# Skyrmion-size dependence of the topological Hall effect: A real-space
calculation

Akira Matsui<EMAIL_ADDRESS>Department of Applied
Physics, The University of Tokyo, Hongo, Bunkyo-ku, Tokyo, 113-8656, Japan
Takuya Nomoto<EMAIL_ADDRESS>Department of Applied Physics, The
University of Tokyo, Hongo, Bunkyo-ku, Tokyo, 113-8656, Japan  Ryotaro Arita
Department of Applied Physics, The University of Tokyo, Hongo, Bunkyo-ku,
Tokyo, 113-8656, Japan  RIKEN Center for Emergent Matter Science (CEMS), Wako
351-0198, Japan

###### Abstract

Motivated by the recent discoveries of magnets harboring short-pitch skyrmion
lattices, we investigate the skyrmion-size dependence of the topological Hall
effect. By means of large-scale real-space calculations, we find that the Hall
conductivity takes its extreme value in the crossover region where both the
real-space and momentum-space Berry curvature play a crucial role. We also
investigate how the optimum skyrmion size ($\lambda_{\rm sk}^{*}$) depends on
the lifetime of itinerant electrons ($\tau$) and coupling constant between
electrons and localized spins ($J$). For the former, we show that
$\lambda_{\rm sk}^{*}$ is proportional to $\sqrt{\tau}$, which indicates that
$\lambda_{\rm sk}^{*}$ is much less sensitive to $\tau$ than the conventional
expectation that $\lambda_{\rm sk}^{*}$ is proportional to the mean-free path
$\propto\tau$. For the latter, we show that the non-adiabaticity considerably
suppresses the topological Hall effect when the time scale determined by the
skyrmion size and Fermi velocity is shorter than $1/J$. However, its effect on
$\lambda_{\rm sk}^{*}$ is not so siginificant and $\lambda_{\rm sk}^{*}$ is
about ten times the lattice constant in a wide range of $J$ and $\tau$.

Introduction. A skyrmion is a vortex-like defect in helimagnets. Its non-
trivial topology is characterized by the number of times spins wind around the
unit sphere. This skyrmion number governs numerous intriguing properties
Nagaosa and Tokura (2013); Back  _et al._ (2020) that hold great promise for
future spintronic applications Fert  _et al._ (2017); Zhang  _et al._ (2020);
Fert  _et al._ (2013). Particularly, a small skyrmion is of great interest in
that it is favorable for high-density data storage and high energy efficiency.
Recently, short-pitch skyrmion lattices have became experimentally accessible
Kurumaji  _et al._ (2019); Hirschberger  _et al._ (2019); Khanh  _et al._
(2020) and are attracting significant attention. Contrary to conventional
skyrmion lattices in non-centrosymmetric materials induced by the
Dzyaloshinskii–Moriya interactions Nagaosa and Tokura (2013); Gayles  _et al._
(2015); Koretsune  _et al._ (2015); Kikuchi  _et al._ (2016), in these
recently discovered centrosymmetric materials, small skyrmion lattices are
stabilized by frustrations Okubo  _et al._ (2012); Leonov and Mostovoy (2015);
Batista  _et al._ (2016); Nomoto  _et al._ (2020) or four spin interactions
originated from the higher-order interactions in itinerant magnets Ozawa  _et
al._ (2017); Hayami  _et al._ (2017); Ozawa  _et al._ (2016); Hayami and
Motome (2014). There, the skyrmion size is even comparable to the mean-free
path of itinerant electrons Hirschberger  _et al._ (2020). The experimental
accessibility to various sizes of skyrmion lattices naturally motivates us to
theoretically investigate how the skyrmion size affects the topological
phenomena.

The non-trivial spin structure of skyrmions generates an emergent magnetic
field acting on itinerant electrons. One notable manifestation of the emergent
field is the topological Hall effect (THE), which was first experimentally
discovered in MnSi Neubauer  _et al._ (2009). While extensive theoretical
efforts have been made so far for the THE Nakazawa  _et al._ (2018); Nakazawa
and Kohno (2019a); Denisov  _et al._ (2016, 2018); Onoda  _et al._ (2004);
Tatara and Kawamura (2002); Nakazawa and Kohno (2019b), one of its interesting
aspects is that we can obtain intuitive insights in the strong coupling limit.
There, the effect of the non-collinear spin structure on itinerant electrons
can be understood in terms of the Berry curvature $\Omega$ in real and
momentum space. When the skyrmion size $\lambda_{\rm sk}$ is large, a real-
space picture suggests that the topological Hall conductivity (THC) should
behave as $\sim 1/\lambda_{\rm sk}^{2}$ since the strength of the real-space
Berry curvature $\Omega(\vec{R})$ is proportional to the scalar spin chilarity
or equivalently, the number of skyrmions in a given area Bruno  _et al._
(2004); Ye  _et al._ (1999). Meanwhile, for small $\lambda_{\rm sk}$, analysis
in momentum-space ($k$-space) is more useful: the THC is given by the sum of
the $k$-space Berry curvature $\Omega(\vec{k})$ of occupied bands. Thus the
THC becomes smaller when the skyrmion size becomes smaller since the size of
the magnetic Brillouin zone (BZ) becomes larger and the number of occupied
bands in the magnetic BZ becomes smaller Hamamoto  _et al._ (2015); Ohgushi
_et al._ (2000); Onoda and Nagaosa (2003); Wang  _et al._ (2020).

This Letter focuses on the skyrmion-size dependence of the THE in the strong
and medium coupling regimes. From the distinct behaviors of the THC in the
limit of large and small $\lambda_{\rm sk}$, we expect that the THC takes its
extreme values in the crossover region where both $\Omega(\vec{R})$ and
$\Omega(\vec{k})$ play a crucial role. To identify the optimum size of
skyrmions ($\lambda_{\rm sk}^{*}$) for the THE, we systematically calculate
the THC for a wide range of $\lambda_{\rm sk}$. We exploit the kernel
polynomial method (KPM) to overcome the difficulty of growing computational
costs for large skyrmion systems. The numerical calculation is particularly
essential in the regions for which analytical approaches are not accessible.

We first confirm that the THC depends on $\lambda_{\rm sk}$ non-monotonically
as expected from the considerations for the limits of large and small
$\lambda_{\rm sk}$. We then show that the THC takes its extreme value for
different values of damping rates of itinerant electrons ($1/2\tau$). We
naively expect that $\lambda_{\rm sk}^{*}$ is characterized by the mean-free
path ($l$) which is proportional to $\tau$. However, we find that
$\lambda_{\rm sk}^{*}$ is proportional to $\sqrt{\tau}$, which can be
understood in terms of the band separations in the magnetic BZ. We further
discuss the effect of non-adiabaticity by changing the coupling between
electrons and spins ($J$), which becomes important when the time scale
determined by the Fermi velocity ($v_{F}$) and $\lambda_{\rm sk}^{*}$ becomes
shorter than $1/J$. We show that while the non-adiabaticity significantly
affects the size of the THC, its effect on $\lambda_{\rm sk}^{*}$ is not so
prominent. Therefore, in a wide range of $J$ and $\tau$, $\lambda_{\rm
sk}^{*}$ is about ten times the lattice constant.

Method.  The simplest model representing the interplay between itinerant
electrons and localized spins is the double exchange model Anderson and
Hasegawa (1955),

$H=\sum_{\langle
i,j\rangle,\sigma}tc_{i,\sigma}^{\dagger}c_{j,\sigma}+J\sum_{i,\sigma,\sigma^{\prime}}(\vec{n}_{i}\cdot\vec{\sigma})^{\sigma,\sigma^{\prime}}c_{i,\sigma}^{\dagger}c_{i,\sigma^{\prime}},$

where $J$ is the coupling between the electrons and spins. $\vec{n}_{i}$
stands for the normalized local spin at site $i$ which is treated as a purely
classical quantity and the quantum fluctuation effects are neglected. We
calculate the THC for the triangular lattice and square lattice, where we
assume triple $Q$ and double $Q$ skyrmion structure Nagaosa and Tokura (2013),

$\vec{n}(\vec{r})\propto
M\vec{e}_{z}+\sum_{i}\vec{m}_{i}\cos{(\vec{q}_{i}\cdot\vec{r}+\phi_{i})}+\vec{e}_{z}\sin{(\vec{q}_{i}\cdot\vec{r}+\phi_{i})}$

with $M=1$, $\phi_{i}=-\pi/2$,
$\vec{m}_{i}=\vec{e}_{z}\times\vec{q}_{i}/|\vec{e}_{z}\times\vec{q}_{i}|$, and
$\vec{q}_{1}=2\pi/\lambda_{\rm sk}(1,0)$, $\vec{q}_{2}=2\pi/\lambda_{\rm
sk}(-1/2,\sqrt{3}/2)$, $\vec{q}_{3}=2\pi/\lambda_{\rm sk}(-1/2,-\sqrt{3}/2)$
for the triangular lattice, wheres $\vec{q}_{1}=2\pi/\lambda_{\rm sk}(1,0)$,
$\vec{q}_{2}=2\pi/\lambda_{\rm sk}(0,1)$ for the square lattice. Hereafter, we
set the hopping parameter $t=1$, the lattice constant $a=1$ and consider only
the nearest-neighbor hoppings for simplicity.

