#  Fermionic steering and its monogamy relations in Schwarzschild spacetime

Shu-Min Wu1, Hao-Sheng Zeng2111Email<EMAIL_ADDRESS>1 Department of
Physics, Liaoning Normal University, Dalian 116029, China
2 Department of Physics, Hunan Normal University, Changsha 410081, China

###### Abstract

Using two different types of quantification for quantum steering, we study the
influence of Hawking radiation on quantum steering for fermionic fields in
Schwarzschild spacetime. The degradation for the steering between physically
accessible observers and the generation for the steering between physically
accessible and inaccessible observers induced by Hawking radiation are
studied. We also reveal the difference between the two types of quantification
for steering, and find some monogamy relations between steering and
entanglement. Furthermore, we show the different properties between fermionic
steering and bosonic steering in Schwarzschild spacetime.

###### pacs:

04.70.Dy, 03.65.Ud,04.62.+v

##  I Introduction

Einstein-Podolsky-Rosen steering, firstly introduced by Schrödinger in 1935,
is an intermediate type of quantum correlation between Bell non-locality and
quantum entanglement Q1 ; Q2 . Quantum steering is a special quantum
phenomenon, which allows an observer by measuring his subsystem to control the
remote subsystem owned by another observer. Since Wiseman $etal.$ in 2007
firstly gave the resource definition of quantum steering via the local hidden
state (LHS) model Q3 , quantum steering has attracted renewed interest Q4 ; Q5
; Q6 ; Q7 ; Q8 ; Q9 ; Q10 ; Q11 ; Q12 ; Q13 . Various inequalities for
detecting quantum steering of quantum states have been proposed Q14 ; Q15 ;
Q16 ; Q17 ; Q18 ; Q19 . Unlike quantum entanglement, the asymmetry of quantum
steering in quantum systems, such as one-way steering and two-way steering,
has been demonstrated experimentally Q20 ; Q21 ; Q22 ; Q23 . One-way steering
and two-way steering are very important quantum resources, which may be used
in quantum information processing Q24 ; Q25 ; Q26 .

General relativity from the Einstein’s theory has made great achievements and
its predictions have been gradually verified in physics and astronomy. One of
the most famous predictions is that black holes exist in our universe. Black
holes may be created by the gravitational collapse of sufficiently massive
stars. Recently, the Advanced LIGO and Virgo detectors detected the
gravitational wave (GW150914) from a binary black hole merger system for the
first time, which indirectly confirms the existence of black holes and
Einstein’s general relativity Q27 . Moreover, the first photo of the
supermassive black hole in the center of the giant elliptical galaxy M87 was
taken by the Event Horizon Telescope Q28 ; Q29 ; Q30 ; Q31 ; Q32 ; Q33 . Soon
after, Holoien $etal.$ shown that a star gets torn apart by a black hole Q34 .
Because black holes are too far away from us and have special properties, it
is covered with a mysterious veil and is the cutting-edge research field, such
as the lost information paradox of black holes Q35 ; Q36 ; Q37 . Hawking
predicted that the vacuum fluctuations near the event horizon cause black
holes to evaporate. In other words, particle-antiparticle pairs near the event
horizon are generated. Hawking speculated that antiparticle falls into the
black hole and particle escapes from the black hole. Obviously, the Hawking
radiation underlies the lost information paradox of black holes.

The combination of relativity theory and another fundamental theory of modern
physics, quantum information, gave birth to relativistic quantum information.
It is believed that the study of quantum information in a relativistic
framework is not only helpful in understanding concepts of quantum correlation
and coherence Q38 ; Q39 ; Q40 ; Q41 ; Q42 ; Q43 ; Q44 ; Q45 ; Q46 ; Q47 ; Q48
; Q49 ; Q50 ; Q51 ; Q52 ; Dong1 ; Dong2 ; Dong3 ; Dong4 ; Dong5 ; Dong6 ;
Dong7 ; MLO ; MLO1 , but also plays an important role in the investigation of
the lost information paradox of black holes Q35 ; Q36 ; Q37 . It has been
already shown that in the relativistic quantum information, the bosonic and
fermionic fields have different behaviors. For example, in the limit of
infinite acceleration, bosonic entanglement vanishes Q38 , while fermionic
entanglement can survive Q39 ; With the increase of acceleration, bosonic
discord appears irreversible decoherence Q46 , while fermionic discord occurs
the phenomenon of freeze Q47 . Compared with quantum entanglement, quantum
steering has richer properties, such as no-way steering, one-way steering,
two-way steering and asymmetric steering. Therefore, studying bosonic and
fermionic steering, and comparing them in the relativistic framework are more
intriguing.

