# Quantum coherence and entanglement in neutral-current neutrino oscillation
in matter

M.M. Ettefaghi<EMAIL_ADDRESS>Department of Physics, University of Qom,
Ghadir Blvd., Qom 371614-6611, I.R. Iran  Z. Askaripour Ravari  Department of
Physics, Islamic Azad University, North Tehran Branch, Tehran, I.R. Iran

###### Abstract

Although neutrino-antineutrino states originating from neutral-current
interactions are blind concerning the flavor state, an oscillation pattern is
predicted provided that both neutrino and antineutrino are detected. This
issue arises from both the coherence and entanglement of the neutrino-
antineutrino states. Based on quantum resource theory, we use the $l_{1}$-norm
and concurrence to quantify quantum coherence and entanglement, respectively.
Considering the localization properties by the wave packet approach and a
matter potential which appears when neutrino and antineutrino propagate in a
material medium, we obtain the $l_{1}$-norm and concurrence. We see that the
neutrino and antineutrino remain entangled for larger baseline lengths when
they propagate in a material medium. In the case of the coherence property,
the $l_{1}$-norm decreases in comparison to the corresponding one in vacuum.
However, its damping occurs for larger distances.

Neutrino oscillation, $Z_{0}$ decay, entanglement, coherence, resonance
potential

##  I Introduction

Quantum coherence and entanglement are two important quantum resources that
play crucial roles in quantum information and quantum computation science and
they are investigated in various branches of condensed matter, atomic physics,
and quantum optics. Additionally, several particle physics phenomena, such as
meson mixing and neutrino oscillation, could potentially provide suitable
opportunities for their explorations LG2016 ; c1 ; b1 ; b2 ; epl ; e2 ; e3 ;
e4 ; epjp ; prd ; 11 ; 12 ; 13 . For instance, violation of the Leggett-Garg
inequality as a manifestation of quantum coherence has been investigated using
MINOS data LG2016 .The $l_{1}$-norm value as a measure of quantum coherence
has also been studied using data from the Day Bay, KamLAND, MINOS and T2K
neutrino experiments c1 . For instance, in the case of the Daya Bay
experiment, in which detectors are near the source and the measured transition
probability is low, the $l_{1}$-norm takes a value much lower than its
maximum, while in the case of KamLAND, in which the neutrino baseline is about
180 km, this coherence measure reaches the maximal value. Certainly, the
neutrino oscillation process, which is a quantum mechanical phenomenon
occurring on the macroscopic scale, is an appropriate medium in which to study
the quantum foundation aspects and quantum correlations such as quantum
coherence and entanglement. Using the wave packet approach, one can give a
meticulous description of neutrino oscillation 43 ; 44 ; 45 ; 46 .
Specifically, the production, propagation and detection of a neutrino should
be considered as localized processes, and this localization is very well
fulfilled using the wave packet approach. To be more precise, we must
emphasize that the observation of neutrino oscillation depends on the
coherence of neutrinos during production, propagation, and detection
processes. The production and detection coherence conditions are satisfied
provided that the intrinsic quantum mechanical energy uncertainties during
these processes are large compared to the energy difference $\Delta E_{ij}$ of
different neutrino mass eigenstates:

$\Delta E_{ij}\sim\frac{\Delta m^{2}_{ij}}{2E}\ll\sigma_{E},$ (1)

where
$\sigma_{E}=\mbox{min}\\{\sigma^{\text{prod}}_{E},\sigma^{\text{det}}_{E}\\}$.
This condition implies that during the production and detection processes, one
cannot discriminate the neutrino mass eigenstates. Conservation of coherence
during propagation means that wave packets describing the mass eigenstates
overlap from the production to the detection regions. The wave packets
describing different neutrino mass eigenstates propagate with different group
velocities. After propagating $L$, the separation of different mass wave
packets is $\frac{\Delta m_{ij}^{2}}{2E^{2}}L$. Consequently, coherent
propagation is guaranteed provided that

$\frac{\Delta
m^{2}_{ij}}{2E^{2}}L\ll\sigma_{x\nu}\simeq\frac{v_{g}}{\sigma_{E}},$ (2)

where $v_{g}$ is the average group velocity of wave packets of different
neutrino mass eigenstates, and $\sigma_{x\nu}$ is their common effective
spatial width. In other words, similar to the double-slit experiment, if one
could determine which mass eigenstate is created or detected, the neutrino
oscillation pattern would disappear. In particular, the conservation of energy
and momentum implies that exact determination of the energy-momentum of
charged leptons leads to determination of the mass eigenstate of the
corresponding neutrino (in fact, exact momentum conservation causes the
neutrino state to be kinematically entangled with the corresponding charged
lepton state), and the neutrino oscillation ceases found ; glashow .

Furthermore, the neutrino oscillation process is observed provided that
neutrinos have a specific initial flavor state. The kinematic analysis shows
that a neutrino state created through charged current (CC) interactions
exhibits this situation. For instance, a muon neutrino is created by pion
decay while muon decay only gives an electron neutrino. In contrast, the
neutral current (NC) processes or the $Z_{0}$ decay are blind with respect to
the neutrino flavor states. In other words, every flavor eigenstate and every
mass eigenstate is created with equal probability. Therefore, one can write,
in general, the state of the created neutrino and antineutrino as follows:

$|\nu_{Z}\rangle=\frac{1}{\sqrt{3}}\sum_{\alpha=e,\mu,\tau}|\nu_{\alpha}\rangle|\bar{\nu}_{\alpha}\rangle=\dfrac{1}{\sqrt{3}}\sum_{i=1}^{3}\left|\nu_{i}\right\rangle\left|\bar{\nu}_{i}\right\rangle.$
(3)

Here, the second equality is satisfied because the transformation matrix
between mass and flavor eigenstates is unitary. Thus, the usual neutrino
oscillation in which either neutrino or antineutrino is detected cannot be
observed for NC neutrinos. However, Eq. (3) shows that there is another
property that is noticeable: the neutrino and antineutrino originating from
the NC interaction are maximally entangled due to the conservation of energy-
momentum and lepton number in electroweak interactions. Hence, if both
neutrino and antineutrino are detected, it is possible that oscillation
patterns will be observed between detectors sza ; 4 ; scrip . However, if only
either neutrino or antineutrino is detected, we will have an entirely
classical ensemble and will be unable to observe any oscillation pattern.
Indeed, this phenomenon can be interpreted as a CPT transformation of half of
the process maps of the propagation of neutrino-antineutrino pairs onto the
traditional neutrino oscillation experiment. Of course, its requirement is
that the initial state is an entangled state according to Eq. (3). Otherwise,
if the initial state is separable as follows:

$\rho=\frac{1}{3}\sum_{\gamma}|\nu_{\gamma}\rangle|\bar{\nu}_{\gamma}\rangle\langle\nu_{\gamma}|\langle\bar{\nu}_{\gamma}|,$
(4)

using the CPT transformation on the antineutrino process, one can show that
the probability of detecting a neutrino with flavor state
$|\nu_{\alpha}\rangle$ at a detector and an antineutrino with flavor state
$|\nu_{\beta}\rangle$ at another detector is

$P_{\alpha\beta}=\frac{1}{3}\sum_{\gamma}P_{\gamma\alpha}P_{\beta\gamma},$ (5)

which is not the traditional neutrino oscillation probability. Although NC
neutrino oscillation is beyond the reach of any experiment, it can be
implemented analogically by the double-double-slit experiment performed in
Ref. dds . In this experiment, the path-entangled photons are passed through
opposite screens with a double slit. Due to entanglement, each photon can
reveal the which-slit-path information of the other photon. Therefore, the
two-photon interference pattern appears if detection locations of photons are
correlated without revealing the which-slit-path information. However, it has
also been shown experimentally and theoretically that two-photon quantum
interference disappears when the which-slit-path of one photon is detected in
the double-double-slit.

When neutrinos (antineutrinos) propagate in a material medium, they might have
a forward coherent scattering off electrons and nucleons of the matter via
both weak CC and weak NC interactions wolf ; mikh . All three neutrino
(antineutrino) flavor states can interact with the matter through NC, but the
electron neutrino (antineutrino) has an additional interaction (i.e.
interaction through the CC) which causes it to feel an additional potential.
In the calculations, we only consider the effective potential due to the CC
interaction because the potentials coming from the NC interactions induce
common phases for all three neutrino (antineutrino) flavor states and do not
alter neutrino oscillating behaviors. Therefore, the effective potential of
matter which must be considered is $V_{CC}=\sqrt{2}G_{F}N_{e}$
($\bar{V}_{CC}=-\sqrt{2}G_{F}N_{e}$), where $G_{F}$ and $N_{e}$ are the Fermi
coupling constant and the density of electrons in the medium, respectively. If
the matter density of the medium is nonuniform, the potential depends on the
coordinate; otherwise, it is constant.

According to Eq. (3), the neutrino and antineutrino produced by NC
interactions are entangled with respect to either their flavor or mass modes.
Indeed, this state has a maximal entanglement similar to the Bell
states111Bell states are the four states that can be created when two qubits
are maximally entangled. The four states are represented as
$|\Phi^{\pm}\rangle=\frac{|00\rangle\pm|11\rangle}{\sqrt{2}}$ and
$|\Psi^{\pm}\rangle=\frac{|10\rangle\pm|10\rangle}{\sqrt{2}}.$. Moreover, they
are entangled due to energy-momentum conservation. In fact, the predicted
oscillation pattern between two detectors is based on these features. Also, as
a result of entanglement, the coherence condition obtained by considering
localization properties via the wave packet approach is stronger than that
obtained by imposing the non-separation mass wave packet constraint analogous
to Eq. (2) 4 ; scrip . In fact, the coherent propagation of both neutrino and
antineutrino is not sufficient because the oscillation pattern ceases if the
distance between the detectors is larger than the coherence length. In this
paper, we consider the matter refraction to reanalyze this problem in the
context of two-flavor neutrino oscillations. In this case, the NC neutrino
oscillation cannot be concluded by a CPT transformation of half the process.
However, since the mixing in the source is independent of matter potential,
the initial state can be written in mass eigenstate according to Eq. (3).
Therefore, as we will show, one can obtain the standard oscillation pattern in
a material medium for the NC neutrinos. Moreover, using quantum resource
theory, we study the quantum entanglement and quantum coherence under these
conditions. For this purpose, concurrence and $l_{1}$-norm are defined as a
measure for entanglement and quantum coherence, respectively. Coherence is the
quantum resource of neutrino oscillation, and its origin during production and
detection processes, as well as its conservation during propagation, has been
discussed extensively and accurately in the literature; for instance, see 43 ;
44 ; 45 ; 46 ; found ; glashow ; a2012 ; guintidensity . Meanwhile, using
quantum resource theory, one can quantify quantum coherence, entanglement, or
another quantum resource by introducing some measure quantities Vedral ;
Horodecki ; hill ; wootters ; bau ; str . In the case of usual neutrino
oscillation, the refraction of the neutrino in a dense medium leads to a
decrease in the quantum coherence in standard neutrino oscillation epjp .
However, as we will see, the value of entanglement of neutrino-antineutrino
pairs shows different behavior in terms of matter potential.

In the next section, treating the neutrino and antineutrino as a wave packet,
we obtain the probability of the NC neutrino oscillation in a material medium
with constant density for the two-flavor schema. In section III, we explore
the variation in entanglement and quantum coherence for this issue in the
quantum resource theory framework. For this purpose, we use concurrence and
$l_{1}$-norm measures for entanglement and coherence, respectively. We
summarize our results in the last section.

##  II Probability of NC neutrino oscillation in matter

According to the quantum wave packet approach, we should describe the neutrino
and antineutrino by a localized wave function at the space-time coordinates
$(x,t)$ and $(\bar{x},\bar{t})$, respectively. Moreover, it is assumed that
the neutrino and antineutrino detectors are located at distances $L$ and
$\bar{L}$ from the source, respectively. The neutrino and antineutrino coming
from the $Z_{0}$ boson must be described by a bipartite entangled state.
Therefore, this state can be written as follows:

$\left|\nu_{Z},{\bf
x},t,{\bf\bar{x}},\bar{t}\right\rangle=\dfrac{1}{\sqrt{2}}\sum_{i}\Psi_{iS}^{m}({\bf
x},t,{\bf\bar{x}},\bar{t})\left|\nu_{i}\right\rangle\left|\bar{\nu}_{i}\right\rangle,$
(6)

where

$\displaystyle\Psi_{iS}^{m}({\bf x},t,{\bf\bar{x}},\bar{t})$ $\displaystyle=$
$\displaystyle\textit{N}\int\dfrac{d^{3}p}{(\sqrt{2\pi})^{3}}\int\dfrac{d^{3}\bar{p}}{(\sqrt{2\pi})^{3}}f_{S}({\bf
p},{\bf
p}^{m}_{i})\bar{f}_{S}({\bf\bar{p}},{\bf\bar{p}}^{m}_{i})\delta^{3}({\bf
p}-{\bf\bar{p}})$ (7) $\displaystyle\times$
$\displaystyle\exp[-iE_{i}^{m}({\bf p})(t-t_{p})+i{\bf p}({\bf x}-{\bf
x_{p}})-i\bar{E}^{m}_{i}({\bf\bar{p}})(\bar{t}-t_{p})+i{\bf\bar{p}}\cdot({\bf\bar{x}}-{\bf
x_{p}})].$