We calculate the THC ($\sigma_{xy}$) based on the linear response theory. We
consider the effect of impurities by introducing the constant elastic
scattering time $\tau$ and neglect the effect of vertex corrections. For non-
interacting Hamiltonian, the conductivity can be obtained by the Kubo-Bastin
formula Bastin  _et al._ (1971) and its Smrcka and Streda’s decomposition
Streda (1982); Crépieux and Bruno (2001). By substituting the non-interacting
retarded and advanced Green functions with a constant finite lifetime $\tau$,
i.e., $G^{\pm}(\varepsilon;x)=\int dx\delta(x-\hat{H})/(\varepsilon-x\pm
i/2\tau)$ into the Kubo-Bastin formula, we obtain

$\begin{split}\sigma^{\alpha}_{\mu\nu}=\frac{e^{2}\hbar}{V}\int\differential
x\differential
yF^{\alpha}(x,y)\mathrm{Tr}[v_{\mu}\delta(x-H)v_{\nu}\delta(y-H)],\end{split}$
(1)

where $\alpha$ is either $\rm{I}$, $\rm{IIa}$, or $\rm{IIb}$, and

$\begin{split}F^{\mathrm{I}}(x,y)=-\frac{1}{2\pi}\frac{1}{x-y-i/\tau}\int\differential\epsilon\partialderivative{f}{\varepsilon}[g^{+}(\varepsilon;x)-g^{-}(\varepsilon;y)]\end{split}$
$\begin{split}F^{\mathrm{IIa}}(x,y)=\frac{1}{2\pi}\frac{1}{x-y}\int\differential\epsilon
f(\varepsilon)\mathrm{Im}[g^{+}(\varepsilon;x)^{2}+g^{+}(\varepsilon;y)^{2}]\end{split}$
$\begin{split}F^{\mathrm{IIb}}(x,y)=\frac{1}{\pi}\frac{1}{(x-y)^{2}}\int\differential\epsilon
f(\epsilon)\mathrm{Im}[g^{+}(\varepsilon;x)-g^{+}(\varepsilon;y)],\end{split}$

with $g^{\pm}(\varepsilon;x)=1/(\varepsilon-x\pm i/2\tau)$. Here,
$f(\varepsilon)$ is the Fermi-Dirac distribution function and
$v_{\mu}=iat/\hbar\sum_{i}(c^{\dagger}_{i+\mu}c_{i}-c^{\dagger}_{i}c_{i+\mu})$
is the $\mu$-th component of the velocity operator. We divided the Fermi sea
term into two terms, $\sigma_{xy}^{\rm{IIa}}$ and $\sigma_{xy}^{\rm{IIb}}$,
following the convention in Kontani  _et al._ (2007). For large $\tau$,
$\sigma_{xy}^{\rm{I}}$ and $\sigma_{xy}^{\rm{IIa}}$ cancel with each other, so
that $\sigma_{xy}\sim\sigma_{xy}^{\rm{IIb}}$. In this limit, the THC can be
obtained by summing up $\Omega(\vec{k})$ for occupied bands (the Karplus-
Luttinger formula) Karplus and Luttinger (1954); Onoda and Nagaosa (2002). We
can confirm this by taking the trace in Eq. (1) explicitly for
$\sigma_{xy}^{\rm IIb}$ using the eigenstates of the Hamiltonian and assuming
that $\tau^{-1}$ in $g^{\pm}(\varepsilon;x)$ is infinitesimally small.
Meanwhile, when $\tau^{-1}$ becomes larger than the typical band separation
$\Delta$ in the magnetic BZ, $\sigma_{xy}^{\rm{IIb}}$ cancels with
$\sigma_{xy}^{\rm{IIa}}$ and, therefore, the total conductivity reduces to
$\sigma_{xy}\sim\sigma_{xy}^{\rm{I}}$. In the actual calculation, we replace
the derivative of the Fermi-Dirac distribution function with the Cauchy
distribution, assuming that the temperature is sufficiently low.

To convert Eq. (1) into a form convenient for KPM Weiße  _et al._ (2006), we
use the following Chebyshev expansion of local density of states;
$\mathrm{Tr}[\delta(x-\hat{H})]\cong\frac{1}{\pi\sqrt{1-x^{2}}}2\sum_{n=0}^{M_{c}}g_{n}h_{n}\mu_{n}T_{n}(x)$,
where $M_{c}$ is the expansion order, $h_{n}=1/(1+\delta_{n,0})$ is a
normalization factor and $\mu_{n}=\mathrm{Tr}[T_{n}(\hat{H})]$ is the
Chebyshev moment. Here, we introduce the Jackson kernel $g_{n}$ to alleviate
the Gibbs oscillations Jackson (1911, 1912); Weiße  _et al._ (2006); Wang  _et
al._ (2018).

Substituting the delta function in the trace directly into Eq. (1), we obtain
the following KPM formula for the conductivity,

$\begin{split}\sigma_{\mu\nu}^{\alpha}=\frac{e^{2}}{h}\frac{2\pi}{N}\sum_{m,n}h_{m}h_{n}g_{m}g_{n}C_{mn}^{\alpha}\mu_{mn,\mu\nu},\end{split}$
(2)

where $N$ is the number of sites, $C_{mn}^{\alpha}=\int\differential
x\differential
y\frac{F^{\alpha}(x,y)}{\pi\sqrt{1-x^{2}}\pi\sqrt{1-y^{2}}}T_{m}(x)T_{n}(y)$,
and
$\mu_{mn,\mu\nu}=\mathrm{Tr}[\tilde{v}_{\mu}T_{m}(\hat{H})\tilde{v}_{\nu}T_{n}(\hat{H})]$
is the two-dimensional Chebyshev moment. Here, we introduced $\tilde{v}=\hbar
v/a$. To reduce the numerical cost for the estimate of the trace, we exploit
the random vector approximation Weiße  _et al._ (2006). Note that
$C_{mn}^{\alpha}$ can be efficiently calculated by the Chebyshev Gauss
quadrature Weiße  _et al._ (2006); Wang  _et al._ (2018); Barros and Kato
(2013).

Results and Discussions. Using the model and method described above, we
compute the conductivity for various $\lambda_{\rm sk}$. Throughout this
paper, we set the system size $N=192^{2}$ and the number of random vectors
$R=100$. The tunable parameters are coupling constant $J$, damping rate
$\tau^{-1}$, and chemical potential $\mu$. $M_{c}=5000$ is enough to achieve
the convergence of $\sigma_{xy}$ for all $\tau^{-1}$ when the skyrmion size
$\lambda_{\rm sk}$ is smaller than 96.

Let us first discuss the $\mu$-dependence of the conductivity for the square
lattice. In Fig. 1, we show the THC for various $\lambda_{\rm sk}$ with
$J=1.0$ and $\tau^{-1}=0.1$. We compare the results with that for a free-
electron system under a uniform magnetic field $b=(e/h)/\lambda_{\rm sk}^{2}$
calculated with the linearized Boltzmann transport theory:

$\displaystyle\sigma_{xy}^{\rm
B}=\frac{e^{2}}{h}b\tau^{2}\sum_{\sigma}\sigma\int\frac{d^{d}k}{(2\pi)^{d}}(-\frac{df(\varepsilon_{\sigma})}{d\varepsilon})(v_{x\sigma}^{2}m_{yy\sigma}^{-1}-v_{x\sigma}v_{y\sigma}m_{xy\sigma}^{-1}),$
(3)

where $\sigma=1~{}(-1)$ for the majority (minority) spin, and
$v_{i\sigma}=\partial\varepsilon_{\sigma}/\partial k_{i}$,
$m_{ij\sigma}^{-1}=(1/\hbar^{2})\partial^{2}\varepsilon_{\sigma}/\partial
k_{i}\partial k_{j}$. This comparison indicates that the system can be mapped
to a free-electron system when $\lambda_{\rm sk}$ is larger than 16. However,
this real-space description is not valid for $\lambda_{\rm sk}<12$. In Fig. 1,
we compare $\sigma_{xy}$ and $\sigma_{xy}^{\rm{IIb}}$ for $\lambda_{\rm
sk}=4$. We see that the agreement is remarkably well, indicating that the main
contribution to the THE can be described in terms of $\Omega({\vec{k}})$
rather than $\Omega({\vec{R}})$.

Figure 1:  $\mu$-dependence of the topological Hall conductivity
($\sigma_{xy}$) for $\tau^{-1}=0.1$ in the unit of $e^{2}/h\cdot
b^{2}\tau^{2}$. For $\lambda_{\rm sk}>16$, the results show good agreement
with that for a free-electron system under a uniform magnetic field $b$. For
$\lambda_{\rm sk}=4$, $\sigma_{xy}^{\rm IIb}$ calculated from the $k$-space
Berry curvature $\Omega({\vec{k}})$ for occupied bands agrees well with
$\sigma_{xy}$.

To identify the optimum skyrmion size ($\lambda_{\rm sk}^{*}$) for the THC, we
investigate the $\lambda_{\rm sk}$-dependence of $\sigma_{xy}$. As shown in
Fig. 2, $\sigma_{xy}$ is not a monotonic function of $\lambda_{\rm sk}$ and
has a peak at $\lambda_{\rm sk}=\lambda_{\rm sk}^{*}$. While $\lambda_{\rm
sk}^{*}$ obviously depends on $\tau^{-1}$, its $\mu$-dependence is not so
significant. According to the real-space description Bruno  _et al._ (2004);
Ye  _et al._ (1999), the conductivity should be promotional to $1/\lambda_{\rm
sk}^{2}$ as the emergent field is stronger for denser skyrmion lattices
Nakazawa  _et al._ (2018). For smaller $\lambda_{\rm sk}$, on the contrary,
the conductivity increases as the skyrmion size becomes large. This is
consistent with the $k$-space description since $\sigma_{xy}$ increases with
the growing number of bands in the magnetic BZ Hamamoto  _et al._ (2015);
Göbel  _et al._ (2017). Such behaviors are observed recently for $J/t\ll 1$
and small $\lambda_{\rm sk}$ Wang  _et al._ (2020).

Figure 2:  $\lambda_{\rm sk}$-dependence of the conductivity for
$\tau^{-1}=0.08,~{}0.4,~{}1.2$. The chemical potential $\mu$ is $0.00$ and
$3.83$ for the left and right panels, respectively.  Figure 3: Optimum
$\lambda_{\rm sk}^{*}$ as a function of $\tau^{-1}$ for various $\mu$’s. The
left and right panels correspond to the square and triangular lattices,
respectively. The solid lines indicate $\lambda_{\rm sk}\propto\sqrt{\tau}$
and $\lambda_{\rm sk}\propto\tau$. Figure 4: (a) The $\lambda_{\rm
sk}$-dependence of the Hall conductivity $\sigma_{xy}$ (red circles) and
$\sigma_{xy}^{\rm IIb}$ (purple circles) for $\tau^{-1}=0.08$ (the upper four)
and for $\tau^{-1}=0.4$ (the lower four). From left to right, $J$ is
$3.0,~{}1.0,~{}0.5,$ and $0.3$, respectively. The blue, red and green solid
lines indicate $\lambda_{\rm sk}=v_{f}/J$, $\Delta=\tau^{-1}$, and
$\lambda_{\rm sk}=l$, respectively. (b) Five types of $\lambda_{\rm
sk}$-dependence of $\sigma_{xy}$ in the plane of $\tau^{-1}$ and $\lambda_{\rm
sk}$. The blue solid line ($\lambda_{\rm sk}=v_{f}/J$) and gray dotted line
($J=\tau^{-1}$) determine the boundary between the adiabatic and non-adiabatic
regime.