In this work, we will study the influence of Hawking radiation on quantum
steering for fermionic fields in Schwarzschild spacetime. We assume that Alice
and Bob share a maximally entangled state of Dirac fields in flat Minkowski
spacetime. Afterwards, Alice continues to stay at an asymptotically flat
region, while Bob hovers near the event horizon of the black hole; at the same
time, the Anti-Bob inside the event horizon is generated. We will calculate
fermionic steering and obtain the analytic expressions in the curved
spacetime. We then study the degradation of steering between Alice and Bob,
the production of steering between Alice and Anti-Bob, Bob and Anti-Bob. We
also study the redistribution of steering between different subsystems and try
to find some monogamy relations for steering and entanglement.

The paper is organized as follows. In Sec.II, we briefly introduce the
quantification of bipartite steering. In Sec.III, we discuss the quantization
of Dirac fields in the background of a Schwarzschild black hole. In Sec.IV, we
study the evolution of steering between different subsystems and their
redistribution rules, and make a comparison with the counterpart of bosonic
fields in the background of a Schwarzschild black hole. In Sec.IV, we study
the monogamy relations between fermionic steering and entanglement in
Schwarzschild spacetime. Finally the summary is arranged in Sec.VI.

##  II Quantification of bipartite steering

###  II.1 Quantification of steering based on entropy uncertainty relation

According to the definition of quantum steering given by Wiseman $etal.$, one
can say that Alice can steer Bob’s state if the results presented by Alice and
Bob have correlations that violate the local-hidden-state (LHS) model Q3 .
Consider a state $\rho_{AB}$ of the discrete-variable compound system $AB$,
with subsystems $A$ and $B$ held by Alice and Bob respectively. Let
$\hat{R}^{A}$ and $\hat{R}^{B}$ are the discrete observables for subsystems
$A$ and $B$, with possible outcomes $\\{R^{A}\\}$ and $\\{R^{B}\\}$. The LHS
model may be written as Q14

$\displaystyle
H(R^{B}|R^{A})\geq\sum_{\lambda}P(\lambda)H_{Q}(R^{B}|\lambda),$ (1)

where $H(R^{B}|R^{A})$ is the conditional entropy of variable $\hat{R}^{B}$
given $\hat{R}^{A}$ to be $R^{A}$, and $H_{Q}(R^{B}|\lambda)$ is the discrete
Shannon entropy of the probability distribution $P_{Q}(R^{B}|\lambda)$ that
measures $\hat{R}^{B}$ to be $R^{B}$ given the details of preparation in the
hidden variable $\lambda$.

In the $N$-dimensional Hilbert space, any pair of discrete observables $R$ and
$S$ with the eigenbases $R_{i}$ and $S_{i}$ ($i=1,2,..,N$) satisfy the entropy
uncertainty relation

$\displaystyle H_{Q}(R)+H_{Q}(S)\geq\log(\mathrm{U}),$ (2)

where $\mathrm{U}\equiv{\rm{min}}_{i,j}\frac{1}{|\langle
R_{i}|S_{j}\rangle|^{2}}$. From above two equations, one can obtain the
entropy-based steering inequality

$\displaystyle H(R^{B}|R^{A})+H(S^{B}|S^{A})\geq\log(\mathrm{U}^{B}),$ (3)

where $\mathrm{U}^{B}$ denotes the value of $\mathrm{U}$ associated with
observables $R^{B}$ and $S^{B}$. This steering inequality that involves a pair
of discrete observables may be generalized to the more general case that
involves arbitrary number of mutually unbiased observables. Especially for the
complete set of pairwise complementary Pauli operators $X$, $Y$ and $Z$, there
exists the steering inequality from Alice to Bob Q14

$\displaystyle I^{A\rightarrow
B}=H(\sigma_{x}^{B}|\sigma_{x}^{A})+H(\sigma_{y}^{B}|\sigma_{y}^{A})+H(\sigma_{z}^{B}|\sigma_{z}^{A})\geq
2.$ (4)

If this steering equality is violated, we say that Alice can steer Bob. To
quantify the ability that Alice steers Bob, one introduce the quantity

$\displaystyle S^{A\rightarrow B}={\rm{max}}\bigg{\\{}0,\frac{I^{A\rightarrow
B}-2}{I_{\rm{max}}-2}\bigg{\\}}.$ (5)

The factor $I_{\rm{max}}$ is to guarantee the normalization of quantum
steerability, which is equal to 6 for the maximally entangled state considered
in our text.