The superscript $m$ refers to the propagation in a material medium, and the
subscript $S$ indicates that this state is related to the source. Here,
$f_{S}({\bf p},{\bf p}^{m}_{i})$ and
$\bar{f}_{S}({\bf\bar{p}},{\bf\bar{p}}^{m}_{i})$ are the momentum distribution
functions of the neutrino and antineutrino with mean momenta ${\bf p}^{m}_{i}$
and ${\bf\bar{p}}^{m}_{i}$. Also, $E^{m}_{i}({\bf p})$ and
$\bar{E}^{m}_{i}({\bf\bar{p}})$ are the energies of neutrino and antineutrino
with mass $m_{i}$, respectively, and $t_{p}$ and ${\bf x}_{p}$ denote the
production coordinates. The Dirac delta factor is added in Eq. (7) to ensure
that the momentum is conserved (here, we consider those $Z_{0}$ bosons as
being almost at rest in the frame where the matter effect is constant).
Indeed, the time evolution of neutrino states in a matter medium is governed
by the following effective Hamiltonian 1 :

$\delta H^{m}=H_{0}+V_{CC}=\frac{1}{2E}[{U}{\text{diag}}(0,\Delta
m_{21}^{2}){U}^{\dagger}+{\text{diag}(2EV(x),0)}].$ (8)

In the case of antineutrino, we denote the corresponding Hamiltonian by
$\delta{\bar{H}}^{m}$, and it is obtained from Eq. (8) by replacing $V(x)$
with $-V(x)$. We consider the density of matter to be constant. The unitary
mixing matrix $U$ for a two-generation platform is parameterized as follows:

$U=\begin{bmatrix}\cos\theta&\sin\theta\\\ -\sin\theta&\cos\theta\\\
\end{bmatrix}.$ (9)

Therefore, one can obtain the following eigenvalues for the Hamiltonian given
by Eq. (8):

$\delta E_{1}^{m}=\dfrac{\Delta m_{21}^{2}+2EV-\sqrt{4E^{2}V^{2}+\Delta
m_{21}^{4}-4EV\Delta m_{21}^{2}\cos 2\theta}}{4E},$ (10)

and

$\delta E_{2}^{m}=\dfrac{\Delta m_{21}^{2}+2EV+\sqrt{4E^{2}V^{2}+\Delta
m_{21}^{4}-4EV\Delta m_{21}^{2}\cos 2\theta}}{4E},$ (11)

and the corresponding mixing angle is given by

$\theta^{m}=\dfrac{1}{2}\arctan\left(\dfrac{\Delta m_{21}^{2}\sin
2\theta}{\Delta m_{21}^{2}\cos 2\theta-2EV}\right).$ (12)

In the case of the antineutrino, we denote the eigenvalues of
$\delta{\bar{H}}^{m}$ and corresponding mixing angle by
${\delta\bar{E}}^{m}_{1,2}$ and ${\bar{\theta}}^{m}$, respectively. These
parameters can be represented by the corresponding expressions given in Eqs.
(10), (11) and (12) with the difference that $V$ must be replaced by $-V$. It
is clear from Eq. (12) that there is a resonance in mixing of neutrinos
provided that $V\simeq\Delta m_{21}^{2}\cos 2\theta/2E$. However, no resonance
exists for antineutrino mixing.

For the momentum distribution function of the neutrino and antineutrino in Eq.
(7), we use the Gaussian momentum wave function as follows:

$f({\bf
p,p}_{i})=\left(\dfrac{2\pi}{\sigma_{p}^{2}}\right)^{\frac{3}{4}}\exp\left[-\dfrac{({\bf
p-p}_{i})^{2}}{4\sigma_{p}^{2}}\right].$ (13)

Hereafter, the momentum uncertainties of the neutrino and antineutrino are
denoted by $\sigma_{p}$ and ${\bar{\sigma}}_{p}$, respectively. It is
appropriate to assume that $\sigma_{p}$ and ${\bar{\sigma}}_{p}$ are much
smaller than the corresponding mean momenta. Therefore, one can expand the
energy up to the first order of the departure from the mean momentum as
follows:

$E_{i}^{m}({\bf p})\simeq E_{i}^{m}+{\bf v}_{i}^{m}({\bf p}-{\bf p}_{i}^{m}),$
(14)

where

$E_{i}^{m}\equiv E_{i}^{m}({\bf p}_{i}^{m})=\sqrt{m_{i}^{2}+{{\bf
p}_{i}^{m}}^{2}},$ (15)

and ${\bf v}^{m}_{i}$ is the group velocity of the $i$th mass eigenstate and
is given by ${\bf v}^{m}_{i}=dE^{m}/d{\bf p}|_{{\bf p}={\bf p}^{m}_{i}}$.
Likewise, we can write a similar expression for antineutrinos. Consequently,
one can write the position wave packet given in Eq. (7) as follows:

$\displaystyle\Psi_{iS}^{m}({\bf x},t,\bar{\bf x},\bar{t})$
$\displaystyle\propto$
$\displaystyle\exp\left[-i(E_{i}^{m}t+\bar{E}_{i}^{m}\bar{t})-\dfrac{(\sigma_{x\nu}^{2}\bar{\sigma}_{x\bar{\nu}}^{2})}{(\sigma_{x\nu}^{2}+\bar{\sigma}_{x\bar{\nu}}^{2})}({\bf
p}_{i}^{m}-\bar{\bf p}_{i}^{m})^{2}-\dfrac{({\bf x}+\bar{\bf x}-{\bf
v}_{i}^{m}t-\bar{\bf
v}_{i}^{m}\bar{t})^{2}}{4(\sigma_{x\nu}^{2}+\bar{\sigma}_{x\bar{\nu}}^{2})}\right.$
(16) $\displaystyle+$
$\displaystyle\left.\dfrac{i}{(\sigma^{2}_{x\nu}+\bar{\sigma}^{2}_{x\bar{\nu}})}({\bf
x}+\bar{\bf x})(\bar{\sigma}^{2}_{x\bar{\nu}}\bar{\bf
p}_{i}^{m}+\sigma^{2}_{x\nu}{\bf
p}_{i}^{m})+\dfrac{i}{(\sigma^{2}_{x\nu}+\bar{\sigma}^{2}_{x\bar{\nu}})}({\bf
p}_{i}^{m}-\bar{\bf p}_{i}^{m})(\bar{\sigma}^{2}_{x\bar{\nu}}{\bf
v}_{i}^{m}t-\sigma^{2}_{x\nu}\bar{\bf v}_{i}^{m}\bar{t})\right].$

where $\sigma^{2}_{x\nu}$ and $\bar{\sigma}^{2}_{x\bar{\nu}}$ are the position
uncertainties of the neutrino and antineutrino in the source, and they are
given by $\sigma_{x}=1/2\sigma_{p}$ and
$\bar{\sigma}_{x}=1/2\bar{\sigma}_{p}$, respectively.

On the other hand, since the detection processes are essentially time-
independent, detected states have no time dependence. Therefore, the wave
function of the detected neutrino and antineutrino states is described by

$\left|\nu_{\alpha},{\bf
x-L},\bar{\nu}_{\beta},{\bf\bar{x}-\bar{L}}\right\rangle=\sum_{i}\sum_{j}U_{\alpha
i}^{m*}\bar{U}_{\beta j}^{m}\Psi_{iD}^{m}({\bf
x-L})\bar{\Psi}^{m}_{j\bar{D}}({\bf\bar{x}-\bar{L}})\left|\nu_{i}\right\rangle\left|\bar{\nu}_{j}\right\rangle,$
(17)

where $\Psi_{iD}^{m}({\bf x-L})$ and
$\bar{\Psi}_{j\bar{D}}^{m}({\bf\bar{x}-\bar{L}})$ are the wave functions of
the detected neutrino and antineutrino at positions ${\bf L}$ and $\bar{\bf
L}$, respectively. Moreover, $U_{\alpha i}^{m}$ and $\bar{U}_{\beta i}^{m}$ in
Eq. (17) are mixing matrices of neutrinos and antineutrinos in a material
medium. In fact, $U^{m}$ ($\bar{U}^{m}$) has a form similar to Eq.(9) with the
difference that $\theta$ is replaced by $\theta^{m}$ ($\bar{\theta}^{m}$)
given by Eq. (12). Again, we assume a localized Gaussian wave function in
momentum space similar to (13) for both the neutrino and antineutrino detected
in the corresponding detectors. Under this scenario, the detected state
Eq.(17) can be written as follows

$\left|\nu_{\alpha},{\bf
x-L},\bar{\nu}_{\beta},{\bf\bar{x}-\bar{L}}\right\rangle=\sum_{i}U_{\alpha
i}^{m*}\bar{U}_{\beta j}^{m}\exp\left[i{\bf p}^{\prime m}_{i}({\bf
x-L})+i{\bf\bar{p}}^{\prime m}_{i}({\bf\bar{x}-\bar{L}})-\dfrac{({\bf
x-L})^{2}}{4\sigma^{2}_{xD}}-\dfrac{({\bf\bar{x}-\bar{L}})^{2}}{4\bar{\sigma}^{2}_{x\bar{D}}}\right]\left|\nu_{i}\right\rangle\left|\bar{\nu}_{j}\right\rangle,$
(18)

where $\sigma^{2}_{xD}$ and $\bar{\sigma}^{2}_{x\bar{D}}$ are the
uncertainties of the detected neutrino and antineutrino processes. In general,
the mean momenta of the produced particles, ${\bf p}_{i}$ and
${\bf\bar{p}}_{i}$, are different from one of the detected particles, ${\bf
p}^{\prime}_{i}$ and ${\bf\bar{p}}^{\prime}_{i}$. However, we assume that they
coincide. Then, the transition amplitude of detecting $\nu_{\alpha}$ and
$\bar{\nu}_{\beta}$ in the corresponding detectors becomes:

$A^{m}_{\alpha\beta}=\int d^{3}x\int d^{3}\bar{x}\langle\nu_{\alpha},{\bf
x-L},\bar{\nu}_{\beta},{\bf\bar{x}-\bar{L}}\mid\nu_{Z},{\bf
x},t,{\bf\bar{x}},\bar{t}\rangle.$ (19)

Substituting Eq.(6) with $\Psi_{iS}^{m}$ given in Eqs. (16) and (18) into the
above equation and calculating the integrals over the position coordinates, we
obtain

$\displaystyle A^{m}_{\alpha\beta}\propto\dfrac{1}{\sqrt{2}}\sum_{i}U_{\alpha
i}^{m*}\bar{U}_{\beta i}^{m}\\!\\!\\!$ exp
$\displaystyle\\!\\!\\!\left[-iE_{i}^{m}t-i\bar{E}_{i}^{m}\bar{t}-\dfrac{({{\bf
L}+\bar{\bf L}}-{\bf
v}_{i}^{m}t-{\bf\bar{v}}_{i}^{m}\bar{t})^{2}}{8\sigma_{x}^{2}}-\dfrac{\sigma^{2}_{x}({\bf
p}_{i}^{m}-\bar{\bf p}_{i}^{m})^{2}}{2}\right.$ (20) $\displaystyle+$
$\displaystyle\\!\\!\\!\\!\left.\dfrac{i({\bf L}+\bar{\bf L})({\bf
p}_{i}^{m}+\bar{\bf p}_{i}^{m})}{2}+\dfrac{i({\bf p}_{i}^{m}-\bar{\bf
p}_{i}^{m})({\bf v}_{i}^{m}t-\bar{\bf v}_{i}^{m}\bar{t})}{2}\right],$

where we assumed that the position uncertainties of neutrinos and
antineutrinos which are defined as

$\sigma_{x}^{2}\equiv\sigma_{x\nu}^{2}+\sigma_{xD}^{2},\,\,\,\,\,\,\bar{\sigma}_{x}^{2}\equiv\bar{\sigma}_{x\bar{\nu}}^{2}+\bar{\sigma}_{x\bar{D}}^{2},$
(21)

are approximately equal $\sigma_{x}^{2}\simeq\bar{\sigma}_{x}^{2}$.

We want to obtain the probability of detecting a neutrino with flavor state
$|\nu_{\alpha}\rangle$ at one detector and the corresponding antineutrino with
flavor state $|\nu_{\beta}\rangle$ at another. In general, the distance of the
two detectors from the production location may be different. In this case, the
arrival times of the neutrino and antineutrino at their corresponding
detectors are not equal. It is suitable that instead of $t$ and $\bar{t}$, we
select the following time variables:

$T=t+\bar{t},\hskip 28.45274pt\text{and}\hskip 28.45274pt\tau=|t-\bar{t}|.$
(22)

Meanwhile, the emission time and the corresponding arrival times of the
neutrino and antineutrino are not measured. Hence, we integrate over $T$,
which is the arrival time of the neutrino or antineutrino from its own
detector to another detector. Therefore, the probability of the oscillation
pattern between detectors can be obtained as follows:

$\displaystyle P^{m}_{\alpha\beta}$ $\displaystyle\propto$
$\displaystyle\dfrac{1}{2}\sum_{i,j}U_{\alpha i}^{m*}U_{\alpha
j}^{m}\bar{U}_{\beta j}^{m*}\bar{U}_{\beta
i}^{m}\,\,\,\text{exp}\Bigg{\\{}-\dfrac{({\bf L}+\bar{\bf
L})^{2}}{8\sigma_{x}^{2}}\dfrac{(\Delta{\bf v}_{ij}^{m}+\Delta\bar{\bf
v}_{ij}^{m})^{2}}{({\bf v}_{i}^{m}+\bar{\bf v}_{i}^{m})^{2}+({\bf
v}_{j}^{m}+\bar{\bf v}_{j}^{m})^{2}}$ (23) $\displaystyle-$
$\displaystyle\dfrac{2\sigma_{x}^{2}(\Delta
E_{ij}^{m}+\Delta\bar{E}_{ij}^{m})^{2}}{({\bf v}_{i}^{m}+\bar{\bf
v}_{i}^{m})^{2}+({\bf v}_{j}^{m}+\bar{\bf
v}_{j}^{m})^{2}}-\dfrac{\sigma_{x}^{2}}{2}\bigg{[}\Big{(}\bar{\bf
p}_{i}^{m}-{\bf p}_{i}^{m}\Big{)}^{2}+\Big{(}\bar{\bf p}_{j}^{m}-{\bf
p}_{j}^{m}\Big{)}^{2}\bigg{]}$ $\displaystyle-$ $\displaystyle i\dfrac{({\bf
L}+\bar{\bf L})(\Delta E_{ij}^{m}+\Delta\bar{E}_{ij}^{m})({\bf v}_{i}^{m}+{\bf
v}_{j}^{m}+\bar{\bf v}_{i}^{m}+\bar{\bf v}_{j}^{m})}{({\bf v}_{i}^{m}+\bar{\bf
v}_{i}^{m})^{2}+({\bf v}_{j}^{m}+\bar{\bf v}_{j}^{m})^{2}}+i\dfrac{({\bf
L}+\bar{\bf L})(\Delta{\bf p}_{ij}^{m}+\Delta\bar{\bf p}_{ij}^{m})}{2}$
$\displaystyle+$ $\displaystyle 2\sigma_{x}^{2}\dfrac{(\Delta
E_{ij}^{m}+\Delta\bar{E}_{ij}^{m})\Big{[}(\bar{\bf v}_{i}^{m}-{\bf
v}_{i}^{m})(\bar{\bf p}_{i}^{m}-{\bf p}_{i}^{m})-(\bar{\bf v}_{j}^{m}-{\bf
v}_{j}^{m})(\bar{\bf p}_{j}^{m}-{\bf p}_{j}^{m})\Big{]}}{({\bf
v}_{i}^{m}+\bar{\bf v}_{i}^{m})^{2}+({\bf v}_{j}^{m}+\bar{\bf
v}_{j}^{m})^{2}}$ $\displaystyle-$
$\displaystyle\dfrac{\sigma_{x}^{2}}{2}\dfrac{\Big{[}({\bf v}_{i}^{m}-\bar{\bf
v}_{i}^{m})(\bar{\bf p}_{i}^{m}-{\bf p}_{i}^{m})-({\bf v}_{j}^{m}-\bar{\bf
v}_{j}^{m})(\bar{\bf p}_{j}^{m}-{\bf p}_{j}^{m})\Big{]}^{2}}{({\bf
v}_{i}^{m}+\bar{\bf v}_{i}^{m})^{2}+({\bf v}_{j}^{m}+\bar{\bf
v}_{j}^{m})^{2}}$ $\displaystyle-$ $\displaystyle i\dfrac{({\bf L}+\bar{\bf
L})}{2}\dfrac{({\bf v}_{i}^{m}+{\bf v}_{j}^{m}+\bar{\bf v}_{i}^{m}+\bar{\bf
v}_{j}^{m})\Big{[}(\bar{\bf p}_{i}^{m}-{\bf p}_{i}^{m})({\bf
v}_{i}^{m}-\bar{\bf v}_{i}^{m})-(\bar{\bf p}_{j}^{m}-{\bf p}_{j}^{m})({\bf
v}_{j}^{m}-\bar{\bf v}_{j}^{m})\Big{]}}{({\bf v}_{i}^{m}+\bar{\bf
v}_{i}^{m})^{2}+({\bf v}_{j}^{m}+\bar{\bf v}_{j}^{m})^{2}}$ $\displaystyle-$
$\displaystyle\dfrac{(\bar{\bf v}_{j}^{m}{\bf v}_{i}^{m}-\bar{\bf
v}_{i}^{m}{\bf v}_{j}^{m})^{2}\tau^{2}}{8\sigma_{x}^{2}(({\bf
v}_{i}^{m}+\bar{\bf v}_{i}^{m})^{2}+({\bf v}_{j}^{m}+\bar{\bf
v}_{j}^{m})^{2})}-i\dfrac{\Delta E_{ij}^{m}\tau}{({\bf v}_{i}^{m}+\bar{\bf
v}_{i}^{m})^{2}+({\bf v}_{j}^{m}+\bar{\bf v}_{j}^{m})^{2}}(\bar{\bf
v}_{i}^{m2}+\bar{\bf v}_{j}^{m2}+\bar{\bf v}_{i}^{m}{\bf v}_{i}^{m}+\bar{\bf
v}_{j}^{m}{\bf v}_{j}^{m})$ $\displaystyle+$ $\displaystyle
i\dfrac{\Delta\bar{E}_{ij}^{m}\tau}{({\bf v}_{i}^{m}+\bar{\bf
v}_{i}^{m})^{2}+({\bf v}_{j}^{m}+\bar{\bf v}_{j}^{m})^{2}}({\bf
v}_{i}^{m2}+{\bf v}_{j}^{m2}+\bar{\bf v}_{i}^{m}{\bf v}_{i}^{m}+\bar{\bf
v}_{j}^{m}{\bf v}_{j}^{m})+\dfrac{({\bf L}+\bar{\bf L})(\bar{\bf
v}_{i}^{m}{\bf v}_{j}^{m}-\bar{\bf v}_{j}^{m}{\bf
v}_{i}^{m})\tau}{4\sigma_{x}^{2}(({\bf v}_{i}^{m}+\bar{\bf
v}_{i}^{m})^{2}+({\bf v}_{j}^{m}+\bar{\bf v}_{j}^{m})^{2})}(\Delta{\bf
v}_{ij}^{m}+\Delta\bar{\bf v}_{ij}^{m})$ $\displaystyle+$ $\displaystyle
i\dfrac{\tau}{2(({\bf v}_{i}^{m}+\bar{\bf v}_{i}^{m})^{2}+({\bf
v}_{j}^{m}+\bar{\bf v}_{j}^{m})^{2})}\bigg{[}\bigg{(}2\bar{\bf v}_{i}^{m}{\bf
v}_{i}^{m}(\bar{\bf v}_{i}^{m}+{\bf v}_{i}^{m})+(\bar{\bf v}_{j}^{m}+{\bf
v}_{j}^{m})(\bar{\bf v}_{j}^{m}{\bf v}_{i}^{m}+\bar{\bf v}_{i}^{m}{\bf
v}_{j}^{m})\bigg{)}({\bf p}_{i}^{m}-\bar{\bf p}_{i}^{m})$ $\displaystyle-$
$\displaystyle\bigg{(}2\bar{\bf v}_{j}^{m}{\bf v}_{j}^{m}(\bar{\bf
v}_{j}^{m}+{\bf v}_{j}^{m})+(\bar{\bf v}_{i}^{m}+{\bf v}_{i}^{m})(\bar{\bf
v}_{j}^{m}{\bf v}_{i}^{m}+\bar{\bf v}_{i}^{m}{\bf v}_{j}^{m})\bigg{)}({\bf
p}_{j}^{m}-\bar{\bf p}_{j}^{m})\bigg{]}\Bigg{\\}},$

in which $\Delta E_{ij}^{m}$ ($\Delta\bar{E}_{ij}^{m}$) and $\Delta{\bf
v}_{ij}^{m}$ ($\Delta\bar{\bf v}_{ij}^{m}$) are the energy and group velocity
differences in the material medium between the two mass eigenstates of the
neutrino (antineutrino), respectively. By using relativistic approximations,
one can write the mean energies as follows:

$E_{i}^{m}\approx E+\xi\delta{E}_{i}^{m},$ (24)

and

$\bar{E}_{i}^{m}\approx E+\xi\delta\bar{E}_{i}^{m},$ (25)

where $E$ is the neutrino and antineutrino energy in the limit of zero mass.
Also, $\xi$ denotes a dimensionless quantity whose value can be estimated from
energy-momentum conservation in the production process 2000qm ; giu02 . In the
case of $Z_{0}$ boson decay in the rest frame, we have $\xi\approx 0$.
Similarly, the corresponding momenta can be given by

${\bf p}_{i}^{m}\approx E+(\xi-1)\delta E_{i}^{m},$ (26)

and

$\bar{\bf p}_{i}^{m}\approx E+(\xi-1)\delta\bar{E}_{i}^{m}.$ (27)

Furthermore, in this approximation, the group velocity of a mass eigenstate is
also written as

${\bf
v}_{i}^{m}\approx\dfrac{dE_{i}^{m}}{dE}=1+\dfrac{d\delta{E}_{i}^{m}}{dE},$
(28)

for neutrinos and

$\bar{\bf
v}_{i}^{m}\approx\dfrac{d\bar{E}_{i}^{m}}{dE}=1+\dfrac{d\delta{\bar{E}}_{i}^{m}}{dE},$
(29)

for antineutrinos. Therefore, Eq. (23) can be simplified as follows:

$\displaystyle
P^{m}_{\alpha\beta}(\mathbf{L},\mathbf{\bar{L}})\propto\dfrac{1}{2}\sum_{i,j}U^{m*}_{\alpha
i}{\bar{U}}^{m}_{\beta i}U^{m}_{\alpha j}{\bar{U}}^{m*}_{\beta j}$
$\displaystyle\\!\\!\\!\exp\\!\\!\\!$ $\displaystyle\Bigg{\\{}-2\pi
i\dfrac{({\bf L}+\bar{\bf L})}{{{\bf
L}^{m}}_{ij}^{\text{osc}}}-\dfrac{1}{2}\Big{(}\dfrac{{\bf L}+\bar{\bf
L}}{{{\bf
L}^{m}}_{ij}^{\text{coh}}}\Big{)}^{2}-\Big{(}\dfrac{2\pi\xi\sigma_{x}}{{{\bf
L}^{m}}_{ij}^{\text{osc}}}\Big{)}^{2}$ (30) $\displaystyle-\\!\\!\\!\\!\\!$
$\displaystyle\dfrac{\sigma_{x}^{2}(\xi-1)^{2}}{2}\bigg{[}(\delta
E_{i}^{m}-\delta\bar{E}_{i}^{m})^{2}+(\delta
E_{j}^{m}-\delta\bar{E}_{j}^{m})^{2}\bigg{]}\Bigg{\\}}$ $\displaystyle\times$
$\displaystyle\exp\Bigg{\\{}\dfrac{\tau}{64}\bigg{[}-32i(\Delta\delta
E_{ij}^{m}-\Delta\delta\bar{E}_{ij}^{m})$ $\displaystyle+$
$\displaystyle\dfrac{1}{\sigma_{x}^{2}}(\Delta\bar{\bf v}_{ij}^{m}-\Delta{\bf
v}_{ij}^{m})\Big{(}2({\bf L}+\bar{\bf L})(\Delta\bar{\bf
v}_{ij}^{m}+\Delta{\bf v}_{ij}^{m})-(\Delta\bar{\bf v}_{ij}^{m}-\Delta{\bf
v}_{ij}^{m})\tau\Big{)}\bigg{]}\Bigg{\\}},$

where the oscillation length ${{\bf L}^{m}}_{ij}^{\text{osc}}$ and the
coherence length ${{\bf L}^{m}}_{ij}^{\text{coh}}$, for $i\neq j$, are defined
by

${{\bf L}^{m}}_{ij}^{\text{osc}}\equiv\dfrac{4\pi}{\Delta\delta
E^{m}_{ij}+\Delta\delta\bar{E}^{m}_{ij}},$ (31)

in which $\Delta\delta E^{m}_{ij}\equiv\delta E^{m}_{i}-\delta E^{m}_{j}$ and
$\Delta\delta\bar{E}^{m}_{ij}\equiv\delta\bar{E}^{m}_{i}-\delta\bar{E}^{m}_{j}$,
and

${{\bf
L}^{m}}_{ij}^{\text{coh}}\equiv\dfrac{4\sqrt{2}\sigma_{x}}{\mid\Delta{\bf
v}^{m}_{ij}+\Delta\bar{\bf v}^{m}_{ij}\mid},$ (32)

respectively. According to Eq. (28), the differences in the neutrino group
velocities appearing in Eq. (32) become

$\Delta{\bf v}_{ij}=-\dfrac{\Delta m_{21}^{2}(\Delta m_{21}^{2}-2EV\cos
2\theta)}{4E^{3}\Delta\delta E^{m}_{ij}}.$ (33)

For $\Delta\bar{\bf v}^{m}_{ij}$, we obtain a similar expression, with the
difference that $\Delta\delta E^{m}_{ij}$ and $V$ must be replaced by
$\Delta\delta\bar{E}^{m}_{ij}$ and $-V$. For neutrinos, $\Delta{\bf v}_{ij}$
becomes zero provided that $V=\Delta m_{21}^{2}/2E\cos 2\theta$. But in the
case of antineutrinos, it does not happen. Therefore, according to Eq. (32),
the greatest value of ${{\bf L}^{m}}_{ij}^{\text{coh}}$ occurs when
$\Delta{\bf v}^{m}_{ij}=0$. The last exponential factor appearing in Eq. (30)
is due to the non-simultaneous neutrino and antineutrino detection processes.
This factor depends on the subtraction of the difference between the energies
and group velocities of the neutrino and antineutrino mass eigenstates. This
difference is zero in vacuum, and in the material medium with usual densities
its effect can be completely ignored compared to the oscillation and
decoherence factors. Therefore, with accuracy such that this factor can be
ignored, the processes of neutrino and antineutrino detection can be
considered to be simultaneous. Furthermore, the first three terms appearing in
the first exponential, in Eq. (30), are the same as the results of Refs. 4 ;
scrip  with the difference that ${{\bf L}^{m}}_{ij}^{\text{osc}}$ and ${{\bf
L}^{m}}_{ij}^{\text{coh}}$ are modified due to the neutrino and antineutrino
scattering off matter. However, the last term in this exponential is coming
absolutely from the difference between the potential of neutrinos and
antineutrinos. Similar to the terms coming from the non-simultaneity of
neutrino and antineutrino detection processes, the effect of this term can
also be ignored.