In Fig. 3, we plot $\lambda_{\rm sk}^{*}$ as a function of $\tau^{-1}$ for the
square lattice and triangular lattice. When the mean-free path $l$ is smaller
than $\lambda_{\rm sk}$, the real-space description is valid Nakazawa  _et
al._ (2018), which suggests $\lambda_{\rm sk}^{*}\propto\tau$. However, we
find $\lambda_{\rm sk}^{*}\propto\sqrt{\tau}$ for all $\mu$, which can be
explained by comparing $\tau^{-1}$ and averaged band separation $\Delta$ in
the magnetic BZ as follows. $\Delta$ can be estimated in analogy to the
formation of the Landau level Onoda  _et al._ (2004), namely, $\Delta\sim
N/\lambda_{\rm sk}^{2}/\mathrm{DOS}\sim t(a/\lambda_{\rm sk})^{2}$ with $N$
denoting the number of sites in the system. As we have mentioned, the
$k$-space description for $\sigma_{xy}$ is valid only when the typical energy
difference $\Delta$ is larger than $\tau^{-1}$. Thus, when $\Delta<\tau^{-1}$,
$\sigma_{xy}$ will not increase as a function of $\lambda_{\rm sk}$. Since
$\Delta\propto\lambda_{\rm sk}^{-2}$, we get the simple relation $\lambda_{\rm
sk}^{*}\propto\sqrt{\tau}$. We further note that we can also estimate
$\lambda_{\rm sk}^{*}$ from $(e\lambda_{\rm sk}^{2}/h)\sim\mu_{\rm{mob}}$,
where $\mu_{\mathrm{mob}}$ is the mobility of itinerant electrons. This
relation will be an useful indicator to see whether a real-space or $k$-space
description is more appropriate to understand the experimental data.

Up to this point, the coupling constant $J$ has been fixed to be $1.0$, for
which the itinerant electrons adiabatically interact with spins. However, for
small $J$, we can no longer map the system to a free-electron system under a
uniform magnetic field even for large $\lambda_{\rm sk}$ Metalidis and Bruno
(2006); Stern (1992). Besides, if the energy scale of $J$ is much smaller than
$\tau^{-1}$, the Karplus-Luttinger formula also becomes inaccurate even when
$\Delta$ is sufficiently smaller than $\tau^{-1}$. Thus, when the electron-
spin coupling is not adiabatic, both the real-space and $k$-space description
for $\sigma_{xy}$ discussed above lose its validity. Whether the electron-spin
coupling is adiabatic or not is determined by the time scale of $1/J$: When
the mean-free path ($l=v_{f}\tau$, where $v_{f}$ is the Fermi velocity) is
smaller than $\lambda_{\rm sk}$, $1/J$ must be shorter than $\tau$. Otherwise,
$1/J$ must be shorter than $\lambda_{\rm sk}/v_{f}$ Metalidis and Bruno
(2006).

In Fig. 4(a), keeping the condition $\Delta>\tau^{-1}$, we show how the
$\sigma_{xy}$-$\lambda_{\rm sk}$ curve changes in the non-adiabatic regime.
For the upper and lower four panels, $\tau^{-1}$ is set to be 0.08 and 0.4,
respectively. From left to right, $J$ is 3.0, 1.0, 0.5, and 0.2. Except for
the case of $J$=0.2 and $\tau^{-1}$=0.4 (the right bottom panel), the
condition $J>\tau^{-1}$ is satisfied. Thus, whether the system is adiabatic or
not is determined by the condition $\lambda_{\rm sk}=v_{f}/J$ indicated by the
blue solid lines. In the right bottom panel, the green line denotes
$\lambda_{\rm sk}=l$, i.e., whether the mean-free path is larger than the
skyrmion size.

We then compare the results in Fig. 4(a) with a simple formula derived from
the Drude theory for a free-electron system under an effective magnetic field
$b\propto 1/\lambda_{\rm sk}^{2}$: $\sigma=K/(\lambda_{\rm
sk}^{2}+\alpha^{2})$. Here, $K$ is determined so that $\sigma$ agrees with
$\sigma_{xy}^{\rm B}$ calculated from the linearized Boltzmann transport
theory for $\lambda_{\rm sk}\gg\alpha$, and $\alpha$ is tuned so that $\sigma$
takes its extreme value at the point where $\sigma^{\rm B}_{xy}$ and
$\sigma^{\rm IIb}_{xy}$ intersect. The result is shown with the solid black
line in each panel. When $\lambda_{\rm sk}$ is larger than $v_{f}/J$ or $l$,
the $\lambda_{\rm sk}$-dependence of $\sigma_{xy}$ is successfully reproduced
by the simple model. However, there are significant deviations for small
$\lambda_{\rm k}$ even in the adiabatic regime when the chemical potential
lies near the edge of the band (see the upper left panel of Fig. 4(a)).

Except for the right bottom panel, we see that the non-adiabatic regime
expands as $J$ decreases. It should be noted that even in the non-adiabatic
regime, the Karplus-Luttinger formula ($\sigma_{xy}^{\rm IIb}$, purple
circles) gives reliable results for small $\lambda_{\rm sk}$, as far as the
condition $\Delta>\tau^{-1}$ is satisfied. Interestingly, while the maximum
value of $\sigma_{xy}$ (red circles) is suppressed in the non-adiabatic
regime, $\lambda_{\rm sk}^{*}$ does not change drastically. Namely, for both
$\tau^{-1}=$0.08 and 0.4, $\lambda_{\rm sk}^{*}$ is always $\sim$ 10.

In Fig. 4(b), we summarize the different types of $\lambda_{\rm
sk}$-dependence in the plane of $\lambda_{\rm sk}$ and $\tau^{-1}$. Region I
and II are in the adiabatic regime, i.e., both the two adiabatic conditions
($J>1/\tau$ and $J>v_{f}/\lambda_{\rm sk}$) are satisfied. In region I and II,
the reap-space and $k$-space description for $\sigma_{xy}$ are valid,
respectively. The red line indicates the condition $\Delta=\tau^{-1}$ at which
$\sigma_{xy}$ takes its extreme value. Region III, IV, and V are in the non-
adiabatic regime. In region V, the $\lambda_{\rm sk}$-dependence of
$\sigma_{xy}$ is considerably suppressed, which is consistent with the
previous perturbative study Nakazawa and Kohno (2019a).

Conclusion. In the present study, we have investigated the $\lambda_{\rm
sk}$-dependence of the THE. While extensive analytical calculations have been
performed so far for different parameter regions, we have performed a large-
scale real-space numerical calculation. We determined the optimum
$\lambda_{\rm sk}^{*}$ for the THE and discussed how it depends on parameters
such as the electron-spin coupling $J$ and electron lifetime $\tau$. For the
$\tau$-dependence of $\lambda_{\rm sk}^{*}$, we found that $\lambda_{\rm
sk}^{*}$ is proportional to $\sqrt{\tau}$ but not to $\tau$ (i.e., mean-free
path). This behavior can be understood in terms of the band separation in the
magnetic BZ. For the $J$-dependence, we found that while the non-adiabticity
suppresses the THE considerablly, its effect on $\lambda^{*}_{\rm sk}$ is not
so significant. Therefore, the size of $\lambda^{*}_{\rm sk}$ is about ten
times the lattice constant in a wide range of $J$ and $\tau$.

Acknowledgements. We would like to thank K. Nakazawa for fruitful discussion.
This work was supported by a Grant-in-Aid for Scientific Research (No.
19K14654 and No. 19H05825), “Program for Promoting Researches on the
Supercomputer Fugaku” (Project ID: hp200132) from MEXT, JST-CREST
(JPMJCR18T3), JST-PRESTO (JPMJPR20L7) and JST-Mirai Program (JPMJMI20A1).

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# Improved Text Classification via Test-Time Augmentation

Helen Lu  Divya Shanmugam  Harini Suresh  John Guttag

###### Abstract

Test-time augmentation—the aggregation of predictions across transformed
examples of test inputs—is an established technique to improve the performance
of image classification models. Importantly, TTA can be used to improve model
performance post-hoc, without additional training. Although test-time
augmentation (TTA) can be applied to any data modality, it has seen limited
adoption in NLP due in part to the difficulty of identifying label-preserving
transformations. In this paper, we present augmentation policies that yield
significant accuracy improvements with language models. A key finding is that
augmentation policy design–for instance, the number of samples generated from
a single, non-deterministic augmentation–has a considerable impact on the
benefit of TTA. Experiments across a binary classification task and dataset
show that test-time augmentation can deliver consistent improvements over
current state-of-the-art approaches.

Machine Learning, Natural Language Processing, Data Augmentation

##  1 Introduction

Designing accurate and robust machine learning models for classification tasks
typically requires large labeled datasets. One standard practice for
artificially expanding a given dataset is data augmentation, in which new data
are generated by transforming existing examples. While data augmentation is
often performed during model training, recent work has shown that the use of
data augmentation at inference, or test-time augmentation (TTA), can improve
model accuracy [12; 25; 15], robustness [17; 24; 5], and uncertainty estimates
[2; 15; 23; 26]. TTA produces a final prediction by aggregating model
predictions (usually by averaging) made from multiple transformed versions of
a given test input. In addition to improved model performance, TTA is popular
because of its low implementation burden—predictions use a pre-trained model,
so no hyperparameter tuning is necessary, and inference on multiple examples
can be easily parallelized. Leveraged successfully, TTA can improve model
performance without requiring additional data or changes to the underlying
model.

Figure 1: Impact of different augmentation policies on binary classification
accuracy. The overall best-performing transformations are depicted and
compared in each policy configuration. The change in accuracy is displayed in
percentage points compared to the baseline, in which no augmentations were
applied to inputs fed to the trained model. Accuracy was calculated by
averaging across 10 random subsamples, each containing a random 80% of the
test inputs.

While TTA is commonly used in computer vision (CV), there is relatively little
research on the potential of TTA in natural language processing (NLP). One
reason for this is that it is not clear which augmentations to apply to text
inputs at test-time. CV enjoys a set of “standard” test-time augmentations,
including flips, crops, and scales, which are often assumed to be label
preserving. In contrast, label-preserving transformations to text are
frequently task-dependent, and as a result, the design of performant test-time
augmentation policies is not straightforward. A number of design choices,
including which augmentations to choose and how to aggregate the resulting
predictions, are involved in TTA. We focus on the first step–understanding
which augmentations to choose–in the context of natural language processing.

The goal of our work is to explore the use of TTA for improving NLP-based
classification First we establish four types of TTA policies, apply these to
text, and use a simple average to aggregate the resulting predictions. We then
use this method to improve accuracy on binary classification of the WILDS
CivilComments dataset. We empirically analyze the four different
configurations and propose optimal augmentation policies that most
significantly improve classification accuracy.

Our results establish that TTA can provide significant improvement to text
classification accuracy. We find that the scale of TTA accuracy increase is
heavily dependent on augmentation policy design. Figure 1 compares the effect
of four different augmentation policy configurations on classification
accuracy. We determine that configurations aggregating across multiple samples
from non-deterministic augmentations yields the greatest benefit and build
intuition as to why this design significantly outperforms configurations using
a single sample. We conclude with practical recommendations for the
application of TTA in NLP.