In this paper, we study symmetric X-state

$\displaystyle\rho_{x}=\left(\\!\\!\begin{array}[]{cccc}\rho_{11}&0&0&\rho_{14}\\\
0&\rho_{22}&\rho_{23}&0\\\ 0&\rho_{23}&\rho_{33}&0\\\
\rho_{14}&0&0&\rho_{44}\end{array}\\!\\!\right),$ (10)

where the real entries satisfy $\rho_{ij}=\rho_{ji}$. This X-state can also be
expressed in the form

$\displaystyle\rho_{x}=\frac{1}{4}\bigg{[}I\otimes I+p\sigma_{z}\otimes
I+qI\otimes\sigma_{z}+\sum_{i=1}^{3}c_{i}\sigma_{i}\otimes\sigma_{i}\bigg{]},$
(11)

where $c_{1}=2(\rho_{14}+\rho_{23})$, $c_{2}=2(\rho_{23}-\rho_{14})$,
$c_{3}=\rho_{11}-\rho_{22}-\rho_{33}+\rho_{44}$,
$p=\rho_{11}+\rho_{22}-\rho_{33}-\rho_{44}$ and
$q=\rho_{11}-\rho_{22}+\rho_{33}-\rho_{44}$. For this state, $I^{A\rightarrow
B}$ in Eq.(5) gives by

$\displaystyle I^{A\rightarrow B}$ $\displaystyle=$
$\displaystyle\frac{1}{2}[(1+c_{3}+p+q)\log(1+c_{3}+p+q)$ (12)
$\displaystyle+$ $\displaystyle(1+c_{3}-p-q)\log(1+c_{3}-p-q)$
$\displaystyle+$ $\displaystyle(1-c_{3}-p+q)\log(1-c_{3}-p+q)$
$\displaystyle+$ $\displaystyle(1-c_{3}+p-q)\log(1-c_{3}+p-q)]$
$\displaystyle+$
$\displaystyle\sum_{i=1,2}[(1+c_{i})\log(1+c_{i})+(1-c_{i})\log(1-c_{i})]$
$\displaystyle-$ $\displaystyle(1+p)\log(1+p)-(1-p)\log(1-p),$

with the base of logarithms being $2$. The steerability from Bob to Alice can
be obtained by exchanging the roles of $A$ and $B$, which is given by

$\displaystyle S^{B\rightarrow A}={\rm{max}}\bigg{\\{}0,\frac{I^{B\rightarrow
A}-2}{I_{\rm{max}}-2}\bigg{\\}},$ (13)

with

$\displaystyle I^{B\rightarrow A}$ $\displaystyle=$
$\displaystyle\frac{1}{2}[(1+c_{3}+p+q)\log(1+c_{3}+p+q)$ (14)
$\displaystyle+$ $\displaystyle(1+c_{3}-p-q)\log(1+c_{3}-p-q)$
$\displaystyle+$ $\displaystyle(1-c_{3}-p+q)\log(1-c_{3}-p+q)$
$\displaystyle+$ $\displaystyle(1-c_{3}+p-q)\log(1-c_{3}+p-q)]$
$\displaystyle+$
$\displaystyle\sum_{i=1,2}[(1+c_{i})\log(1+c_{i})+(1-c_{i})\log(1-c_{i})]$
$\displaystyle-$ $\displaystyle(1+q)\log(1+q)-(1-q)\log(1-q).$

Unlike quantum entanglement, quantum steering may not be symmetrical, i.e.
$S^{A\rightarrow B}\neq S^{B\rightarrow A}$. Based on the asymmetry of quantum
steering, we distinguish the quantum steering into three cases: (i) no-way
steering $S^{A\rightarrow B}=S^{B\rightarrow A}=0$; (ii) one-way steering
$S^{A\rightarrow B}>0$ and $S^{B\rightarrow A}=0$, or vice versa
$S^{B\rightarrow A}>0$ and $S^{A\rightarrow B}=0$; (iii) two-way steering
$S^{A\rightarrow B}>0$ and $S^{B\rightarrow A}>0$. This means that, compared
with other quantum correlations, quantum steering has richer properties.