Since the lifetime of $Z_{0}$ is so small (about $\tau_{Z_{0}}\simeq 3\times
10^{-25}s$), the interval between two nearest collisions becomes smaller than
the mean distance between particles in the nucleus. Therefore, $\sigma_{x}$ is
determined by the $Z_{0}$ decay width $\Gamma_{Z_{0}}$ as follows 45 :

$\sigma_{x}\simeq\frac{p}{E}\Gamma_{Z_{0}}^{-1}.$ (34)

The energy of the neutrino and antineutrino coming from the decay of a rest
$Z_{0}$ is about $E=46\text{GeV}$. If we take $\Delta m^{2}_{21}=7.53\times
10^{-5}eV^{2}$, and $\theta=33.46^{\circ}$ 2020e , $\sigma_{x}$ of
approximately $10^{-16}m$ is obtained. With these values for relevant
parameters, using (32) and (31), coherence and oscillation length of about
$10^{10}m$ and $10^{15}m$ are obtained, respectively. Therefore, under this
condition, the oscillation pattern ceases due to the separation of mass
eigenstate wave packets. This situation is not improved by choosing the
oscillation parameters between the second and third generations, and the
oscillation length will still be several orders of magnitude larger than the
coherence length. In the following, however, we want to investigate quantum
coherence and entanglement as a resource of the oscillation pattern in this
problem. Therefore, we suppose $\sigma_{x}$ to be about the atomic distance
($5\times 10^{-10}\text{m}$) so that the oscillating behavior does not cease.

Figure 1: Probabilities of the survival (a) and transition (b) patterns
between detectors versus their distance $L+\bar{L}$. The matter potential is
taken as $V=0$ (blue), $V=3.208\times 10^{-16}\text{eV}$ (pink),
$V=2.087\times 10^{-15}\text{eV}$ (green) and $V=5\times 10^{-14}\text{eV}$
(brown).

For greater clarification, we illustrate the probabilities of the survival (a)
and transition patterns (b) between detectors versus their distance
$L+\bar{L}$ in Fig. 1. Specifically, we plot the probability in terms of
$L+\bar{L}$ for four cases: vacuum ($V=0$, blue curve), for a value of the
potential which corresponds to the neutrino resonance mixing ($V=3.208\times
10^{-16}\text{eV}$, pink curve), for a value of the potential in which ${{\bf
L}^{m}}_{12}^{\text{coh}}$ takes the maximum value ($V=2.087\times
10^{-15}\text{eV}$, green curve) and for a matter-dominated potential
($V=5\times 10^{-14}\text{eV}$, brown curve).

##  III Quantum Correlations

As mentioned, the NC neutrino oscillation originates from the neutrino and
antineutrino entanglement and quantum coherence due to the overlap of mass
eigenstate wave functions. Both of these correlations are affected by the
disentangling and decoherence due to the separation of wave packets as well as
the scattering of neutrinos and antineutrinos off material medium. Therefore,
in this section, we use concurrence and $l_{1}\text{-norm}$, respectively, as
a measure for entanglement and quantum coherence in the framework of quantum
resource theory, in order to investigate the mentioned effects.

Entanglement as a quantum correlation is studied extensively in the framework
of the quantum resource theory Vedral ; Horodecki ; hill ; wootters .
Concurrence is one of the measures suggested to quantify entanglement. First,
let us define $\tilde{\rho}$ as follows:

$\tilde{\rho}=(\sigma_{y}\otimes\sigma_{y})\rho^{*}(\sigma_{y}\otimes\sigma_{y}).$
(35)

The concurrence measure is generally defined by hill ; wootters :

${\cal
C}(\rho)=\text{max}(\lambda_{1}-\lambda_{2}-\lambda_{3}-\lambda_{4},0),$ (36)

where $\lambda_{i}$’s are the square roots of the four eigenvalues of the non-
Hermitian matrix $\rho\tilde{\rho}$ in decreasing order. Accordingly,
considering a general two-qubit state, we have

$\left|\phi\right\rangle=\alpha_{00}\left|0,0\right\rangle+\alpha_{01}\left|0,1\right\rangle+\alpha_{10}\left|1,0\right\rangle+\alpha_{11}\left|1,1\right\rangle.$
(37)

The concurrence for this state is obtained as follows

${\cal C}(\rho)=2\mid\alpha_{00}\alpha_{11}-\alpha_{01}\alpha_{10}\mid.$ (38)

Clearly, for Bell states, we have ${\cal C}(\rho)=1$. Otherwise, if the
coefficients in Eq. (37) are such that $\left|\phi\right\rangle$ is separable,
${\cal C}(\rho)$ will be zero. For other cases, one obtains a value between 0
and 1. The more entangled the state, the closer the value of ${\cal C}(\rho)$
is to 1.

For an entangled neutrino and antineutrino state originating from the $Z_{0}$
decay, the corresponding bipartite state is given in Eq. (6) which is written
based on the wave packet approach. The density matrix operator corresponding
to this state, after integrating over the related momenta and time 222Similar
to the calculation of the oscillation probability, we integrate over the
propagation time because in all existing neutrino oscillation experiments only
the source detector distance is known., is obtained as

$\displaystyle\rho({\bf x},\bar{\bf x})$ $\displaystyle\propto$
$\displaystyle\dfrac{1}{2}\sum_{i,j}\exp\Bigg{[}-\dfrac{1}{8\sigma_{x}^{2}(({\bf
v_{i}}+\bar{\bf v_{i}})^{2}+({\bf v_{j}}+\bar{\bf v_{j}})^{2})}({\bf
x}+\bar{\bf x})^{2}(\Delta{\bf v}_{ij}+\Delta\bar{\bf v}_{ij})^{2}$ (39)
$\displaystyle+$ $\displaystyle\dfrac{i}{2}(\Delta{\bf p}_{ij}+\Delta\bar{\bf
p}_{ij})({\bf x}+\bar{\bf x})-\dfrac{\sigma_{x}^{2}}{2}((\bar{\bf p_{i}}-{\bf
p_{i}})^{2}+(\bar{\bf p_{j}}-{\bf p_{j}})^{2})$ $\displaystyle-$
$\displaystyle\dfrac{1}{8\sigma_{x}^{2}(({\bf v_{i}}+\bar{\bf
v_{i}})^{2}+({\bf v_{j}}+\bar{\bf v_{j}})^{2})}[({\bf x}+\bar{\bf x})({\bf
v_{i}}+{\bf v_{j}}+\bar{\bf v_{i}}+\bar{\bf v_{j}})]^{2}$ $\displaystyle+$
$\displaystyle\dfrac{1}{8\sigma_{x}^{2}(({\bf v_{i}}+\bar{\bf
v_{i}})^{2}+({\bf v_{j}}+\bar{\bf v_{j}})^{2})}[-4i\sigma_{x}^{2}(\Delta
E_{ij}+\Delta\bar{E}_{ij})+({\bf x}+\bar{\bf x})({\bf v_{i}}+{\bf
v_{j}}+\bar{\bf v_{i}}+\bar{\bf v_{j}})$ $\displaystyle+$ $\displaystyle
2i\sigma_{x}^{2}(\bar{\bf p_{j}}-{\bf p_{j}})(\bar{\bf v}_{j}-{\bf
v}_{j})-2i\sigma_{x}^{2}(\bar{\bf p_{i}}-{\bf p_{i}})(\bar{\bf v}_{i}-{\bf
v}_{i})]^{2}\Bigg{]}\mid\nu_{i},\bar{\nu}_{i}\rangle\langle\nu_{j},\bar{\nu}_{j}\mid.$

Taking into account the issues related to energies, momenta and group
velocities, given in Eqs. (24)-(29), and using the definitions given in Eqs.
(31) and (32), one can simplify this relation as follows:

$\displaystyle\rho({\bf x},\bar{\bf x})\propto\dfrac{1}{2}\sum_{i,j}$
$\displaystyle\\!\\!\\!\exp\\!\\!\\!$ $\displaystyle\Bigg{[}-2\pi
i\dfrac{({\bf x}+\bar{\bf x})}{{{\bf
L}^{m}}_{ij}^{\text{osc}}}-\dfrac{1}{2}\Bigg{(}\dfrac{{\bf x}+\bar{\bf
x}}{{{\bf
L}^{m}}_{ij}^{\text{coh}}}\Bigg{)}^{2}-\Bigg{(}\dfrac{2\pi\xi\sigma_{x}}{{{\bf
L}^{m}}_{ij}^{\text{osc}}}\Bigg{)}^{2}$ (40) $\displaystyle-\\!\\!\\!\\!\\!$
$\displaystyle\dfrac{\sigma_{x}^{2}}{2}(\xi-1)^{2}[(\delta
E_{i}^{m}-\delta\bar{E}_{i}^{m})^{2}+(\delta
E_{j}^{m}-\delta\bar{E}_{j}^{m})^{2}]\Bigg{]}\mid\nu_{i},\bar{\nu}_{i}\rangle\langle\nu_{j},\bar{\nu}_{j}\mid,$

The behavior of this density matrix in terms of ${\bf x}+\bar{\bf x}$ is
analogous to that obtained in Ref. guintidensity  with some differences coming
from the distinct propagation of neutrinos and antineutrinos in a material
medium.

Now we can calculate the concurrence whose definition is given in Eq. (36) as
a measure of the neutrino and antineutrino entanglement. By direct
calculation, one can obtain the following expression for concurrence:

${\cal C}(\rho)=\exp\Bigg{[}-\dfrac{1}{2}\Bigg{(}\dfrac{{\bf x}+\bar{\bf
x}}{{{\bf
L}^{m}}_{12}^{\text{coh}}}\Bigg{)}^{2}-\Bigg{(}\dfrac{2\pi\xi\sigma_{x}}{{{\bf
L}^{m}}_{12}^{\text{osc}}}\Bigg{)}^{2}-\dfrac{\sigma_{x}^{2}}{2}(\xi-1)^{2}[(\delta
E_{1}^{m}-\delta\bar{E}_{1}^{m})^{2}+(\delta
E_{2}^{m}-\delta\bar{E}_{2}^{m})^{2}]\Bigg{]}.$ (41)

In fact, the first term in the exponential has a decisive role in the behavior
of ${\cal C}(\rho)$ when the propagation length is of the order of the
coherence length. Furthermore, we should emphasis that if one uses a plane
wave description for neutrinos, the value of concurrence remains equal to 1
for every value of ${\bf x}+\bar{\bf x}$ and $V$. Meanwhile, in the wave
packet approach, the concurrence depends on $V$ as well as ${\bf x}+\bar{\bf
x}$; it becomes zero when ${\bf x}+\bar{\bf x}$ exceeds the coherence length,
and its falling is postponed for higher matter density (see Fig. 2).

Figure 2: Concurrence versus propagation length for various matter potentials.
The second and third values of potential have been chosen to correspond to the
resonance and the infinite coherence length of a neutrino, respectively.

Specifically, according to Eq. (41), when the localization properties are
considered by the wave packet approach, the overlap of mass eigenstate wave
packets is diminished for a propagation length larger than the coherence
length because the corresponding velocities are different. However, it can be
compensated by propagation of the neutrino in the material medium. In other
words, the refraction of neutrinos in a dense medium causes the difference in
wave packet velocities to decrease. While the concurrence vanishes for smaller
matter potentials for a specific propagation length, it becomes nonzero for
larger potentials; for example, see Fig. 3.

Figure 3: Concurrence versus matter potential for various propagation lengths.

Now let us study the coherence of neutrinos and antineutrinos originating from
the $Z_{0}$ decay process. As was noted, in order for oscillation to occur,
neutrinos must be coherent during the production, propagation, and detection
processes. In the framework of the quantum resource theory, quantum coherence
is a quantum correlations that can be quantified by appropriate measure
quantities bau ; str . Among the several measures for quantum coherence, a
very intuitive one is related to the off-diagonal elements of the considered
quantum state. Hence, for a given density matrix, $\rho$, the
$l_{1}\text{-norm}$ as a measure of quantum coherence is defined by

$\textit{c}(\rho)=\sum_{i\neq j}|\rho_{ij}|.$ (42)

In general, the maximum possible value for $\textit{c}(\rho)$ is $d-1$, where
$d$ is the dimension of the corresponding density matrix str . Therefore, the
neutrino oscillation shows that the off-diagonal elements of the corresponding
density matrix in the basis of the flavor state are nonzero, and consequently
we have no vanishing $l_{1}-\text{norm}$ epjp .

In the case of NC neutrinos, the dimension of the density matrix given in Eq.
(40) is four in two-flavor schema. In this case, therefore, the maximum value
of the $l_{1}$-norm is 3. According to the definition given in Eq. (42), one
can write this parameter in terms of the transition amplitudes as follows:

$\displaystyle\textit{c}(\rho)$ $\displaystyle=$ $\displaystyle
2\Bigg{\\{}|A_{ee}^{m}({\bf x},\bar{\bf x})A_{e\mu}^{m*}({\bf x},\bar{\bf
x})|+|A_{ee}^{m}({\bf x},\bar{\bf x})A_{\mu e}^{m*}({\bf x},\bar{\bf
x})|+|A_{ee}^{m}({\bf x},\bar{\bf x})A_{\mu\mu}^{m*}({\bf x},\bar{\bf x})|$
(43) $\displaystyle+$ $\displaystyle|A_{e\mu}^{m}({\bf x},\bar{\bf x})A_{\mu
e}^{m*}({\bf x},\bar{\bf x})|+|A_{e\mu}^{m}({\bf x},\bar{\bf
x})A_{\mu\mu}^{m*}({\bf x},\bar{\bf x})|+|A_{\mu e}^{m}({\bf x},\bar{\bf
x})A_{\mu\mu}^{m*}({\bf x},\bar{\bf x})|\Bigg{\\}}.$

Here, the first and second terms and the fifth and sixth terms are identical.
An explicit form of $\textit{c}(\rho^{m})$ in terms of neutrino oscillation
parameters is given in the Appendix. We illustrate the behavior of
$\textit{c}(\rho^{m})$ versus the distance of detectors ${\bf x}+\bar{\bf x}$
through Fig. 4. In this plot, we consider three values for matter potential:
vacuum ($V=0$), the value corresponding to the infinite coherence length for
neutrinos, and a matter-dominated value. We see that although for baseline
less than the coherence length the matter effect causes $\textit{c}(\rho^{m})$
to drop relative to vacuum, the coherence length in a material medium becomes
larger than one in the vacuum.