##  2 Related Work

Text augmentation at test-time has been applied to several tasks, including
abstractive summarization [8; 16; 32], question answering [13; 29; 22],
sequence tagging [7; 18; 6], parsing [10; 30; 1], grammatical error correction
[4; 31; 28], and neural machine translation [20; 8; 27]. Because our focus is
test-time augmentation for the purpose of improving language model
classification accuracy, we limit our discussion to work considering this
problem.

Our work focuses on augmentation policy design choices and why specific
policies outperform others.

##  3 Method

We borrow terminology from Shanmugam et al. [21] and assume two inputs to our
method:

  1. 1.

A pre-trained black-box classifier $f:X\rightarrow R^{C}$ that maps text
inputs to a vector of class logit predictions. We use $X$ to denote the space
of inputs on which the classifier can operate and $C$ to denote the number of
classes. We assume that $f$ is not fully invariant with respect to
augmentations (i.e., predictions on transformations of a test input are not
guaranteed to be the same).

  2. 2.

A list of $M$ augmentation transforms, $[a_{1},…a_{m}]$ which we deem an
augmentation policy $A$. Each transform $a_{m}:X\rightarrow X$ is a stochastic
transform that modifies the given input at a character- or word-level based on
parameters defining the probability the transform is applied and the magnitude
of transformation (e.g. percent of words changed, percent of characters
changed), where larger magnitudes translate to larger changes to the example.

Given a text input $t$, we apply an augmentation policy $A$ containing $M$
augmentation transforms to generate $M$ transformed inputs. All $M+1$ inputs
(transformed and the original) are then passed into the pre-trained classifier
$f$ to generate $(M+1)$ $R^{C}$ vectors containing the class logit
predictions. We then generate a single prediction by applying a simple average
to the $M+1$ logit predictions. While there are more complex means of
aggregating predictions that may maximize prediction performance, we choose to
use averaging because it is the simplest version of TTA and suits our goal of
understanding the baseline value of TTA in NLP.

##  4 Experimental Set-Up

We evaluate the performance of our method across a dataset and model
architecture laid out in the following sections.

#### Datasets

The WILDS CivilComments dataset [11], a modification of the original
CivilComments dataset [3], consists of 448,000 comments made on online
articles. Each comment is labeled as either toxic or non-toxic using a binary
indicator. Further details on the dataset can be found in Appendix A. We use
the pre-defined test set to report accuracy because the default TTA approach
to aggregation (i.e. averaging) requires no training data, and assumes access
to a pre-trained model.

#### Augmentation Policies

We consider two classes of augmentation policies from the nlpaug library,
consisting of fifteen different augmentation transformations. Insertion,
deletion, and substitution of randomly-selected characters are categorized as
character-based and of randomly-selected words as word-based.

For each class of augmentation policies, we chose parameters that maximized
accuracy on a held-out validation set. Character-based augmentation policies
transformed a random character from 10% of randomly-selected words of an
input, with the constraint that any selected word less than 4 characters long
would remain unmodified. Word-based augmentation policies modified one random
word from a given input, and sentence-based augmentation policies use the
default parameters of the nlpaug library.

#### Models

For all experiments, we used a pre-trained network made available by Koh et
al.[2021]. Hyperparameters and training details can be found in Appendix B.
The predictions generated from these models on unaugmented test inputs served
as our baseline accuracy.

#### Augmentation Configurations

Augmentation policies can vary in terms of the type of augmentations, and the
number of samples drawn from a particular augmentation. Using these two
dimensions, we constructed four representative test-time augmentation policy
configurations and compared their effects on classification accuracy. The
configurations are as follows:

1-sample, 1-augmentation: Averages predictions over the original, unaugmented
input and predictions generated following augmentation policy $A=[a_{m}]$.

1-sample, 4-augmentations: Averages predictions over the original, unaugmented
input and four prediction vectors, each generated by a different augmentation
transform in policy $A=[a_{m},a_{n},a_{p},a_{q}]$. All transforms were of the
same class (i.e. either all word-based or all character-based).

4-samples, 1-augmentation: Averages predictions over the original, unaugmented
input and four prediction vectors, each generated by the same augmentation
transform $A=[a_{m},a_{m},a_{m},a_{m}]$.

4-samples, 4-augmentations: Averages of the class logit prediction vector of
the original, unaugmented input and sixteen class logit prediction vectors:
four samples generated by each of four different augmentation transforms
$a_{m}$ , $a_{n}$ , $a_{p}$, and $a_{q}$. All transforms were of the same
class (i.e. either all word-based or all character-based).

Since each augmentation transform (e.g., word deletion) is stochastic,
multiple augmented samples were created. Specifically four samples were
generated to maximize accuracy based on analysis with a held-out validation
set. Policies aggregating four distinct augmentation transforms were designed
to complement the optimal four sample configurations outlined above.

#### Statistical Significance

We use a pairwise t-test to measure the statistical significance (with respect
to the baseline model) of our accuracy results, which were obtained by
averaging 10 random subsamples containing 80% of the test set.

##  5 Results

Our results demonstrate that the multi-sample single augmentation
configuration performs best. We walk through each configuration’s results in
detail below. We focus on word-based augmentation transforms in the following
sections after finding that character-based augmentation transforms were
consistently unsuccessful in yielding accuracy improvement.

For the 1-sample, 1-augmentation configuration, only the augmentation policy
using the transform ppdb_synonym (substitution of randomly chosen words with a
synonym determined by the PPDB database) outperformed the baseline model’s
classifications on average, as shown in Figure 1. However, the increase in
accuracy was found to be statistically insignificant.

For the 4-samples, 1-augmentation configuration, all nine word-based policies
yielded statistically-significant improvements to classification accuracy,
from 0.6% to 0.9%. (Figure 1).

For the 1-sample, 4-augmentations configuration, four augmentation policies
slightly outperformed the baseline model’s classifications, but not
significantly (Figure 1). All four augmentation policies consisted of four
distinct word-based transforms.

For the 4-samples, 4-augmentations configuration, all 126 augmentation
policies–combinations of four out of nine word-based augmentation
transforms–significantly outperformed the baseline model’s classifications,
yielding improvements from 0.8% to 1.0% (Figure 1).

###  5.1 Comparing 4-Sample and 1-Sample Policies

Our results show that multi-sample single-augmentation TTA configurations
offer significant improvement in accuracy, while single-sample single-
augmentation configurations do not. This is surprising, because any individual
augmentation _decreases_ the pre-trained model’s classification accuracy on
average (Fig. 2), where each augmentation introduces more corruptions (red)
than corrections (green). Here, we explore the mechanism that drives this
difference in performance.

We first examined the count of corrections (i.e. inputs initially
misclassified by the pre-trained model but correctly classified after applying
an augmentation transform) and corruptions (i.e. inputs initially correctly
classified by the pre-trained model but misclassified after applying an
augmentation transform). Figure 2 shows that that the 4-sample, 1-augmentation
policy changes many more examples than the 1-sample policy. Importantly, it
consistently creates more corrections than corruptions, leading to an overall
gain in accuracy.

Figure 2: Corrections and corruptions of 1-sample, 1-augmentation and
4-samples, 1-augmentation policies. The change in accuracy is in percentage
points compared to the baseline. Accuracy was calculated by averaging across
10 samples, each containing a random 80% of the test inputs.

Why do multiple samples lead to more corrections than corruptions? We
hypothesized that for corrections, there may be a greater overlap between
individual samples — i.e., when an augmentation improves the model’s
prediction, multiple samples are likely to agree, and their average will
result in a strong correction. On the other hand, corruptions may be more
randomly distributed — and averaging across them washes out any individual
decrease in accuracy. We tested this hypothesis by calculating how often
sample predictions overlapped for corrections vs corruptions. In line with our
hypothesis, we found that for corrected examples, there was an average overlap
of around 0.51 across samples, as compared to only 0.23 for corrupted examples
(see Appendix C for supporting figure).

##  6 Discussion

In this paper, we investigate accuracy improvements when TTA is combined with
a large pre-trained language model. Through an analysis of the WILDS
CivilComments dataset, we show that TTA can introduce significant improvements
in classification accuracy.

Our results show that policy design significantly influences accuracy
improvements of TTA. Fixing the number of augmentations in the policy, we
compare results using two approaches: one which produces multiple samples from
a single augmentation, and another which uses single samples from different
augmentations. We find that drawing multiple samples from a single transform
produces significantly larger improvements in accuracy: from 0.06% to 1%.

A natural extension to the multi-sample single-augmentation approach is to
replicate it with multiple augmentations. This produces the multi-sample
multi-augmentation configuration and we find that it yields a marginal, often
insignificant benefit (at most a .04% improvement in accuracy). Given how the
multi-augmentation multi-sample configuration comes with the computational
cost of many more predictions at test-time, this suggests that multi-sample
single-augmentation approaches offer a better tradeoff between computational
efficiency and accuracy.

We also observe that word-based transforms consistently outperform character-
based transforms. This is likely due to how word-based transforms modify
inputs. The best-performing word-based transforms incorporated semantic
meaning via predefined libraries (WordNet Synonym, PPDB Synonym) or embedding
models. It is likely that the alterations made by character-based transforms,
on the other hand, are altering fundamental word stems. These stems are often
used by natural language models to encode words (and their meanings).
Modifying a single character may generate a new stem with unrelated (or
opposite) meaning, resulting in misclassification of the transformed input.

The accuracy improvements generated from applying TTA are significant for
three key reasons. First, TTA can outperform a model fine-tuned to perform
well on test data. Despite the fact that the original accuracy is 92.3%, TTA
can offer improvements of up to 93.7%. Secondly, the baseline model was
trained without training-time augmentation [11]. This indicates that TTA may
have even greater potential in NLP contexts than in CV, where augmentations
used in test-time often parallel those used by the model in train-time.
Thirdly, our approach yielded improvements even when using a simple
aggregation scheme of averaging. It is likely that, similar to computer
vision, greater improvements are possible by weighting augmentations
differently.

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##  Appendix A CivilComments Dataset

The WILDS CivilComments dataset (Koh et. al 2021), a modification of the
original CivilComments dataset (Borkan et. al., 2019), consists of 448,000
comments made on online articles. Ranging from word lengths of 1 to 315, the
comments are labeled as either toxic or non-toxic using a binary indicator
obtained from crowdsourcing and majority voting among at least 10
crowdworkers. Each comment is also annotated with binary indicators regarding
content mention of nine demographic identities (male, female, LGBTQ,
Christian, Muslim, other religions, black, white, other) and six
characteristics (severely toxic, obscene, threatening, insulting, identity
attacking, and sexually explicit). See Table 1 for examples of a toxic and
non-toxic comment.

