of the label reliability estimation
to suppress the contribution of samples with error-prone weak labels during
model training.

##  5 Conclusion

In this paper, we present a novel label noise-resistant mean teaching approach
(LNMT) for weakly supervised fake news detection. LNMT automatically assigns
initial weak labels to unlabeled samples based on the semantic correlation and
emotional associations between news contents and the reports. Moreover, LNMT
establishes a mean teacher framework with different learning ability and
information interaction. The label propagation and label reliability
estimation within the framework is able to refine the weak labels and
alleviate the negative effect of noisy weakly labeled samples within the loss
function. Extensive experiments show that the proposed LNMT achieves
significant improvement over the state-of-the-art methods.

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