Furthermore, in Fig. 5, the behavior of $\textit{c}(\rho^{m})$ in terms of the
matter potential is depicted for the results obtained based on both plane wave
(a) and wave packet (b) approaches. We consider two values for the baseline: a
value smaller than the coherence length, $10^{16}$m (blue curve), and a value
greater than that, $10^{18}$m (pink curve). In the case of the former (blue
curve), $\textit{c}(\rho^{m})$ obtained in both approaches behaves similarly.
Although for some values of the potential a slight increase in the value of
$\textit{c}(\rho^{m})$ is seen, as the value of the potential increases,
$\textit{c}(\rho^{m})$ eventually decreases to a value less than the
corresponding one in the vacuum. However, in the case of $10^{18}$m baseline
(pink curve), we see that in the plane wave approach, $\textit{c}(\rho^{m})$
achieves the maximum value for a region of the potential value, while in the
wave packet approach, $\textit{c}(\rho^{m})$ is certainly smaller than the
maximum value. This is due to the decoherence effects coming from the
separation of neutrino wave packets.

Figure 4: $l_{1}$-norm versus the baseline $\bar{x}+x$ for three values of
matter potential.

Figure 5: $l_{1}$-norm versus the matter potential in both the plane wave (a)
and wave packet approaches (b). Two values are taken for the baseline; a value
smaller and a value larger than the coherence length.

##  IV Conclusion

According to the theory of neutrino oscillation, it is possible to observe an
oscillation pattern between the detectors of neutrinos and antineutrinos
coming from $Z_{0}$ boson decay when the neutrino production, propagation, and
detection coherence conditions are satisfied. We see that under a realistic
condition for $Z_{0}$ boson decay, $\sigma_{x}$ is estimated to be about
$10^{-16}$m. Thus the coherence length is smaller than the oscillation length,
and the propagation coherence conditions are not satisfied. Nevertheless, we
have assumed $\sigma_{x}=5\times 10^{-10}$ so that the coherence conditions
are satisfied. However, if only one of them is detected, neutrino oscillation
cannot be observed. This phenomenon results from the fact that the neutrino
and antineutrino arising from the $Z_{0}$ decay are both entangled and
coherent. These quantum properties can be quantified according to the quantum
resource theory. In this paper, we use the concurrence and $l_{1}$-norm to
quantify entanglement and quantum coherence, respectively. On the other hand,
considering localization properties by using the wave packet approach, one can
see that the wave packet separation of neutrino mass eigenstates due to the
difference in the corresponding velocities suppresses the amount of
entanglement and coherence measures. Another factor that can affect these
quantum correlations is the propagation of neutrinos and antineutrinos in
material media. Hence, in this paper, we have reanalyzed the $Z_{0}$ decay
neutrino oscillation by considering the localization properties and matter
potential. In particular, we have obtained the concurrence and $l_{1}$-norm in
two-flavor schema. If the localization effects are ignored and in other words,
the plane wave approach is used to obtain the transition probability, the
concurrence always remains 1. Otherwise, the concurrence ceases when the
propagation length exceeds the coherence length. However, the matter potential
causes the coherence length to increase, and as a result, this damping occurs
in larger propagation lengths (see Fig. 2). Indeed, the propagation in
material medium causes wave packet separation, with the result that the
damping of entanglement is compensated to some extent (see Fig. 3).
Furthermore, we have shown that although the matter potential causes the
$l_{1}$-norm value to be small compared to the corresponding one in the
vacuum, the damping due to the wave packet separation occurs in larger
propagation lengths as well (see Fig. 4). As another point, we have compared
the behavior of the $l_{1}$-norm in terms of matter potential for the case
when using the plane wave approach and when using the wave packet approach.
Obviously, when the propagation length is smaller than the coherence length,
both approaches give the same result. Otherwise, in the plane wave approach,
the coherence value is maintained, but it is ultimately suppressed due to the
interaction with the material medium. In the wave packet approach, its value
is suppressed before the significant impact of the potential (see Fig. 5).

## Appendix A

Using Eq. (40) for density matrix, one can obtain the following expression for
the $l_{1}$-norm in the limit of a large baseline:

$\displaystyle c(\rho)$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{2}\sqrt{h^{2}_{12}\Omega_{i}^{2}-\dfrac{1}{4}\gamma_{1}^{2}(h_{12}^{2}\sin^{2}(2\pi
f_{12})-1)-\gamma_{1}\Omega_{i}h_{12}\cos(2\pi f_{12})}$ (44) $\displaystyle+$
$\displaystyle\sqrt{\gamma_{2}^{2}+h_{12}^{2}\sin^{2}2\theta^{m}-\gamma_{1}^{2}h_{12}^{2}\cos^{2}(2\pi
f_{12})-2\gamma_{1}\gamma_{4}h_{12}\cos(2\pi f_{12})}$ $\displaystyle+$
$\displaystyle\sqrt{\gamma_{3}^{2}+h_{12}^{2}\sin^{2}2\bar{\theta}^{m}-\gamma_{1}^{2}h_{12}^{2}\cos^{2}(2\pi
f_{12})-2\gamma_{1}\gamma_{4}h_{12}\cos(2\pi f_{12})},$

in which

$h_{12}=\exp[-\dfrac{1}{2}(\dfrac{{\bf x}+\bar{\bf x}}{{{\bf
L}^{m}}_{12}^{\text{coh}}})^{2}-(\dfrac{2\pi\xi\sigma_{x}}{{{\bf
L}^{m}}_{12}^{\text{osc}}})^{2}]$ (45) $f_{12}=\dfrac{{\bf x}+\bar{\bf
x}}{{{\bf L}^{m}}_{12}^{\text{osc}}}$ (46)

and

$\Omega_{1}=\cos^{2}\theta^{m}\sin^{2}\bar{\theta}^{m}+\sin^{2}\theta^{m}\cos^{2}\bar{\theta}^{m},$
(47)
$\Omega_{2}=-\left(\cos^{2}\theta^{m}\cos^{2}\bar{\theta}^{m}+\sin^{2}\theta^{m}\sin^{2}\bar{\theta}^{m}\right),$
(48)

$\gamma_{1}=\sin 2\theta^{m}\sin 2\bar{\theta}^{m},$ (49) $\gamma_{2}=\cos
2\theta^{m}\sin 2\bar{\theta}^{m},$ (50) $\gamma_{3}=\sin 2\theta^{m}\cos
2\bar{\theta}^{m},$ (51) $\gamma_{4}=\cos 2\theta^{m}\cos 2\bar{\theta}^{m}.$
(52)

## References

  * (1) J.A. Formaggio, D.I. Kaiser, M.M. Murskyj, T.E. Weiss, Phys. Rev. Lett. 117, 050402 (2016).
  * (2) X.-K. Song, Y. Huang, J. Ling, M.-H. Yung, Phys. Rev. A 98, 050302(R) (2018).
  * (3) M. Blasone, F. Dell’Anno, S. De Siena, F. Illuminati, Europhys. Lett. 85, 50002 (2009).
  * (4) M. Blasone, F. Dell’Anno, S. De Siena, F. Illuminati, Europhys. Lett. 112, 20007 (2015).
  * (5) M.M. Ettefaghi, Z.S. Tabatabaei Lotfi, R. Ramezani Arani, Europhys. Lett. 132, 31002 (2020).
  * (6) A.K. Alok, S. Banerjee, S. U. Sankar, Nucl. Phys. B 909, 65 (2016).
  * (7) S. Banerjee, A.K. Alok, R. Srikanth, B.C. Hiesmayr, Eur. Phys. J. C 75, 1 (2015).
  * (8) J. Naikoo, A.K. Alok, S. Banerjee, S.U. Sankar, G. Guarnieri, C. Schultze, B.C. Hiesmayr, Nucl. Phys. B 951, 114872 (2020).
  * (9) Z. Askaripour Ravari, M.M. Ettefaghi, S. Miraboutalebi, Eur. Phys. J. Plus 137, 488 (2022).
  * (10) M.M. Ettefaghi, R. Ramazani Arani, Z.S. Tabatabaei Lotfi, Phys. Rev. D 105, 095024 (2022).
  * (11) Y.W. Li, L.J. Li, X.K. Song, D. Wang, L.Ye, Eur. Phys. J. C 82, 799(2022).
  * (12) A.K. Jha, A. Chatla, Eur. Phys. J. Spec. Top. 231, 141 (2022).
  * (13) B. Yadav, T. Sarkar, K. Dixit, A.K. Alok, Eur. Phys. J. C 82, 1 (2022).
  * (14) B. Kayser, Phys. Rev. D 24, 110 (1981).
  * (15) M. Beuthe, Phys. Rep. 375, 105 (2003).
  * (16) E.Kh. Akhmedov, A.Yu. Smirnov, Phys. At. Nucl. 72, 1363 (2009).
  * (17) C. Giunti, C.W. Kim, Phys. Rev. D 58, 017301 (1998).
  * (18) E.Kh. Akhmedov, A.Yu. Smirnov, Found. Phys. 41, 1279 (2011).
  * (19) A.G. Cohen, S.L. Glashow, Z. Ligeti, Phys. Lett. B 678, 191 (2009).
  * (20) E.Kh. Akhmedov, D.Hernandez, A.Yu Smirnov, J. High Energy Phys. 04, 1 (2012).
  * (21) C. Giunti, Found. Phys. Lett. 17, 103 (2004).
  * (22) A.Yu. Smirnov, G.T. Zatsepin, Mod. Phys. Lett. A7 1272 (1992).
  * (23) M.M. Ettefaghi, Z. Askaripour Ravari, Phys. Lett. B 59 747 (2015).
  * (24) M.M. Ettefaghi, Z. Askari Ravari. Phys. Scr. 95 035301 (2020).
  * (25) M. Kaur, M. Singh, Sci. Rep. 10, 11427 (2020).
  * (26) L. Wolfenstein, Phys. Rev. D 17, 2369 (1978).
  * (27) S.P. Mikheyev, A.Yu. Smirnov, Sov. J. Nucl. Phys. 42, 913 (1985).
  * (28) P.B. Denton, H. Minakata, S.J. Parke, J. High Energy Phys. 1606, 051 (2016).
  * (29) C. Giunti, C.W. Kim, Found. Phys. Lett. 14, 213 (2001).
  * (30) C. Giunti, J. High Energy Phys. 0211, 017 (2002).
  * (31) V. Vedral, M.B. Plenio, M.A. Rippin, P.L. Knight, Phys. Rev. Lett. 78, 2275 (1997).
  * (32) R. Horodecki, P. Horodecki, M. Horodecki, K. Horodecki, Rev. Mod. Phys. 81, 865 (2009).
  * (33) S.A. Hill, W.K. Wootters, Phys. Rev. Lett. 78, 5022 (1997).
  * (34) W.K. Wootters, Phys. Rev. Lett. 80, 2245 (1998).
  * (35) T. Baumgratz, M. Cramer, M.B. Plenio. Phys. Rev. Lett. 113, 140401 (2014).
  * (36) A. Streltsov, G. Adesso, M.B. Plenio. Rev. Mod. Phys. 89, 041003 (2017).
  * (37) K. Abe et al., Phys. Rev. Lett. 124, 161802 (2020).


11footnotetext: The first two authors contributed equally.22footnotetext:
Corresponding author. This work was supported by the National Natural Science
Foundation of China under Grant 61901454. 00footnotetext: Project page:
https://github.com/AceCHQ/TransMatting11institutetext: University of Chinese
Academy of Sciences, Beijing, China
Key Laboratory of Space Utilization, Technology and Engineering Center
for Space Utilization, Chinese Academy of Sciences, Beijing, China
11email<EMAIL_ADDRESS><EMAIL_ADDRESS>
# TransMatting: Enhancing Transparent Objects Matting with Transformers

Huanqia Cai∗ Fanglei Xue∗ Lele Xu∗∗ Lili Guo

###### Abstract

Image matting refers to predicting the alpha values of unknown foreground
areas from natural images. Prior methods have focused on propagating alpha
values from known to unknown regions. However, not all natural images have a
specifically known foreground. Images of transparent objects, like glass,
smoke, web, etc., have less or no known foreground. In this paper, we propose
a Transformer-based network, TransMatting, to model transparent objects with a
big receptive field. Specifically, we redesign the trimap as three learnable
tri-tokens for introducing advanced semantic features into the self-attention
mechanism. A small convolutional network is proposed to utilize the global
feature and non-background mask to guide the multi-scale feature propagation
from encoder to decoder for maintaining the contexture of transparent objects.
In addition, we create a high-resolution matting dataset of transparent
objects with small known foreground areas. Experiments on several matting
benchmarks demonstrate the superiority of our proposed method over the current
state-of-the-art methods.

###### Keywords:

Image Matting; Vision Transformer; Deep Learning

##  1 Introduction

Image matting is a technique to separate the foreground object and the
background from an image by predicting a precise alpha matte as a result. It
has been widely used in many applications, such as image and video editing,
background replacement, and virtual reality [5, 46, 18]. Image matting assumes
that every pixel in the image $I$ is a linear combination of the foreground
object $F$ and the background $B$ by an alpha matte $\alpha$:

$I=\alpha F+(1-\alpha)B,\alpha\in[0,1]$ (1)

As only the image $I$ is known in this equation, the image matting is an ill-
posed problem. So many existing methods [5, 18, 38, 43, 46, 29, 22] take a
trimap as an auxiliary input. The trimap segments the image into three parts:
known foreground and background, and unknown area, indicated as white, black,
and gray, separately.