Table 1: Sample of Toxic and Non-Toxic Comment from the WILDS CivilComments
dataset.
Toxicity |  Comment |  Male |  Female |  … |  Identity Attack
---|---|---|---|---|---
0 |  No, he was accused of being a racist white man. |  1 |  0 |  … |  0
1 |  The truth hurts so you have to reply with your idiotic comments. You really do not realize how ignorant you are. |  0 |  0 |  … |  0

##  Appendix B Model Hyperparameter Settings

For all experiments, we used a state-of-the-art classifier developed for the
WILDS CivilComments. We downloaded the pre-trained models, which were fine-
tuned DistilBERT-base-uncased models using implementation from Sanh et al.[19]
and hyperparameter settings from Koh et al [11]: batch size $16$; learning
rate $10^{-5}$ using the AdamW optimizer [14] for $5$ epochs with early
stopping; an L2-regularization strength of $10^{-2}$; a maximum number of
tokens of $300$; a learning rate chosen through a grid search over
${10^{-6},2\times 10^{-6},10^{-6},2\times 10^{-5}}$; and default values for
all other hyperparameters.

##  Appendix C 4-Sample Corrections Corruptions Analysis

Figure 3: Fractional overlap of corrections and corruptions across pairs of
samples generated by applying the 4-sample, 1-augmentation policy with the
Synonym (PPDB) transform. For two samples (without loss of generality, Sample
1 and Sample 2), overlap was calculated by taking the results from dividing
the number of inputs that were corrected/corrupted by both Sample 1 and Sample
2 by the total unique inputs that were corrected/corrupted by either Sample 1
or Sample 2. The Synonym (PPDB) transform was selected without loss of
generality.


Improving the bidirectional steerability between two accelerated partners via
filtering process

M.Y.Abd-Rabboua 111e-mail<EMAIL_ADDRESS>N.
<EMAIL_ADDRESS>M. M. A. Ahmed a, and A.-S. F. Obada
a

aMathematics Department, Faculty of Science, Al-Azhar University, Nasr City
11884, Cairo, Egypt
bMath. Dept., College of Science, University of Bahrain, Bahrain.

cDepartment of Mathematics, Aswan University Aswan, Sahari 81528, Egypt.

###### Abstract

The bidirectional steering between two accelerated partners sharing initially
different classes of entangled states is discussed. Due to the decoherence,
the steerability and its degree decrease either as the acceleration increases
or the partners share initially a small amount of quantum correlations. The
possibility of increasing the steerability is investigated by applying the
filtering process. Our results show that by increasing the filtering strength,
one can improve the upper bounds of the steerability and the range of
acceleration at which the steerability is possible. Steering large coherent
states is much better than steering less coherent ones.

Keywords:Steering; Accelerated system; Steerability; Decoherence.

##  1 Introduction

The concept of Einstein–Podolsky–Rosen (EPR) steering is suggested by
Schrödinger in 1935 to discuss the EPR paradox [1, 2]. EPR steering is a
significant quantum phenomenon in which a quantum state can be non-locally
changed or steered another state remotely by performing some local
measurements [3]. Besides, the violation of EPR steering violation has been
set as a measure of quantum correlations, which is laid out between the non-
separability and Bell non-locality [4, 5]. Several experimental studies deal
with interpreting the dynamics of quantum steering, such as; EPR steering game
employing an all-versus-nothing criterion that depends on obtaining two
different pure normalized conditional states has been explored [6].
Measurement-device-independent steering protocols for optical polarization
qubits and states that do not violate Bell inequality have been demonstrated
[7]. By polarizing the photons in a linear-optical setup, the temporal quantum
steering has been applied for testing and securing the quantum communications
[8]. The convex steering monotone and measure of steerability depend on the
observed statistics and the quantum inputs have been discovered [9]. Quantum
steering non-locality of high-dimensional quantum systems with isotropic noise
fraction has been retrieved [10]. Theoretically, the steering inequality and
steerability have been reconstructed by using different relations as
Heisenberg uncertainty principle [11], the standard geometric Bell
inequalities [12, 13], a version of the Clauser-Horne-Shimony-Holt inequality
[14, 15], fine-grained uncertainty relation[16], Abner Shimony inequalities
[17], and steering witnesses[18]. The violating of EPR inequalities indicates
the possibility of steering process [19, 20].

Due to the importance of steering in determining the quantum correlation, it
has been used in many practical applications of quantum informatics. For
example, it has been applied to investigate the security and feasibility of a
one-sided device of the standard quantum key distribution [21]. It also has
been employed in quantum computation [22] and securing quantum teleportation
[23]. Moreover, quantum steering has been discussed for some quantum systems
such as, bipartite two-qubit X-state [24, 25], Heisenberg chain models [26],
two-level or three-level detectors [27]. As quantum systems inevitably
interact with the different environments, it is necessary to discuss the
influence of these environments on the steering process. For instance, the
effect of asymmetric dissipation relativistic motion [28], finite temperature
[29], non-Markovian environment on the steering [30], and noise channel [31]
have been discussed. The steering entropic uncertainty of the qutrit system
under amplitude damping decoherence has been examined [32].

In addition, the possibility of improving of quantum correlations via local
filtering operation has been widely investigated, such as, improving the
steering and nonlocality in Heisenberg XY mode [33], also protecting
entanglement from the decoherence due to amplitude damping and
acceleration[34, 35]. On the other hand, investigating the quantum correlation
in an accelerated frame is one of the considerable areas in theoretical
quantum processing, where the actual quantum systems are essentially non-
inertial [36, 37]. The acceleration process may cause a dissipation of quantum
correlations between an inertial observer and an accelerated one [38]. Under
the non-inertial frame, the entanglement of a general two-qubit system has
been studied [39]. The classicality and quantumness under different
decoherence noisy channels in accelerated frame were investigated in [40]. The
general behaviour of EPR steering inequality and steerability between two
users have been investigated, where the first in flat space-time and another
in the non-inertial frame [41].

Our motivation in this work is to derive a generic version of the steering
inequality for two quantum subsystems with two dimensions (two-qubit). We then
apply the new inequality on a general form of a two-qubit system, as it is
possible to generate some special classes of them [42]. We assume that the
actual quantum regimes are essentially in the non-inertial frame, where the
relativistic framework is taken into account, whether we accelerate only one
subsystem or accelerate the two subsystems collectively.

The paper is organized as follows: In Sec.2, we derive a general form of
steering inequality for a bipartite qubit system. In Sec. (3) we are defined
the filtering process of the two-qubit state to improve the efficiency of
shared entangled state. In Sec. (4) the EPR steering under the acceleration
process of the generalized Werner state and the generic pure state has
proposed. Additionally, the degree of steerability is enhanced via using the
local filtering operation. Bidirectional steerability for the generic pure
state from Alice to Bob is studied in Sec.5, where Alice’s particle is
accelerated and Bob’s particle is at rest. Finally, we conclude our results in
Sec.(6).

##  2 Entropic Steering:

In this section, we derive a generalized form of quantum steering for a
bipartite system. According to the definition in [11], the entropic
uncertainty steering inequality for the discrete observable in the even
N-dimensional Hilbert space is defined as,

$\sum_{i=1}^{N+1}H(R_{b_{i}}|R_{a_{i}})\geq\frac{N}{2}Log_{2}(\frac{N}{2})+(1+\frac{N}{2})Log_{2}(1+\frac{N}{2}),$
(1)

where $H(R_{b}|R_{a})=H(\rho_{ab})-H(\rho_{a})$ is the conditional Shannon
entropy for the subsystems $a$ and $b$. By applying the Pauli spin operator
($\sigma_{x},\sigma_{y},\sigma_{z}$) as measurements, the EPR inequality of
steering from A to B is given by,

$\mathcal{I}_{ab}=H(\sigma_{x}^{(b)}|\sigma_{x}^{(a)})+H(\sigma_{y}^{(b)}|\sigma_{y}^{(a)})+H(\sigma_{z}^{(b)}|\sigma_{z}^{(a)}\geq
2.$ (2)

In the computational basis $|00\rangle,\ |01\rangle,\ |10\rangle$, and
$|11\rangle$, the general form of an arbitrary two-qubit system may be written
as:

$\hat{\rho}_{ab}=\begin{pmatrix}\rho_{11}&\rho_{12}&\rho_{13}&\rho_{14}\\\
\rho_{12}^{*}&\rho_{22}&\rho_{23}&\rho_{24}\\\
\rho_{13}^{*}&\rho_{23}^{*}&\rho_{33}&\rho_{34}\\\
\rho_{14}^{*}&\rho_{24}^{*}&\rho_{34}^{*}&\rho_{44}\end{pmatrix},$ (3)

By using Pauli matrices $\sigma_{x}$, $\sigma_{y}$, and $\sigma_{z}$
measurements, one can simplify the expression of EPR steering inequality in
Eq.(2) reads,

$\begin{split}\mathcal{I}_{ab}=&\frac{1}{2}\sum_{i=1}^{4}\bigg{\\{}P^{ab}_{x_{i}}\log_{2}[P^{ab}_{x_{i}}]+P^{ab}_{y_{i}}\log_{2}[P^{ab}_{y_{i}}]+P^{ab}_{z_{i}}\log_{2}[P^{ab}_{z_{i}}]\bigg{\\}}\\\
&-\sum_{i=1}^{2}\bigg{\\{}P^{a}_{x_{i}}\log_{2}[P^{a}_{x_{i}}]+P^{a}_{y_{i}}\log_{2}[P^{a}_{y_{i}}]+P^{a}_{z_{i}}\log_{2}[P^{a}_{z_{i}}]\bigg{\\}}\end{split},$
(4)

where $P^{ab}_{x_{i}},\ P^{ab}_{y_{i}}$, and $P^{ab}_{z_{i}}$ represent the
eigenvalues of the density operator $\hat{\rho}_{ab}$. Explicitly, they are
given by