###  II.2 Quantification of steering based on quantum entanglement

Firstly, we mention that quantum entanglement of bipartite states can be
effectively identified by the concurrence. For the X-state $\rho_{x}$ given by
Eq.(10), the concurrence can be expressed as LLL45

$\displaystyle
C(\rho_{x})=2\max\\{|\rho_{14}|-\sqrt{\rho_{22}\rho_{33}},|\rho_{23}|-\sqrt{\rho_{11}\rho_{44}}\\}.$
(15)

Next, for any two-qubit state $\rho_{AB}$ shared by Alice and Bob, the
steering from Bob to Alice can be witnessed if the density matrix
$\tau_{AB}^{1}$ defined as LLL46 ; LLLL46

$\displaystyle\tau_{AB}^{1}=\frac{\rho_{AB}}{\sqrt{3}}+\frac{3-\sqrt{3}}{3}(\rho_{A}\otimes\frac{I}{2}),$
(16)

is entangled. Here $\rho_{A}$ is Alice’s reduced density matrix, namely
$\rho_{A}=\rm{Tr}_{B}(\rho_{AB})$, and $I$ denotes the two-dimension identity
matrix for Bob’s qubit. Similarly, the corresponding steering from Alice to
Bob can be witnessed if the state $\tau_{AB}^{2}$ defined as

$\displaystyle\tau_{AB}^{2}=\frac{\rho_{AB}}{\sqrt{3}}+\frac{3-\sqrt{3}}{3}(\frac{I}{2}\otimes\rho_{B}),$
(17)

is entangled, where $\rho_{B}=\rm{Tr}_{A}(\rho_{AB})$.

Now, for the X-state $\rho_{x}$ given by Eq.(10), the matrix $\tau_{AB}^{1}$
can be written as

$\displaystyle\tau_{AB}^{1,x}=\left(\\!\\!\begin{array}[]{cccc}\frac{\sqrt{3}}{3}\rho_{11}+r&0&0&\frac{\sqrt{3}}{3}\rho_{14}\\\
0&\frac{\sqrt{3}}{3}\rho_{22}+r&\frac{\sqrt{3}}{3}\rho_{23}&0\\\
0&\frac{\sqrt{3}}{3}\rho_{23}&\frac{\sqrt{3}}{3}\rho_{33}+s&0\\\
\frac{\sqrt{3}}{3}\rho_{14}&0&0&\frac{\sqrt{3}}{3}\rho_{44}+s)\end{array}\\!\\!\right),$
(22)

with $r=\frac{(3-\sqrt{3})}{6}(\rho_{11}+\rho_{22})$ and
$s=\frac{(3-\sqrt{3})}{6}(\rho_{33}+\rho_{44})$. According to Eq.(15), as long
as one of the following inequalities,

$\displaystyle|\rho_{14}|^{2}>Q_{a}-Q_{b},$ (23)

or

$\displaystyle|\rho_{23}|^{2}>Q_{c}-Q_{b},$ (24)

is satisfied, then the state $\tau_{AB}^{1,x}$ is entangled, where

$\displaystyle
Q_{a}=\frac{2-\sqrt{3}}{2}\rho_{11}\rho_{44}+\frac{2+\sqrt{3}}{2}\rho_{22}\rho_{33}+\frac{1}{4}(\rho_{11}+\rho_{44})(\rho_{22}+\rho_{33}),$
$\displaystyle Q_{b}=\frac{1}{4}(\rho_{11}-\rho_{44})(\rho_{22}-\rho_{33}),$
$\displaystyle
Q_{c}=\frac{2+\sqrt{3}}{2}\rho_{11}\rho_{44}+\frac{2-\sqrt{3}}{2}\rho_{22}\rho_{33}+\frac{1}{4}(\rho_{11}+\rho_{44})(\rho_{22}+\rho_{33}).$

The steering from Bob to Alice is thus witnessed. In a similar way, the
steering from Alice to Bob can be witnessed through one of the inequalities,

$\displaystyle|\rho_{14}|^{2}>Q_{a}+Q_{b},$ (25)

or

$\displaystyle|\rho_{23}|^{2}>Q_{c}+Q_{b}.$ (26)

Further, we introduce the quantities

$\displaystyle T^{B\rightarrow
A}={\rm{max}}\bigg{\\{}0,\frac{8}{\sqrt{3}}(|\rho_{14}|^{2}-Q_{a}+Q_{b}),\frac{8}{\sqrt{3}}(|\rho_{23}|^{2}-Q_{c}+Q_{b})\bigg{\\}},$
(27)

and

$\displaystyle T^{A\rightarrow
B}={\rm{max}}\bigg{\\{}0,\frac{8}{\sqrt{3}}(|\rho_{14}|^{2}-Q_{a}-Q_{b}),\frac{8}{\sqrt{3}}(|\rho_{23}|^{2}-Q_{c}-Q_{b})\bigg{\\}},$
(28)

to quantify the steerability from Bob to Alice and from Alice to Bob
respectively. The factor $\frac{8}{\sqrt{3}}$ guarantees that the steerability
of the maximum entangled state is $1$. Note that we here use the capital
letter $T$ to describe the entanglement-based steerability, so as to
distinguish from marker $S$ of the steerability based on entropy uncertainty
relation.