Most traditional methods, including sampling-based [2, 7, 43, 12, 15, 36] and
propagation-based methods [5, 17, 38, 18], utilize the known area samples to
find candidate colors or propagate the known alpha value. They heavily rely on
the information from known areas, especially the known foreground areas.
Recently, learning-based methods directly predict alpha mattes by neural
network learning from well-annotated datasets. Although these methods take a
great improvement in image matting, they also need specific information from
known areas to predict unknown areas. However, according to [25], more than
50% pixels in the unknown areas cannot be correlated to pixels in the known
regions due to the limited reception field of deep learning methods. LFPNet
[25] proposes a Center-Surround Pyramid Pooling module to propagate the
context feature from the known regions to the near unknown regions. However,
not all natural images have a salient and opaque object as the known
foreground [21]. Images of the glass, bonfires, plastic bags, etc., have
salient foregrounds but with transparent or meticulous interiors; images of
the web, smoke, water drops, etc., have non-salient foregrounds. The
corresponding trimaps of these kinds of images will have very few or even no
foreground areas. Most of the areas will be divided into the unknown regions.
It is very challenging for existing models to learn long-range features with
little known information. Furthermore, with the development of modern cameras,
picture resolution is becoming higher and higher. However, the reception
fields of existing models could not increase as the resolution of input images
do, which makes the problem even worse.

To address this issue, we make the first attempt to introduce Vision
Transformer (ViT) [9] to extract features with a large receptive field. The
Transformer model is first proposed in natural language processing (NLP) and
has achieved great performance in computer vision tasks, such as
classification [9, 41, 27], segmentation [53, 30], and detection [4, 47]. It
mainly consists of a multi-head self-attention (MHSA) module and a multi-layer
perception module. The MHSA module could mine information in a global scope.
Thus, the ViT model could learn global semantic features of the foreground
object with high-level position relevance. To further help the model integrate
the low-level appearance features (e.g., texture) with high-level semantic
features (e.g., shape), a Multi-scale Global-guided Fusion (MGF) module is
proposed. The MGF takes three adjacent scales of features as input, uses the
non-background mask to guide the low-level feature, and employs the high-level
feature to guide the information integration. With this new MGF module, only
foreground features could be transmitted to the decoder, reducing the
influence of background noises.

Since the DIM [46] concatenates the trimap and RGB image to feed into the
network, almost all subsequent trimap-based methods follow this strategy.
However, compared with the RGB image, the trimap is very sparse and has some
high-level positional relevance [26]. Most areas in the trimap have the same
value, making convolution neural networks with small kernels inefficient in
extracting features. Inspired by the [cls] token in ViT, we propose a new form
of trimap named the tri-token map. Three learnable tokens are used to indicate
the foreground, background, and unknown categories. We denote them as tri-
tokens. Based on these tri-tokens, we propose a Tri-token Guided Transformer
Block (TGTB), which adds the query with the corresponding tri-tokens for
introducing the trimap information into the self-attention mechanism. With
this high-level position information, the Transformer module could identify
which features are from the known areas and which are from the unknown areas.

Besides, there has not been any testbed for images with transparent or non-
salient foreground objects. Previous datasets mainly focus on salient and
opaque foregrounds, like animals [20] and portraits [37, 26], which have
significantly been investigated. To further help the community to dig into the
transparent and non-salient cases, we collect 460 high-solution natural images
with large unknown areas and manually label their alpha mattes.

Our main contributions can be summarized as follows:

  1. 1.

We propose a TGTB module, introducing the Vision Transformer module to extract
global semantic features with a big receptive field. We also redesign the
trimap as a tri-token map to directly bring location information to the self-
attention mechanism.

  2. 2.

A MGF module is proposed to integrate multi-scale features, and the global
information is well organized to guide the integration with low-level
Transformer features.

  3. 3.

We build a high-resolution matting dataset with 460 images of the transparent
or non-salient foreground. The dataset will be released to promote the
development of matting technology.

  4. 4.

Experiments on three matting datasets demonstrate that the proposed
TransMatting method outperforms the current SOTA methods, indicating the
effectiveness of our proposed modules.

##  2 Related Works

In this section, we first briefly review matting from two perspectives:
traditional methods and deep-learning methods. Then, we further give an
overview of Vision Transformer models, as the Tri-token Guided Transformer
Block (TGTB) is one of the main contributions of this work.

###  2.1 Traditional Matting

Traditional matting methods can be divided into two categories: sampling-based
and propagation-based methods. These methods mainly rely on low-level
features, like color, location, etc. The sampling-based methods [2, 7, 43, 12,
15, 36] first predict the colors of the foreground and background by
evaluating the similarity of colors between the known foreground, background,
and unknown area in samples, and then predict alpha mattes. Various sampling
techniques have been investigated, including color cluster sampling [36], edge
sampling [15], ray casting [12], etc. The propagation-based methods [5, 17,
38] propagate the information from the known foreground and background to the
unknown area by solving the sparse linear equation system [18], the Poisson
equation system [13], etc., to obtain the best global optimal alpha.

###  2.2 Deep-Learning Matting

In recent decades, deep learning technologies have boomed in various fields of
computer vision. The same goes for the image matting task. [40] combines the
sampling and deep neural network to improve the accuracy of alpha matting
prediction. The Indices matter method [29] proposes an index-guided method for
up-sampling and down-sampling to make the detailed information in the
prediction graph more complete. Based on providing a larger dataset
Composition-1k [46], DIM utilizes an encoder-decoder model to directly predict
alpha mattes, which effectively improves the accuracy. [39] introduces
semantic classification information of the matting region and uses learnable
weights and multi-class discriminators to revise the prediction results. [51]
proposes a general matting framework, which is conducive to obtaining better
results under the guidance of different qualities and forms. [26] further
mines the information of the RGB map and trimap and fuses the global
information from these maps for obtaining better alpha mattes. All of the
above methods use trimap as guidance. Some trimap-free methods can predict
alpha mattes without using trimap. However, the accuracy of these trimap-free
methods still has a big gap compared to that of the trimap-guided ones [6, 49,
35, 48], indicating that the trimap could help the model to capture
information efficiently.

###  2.3 Vision Transformer

The Transformer is firstly proposed in [42] to model long-range dependencies
for machine translation and has demonstrated impressive performance on NLP
tasks. Inspired by this, numerous attempts have been made to adapt
transformers for vision tasks, and promising results have been shown for
vision fields such as image classification, objection detection, semantic
segmentation, etc. In particular, ViT [9] divides the input image into patches
with a size of 16 $\times$ 16 and feeds the patch sequences to the vanilla
Transformer model. To help the training process and improve the performance,
DeiT [41] proposes a teacher-student strategy, which includes a distillation
token for the student to learn from the teacher. Later, Swin [27], PVT [44],
Crossformer [45], and HVT [32] combine the Transformer and pyramidal structure
to decrease the number of patches progressively for obtaining multi-scale
feature maps. To reduce computing and memory complexity, Swin, HRFormer [52],
and CrossFormer apply local-window self-attention in Transformer, which also
shows superior or comparable performance compared to the counterpart CNNs. The
powerful self-attention mechanism in Transformer shows great advantages over
CNN by capturing global attention of the whole image. However, some
researchers [23] argue that locality and globality are both essential for
vision tasks. Therefore, various researchers have tried combining the locality
of CNN with the globality of Transformer to improve performance further.
LocalViT [23] brings depth-wise convolutions to vision transformer to combine
self-attention mechanism with locality, and shows great improvement compared
to the pure Transformer, like DeiT, PVT, and TNT [14].

##  3 Matting Dataset

According to the transparency of foregrounds, we could divide the images of
matting into two types: 1) Transparent partially (TP): TP refers to that there
are significant foreground and uncertainty areas, and the foreground areas can
provide information for the prediction of uncertainty areas. For example, when
the foreground is human, the opaque and unknown regions are the hair or
clothes. 2) Transparent totally (TT): there are minor or non-salient
foreground areas, and the entire image is semi-transparent or high
transparent. These images include glass, plastic bags, fog, water drops, etc.

As illustrated in Tab. 1, we select four popular image matting datasets for
comparison, including DAPM [37], Composition-1k, Distinctions-646 [34], and
AIM-500 [21]. The DAPM dataset consists only of portraits with no translucent
or transparent objects. The Composition-1k dataset contains multiple
categories, while most images are portraits (227 out of 481, TP-type). The
Distinctions-646 dataset also mainly consists of portraits (343 out of 646,
TP-type) [26]. The AIM-500 dataset contains only 76 TT-type images (correspond
to the Salient Transparent/Meticulous type and the Non-Salient type in the
original dataset) but 424 TP-type images.

Table 1: Comparison between different public matting datasets.  Image Matting Dataset | total num | TT num | resolution
---|---|---|---
DAPM [37] | 2000 | 0 | 800$\times$600
Composition-1k [46] | 481 | 86 | 1297$\times$1082
Distinction-646 [34] | 646 | 79 | 1727$\times$1565
AIM-500 [21] | 500 | 76 | 1260$\times$1397
Transparent-460 (Ours) | 460 | 460 | 3820$\times$3766

As we can see, the transparent objects in the above datasets only occupy a
small portion. This may be because it is much more difficult to label
transparent objects than other objects, limiting the progress of transparent
objects in the matting field. In this work, we propose the first large-scale
dataset targeting various high transparent objects called Transparent-460
dataset. Our Transparent-460 dataset includes 460 high-quality manually-
annotated alpha mattes, where 410 images are for training and 50 for testing.
Furthermore, to our best knowledge, the resolution of our Transparent-460 is
the highest (the average resolution is up to 3820 $\times$ 3766) among all
datasets with high transparent objects. We believe this new matting dataset
will greatly advance the matting research on objects with massive transparent
areas.

##  4 Methodology

Figure 1: The structure of our TransMatting.

###  4.1 Motivation

By evaluating the results of some SOTA methods on TT and TP objects separately
on the Composition-1k dataset (Tab. 2), we find that the results of TT, which
denotes the total transparent objects, are much worse than TP, indicating that
TT objects are the key to affecting the overall evaluation results.
Furthermore, we find that most of the existing methods rely on the information
of the foreground region for predicting the unknown region [5, 17, 19, 22, 1].
However, such methods will become useless or ineffective when facing images
with no definite known regions. For example, [22] borrows features from both
the known and unknown regions; when the unknown region is overwhelming in the
images, the opacity propagation and the mattes prediction will face
difficulties. Therefore, global information with a large or global receptive
field and local features with inherent representation are needed to enhance
the understanding and recognition capacity for objects with totally unknown
regions. Although we can stack CNN layers to enlarge the receptive field, the
information that covers the whole image is still hard to be obtained [25].
Besides, CNN also lacks global connectivity [23]. By contrast, the Transformer
is good at modeling long-range connectivity with its attention mechanism.

Table 2: Performance of TT and TP objects on the Composition-1k dataset. Methods |  MSE$\downarrow$ |  SAD$\downarrow$
---|---|---
TT | TP | TT+TP | TT | TP | TT+TP
IndexNet [29] | 22.87 | 8.9 | 13 | 110.3 | 18.08 | 45.8
GCAMatting [22] | 15.89 | 6.2 | 9.1 | 85.72 | 13.68 | 35.3
MGMatting [51] | 13.01 | 4.65 | 7.18 | 77.88 | 11.87 | 31.76
TransMatting(Ours) | 7.49 | 3.4 | 4.58 | 59.37 | 10.35 | 24.96

Moreover, most existing SOTA trimap-guided methods directly concatenate the
trimap with RGB image as input. However, the huge gap between the two
modalities of RGB image and trimap brings great difficulties in semantic
feature extraction. At the same time, the trimap cannot effectively help the
model focus on the region of interest. Therefore, a more efficient way to
promote the guiding role of trimap is needed.

In short, to improve the performance of TT objects, more global and local
features should be captured, and an effective guidance method for the trimap
should be developed.

###  4.2 Baseline Structure

To extract both the local and global features, we combine CNN and the
Transformer model as our encoder. Specifically, the first part, like [51, 22]
is the same as the first two stages of ResNet34-UNet (denoted as CNN Local
Extractor in Fig. 1). The second part consists of a stack of our proposed Tri-
token Guided Transformer Block (TGTB) based on the Swin Transformer [27]. As
the decoder, we adopt the original ResNet34-UNet, a widespread network in the
matting field [51, 22].

###  4.3 Trimap Guided Methods

Almost all SOTA methods [26, 51, 22, 46, 50, 39, 3] use trimap as a guide and
directly concatenate the RGB image and the annotated trimap as the model’s
input. However, the modalities of the RGB image and trimap are quite
different. The RGB image scales from 0 to 255 and shows fine low-level
features like texture, color similarity, etc. The trimap includes three
values, containing high-level semantic information, like shape, location,
etc., [26]. Thus, the direct concatenation between them is not the most
efficient way to extract features.

Although trimap can explicitly indicate the region of interest, it is still
hard to take full advantage of this information. To the best of our knowledge,
we are the first to attempt to harmonize the RGB image and trimap rather than
simply concatenating them. We insert a learnable trimap into the Transformer
module to guide the model to concentrate on the valuable area, making the
network learning more efficient and robust.

###  4.4 Tri-token

Inspired by the [cls] token in Vision Transformer, we design a new tri-token
(shown in Fig. 1) structure, aiming to introduce the high-level semantic
information directly into the self-attention mechanism to replace the
inefficient concatenation methods. Given a vanilla
$Trimap\in\mathbb{R}^{H\times W}$, we generate three learnable tri-tokens
(denoted as $Token_{i}$, $i=\\{0,1,2\\}$) with different initialization to
represent the known foreground, known background, and unknown areas,
respectively. Every tri-token is a 1D vector, that is,
$Token_{i}\in\mathbb{R}^{C}$. Then we replace every pixel in the trimap with
the corresponding tri-token to generate the tri-token map, formulated as:

$Trimap[Trimap==i]=Token_{i},i=\\{0,1,2\\}$ (2)

In this manner, the tri-token map can directly guide the self-attention
process in the Transformer to pay more attention to the unknown areas for
self-updating.