$\begin{split}&P^{ab}_{x_{1}}=(1+2\
Re[\rho_{12}+\rho_{13}+\rho_{14}+\rho_{23}+\rho_{24}+\rho_{34}]),\\\
&P^{ab}_{x_{2}}=(1-2\
Re[\rho_{12}+\rho_{13}-\rho_{14}-\rho_{23}+\rho_{24}+\rho_{34}]),\\\
&P^{ab}_{x_{3}}=(1-2\
Re[\rho_{12}-\rho_{13}+\rho_{14}+\rho_{23}-\rho_{24}+\rho_{34}]),\\\
&P^{ab}_{x_{4}}=(1+2\
Re[\rho_{12}-\rho_{13}-\rho_{14}-\rho_{23}-\rho_{24}+\rho_{34}]),\\\
&P^{ab}_{y_{1},y_{2}}=(1+2\ Re[\rho_{23}-\rho_{14}]\pm 2\
Im[\rho_{12}+\rho_{13}+\rho_{24}+\rho_{34}]),\\\ &P^{ab}_{y_{3},y_{4}}=(1-2\
Re[\rho_{23}-\rho_{14}]\pm 2\ Im[\rho_{12}-\rho_{13}-\rho_{24}+\rho_{34}]),\\\
&P^{ab}_{z_{i}}=4\rho_{ii}.\end{split}$ (5)

Likewise, $P^{a}_{x_{i}},\ P^{a}_{y_{i}}$, and $P^{a}_{z_{i}}$ are the
eigenvalues of the reduced density operator $\hat{\rho}_{A}$, where,

$\begin{split}&P^{a}_{x_{1},x_{2}}=(1\pm 2\ Re(\rho_{13}+\rho_{24})),\quad
P^{a}_{y_{1},y_{2}}=(1\pm 2\ Im(\rho_{13}+\rho_{24})),\\\ &\text{and}\quad
P^{a}_{z_{1},z_{2}}=(1\pm(\rho_{11}+\rho_{22}-\rho_{33}-\rho_{44})).\end{split}$
(6)

The degree of steerability is quantified based on Alice’s measurements as
follows [22],

$\mathcal{S}^{A\longrightarrow
B}=\max\bigg{\\{}0,\frac{\mathcal{I}_{ab}-2}{\mathcal{I}_{max}-2}\bigg{\\}},$
(7)

where $\mathcal{I}_{max}$=6 is calculated for a system initially prepared in
Bell states. The factor $(\mathcal{I}_{max}-2)$ is introduced to ensure that
the steering process is normalized. By exchanging the roles of A and B, the
possibility of the steering by performing measurements on the subsystem B is
given by,

$\mathcal{S}^{B\longrightarrow
A}=\max\bigg{\\{}0,\frac{I_{ba}-2}{I_{max}-2}\bigg{\\}},$ (8)

where $\mathcal{I}_{ba}$ quantifies the steering from Bob to Alice, it is
given by,

$\displaystyle\mathcal{I}_{ba}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\sum_{i=1}^{4}\bigg{\\{}P^{ab}_{x_{i}}\log_{2}[P^{ab}_{x_{i}}]+P^{ab}_{y_{i}}\log_{2}[P^{ab}_{y_{i}}]+P^{ab}_{z_{i}}\log_{2}[P^{ab}_{z_{i}}]\bigg{\\}}$
(9) $\displaystyle-$
$\displaystyle\sum_{i=1}^{2}\bigg{\\{}P^{b}_{x_{i}}\log_{2}[P^{b}_{x_{i}}]+P^{b}_{y_{i}}\log_{2}[P^{b}_{y_{i}}]+P^{b}_{z_{i}}\log_{2}[P^{b}_{z_{i}}]\bigg{\\}},$

with,

$\displaystyle P^{b}_{x_{1},x_{2}}$ $\displaystyle=$ $\displaystyle(1\pm 2\
Re(\rho_{12}+\rho_{34})),\quad P^{b}_{y_{1},y_{2}}=(1\pm 2\
Im(\rho_{12}+\rho_{34})),$ $\displaystyle\quad P^{b}_{z_{1},z_{2}}$
$\displaystyle=$
$\displaystyle(1\pm(\rho_{11}-\rho_{22}+\rho_{33}-\rho_{44})).$ (10)

Hereinafter, we shall discuss the Unruh effect on the steering process of the
system $\hat{\rho_{ab}}$ and the possibility of improving this process via
local filter operations.

##  3 Filter Process:

Filtering process is a method that may be used to improve the efficiency of a
shared entangled state between two users Alice and Bob to perform some quantum
information tasks. As we shall see in Sec.4, the acceleration process
decreases the efficiency of steerability, therefore the users need to apply
the filtering process on the accelerated shared state to improve its
efficiency. The final output filtered state $\hat{\rho}_{ab}^{\mathcal{F}}$ is
given by [43],

$\hat{\rho}_{ab}^{\mathcal{F}}=\frac{1}{\mathcal{N}}\mathcal{W}_{a}\mathcal{W}_{b}.\hat{\rho}_{ab}.\big{(}\mathcal{W}_{a}\mathcal{W}_{b}\big{)}^{\dagger},$
(11)

with
$\mathcal{N}=Tr[\mathcal{W}_{a}\mathcal{W}_{b}.\hat{\rho}_{ab}.\big{(}\mathcal{W}_{a}\mathcal{W}_{b}\big{)}^{\dagger}]$
and,

$\mathcal{W}_{i}=diag\\{\sqrt{\alpha_{i}},\sqrt{1-\alpha_{i}}\\},\quad i=a\
\text{or}\ b.$ (12)

The parameters $\alpha_{i}$ are the strengths of the filtering process. After
applying the operator $\mathcal{W}_{i}$, the final output state is given by,

$\hat{\rho}_{ab}^{\mathcal{F}}=\frac{1}{\mathcal{N}}\begin{pmatrix}\rho_{11}\alpha^{2}&\rho_{12}\sqrt{1-\alpha}\alpha^{\frac{3}{2}}&\rho_{13}\sqrt{1-\alpha}\alpha^{\frac{3}{2}}&\rho_{14}\alpha(1-\alpha)\\\
\rho_{12}^{*}\sqrt{1-\alpha}\alpha^{\frac{3}{2}}&\rho_{22}\alpha(1-\alpha)&\rho_{23}\alpha(1-\alpha)&\rho_{24}\sqrt{\alpha(1-\alpha)^{3}}\\\
\rho_{13}^{*}\sqrt{1-\alpha}\alpha^{\frac{3}{2}}&\rho_{23}^{*}\alpha(1-\alpha)&\rho_{33}\alpha(1-\alpha)&\rho_{34}\sqrt{\alpha(1-\alpha)^{3}}\\\
\rho_{14}^{*}\alpha(1-\alpha)&\rho_{24}^{*}\sqrt{\alpha(1-\alpha)^{3}}&\rho_{34}^{*}\sqrt{\alpha(1-\alpha)^{3}}&\rho_{44}(1-\alpha)^{2}\end{pmatrix},$
(13)

where it is assumed that, the strength parameters of the filtering process are
equal.

##  4 EPR steering under the Unruh thermal framework:

Let us consider that, the users Alice and Bob share a general two-qubit system
on the following form,

$\hat{\rho}_{ab}=\frac{1}{4}\bigg{(}I^{(a)}_{2}\otimes
I^{(b)}_{2}+\vec{s}.\sigma\otimes
I^{(b)}_{2}+I^{(a)}_{2}\otimes\vec{t}.\tau+\sum_{i,j}^{3}c_{ij}\sigma_{i}\otimes\tau_{j}\bigg{)},i,j=x,y,z,$
(14)

where $I_{2}^{a(b)}$ represents the identity operators of Alice’s (a) and
Bob’s (b) qubits. The vectors $\vec{s}=(s_{x}~{},s_{y},~{}s_{z})$, and
$\vec{t}=(t_{x},t_{y},t_{z})$ are the Bloch vectors of Alice’s and Bob’s
qubit, respectively. The operators $\sigma_{i},\tau_{i},i=1..3$, are the
standard Pauli spin operators, where
$\sigma_{x}=\tau_{x}=\bigl{|}0\bigr{\rangle}\bigl{\langle}1\bigr{|}+\bigl{|}1\bigr{\rangle}\bigl{\langle}0\bigr{|}$,
$\sigma_{y}=\tau_{y}=i(\bigl{|}0\bigr{\rangle}\bigl{\langle}1\bigr{|}-\bigl{|}1\bigr{\rangle}\bigl{\langle}0\bigr{|})$
and
$\sigma_{z}=\tau_{z}=\bigl{|}0\bigr{\rangle}\bigl{\langle}0\bigr{|}-\bigl{|}1\bigr{\rangle}\bigl{\langle}1\bigr{|}$.
The correlation matrix (dyadic), $c_{ij}$ is a $3\times 3$ matrix, where its
elements are defined by
$c_{ij}=Tr\\{\sigma_{i}\tau_{j}\rho_{ab}\\},~{}i,j=1,2,3$.

Since our aim is to discuss the phenomena of the quantum steering by using an
accelerated state between Alice and Bob, then it is important to review Unruh
effect on Minkowski’s particles. However, in the inertial frames, Minkowsik
coordinates $(t,x)$ are used to describe the dynamics of these particles,
while in the non-inertial frames Rindler coordinates $(\zeta,\chi)$ are used.
The relations between the Minkowski and Rindler’s spaces are given by,

$\zeta=r~{}tanh\left(\frac{t}{x}\right),\quad\chi=\sqrt{x^{2}-t^{2}},$ (15)

where $-\infty<\zeta<\infty$, $-\infty<\chi<\infty$ and $r$ is the
acceleration of the moving particle. The transformation (15) defines two
regions in Rindler’s space: the first region $I$ for $|t|<x$ and the second
region $II$ for $x<-|t|$.