##  III Quantization of Dirac fields in Schwarzschild spcetime

Let’s consider a Schwarzschild black hole that is given by the metric Q42

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle-(1-\frac{2M}{r})dt^{2}+(1-\frac{2M}{r})^{-1}dr^{2}$ (29)
$\displaystyle+r^{2}(d\theta^{2}+\sin^{2}\theta d\varphi^{2}),$

where $r$ and $M$ are respectively the radius and mass of the black hole. For
simplicity, we take $\hbar,G,c$ and $k$ as unity in this paper. The Dirac
equation Q55  $[\gamma^{a}e_{a}{}^{\mu}(\partial_{\mu}+\Gamma_{\mu})]\Phi=0$
in Schwarzschild spacetime can be written as

$\displaystyle-\frac{\gamma_{0}}{\sqrt{1-\frac{2M}{r}}}\frac{\partial\Phi}{\partial
t}+\gamma_{1}\sqrt{1-\frac{2M}{r}}\bigg{[}\frac{\partial}{\partial
r}+\frac{1}{r}+\frac{M}{2r(r-2M)}\bigg{]}\Phi$
$\displaystyle+\frac{\gamma_{2}}{r}(\frac{\partial}{\partial\theta}+\frac{\cot\theta}{2})\Phi+\frac{\gamma_{3}}{r\sin\theta}\frac{\partial\Phi}{\partial\varphi}=0,$
(30)

where $\gamma_{i}$ ($i=0,1,2,3$) are the Dirac matrices Q56 ; Q57 .

Solving the Dirac equation near the event horizon, we obtain a set of positive
(fermions) frequency outgoing solutions inside and outside regions of the
event horizon Q56 ; Q57

$\displaystyle\Phi^{+}_{{\mathbb{k}},{\rm in}}\sim\phi(r)e^{i\omega u},$ (31)
$\displaystyle\Phi^{+}_{{\mathbb{k}},{\rm out}}\sim\phi(r)e^{-i\omega u},$
(32)

where $\phi(r)$ denotes four-component Dirac spinor, $u=t-r_{*}$ with
$r_{*}=r+2M\ln\frac{r-2M}{2M}$ is the tortoise coordinate. $\mathbb{k}$ and
$\omega$ are the wave vector and frequency which fulfill $|\mathbf{k}|=\omega$
for the massless Dirac field. The Dirac field $\Phi$ can be expanded as

$\displaystyle\Phi$ $\displaystyle=$ $\displaystyle\int
d\mathbb{k}[\hat{a}^{\rm
in}_{\mathbb{k}}\Phi^{+}_{{\mathbb{k}},\text{in}}+\hat{b}^{\rm
in{\dagger}}_{\mathbb{k}}\Phi^{-}_{{\mathbb{k}},\text{in}}$ (33)
$\displaystyle+$ $\displaystyle\hat{a}^{\rm
out}_{\mathbb{k}}\Phi^{+}_{{\mathbb{k}},\text{out}}+\hat{b}^{\rm
out{\dagger}}_{\mathbb{k}}\Phi^{-}_{{\mathbb{k}},\text{out}}],$

where $\hat{a}^{\rm in}_{\mathbb{k}}$ and $\hat{b}^{\rm
in{\dagger}}_{\mathbb{k}}$ are the fermion annihilation and antifermion
creation operators for the quantum field in the interior of the event horizon,
and $\hat{a}^{\rm out}_{\mathbb{k}}$ and $\hat{b}^{\rm
out{\dagger}}_{\mathbb{k}}$ are the fermion annihilation and antifermion
creation operators for the quantum field of the exterior region, respectively.
These annihilation and creation operators satisfy canonical anticommutation
$\\{\hat{a}^{\rm out}_{\mathbf{k}},\hat{a}^{\rm
out\dagger}_{\mathbf{k^{\prime}}}\\}=\\{\hat{b}^{\rm
in}_{\mathbf{k}},\hat{b}^{\rm
in\dagger}_{\mathbf{k^{\prime}}}\\}=\delta_{\mathbf{k}\mathbf{k^{\prime}}}.$
One can define the Schwarzschild vacuum through expression $\hat{a}^{\rm
in}_{\mathbb{k}}|0\rangle_{S}=\hat{a}^{\rm out}_{\mathbb{k}}|0\rangle_{S}=0$.
Therefore, the modes $\Phi^{\pm}_{{\mathbb{k}},{\rm in}}$ and
$\Phi^{\pm}_{{\mathbb{k}},{\rm out}}$ are usually called Schwarzschild modes.