###  4.5 Tri-token Guided Transformer Block

Global connectivity is much more important for the prediction of total
transparent objects. CNN does not have global attention, and its receptive
field cannot cover the whole image [25], which leads to poor estimation for
pixels outside receptive fields, while Transformer has global attention, and
its receptive field can cover every pixel at the first layer.

The Transformer consists of multi-head self-attention (MHSA) and Multilayer
Perceptron (MLP) blocks. The self-attention mechanism can be thought of as a
mapping between a query and a collection of key-value pairs. The output is a
weighted sum of the values, and the weights are assigned by the compatibility
function between the query and the relevant key. This can be implemented by
Scaled Dot-Product Attention [42], in which a softmax function is used to
activate the dot products of query and all keys for obtaining the weights.
MHSA means that more than one self-attention is performed in parallel.

Like [27, 45, 52], we use non-overlapping windows whose size is $M\times M$ to
divide the feature maps. The MHSA is performed within each window. The
formulations of vanilla attention and our tri-token attention in a specific
window are shown as follows:

$Attention(Q,K,V)=Softmax(QK^{T}/\sqrt{d})V$ (3)

$Tri\mbox{-}token\
Attention(Q,K,V)=Softmax((Q+Tri\mbox{-}token)K^{T}/\sqrt{d})V$ (4)

where $Q,K,V\in R^{M^{2}\times d}$ represent the query, key, and value in the
attention mechanism, respectively. $d$ is the query/key dimension. In the Tri-
token Attention formulation, $Q$, $K$, and $V$ are the same as that in the
standard self-attention. The $Tri\mbox{-}token$ is our proposed learnable
_trimap_ that adds to the query for forming a new tri-token query. In this
way, our tri-token attention mechanism can selectively aggregate contexts and
evaluate which region should be paid more attention to with the guidance of
our learnable tri-tokens.

In this way, we combine the self-attention and tri-tokens to focus on more
valuable regions by considering the relationship between non-background and
background areas, and finally achieve the best performance. We use our tri-
token attention every five blocks in each Tri-token Guided Transformer Block
(TGTB).

###  4.6 Multi-scale Global-guided Fusion Module

In the multi-scale feature pyramid structure, in-depth features contain more
global information, while shallow features have rich local information like
texture, color similarity, etc. Fusing these features is vital for accurately
predicting alpha mattes for high transparent objects [35]. Although the direct
sum operation can realize feature fusion, the details in the shallow features
may attenuate the impact of the advanced semantics, resulting in some subtle
regions missing [35]. To address this issue, we propose a Multi-scale Global-
guided Fusion (MGF) module in the decoder process (see Fig. 1 for details),
with both the non-background information and the advanced semantic features as
guidance, to fuse the high-level semantic information and the lower ones
effectively.

Specifically, we denote three adjacent features from shallow to deep as
$T_{n-1}$, $T_{n}$, and $T_{n+1}$. The $T_{n-1}$ is first down-sampled, then
the Hadamard product is employed between the non-background mask and $T_{n-1}$
to extract the low-level features of non-background, which helps to reduce the
impact of complex background influence. This can guide the network to pay more
attention to the foreground and unknown areas. After that, the $T_{n-1}$ is
concatenated with $T_{n}$, and a convolution layer is performed to align the
channel of fused features. We mark this feature as $T_{f}$.

For the $T_{n+1}$, we first perform a global average pooling to generate
channel-wise statistics and then use two fully connected (FC) layers to
squeeze channels. As shown in Fig. 1, features output from the two FC layers
are denoted as $\gamma$ and $\beta$, separately.To fully capture channel-wise
dependencies, we add a sigmoid function to activate $\gamma$ and perform
broadcast multiplication with $T_{f}$ for channel re-weighting. After that,
broadcast addition is performed between the channel-weighted feature and
$\beta$. A convolution layer is used to fuse information from different
groups. Notably, a skip connection from $T_{n}$ is employed for obtaining the
final fused features of MGF.

In short, considering that fusing low-level features directly may cause a
negative impact on the advanced semantics [35], two techniques are proposed
here. Firstly, the non-background mask is introduced into the fusion process
to filter out the complex background information and further help to
concentrate more attention on the foreground and unknown areas. Secondly, the
global channel-wise attention from higher-level features is used for re-
weighting and enhancing the important information in the fused features.

###  4.7 Loss Function

Following [51], we use three losses, including the alpha loss
($\mathcal{L}_{\alpha}$), Compositional loss [46] ($\mathcal{L}_{comp}$), and
Laplacian loss [16] ($\mathcal{L}_{lap}$). As formulated below, their weights
are set as 0.4, 1.2, and 0.16, respectively.

$\mathcal{L}_{final}=0.4*\mathcal{L}_{\alpha}+1.2*\mathcal{L}_{comp}+0.16*\mathcal{L}_{lap}$
(5)

##  5 Experiments

In this section, we show our experimental settings and compare our evaluation
results on the test set of Composition-1k [46], Distinction-646 [34], and our
Transparent-460 datasets with other state-of-the-art methods.

###  5.1 Dataset

Composition-1k contains 431 and 50 unique foreground objects and manually
labeled alpha mattes as training and test sets, respectively. Every foreground
object is composited with 100 (for training set) and 20 (for test set)
background images from COCO [24] and Pascal VOC [10]. As a result, there are
43,100 images for training and 1,000 images for testing.

Distinction-646 comprises 646 distinct foreground objects. Similar to the
Composition-1k, 50 objects are divided as the test set. Following the same
composition rule, there are 59,600 and 1000 images for training and testing,
respectively.

Our Transparent-460  mainly consists of transparent and non-salient objects as
the foreground, like water drops, jellyfish, plastic bags, glass, crystals,
etc. We collect 460 high-resolution images and carefully annotate them with
Photoshop. Considering the transparent objects are very meticulous, we keep
the original resolution of all collected images (up to 3820 $\times$ 3766
pixels on average). To our best knowledge, this is the first transparent
object matting dataset in such a high resolution.

###  5.2 Evaluation Metrics

Following [16, 3, 29, 26], we use four metrics for evaluation, including the
Sum of Absolute Differences (SAD), Mean Squared Error (MSE), Gradient error
(Grad.) and Connectivity error (Conn). It is notable that the unit of MSE
value is set to 1e-3 for easy reading.

###  5.3 Implementation Details

We use PyTorch [33] to implement our proposed method. All the experiments are
trained for 200,000 iterations. We initialize our network with ImageNet [8]
pre-trained weights. The ablation experiments in Tab. 3, 4, 5 are done with 2
NVIDIA Tesla V100 GPU with a batch size of 32. Moreover, to compare our method
with the existing SOTA methods, we use a batch size of 64 with 4 NVIDIA Tesla
V100 GPU to train our proposed method in Tab. 6, 7, 8. The Adam optimizer is
utilized, and the initial learning rate is set to 1e-4 with the same learning
rate decay strategy as [51, 28]. For a fair comparison, we follow the data
augmentation methods used in [22], like random crop, rotation, scaling,
shearing, etc. Moreover, the trimaps for training are generated using dilation
and erosion ways on alpha images by random kernel sizes from 1 to 30. Finally,
we crop 512×512 patches on the center of the unknown area of alpha and
composite them with the background from COCO. We use the same training
conditions on the Composition-1k and Distinction-646 datasets.

###  5.4 Ablation Study

To evaluate the effectiveness of our new proposed modules of TGTB and MGF, and
the performance with different hyper-parameters, we design the ablation study
on the Composition-1k dataset.

Evaluate the effectiveness of our proposed modules. The quantitative results
under the SAD, MSE, Gradient, and Connectivity errors with and without our
proposed TGTB and MGF modules are illustrated in Tab. 3. As we can see, with
the TGTB module, the four metrics listed above decrease to 27.45, 5.66, 11.77,
and 24.30, respectively. The main reason is that our redesigned tri-token map
is more suitable for propagating location information than simply
concatenating to the input image. The MGF module could solely achieve similar
performance, indicating that our proposed multi-scale feature fusion strategy
can also help the decoder to make better use of the local and global
information. When combined with the TGTB and MGF modules, the model achieves
the best performance, indicating the effectiveness of the two new proposed
modules.

Table 3: The effectiveness of our proposed TGTB and MGF modules on the Composition-1k dataset. TGTB | MGF | SAD $\downarrow$ | MSE $\downarrow$ | Grad. $\downarrow$ | Conn.$\downarrow$
---|---|---|---|---|---
|  | 29.14 | 6.34 | 12.06 | 25.21
$\checkmark$ |  | 27.45 | 5.66 | 11.77 | 24.30
| $\checkmark$ | 27.21 | 5.57 | 11.23 | 23.25
$\checkmark$ | $\checkmark$ | 26.83 | 5.22 | 10.62 | 22.14

Determine where to introduce tri-tokens. There are four TGTB stages in our
encoder model. Tab. 4 reports the performance with different positions to
introduce tri-tokens. As the position goes deep, the feature map size
decreases, making more position information lose. On the other hand, deep
stages have learned more abstract semantic features, which is suitable for
mutual learning with tri-tokens. As shown in Tab. 4, both shallow and deep
stages benefit from tri-tokens, indicating that the tri-tokens in TGTB modules
could guide the encoder to focus on the right regions.

Table 4: Ablation results on the Composition-1k dataset with different
positions to introduce the proposed tri-tokens.
Position | SAD$\downarrow$ | MSE$\downarrow$ | Grad.$\downarrow$ | Conn.$\downarrow$
---|---|---|---|---
1 | 31.68 | 7.24 | 14.20 | 27.42
4 | 29.50 | 6.20 | 13.18 | 25.23
1,2,3,4 | 26.83 | 5.22 | 10.62 | 22.14

Table 5: Ablation results on the Composition-1k dataset with local or (and) global features in the proposed MGF module. local | global | SAD$\downarrow$ | MSE$\downarrow$ | Grad.$\downarrow$ | Conn.$\downarrow$
---|---|---|---|---|---
|  | 27.45 | 5.66 | 11.77 | 24.30
$\checkmark$ |  | 27.16 | 5.34 | 11.03 | 22.60
| $\checkmark$ | 27.39 | 5.46 | 11.43 | 23.20
$\checkmark$ | $\checkmark$ | 26.83 | 5.22 | 10.62 | 22.14

The impact of local and global features in MGF. Tab. 5 reports the
effectiveness of our MGF module with and without local or global branches. The
local branch is proposed to integrate $T_{n-1}$ with the non-background mask,
and the global branch is responsible for introducing global features from
$T_{n+1}$ to guide the feature flow. As we can see from Tab. 5, combining
local and global branches could achieve the best performance compared with
using one of them solely. The main reason is the effectiveness of our MGF in
fusing local (texture, border) and global (semantic, location) features for
modeling unknown regions.

Figure 2: Visual comparison of our TransMatting against SOTA methods on the
Composition-1k test set.

###  5.5 Comparison with Prior Work

To evaluate our method’s performance, we compare it with other state-of-the-
art models on the following three datasets. Notably, we achieve the best
performance on all three datasets.

Testing on Composition-1k. We show the quantitative and visual results on Tab.
6 and Fig. 2. Without any test-time augmentations, our proposed TransMatting
outperforms other SOTA methods on all four evaluation metrics by only using
the Composition-1k training set for training. As illustrated in Tab. 6, our
model decreases the MSE and Grad metrics heavily: from 5.2, 10.6 to 4.58 and
9.72, respectively, indicating the effectiveness of our TransMatting.

Table 6: The quantitative results on the Composition-1k test set [46]. † denotes results with test-time augmentation. Methods | SAD$\downarrow$ | MSE$\downarrow$ | Grad.$\downarrow$ | Conn.$\downarrow$
---|---|---|---|---
AlphaGAN [31] | 52.4 | 30 | 38 | 53
DIM [29] | 50.4 | 14 | 31.0 | 50.8
IndexNet [29] | 45.8 | 13 | 25.9 | 43.7
AdaMatting [3] | 41.7 | 10 | 16.8 | -
ContextNet [16] | 35.8 | 8.2 | 17.3 | 33.2
GCAMatting [22] | 35.3 | 9.1 | 16.9 | 32.5
MGMatting [51] | 31.5 | 6.8 | 13.5 | 27.3
TIMI-Net [26] | 29.08 | 6.0 | 12.9 | 27.29
FBAMatting [11] † | 25.8 | 5.2 | 10.6 | 20.8
TransMatting(Ours) | 24.96 | 4.58 | 9.72 | 20.16
Figure 3: Visual comparison of our TransMatting against SOTA methods on our
Transparent-460 test set.

Testing on Distinction-646. Tab. 7 compares the performance of our
TransMatting with other state-of-the-art methods on Distinction-646. For a
fair comparison, we follow the whole inference protocol in [34, 51] to
calculate the metrics based on the whole image. Without any additional tuning,
our method outperforms all the SOTA methods.

Testing on our Transparent-460 . Based on their release codes, we train
IndexNet and MGMatting methods on our dataset and compare them with ours in
Tab. 8. Our Transparent-460 dataset mainly focuses on transparent and non-
salient foregrounds, which is very difficult for existing image matting
methods. Surprisingly, as illustrated in Tab. 8, our TransMatting achieves
promising results with only a 4.02 MSE error. Furthermore, to evaluate the
generalization performance of our model. We train our TransMatting on the
Composition-1k training set and directly test it on the Transparent-460 test
set. The results are shown in Tab. 9. Thanks to the big receptive field and
well-designed multi-scale fusion module, our model reduces nearly half of the
SAD, MSE, and Conn. errors compared to the SOTA methods.