A single mode $k$ of fermions and anti-fermions in Minkowski space is
described by the annihilation operators $a_{k}$ and $b_{-k}$ respectively,
where $a_{k}\bigl{|}0_{k}\bigr{\rangle}=0$ and
$b^{\dagger}_{-k}\bigl{|}0_{k}\bigr{\rangle}=0$. Based on the approximation by
Bruschi et. al [36, 44], the relations between Minkowski and Rindler’s
operators are given by Bogoliubov transformation,

$a_{k}=\cos rc^{(I)}_{k}-\exp(-i\phi)\sin rd^{(II)}_{-k},\quad
b^{\dagger}_{-k}=\exp(i\phi)\sin rc^{(I)}_{k}+\cos rd^{(II)}_{k},$ (16)

where $c^{(I)}_{k},d^{(II)}_{-k}$ are the annihilation operators of Rindler’s
space in the regions $I$ ( for fermions) and $II$ (for anti-fermions)
respectively, $tanr=e^{-\pi\omega\frac{c}{a}}$, $0\leq r\leq\pi/$4, $a$ is the
acceleration such that $0\leq a\leq\infty$, $\omega$ is the frequency of the
travelling qubits, $c$ is the speed of light and $\phi$ is an unimportant
phase that can be absorbed into the definition of the operators [45, 46]. The
transformation (16) mixes a particle in first region $I$ and an anti particle
in the second region $II$[47]. In terms of Rindler’s modes, the Minkowski
vacuum $\bigl{|}0_{k}\bigr{\rangle}_{M}$ and the one particle state
$\bigl{|}1_{k}\bigr{\rangle}_{M}$ take the forms,

$\displaystyle\bigl{|}0_{k}\bigr{\rangle}_{M}$ $\displaystyle=$
$\displaystyle\cos
r\bigl{|}0_{k}\bigr{\rangle}_{I}\bigl{|}0_{-k}\bigr{\rangle}_{II}+\sin
r\bigl{|}1_{k}\bigr{\rangle}_{I}\bigl{|}1_{-k}\bigr{\rangle}_{II},$
$\displaystyle\bigl{|}1_{k}\bigr{\rangle}_{M}$ $\displaystyle=$ $\displaystyle
a^{\dagger}_{k}\bigl{|}0_{k}\bigr{\rangle}_{M}=\bigl{|}1_{k}\bigr{\rangle}_{I}\bigl{|}0_{k}\bigr{\rangle}_{II}.$
(17)

Then, by using these transformations and after tracing out the degree of
freedom in the second region $II$, the initial state (3) may be written as,

$\hat{\rho}_{ab}^{acc}=\begin{pmatrix}\mathcal{C}^{2}_{a}\mathcal{C}^{2}_{b}\rho_{11}&\mathcal{C}^{2}_{a}\mathcal{C}_{b}\rho_{12}&\mathcal{C}_{a}\mathcal{C}^{2}_{b}\rho_{13}&\mathcal{C}_{a}\mathcal{C}_{b}\rho_{14}\\\
\mathcal{C}^{2}_{a}\mathcal{C}_{b}\rho_{12}^{*}&\mathcal{C}^{2}_{a}\big{(}\rho_{22}+S^{2}_{b}\rho_{11}\big{)}&\mathcal{C}_{a}\mathcal{C}_{b}\rho_{23}&\mathcal{C}_{a}\big{(}\rho_{24}+S^{2}_{b}\rho_{13}\big{)}\\\
\mathcal{C}_{a}\mathcal{C}^{2}_{b}\rho_{13}^{*}&\mathcal{C}_{a}\mathcal{C}_{b}\rho_{23}^{*}&\mathcal{C}^{2}_{b}\big{(}\rho_{33}+S^{2}_{a}\rho_{11}\big{)}&\mathcal{C}_{b}\big{(}\rho_{34}+S^{2}_{a}\rho_{12}\big{)}\\\
\mathcal{C}_{a}\mathcal{C}_{b}\rho_{14}^{*}&\mathcal{C}_{a}\big{(}\rho_{24}^{*}+S^{2}_{b}\rho_{13}^{*}\big{)}&\mathcal{C}_{b}\big{(}\rho_{34}^{*}+S^{2}_{a}\rho_{12}^{*}\big{)}&S^{2}_{a}\big{(}\rho_{22}+S^{2}_{b}\rho_{11}\big{)}+S^{2}_{b}\rho_{33}+\rho_{44}\end{pmatrix},$
(18)

where $\mathcal{C}_{i}=\cos r_{i}$, and $S_{i}=\sin r_{i},i=a,b$.

Now, we are ready to investigate the steerability process between Alice and
Bob who share the accelerated state. This idea will be clarified by using two
different initial states namely; a generalized Werner state and a generic pure
state.

###  4.1 Generalized Werner state:

This state is obtained by setting $\vec{r}=\vec{0}=\vec{s}$, and the cross
dyadic $C$ is defined by a diagonalize $3\times 3$ matrix, where
$\\{c_{ij}\\}=diag\\{c_{11},c_{22},c_{33}\\}$. Then, Eq. (14) reduces to be;

$\hat{\rho}_{ab}=\frac{1}{4}\bigg{(}I^{(a)}_{2}\otimes
I^{(b)}_{2}+\sum_{i,j}^{3}c_{ij}\sigma_{i}\otimes\tau_{j}\bigg{)},i,j=x,y,z.$
(19)

Hence, the output accelerated system according to Eq.(18) is given by,

$\begin{split}\hat{\rho}_{ab}^{acc}=&\mathcal{C}^{2}_{a}\mathcal{C}^{2}_{b}\mathcal{A}_{11}|00\rangle\langle
00|+\mathcal{C}^{2}_{a}\big{(}\mathcal{A}_{22}+S^{2}_{b}\mathcal{A}_{11}\big{)}|01\rangle\langle
01|\\\
+&\mathcal{C}^{2}_{b}\big{(}\mathcal{A}_{22}+S^{2}_{a}\mathcal{A}_{11}\big{)}|10\rangle\langle
10|\\\
+&(S^{2}_{a}\big{(}\mathcal{A}_{22}+S^{2}_{b}\mathcal{A}_{11}\big{)}+S^{2}_{b}\mathcal{A}_{22}+\mathcal{A}_{11})|11\rangle\langle
11|\\\
+&\mathcal{C}_{a}\mathcal{C}_{b}\big{\\{}\mathcal{A}_{14}|00\rangle\langle
11|+\mathcal{A}_{23}|01\rangle\langle 10|+h.c.\big{\\}},\end{split}$ (20)

where,

$\mathcal{A}_{11}=\frac{1+c_{33}}{4},\quad\mathcal{A}_{22}=\frac{1-c_{33}}{4},\quad\mathcal{A}_{14}=\frac{c_{11}+c_{22}}{4},\quad\mathcal{A}_{23}=\frac{c_{11}-c_{22}}{4},$
(21)

Figure 1: Steering from Alice to Bob with $c_{11}=c_{22}=c_{33}=-1$ (a)
$\alpha=0.1$, (b) $\alpha=0.4$ .(c)$\alpha=0.7$. (d,e,f) the same as (a,b,c)
respectively, but $c_{11}=c_{22}=c_{33}=-0.8$.

In Fig.(1), the behavior of steerability is displayed, where we consider that
the two qubit systems are initially prepared in the singlet state, namely we
set $c_{11}=c_{22}=c_{33}=-1$. Due to the symmetry, the behavior of
$\mathcal{I}_{ab}$ is the same as $\mathcal{I}_{ba}$, thus we consider only
$\mathcal{I}_{ab}$. In this discussion, it is assumed that, both qubits are
accelerated. Fig.(1.a), shows the behavior of the steering inequality
$\mathcal{I}_{ab}$ before applying the filtering process. The general
behaviour shows that, the steerability is violated as the acceleration
increases. However, it is well known that, the decoherence of the singlet
state due to the acceleration is bounded, where the minimum entanglement is
$\sim 0.3$ [48]. Therefore the possibility that Alice steers Bob’s state is
violated at $r_{a}(r_{b})>0.4$. The effect of the filtering process on the
steerability is displayed in Fig.(1b), and Fig.(1c), where we set $\alpha=0.4$
and $0.7$, respectively. The behavior of $\mathcal{I}_{ab}$ shows that, the
steerability could be implemented at any value of the acceleration.

The effect of the initial state settings is displayed in Figs. (1d-1f), where
it is assumed that the users share Werner state, such that
$c_{11}=c_{22}=c_{33}=-0.8$. The behavior of the steerability is similar to
that displayed for the singlet state, but it is violated at smaller values of
the accelerations. As it is displayed from Fig.(1d), the inequality of
steering is violated at $r_{a}=r_{b}<0.2$. By applying the filtering process,
the steerability could be implemented at large values of the accelerations as
it is displayed from Figs.(1e) and (1f), where the filtering strength is $0.4$
and $0.7$, respectively.

Figure 2:  The same as Fig.(1), but for the degree of steerability
$\mathcal{S}^{A\longrightarrow B}$.

The degree of steerability $\mathcal{S}^{A\longrightarrow B}$ that Alice can
steer Bob’s qubit is displayed in Fig.(2), where it is assumed that, the
steerer and the steered partner share a singlet state. The general behavior of
$\mathcal{S}^{A\longrightarrow B}$ shows that, the degree of steerability
decreases as the acceleration increases. In the absence of the filtering
process, the degree of the steerability decreases fast, while when the
filtering process is applied, the steerability decreases slowly as the
acceleration of both qubits increases. As it is displayed from Fig.(2b) and
Fig.(2c), where we set $\alpha=0.4$, and $0.7$, respectively, the degree of
steerability that is depicted at $r_{a}=r_{b}=0.6$ is given by
$\mathcal{S}^{A\longrightarrow B}=0.2$ and $0.7$, respectively.

In Figs.(2d-2f) we quantify the degree of steerability
$\mathcal{S}^{A\longrightarrow B}$, where the partners share a Werner state.
It is clear that, its upper bounds are smaller than those displayed for the
singlet state. The filtering process has the same effect, namely it increases
as one increases the filtering strength $\alpha$.

###  4.2 Generic Pure state:

For this state, the coherence (Bloch) vectors of both qubits are equals,
namely, $|\vec{r}|=p=|\vec{s}|$. The non-zero elements in the correlation
matrix are defined by $c_{11}=-1$, and $c_{22}=-q=c_{33}$. In an explicit
form, the state (14) may be written as:

$\hat{\rho}_{ab}=\frac{1}{4}\bigg{(}I^{(a)}_{2}\otimes
I^{(b)}_{2}-\sigma_{x}\otimes\tau_{x}+p(\sigma_{x}\otimes
I^{(b)}_{2}-I^{(a)}_{2}\otimes\tau_{x})-q(\sigma_{y}\otimes\tau_{y}+\sigma_{z}\otimes\tau_{z})\bigg{)},$
(22)

where, $p=\sqrt{1-q^{2}}$. After the acceleration process, the generic
accelerated pure state is given by,

$\begin{split}\hat{\rho}_{AB}^{acc}=&\mathcal{C}^{2}_{a}\mathcal{C}^{2}_{b}\mathcal{B}_{11}|00\rangle\langle
00|+\mathcal{C}^{2}_{a}\big{(}\mathcal{B}_{22}+S^{2}_{b}\mathcal{B}_{11}\big{)}|01\rangle\langle
01|+\mathcal{C}^{2}_{b}\big{(}\mathcal{B}_{22}+S^{2}_{a}\mathcal{B}_{11}\big{)}|10\rangle\langle
10|+\\\
&(S^{2}_{a}\big{(}\mathcal{B}_{22}+S^{2}_{b}\mathcal{B}_{11}\big{)}+S^{2}_{b}\mathcal{B}_{22}+\mathcal{B}_{11})|11\rangle\langle
11|+\mathcal{C}_{a}\mathcal{C}_{b}\big{\\{}\mathcal{B}_{14}|00\rangle\langle
11|+\mathcal{B}_{23}|01\rangle\langle 10|+\\\
&\mathcal{B}_{12}\mathcal{C}_{a}\big{(}-|00\rangle\langle
01|+(1+S_{b}^{2})|01\rangle\langle
11|\big{)}+\mathcal{B}_{12}\mathcal{C}_{b}\big{(}|00\rangle\langle
10|-(1+S_{a}^{2})|10\rangle\langle 11|\big{)}+h.c.\big{\\}},\end{split}$ (23)

with,

$\mathcal{B}_{11}=\frac{1-q}{4}=-\mathcal{B}_{14},\quad\mathcal{B}_{22}=\frac{1+q}{4}=-\mathcal{B}_{23},\quad\mathcal{B}_{12}=\frac{p}{4}.$
(24)

Figure 3:  quantum steering $\mathcal{I}_{ab}$ (green-solid curve), and
$\mathcal{I}_{ba}$(red-dot dash curve), (a)$r_{a}=0=r_{b}$, (b)$r_{a}=0.5,\
r_{b}=0$, and (c) $r_{a}=0.5,\ r_{b}=0.5$.