Making an analytic continuation of Eqs.(31) and (32) in the light of Domour
and Ruffini’s suggestion Q58 , one find a complete basis for positive energy
modes, i.e., the Kruskal modes,.

$\displaystyle\Psi^{+}_{{\mathbb{k}},{\rm out}}=e^{-2\pi
M\omega}\Phi^{-}_{{-\mathbb{k}},{\rm in}}+e^{2\pi
M\omega}\Phi^{+}_{{\mathbb{k}},{\rm out}},$ (34)
$\displaystyle\Psi^{+}_{{\mathbb{k}},{\rm in}}=e^{-2\pi
M\omega}\Phi^{-}_{{-\mathbb{k}},{\rm out}}+e^{2\pi
M\omega}\Phi^{+}_{{\mathbb{k}},{\rm in}}.$ (35)

Thus, we can also use the Kruskal modes to expand the Dirac fields in the
Kruskal spacetime

$\displaystyle\Phi$ $\displaystyle=$ $\displaystyle\int
d\mathbb{k}[2\cosh(4\pi M\omega)]^{-\frac{1}{2}}[\hat{c}^{\rm
in}_{\mathbb{k}}\Psi^{+}_{{\mathbb{k}},\text{in}}+\hat{d}^{\rm
in{\dagger}}_{\mathbb{k}}\Psi^{-}_{{\mathbb{k}},\text{in}}$ (36)
$\displaystyle+$ $\displaystyle\hat{c}^{\rm
out}_{\mathbb{k}}\Psi^{+}_{{\mathbb{k}},\text{out}}+\hat{d}^{\rm
out{\dagger}}_{\mathbb{k}}\Psi^{-}_{{\mathbb{k}},\text{out}}],$

where $\hat{c}^{\sigma}_{\mathbb{k}}$ and
$\hat{d}^{\sigma{\dagger}}_{\mathbb{k}}$ with $\sigma=(\rm in,\rm out)$ are
the fermion annihilation and antifermion creation operators acting on the
Kruskal vacuum.

Eqs.(33) and (36) represent the different decompositions of the same Dirac
field in Schwarzschild and Kruskal modes, respectively, which lead to the
well-known Bogoliubov transformation between the Kruskal and Schwarzschild
operators,

$\displaystyle\hat{c}^{\rm out}_{\mathbb{k}}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{e^{-8\pi M\omega}+1}}\hat{a}^{\rm
out}_{\mathbb{k}}-\frac{1}{\sqrt{e^{8\pi M\omega}+1}}\hat{b}^{\rm
out{\dagger}}_{\mathbb{k}},$ (37) $\displaystyle\hat{c}^{\rm
out{\dagger}}_{\mathbb{k}}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{e^{-8\pi M\omega}+1}}\hat{a}^{\rm
out{\dagger}}_{\mathbb{k}}-\frac{1}{\sqrt{e^{8\pi M\omega}+1}}\hat{b}^{\rm
out}_{\mathbb{k}}.$ (38)

The Kruskal vacuum and excited states thus can be expressed in the
Schwarzschild Fock space as

$\displaystyle|0\rangle_{K}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{e^{-\frac{\omega}{T}}+1}}|0\rangle_{\rm
out}|0\rangle_{\rm in}+\frac{1}{\sqrt{e^{\frac{\omega}{T}}+1}}|1\rangle_{\rm
out}|1\rangle_{\rm in},$ $\displaystyle|1\rangle_{K}$ $\displaystyle=$
$\displaystyle|1\rangle_{\rm out}|0\rangle_{\rm in},$ (39)

where $T=\frac{1}{8\pi M}$ is the Hawking temperature, $\\{|n\rangle_{\rm
out}\\}$ and $\\{|n\rangle_{\rm in}\\}$ are the Schwarzschild number states
for the fermion outside the region and the antifermion inside the region of
the event horizon, respectively.

For the Schwarzschild observer Bob who hovers outside the event horizon, the
Hawking radiation spectrum from the viewpoint of his detector is given by Q57

$\displaystyle N_{F}^{2}=\sideset{{}_{K}}{}{\mathop{\langle}}0|\hat{a}^{\rm
out{\dagger}}_{\mathbb{k}}\hat{a}^{\rm
out}_{\mathbb{k}}|0\rangle_{K}=\frac{1}{e^{\frac{\omega}{T}}+1}.$ (40)

This equation means that a Kruskal vacuum observed by the Kruskal observer
Alice would be detected as a number of the generated fermions $N_{F}^{2}$ from
Bob’s viewpoint. In other words, the Schwarzschild observer Bob in the
exterior of the black hole can detect a thermal Fermi-Dirac statistic of
fermions.