Table 7: The quantitative results on the Distinction-646 test set.  Methods | SAD$\downarrow$ | MSE$\downarrow$ | Grad.$\downarrow$ | Conn.$\downarrow$
---|---|---|---|---
KNNMatting [5] | 116.68 | 25 | 103.15 | 121.45
DIM [46] | 47.56 | 9 | 43.29 | 55.90
HAttMatting [34] | 48.98 | 9 | 41.57 | 49.93
GCAMatting [22] | 27.43 | 4.8 | 18.7 | 21.86
MGMatting [51] | 33.24 | 4.51 | 20.31 | 25.49
TransMatting (Ours) | 25.65 | 3.4 | 16.08 | 21.45
Table 8: The quantitative results on our proposed Transparent-460 test set. Methods | SAD$\downarrow$ | MSE$\downarrow$ | Grad.$\downarrow$ | Conn.$\downarrow$
---|---|---|---|---
IndexNet [29] | 573.09 | 112.53 | 140.76 | 327.97
MGMatting [51] | 111.92 | 6.33 | 25.67 | 103.81
TransMatting (Ours) | 88.34 | 4.02 | 20.99 | 82.56
Table 9: The generalization results on our proposed Transparent-460 test set. Methods | SAD$\downarrow$ | MSE$\downarrow$ | Grad.$\downarrow$ | Conn.$\downarrow$
---|---|---|---|---
DIM [46] | 356.2 | 49.68 | 146.46 | 296.31
IndexNet [29] | 434.14 | 74.73 | 124.98 | 368.48
MGMatting [51] | 344.65 | 57.25 | 74.54 | 282.79
TIMI-Net [26] | 328.08 | 44.2 | 142.11 | 289.79
TransMatting(Ours) | 192.36 | 20.96 | 41.8 | 158.37

##  6 Conclusion

In order to generalize to transparent and non-salient foregrounds, matting
algorithms must have the ability to mine long-range features and utilize the
semantic features in trimap. In this paper, we propose a novel Transformer-
based network by redesigning a tri-token map to introduce the trimap semantic
features into the long-range dependencies of the self-attention mechanism.
Furthermore, a multi-scale global-guided fusion module is proposed to take the
global information and local non-background mask as a guide to fuse multi-
scale features for better modeling the unknown regions in transparent objects.
Experiments on the Composition-1k, Distinctions-646, and our proposed
Transparent-460 datasets demonstrate that our TransMatting outperforms the
state-of-the-art methods.

## References

  * [1] Aksoy, Y., Aydın, T.O., Pollefeys, M.: Information-flow matting. arXiv preprint arXiv:1707.05055 (2017)
  * [2] Berman, A., Dadourian, A., Vlahos, P.: Method for removing from an image the background surrounding a selected object (Oct 17 2000), uS Patent 6,134,346
  * [3] Cai, S., Zhang, X., Fan, H., Huang, H., Liu, J., Liu, J., Liu, J., Wang, J., Sun, J.: Disentangled image matting. In: Proceedings of the IEEE/CVF International Conference on Computer Vision. pp. 8819–8828 (2019)
  * [4] Carion, N., Massa, F., Synnaeve, G., Usunier, N., Kirillov, A., Zagoruyko, S.: End-to-End Object Detection with Transformers. In: European Conference on Computer Vision. http://arxiv.org/abs/2005.12872
  * [5] Chen, Q., Li, D., Tang, C.K.: Knn matting. IEEE transactions on pattern analysis and machine intelligence 35(9), 2175–2188 (2013)
  * [6] Chen, Q., Ge, T., Xu, Y., Zhang, Z., Yang, X., Gai, K.: Semantic human matting. In: Proceedings of the 26th ACM international conference on Multimedia. pp. 618–626 (2018)
  * [7] Chuang, Y.Y., Curless, B., Salesin, D.H., Szeliski, R.: A bayesian approach to digital matting. In: Proceedings of the 2001 IEEE Computer Society Conference on Computer Vision and Pattern Recognition. CVPR 2001. vol. 2, pp. II–II. IEEE (2001)
  * [8] Deng, J., Dong, W., Socher, R., Li, L.J., Li, K., Fei-Fei, L.: Imagenet: A large-scale hierarchical image database. In: 2009 IEEE conference on computer vision and pattern recognition. pp. 248–255. Ieee (2009)
  * [9] Dosovitskiy, A., Beyer, L., Kolesnikov, A., Weissenborn, D., Zhai, X., Unterthiner, T., Dehghani, M., Minderer, M., Heigold, G., Gelly, S., et al.: An image is worth 16x16 words: Transformers for image recognition at scale. arXiv preprint arXiv:2010.11929 (2020)
  * [10] Everingham, M., Van Gool, L., Williams, C.K., Winn, J., Zisserman, A.: The pascal visual object classes (voc) challenge. International journal of computer vision 88(2), 303–338 (2010)
  * [11] Forte, M., Pitié, F.: $f$, $b$, alpha matting. arXiv preprint arXiv:2003.07711 (2020)
  * [12] Gastal, E.S., Oliveira, M.M.: Shared sampling for real-time alpha matting. In: Computer Graphics Forum. vol. 29, pp. 575–584. Wiley Online Library (2010)
  * [13] Grady, L., Schiwietz, T., Aharon, S., Westermann, R.: Random walks for interactive alpha-matting. In: Proceedings of VIIP. vol. 2005, pp. 423–429 (2005)
  * [14] Han, K., Xiao, A., Wu, E., Guo, J., Xu, C., Wang, Y.: Transformer in transformer. Advances in Neural Information Processing Systems 34 (2021)
  * [15] He, K., Rhemann, C., Rother, C., Tang, X., Sun, J.: A global sampling method for alpha matting. In: CVPR 2011. pp. 2049–2056. IEEE (2011)
  * [16] Hou, Q., Liu, F.: Context-aware image matting for simultaneous foreground and alpha estimation. In: Proceedings of the IEEE/CVF International Conference on Computer Vision. pp. 4130–4139 (2019)
  * [17] Lee, P., Wu, Y.: Nonlocal matting. In: CVPR 2011. pp. 2193–2200. IEEE (2011)
  * [18] Levin, A., Lischinski, D., Weiss, Y.: A closed-form solution to natural image matting. IEEE transactions on pattern analysis and machine intelligence 30(2), 228–242 (2007)
  * [19] Levin, A., Rav-Acha, A., Lischinski, D.: Spectral matting. IEEE transactions on pattern analysis and machine intelligence 30(10), 1699–1712 (2008)
  * [20] Li, J., Zhang, J., Maybank, S.J., Tao, D.: End-to-end animal image matting. arXiv e-prints pp. arXiv–2010 (2020)
  * [21] Li, J., Zhang, J., Tao, D.: Deep automatic natural image matting. arXiv preprint arXiv:2107.07235 (2021)
  * [22] Li, Y., Lu, H.: Natural image matting via guided contextual attention. In: Proceedings of the AAAI Conference on Artificial Intelligence. vol. 34, pp. 11450–11457 (2020)
  * [23] Li, Y., Zhang, K., Cao, J., Timofte, R., Van Gool, L.: Localvit: Bringing locality to vision transformers. arXiv preprint arXiv:2104.05707 (2021)
  * [24] Lin, T.Y., Maire, M., Belongie, S., Hays, J., Perona, P., Ramanan, D., Dollár, P., Zitnick, C.L.: Microsoft coco: Common objects in context. In: European conference on computer vision. pp. 740–755. Springer (2014)
  * [25] Liu, Q., Xie, H., Zhang, S., Zhong, B., Ji, R.: Long-range feature propagating for natural image matting. In: Proceedings of the 29th ACM International Conference on Multimedia. pp. 526–534 (2021)
  * [26] Liu, Y., Xie, J., Shi, X., Qiao, Y., Huang, Y., Tang, Y., Yang, X.: Tripartite information mining and integration for image matting. In: Proceedings of the IEEE/CVF International Conference on Computer Vision. pp. 7555–7564 (2021)
  * [27] Liu, Z., Lin, Y., Cao, Y., Hu, H., Wei, Y., Zhang, Z., Lin, S., Guo, B.: Swin transformer: Hierarchical vision transformer using shifted windows. In: Proceedings of the IEEE/CVF International Conference on Computer Vision. pp. 10012–10022 (2021)
  * [28] Loshchilov, I., Hutter, F.: Sgdr: Stochastic gradient descent with warm restarts. arXiv preprint arXiv:1608.03983 (2016)
  * [29] Lu, H., Dai, Y., Shen, C., Xu, S.: Indices matter: Learning to index for deep image matting. In: Proceedings of the IEEE/CVF International Conference on Computer Vision. pp. 3266–3275 (2019)
  * [30] Lu, Z., He, S., Zhu, X., Zhang, L., Song, Y.Z., Xiang, T.: Simpler Is Better: Few-Shot Semantic Segmentation With Classifier Weight Transformer. In: ICCV 2021. pp. 8741–8750
  * [31] Lutz, S., Amplianitis, K., Smolic, A.: Alphagan: Generative adversarial networks for natural image matting. arXiv preprint arXiv:1807.10088 (2018)
  * [32] Pan, Z., Zhuang, B., Liu, J., He, H., Cai, J.: Scalable visual transformers with hierarchical pooling. arXiv e-prints pp. arXiv–2103 (2021)
  * [33] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library. Advances in neural information processing systems 32 (2019)
  * [34] Qiao, Y., Liu, Y., Yang, X., Zhou, D., Xu, M., Zhang, Q., Wei, X.: Attention-guided hierarchical structure aggregation for image matting. In: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition. pp. 13676–13685 (2020)
  * [35] Qiao, Y., Liu, Y., Zhu, Q., Yang, X., Wang, Y., Zhang, Q., Wei, X.: Multi-scale information assembly for image matting. In: Computer Graphics Forum. vol. 39, pp. 565–574. Wiley Online Library (2020)
  * [36] Shahrian, E., Rajan, D., Price, B., Cohen, S.: Improving image matting using comprehensive sampling sets. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition. pp. 636–643 (2013)
  * [37] Shen, X., Tao, X., Gao, H., Zhou, C., Jia, J.: Deep automatic portrait matting. In: European conference on computer vision. pp. 92–107. Springer (2016)
  * [38] Sun, J., Jia, J., Tang, C.K., Shum, H.Y.: Poisson matting. In: ACM SIGGRAPH 2004 Papers, pp. 315–321 (2004)
  * [39] Sun, Y., Tang, C.K., Tai, Y.W.: Semantic image matting. In: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition. pp. 11120–11129 (2021)
  * [40] Tang, J., Aksoy, Y., Oztireli, C., Gross, M., Aydin, T.O.: Learning-based sampling for natural image matting. In: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition. pp. 3055–3063 (2019)
  * [41] Touvron, H., Cord, M., Douze, M., Massa, F., Sablayrolles, A., Jégou, H.: Training data-efficient image transformers & distillation through attention. In: International Conference on Machine Learning. pp. 10347–10357. PMLR (2021)
  * [42] Vaswani, A., Shazeer, N., Parmar, N., Uszkoreit, J., Jones, L., Gomez, A.N., Kaiser, Ł., Polosukhin, I.: Attention is all you need. Advances in neural information processing systems 30 (2017)
  * [43] Wang, J., Cohen, M.F.: Optimized color sampling for robust matting. In: 2007 IEEE Conference on Computer Vision and Pattern Recognition. pp. 1–8. IEEE (2007)
  * [44] Wang, W., Xie, E., Li, X., Fan, D.P., Song, K., Liang, D., Lu, T., Luo, P., Shao, L.: Pyramid vision transformer: A versatile backbone for dense prediction without convolutions. In: Proceedings of the IEEE/CVF International Conference on Computer Vision. pp. 568–578 (2021)
  * [45] Wang, W., Yao, L., Chen, L., Cai, D., He, X., Liu, W.: Crossformer: A versatile vision transformer based on cross-scale attention. arXiv e-prints pp. arXiv–2108 (2021)
  * [46] Xu, N., Price, B., Cohen, S., Huang, T.: Deep image matting. In: Proceedings of the IEEE conference on computer vision and pattern recognition. pp. 2970–2979 (2017)
  * [47] Yang, J., Li, C., Zhang, P., Dai, X., Xiao, B., Yuan, L., Gao, J.: Focal self-attention for local-global interactions in vision transformers. arXiv preprint arXiv:2107.00641 (2021)
  * [48] Yang, X., Qiao, Y., Chen, S., He, S., Yin, B., Zhang, Q., Wei, X., Lau, R.W.: Smart scribbles for image matting. ACM Transactions on Multimedia Computing, Communications, and Applications (TOMM) 16(4), 1–21 (2020)
  * [49] Yang, X., Xu, K., Chen, S., He, S., Yin, B.Y., Lau, R.: Active matting. Advances in Neural Information Processing Systems 31 (2018)
  * [50] Yu, H., Xu, N., Huang, Z., Zhou, Y., Shi, H.: High-resolution deep image matting. arXiv preprint arXiv:2009.06613 (2020)
  * [51] Yu, Q., Zhang, J., Zhang, H., Wang, Y., Lin, Z., Xu, N., Bai, Y., Yuille, A.: Mask guided matting via progressive refinement network. In: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition. pp. 1154–1163 (2021)
  * [52] Yuan, Y., Fu, R., Huang, L., Lin, W., Zhang, C., Chen, X., Wang, J.: Hrformer: High-resolution transformer for dense prediction. arXiv preprint arXiv:2110.09408 (2021)
  * [53] Zheng, S., Lu, J., Zhao, H., Zhu, X., Luo, Z., Wang, Y., Fu, Y., Feng, J., Xiang, T., Torr, P.H., Zhang, L.: Rethinking Semantic Segmentation from a Sequence-to-Sequence Perspective with Transformers. In: 2021 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR). pp. 6877–6886. IEEE. https://doi.org/10.1109/CVPR46437.2021.00681