The inequalities of steerability from Alice to Bob and vise versa, are
displayed in Fig.(3), where they decrease gradually as $p$ increases. Fig.(3a)
shows the bidirectional steering’s inequalities when both the steerer and the
steered are in the inertial frame, namely $r_{a}=r_{b}=0$. It is clear that,
the behavior of both inequalities coincides and decreases gradually as the
parameter $p$ increases. The steerability from both directions vanishes
completely at $p=1$, where the initial shared state turns into a separable
state. As it is displayed from Fig.(3.b), the steerability increases slightly
if the steerer is accelerated and the steered particle is on the inertial
frame, where $\mathcal{I}_{ab}$ satisfies the steerability criteria at large
values of $p$. The consistent behavior of both inequalities is displayed in
Fig.(3c), where both users are accelerated with the same acceleration.

Figure 4: Steerability for generic pure state from Alice to Bob with zero
acceleration (green curve), $r_{a}=0.3$, $r_{b}=0$ (red curve), $r_{a}=0.3$,
$r_{b}=0.3$ (blue curve), under weak measurement, when we set
$q=\sqrt{1-p^{2}}$. (a)$\alpha=0.1$, (b)$\alpha=0.4$, (c)$\alpha=0.8$.

Fig.(4) shows the degree of steerability $\mathcal{S}^{A\longrightarrow B}$,
when the partners share a generic pure state. Different cases are considered,
either both partners in the stationary frame, or only one partner or both
partners qubits are accelerated. It is clear that, the steerability increases
gradually to reach its maximum bound at $p=0$, namely the shared state is a
maximum entangled state. However, as $p$ increases further, the steerability
decreases gradually to vanish completely. As it is displayed in Fig.(4a), the
largest upper bounds are reached when the partners’ qubits are in the
stationary frame, while the smallest ones are depicted when both partners are
accelerated. Figs. (4b) and (4c) display the effect of the filtering process
on the degree of steerability. It is clear that, as one increases the
filtering strength, the upper bounds of the steerability increase.

Figure 5: Steerability from Alice to Bob by using the generic pure state, when
we set $p=0.5$, (a) $\alpha=0.1$ .(b) $\alpha=0.4$, (c) $\alpha=0.7$.

The effect of the filtering process on an initial state of a generic pure
state with $p=0.5$ is displayed in Fig.(5), where it is assumed that, both
qubits of the shared state are accelerated. In this situation, the decoherence
arises only from the acceleration process, and consequently the degree of
steerability decreases as the acceleration of both partners increases.
However, as one switches on the filtering process, the possibility of
achieving the steering process increases, namely the steering inequality is
satisfied at large values of the acceleration. This phenomenon is displayed by
comparing Figs.(5a), (5b), and (5c), where the steerable areas are enlarged as
the filtering parameter $\alpha$ increases. It is worth to mention that, the
filtering process does not increase the degree of steerability, but increase
the range of steerability.

Figure 6: Steerability from Alice to Bob for generic pure state, where Alice
is accelerated with $r_{a}$, and Bob at rest $r_{b}=0$. (a) $\alpha=0.1$ .(b)
$\alpha=0.4$, (c) $\alpha=0.8$.

To complete the discussion on the effect of the filtering process on the
steerability and its degree, we consider several classes of initial state
settings, where we plot the function $\mathcal{S}^{A\longrightarrow
B}(p,r_{a},r_{b}=0)$ in Fig.(6). In this case, there are two sources of
decoherence: one due to the initial state settings, where the entanglement of
the shared state decreases as $p$ increases, and the second due to the
acceleration process. Therefore, as it is expected, the degree of steerability
decreases as one increases the decoherence parameter $p$, and the acceleration
$r_{a}$. Figs.(6a-6c) show that, the filtering process has no effect on the
upper bounds of $\mathcal{S}^{A\longrightarrow B}$. The comparison of these
figures shows that, the range of steerability is enlarged as the filtering
parameter increases. This means that, the steering inequality is satisfied at
larger range of the acceleration.

##  5 Bidirectional steerability

In this section, we show that for the generic pure state the degree of
steerability from Alice to Bob and vise versa, namely
$\mathcal{S}^{B\longrightarrow A}$ depends on the type of steered information.
Fig(7) shows the behavior of the bidirectional steerability when only Alice is
accelerated and different initial state settings are considered. The general
behavior shows that, the degree of steerability decreases as the acceleration
increases. However, as it is displayed in Fig.(7a), if the partners share a
maximum entangled state (singlet), both degrees of steerability coincide at
small values of the acceleration, i.e. $r_{a}\in[0,0.4]$. However, at larger
values of $r_{a}$, the degree of steerability from Bob to Alice,
$\mathcal{S}^{B\longrightarrow A}$ is larger than that displayed from Alice to
Bob, $\mathcal{S}^{A\longrightarrow B}$. On the other hand, since Alice
particle is accelerated, then its decoherence increases and, consequently its
local information decreases. Therefore, one can say that the possibility of
steering states that coded coherent information is much larger than that coded
mixed information.

Figure 7:  Quantum steering for Alice is accelerated $r_{a}$, and Bob is fixed
($r_{b}=0$) where $\mathcal{S^{A\longrightarrow B}}$(green-solid
curve),$\mathcal{S^{B\longrightarrow A}}$(red- dot dash curve),
$\Delta=|\mathcal{S^{A\longrightarrow B}}-\mathcal{S^{A\longrightarrow B}}|$
(blue-dash curve). (a)$p\longrightarrow 0$, (b)$p=0.6$, and (b)$p=0.9$ .

In Fig.(7.b), it is assumed that the partners share a partially entangle state
with $p=0.6$. A similar behavior is depicted in Fig.(7a), where the degree of
steerability decreases as the acceleration $r_{a}$ increases. The displayed
behavior shows that $\mathcal{S}^{A\longrightarrow B}$ and
$\mathcal{S}^{B\longrightarrow A}$ have a slight difference,such that
$\mathcal{S}^{A\longrightarrow B}>\mathcal{S}^{B\longrightarrow A}$. This
means that, the decoherence due to the acceleration is smaller than that due
to the decoherence parameter $p$. Consequently, the encoded local information
on Bob’s qubit is more decoherent. Fig.(7c), shows the bidirectional
steerability from both sides at larger value of the characteristic parameter
$p$, namely the shared state has small degree of entanglement. The behavior of
the steerability $\mathcal{S}^{A\longrightarrow B}$, is larger than that
displayed for $\mathcal{S}^{B\longrightarrow A}$ and the difference between
them $\Delta$, increases as $r_{a}$ increases. In the interval
$r_{a}\in[0.1,0.8]$, Bob’s information is more coherent, and consequently the
degree of Alice steerability $\mathcal{S}^{A\longrightarrow B}$ is larger than
that displayed for $\mathcal{S}^{B\longrightarrow A}$. From Fig.(7), it is
clear that the bidirectional degree of steerability decreases as one increases
the parameter $p$. The rate of decreasing due to the acceleration is much
smaller than that displayed for the less entangled shared state between the
steerer and the steered.

Figure 8: decoherence for generic pure state where Alice is accelerated,
decoherence $\rho_{AB}$ (Blue curve), decoherence $\rho_{A}$ (green curve),
decoherence $\rho_{B}$(red curve), (a)$p=0.0$, (b)$p=0.6$, (c)$p=0.9$.

##  6 Conclusion

In this contribution, we investigated the bidirectional steerability between
two partners who share different classes of initial states. It is assumed
that, either the steerer or the steered partners are in inertial/ non-inertial
frames. The effect of the initial state settings and the acceleration
parameters on the steerability and its degree are discussed, where analytical
forms are obtained for both quantities. However, the possibility of enhancing
the degree of steerability is discussed by applying the filtering process.

The general behaviour of the steerability and its degree decreases, if both
partners’s qubits are accelerated. The decay rate depends on whether one or
both qubits are accelerated. It is shown that, the bidirectional steering is
completely consistent if both partners are in the inertial frame or in the
non-inertial frame with the same value of the acceleration. The bidirectional
steerability and its degree are slightly different when one of the partners’s
qubit is at rest and the other is accelerated. The initial state settings play
a significant role on maximizing/minimizing the steerability and its degree.
It is shown that, the violation of the steering inequality appears as the
shared state loses its quantum correlation. The largest upper bounds of the
steerability and its degree are exhibited for the maximum entangled state. The
two quantities are investigated for a class of a generic pure state. The
violation of the steerability is displayed at larger values of this parameter,
where the shared state turns into a separable state when one increases the
decoherence parameter. However, due to the acceleration, the violation of
steerability is depicted at small values of this decoherence parameter.

The possibility of improving the steerability and its degree is discussed by
applying the filtering process. The results show that, as one increases the
strength of the filtering process, the steerability increases. The filtering
process restrains the decoherence that may be arisen from the accelerating
process, and consequently, one enlarges the range of the acceleration in which
the steerability could be achieved. At fixed acceleration, the filtering
process increases the upper bounds of the degree of steerability.

The bidirectional steerability between the two users is discussed for the non-
symmetric cases. It is clear that, if the users’ qubits are accelerated, then
the degree of steerability between the two users coincide. It is shown that,
for non-symmetric cases, the degree of steerability depends on the types of
steerable information. However, due to the acceleration the coherence of the
qubit decreases and the possibility of its steering it decreases. In this
case, the difference between the steerability from both directions increases.

Finally, one may conclude that, the decay of the steerability and its degree
is due to the decoherence from the initial state settings and the acceleration
process. The filtering process improves the upper bounds of the degree of
steerability and enlarges the range of acceleration in which the steerability
could be achieved. The most information holds a great degree of steerability.

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