##  IV Fermionic steering in Schwarzschild spacetime

Consider two maximally entangled fermionic modes in the asymptotically flat
region of the Schwarzschild black hole

$\displaystyle|\phi_{AB}\rangle=\frac{1}{\sqrt{2}}(|0\rangle_{A}|0\rangle_{B}+|1\rangle_{A}|1\rangle_{B}),$
(41)

where the subscripts $A$ and $B$ denote the modes which are associated with
the observers Alice and Bob, respectively. After the coincidence of Alice and
Bob, Alice stays stationary at the asymptotically flat region, while Bob
hovers outside the event horizon of the black hole. Bob will detects a thermal
Fermi-Dirac statistic of fermions and his detector is found to be excited.
Using Eq.(III), we can rewrite Eq.(41) in terms of Kruskal modes for Alice and
Schwarzschild modes for Bob

$\displaystyle|\phi_{AB\bar{B}}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}(\frac{1}{\sqrt{e^{-\frac{\omega}{T}}+1}}|0\rangle_{A}|0\rangle_{B}|0\rangle_{\bar{B}}+\frac{1}{\sqrt{e^{\frac{\omega}{T}}+1}}|0\rangle_{A}|1\rangle_{B}|1\rangle_{\bar{B}}$
(42) $\displaystyle+$
$\displaystyle|1\rangle_{A}|1\rangle_{B}|0\rangle_{\bar{B}}),$

where the mode $\bar{B}$ is observed by a hypothetical observer Anti-Bob
inside the event horizon of the black hole. We write its density matrix as

$\displaystyle\rho_{AB\bar{B}}=\frac{1}{2}\left(\\!\\!\begin{array}[]{cccccccc}\mathcal{C}^{2}&0&0&\mathcal{C}\mathcal{S}&0&0&\mathcal{C}&0\\\
0&0&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0&0\\\
\mathcal{C}\mathcal{S}&0&0&\mathcal{S}^{2}&0&0&\mathcal{S}&0\\\
0&0&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0&0\\\ \mathcal{C}&0&0&\mathcal{S}&0&0&1&0\\\
0&0&0&0&0&0&0&0\end{array}\\!\\!\right).$ (51)

in the orthonormal basis
$\\{|0,0,0\rangle,|0,0,1\rangle,|0,1,0\rangle,|0,1,1\rangle,|1,0,0\rangle,|1,0,1\rangle,|1,1,0\rangle,|1,1,1\rangle\\}$,
where we have defined
$|abc\rangle=|a\rangle_{A}|b\rangle_{B}|c\rangle_{\bar{B}}$, and
$\mathcal{C}=\frac{1}{\sqrt{e^{-\frac{\omega}{T}}+1}}$,
$\mathcal{S}=\frac{1}{\sqrt{e^{\frac{\omega}{T}}+1}}$ for simplicity.

###  IV.1  Physically accessible quantum steering

Since Bob is causally disconnected from the region inside the event horizon,
the only information that is physically accessible to the observers is encoded
in the mode $A$ described by Alice and the mode $B$ outside the event horizon
described by Bob. Taking the trace over the $\bar{B}$ mode inside the event
horizon, we obtain a mixed density matrix for Alice and Bob

$\displaystyle\rho_{AB}=\frac{1}{2}\left(\\!\\!\begin{array}[]{cccccccc}\mathcal{C}^{2}&0&0&\mathcal{C}\\\
0&\mathcal{S}^{2}&0&0\\\ 0&0&0&0\\\
\mathcal{C}&0&0&1\end{array}\\!\\!\right),$ (56)

in the basis $\\{|00\rangle,|01\rangle,|10\rangle,|11\rangle\\}$. In the
following, we use two types of methods to measure steering in curved
spacetime.

Firstly, according to Eqs.(5) and (13), we obtain the analytic expressions for
the steerability of the $S^{A\rightarrow B}$ and $S^{B\rightarrow A}$ based on
entropy uncertainty relation as

$\displaystyle S^{A\rightarrow B}(\rho_{AB})$ $\displaystyle=$
$\displaystyle\frac{1}{4}\big{[}2(1+\mathcal{C})\log(1+\mathcal{C})+2(1-\mathcal{C})\log(1-\mathcal{C})$
(57) $\displaystyle+$
$\displaystyle\mathcal{C}^{2}\log(\mathcal{C}^{2})+\mathcal{S}^{2}\log(\mathcal{S}^{2})\big{]},$

and

$\displaystyle S^{B\rightarrow A}(\rho_{AB})$ $\displaystyle=$
$\displaystyle\frac{1}{4}\big{[}2(1+\mathcal{C})\log(1+\mathcal{C})+2(1-\mathcal{C})\log(1-\mathcal{C})$
(58) $\displaystyle-$
$\displaystyle(1+\mathcal{S}^{2})\log(1+\mathcal{S}^{2})+\mathcal{S}^{2}\log(\mathcal{S}^{2})].$

We see that the steerability depends on the Hawking temperature $T$, i.e.,
Hawking radiation of the black hole influences the fermionic steerability.
Under the influence of Hawking radiation, the steering becomes asymmetric,
i.e., the steerability from Alice to Bob is different from the steerability
from Bob to Alice. In order to measure the degrees of asymmetry, we introduce
the steering difference

$\displaystyle S_{AB}^{\Delta}$ $\displaystyle=$
$\displaystyle|S^{A\rightarrow B}(\rho_{AB})-S^{B\rightarrow A}(\rho_{AB})|$
$\displaystyle=$
$\displaystyle\frac{1}{4}[\mathcal{C}^{2}\log(\mathcal{C}^{2})+(1+\mathcal{S}^{2})\log(1+\mathcal{S}^{2})].$

Secondly, we study another quantification of steering based on quantum
entanglement in Schwarzschild spacetime. The analytic expressions for the
entanglement-based steerability $T^{A\rightarrow B}$, $T^{B\rightarrow A}$ and
the corresponding steering difference read

$\displaystyle T^{A\rightarrow B}(\rho_{AB})$ $\displaystyle=$
$\displaystyle\mathcal{C}^{2}-\frac{1}{\sqrt{3}}\mathcal{C}^{2}\mathcal{S}^{2},$
(60) $\displaystyle T^{B\rightarrow A}(\rho_{AB})$ $\displaystyle=$
$\displaystyle\mathcal{C}^{2}-\frac{1}{\sqrt{3}}\mathcal{S}^{2},$ (61)

and

$\displaystyle T_{AB}^{\Delta}=|T^{A\rightarrow B}(\rho_{AB})-T^{B\rightarrow
A}(\rho_{AB})|=\frac{1}{\sqrt{3}}\mathcal{S}^{4},$ (62)

respectively.

Figure 1: The fermionic steerability and steering asymmetry between Alice and
Bob as functions of the Hawking temperature $T$. The parameter $\omega$ is
fixed as $\omega=1$.

In Fig.1, we plot the $A\rightarrow B$ steering, $B\rightarrow A$ steering and
steering asymmetry as functions of the Hawking temperature $T$. We find that
the results for the two types of measures are consistent. With the increase of
the Hawking temperature $T$, the fermionic steerability between Alice and Bob
decreases firstly and then approaches to the nonzero asymptotic value in the
infinite Hawking temperature. The $A\rightarrow B$ fermionic steerability is
always bigger than the $B\rightarrow A$ fermionic steerability. The steering
asymmetry increases with Hawking temperature and approaches to the asymptotic
values,

$\lim_{T\rightarrow\infty}S_{AB}^{\Delta}\approx 0.0944,\hskip
28.45274pt\lim_{T\rightarrow\infty}T_{AB}^{\Delta}=\frac{\sqrt{3}}{12},$

for infinite Hawking temperature. These results contrast sharply with case of
bosonic fields Q53 ; Q54 ; Q59 . With the increase of Hawking temperature, the
bosonic steerability reduces quickly and suffers from a“sudden death”. The
$A\rightarrow B$ bosonic steerability is always smaller than the $B\rightarrow
A$ bosonic steerability, and the bosonic steering asymmetry increases firstly
and then decreases to zero when Hawking temperature increases. These clearly
different results between bosonic steerability and fermionic steerability
originate from the difference between the Fermi-Dirac statistic and the Bose-
Einstein statistic, which perhaps are available in practice. For example, if
we need the steerability from Alice to Bob over the steerability from Bob to
Alice in curved spacetime, then we should use fermionic steering rather than
bosonic steering.

###  IV.2  Physically inaccessible quantum steering

Besides the steering between Alice and Bob, we can also discuss the steering
between Alice and Anti-Bob, and the steering between Bob and Anti-Bob. As
Anti-Bob is inside of the event horizon, we use the term “inaccessible
steering”.

(i) The fermionic steering between Alice and Anti-Bob. Tracing over the mode
$B$ held by Bob, we obtain the density matrix for subsystem Alice and Anti-Bob

$\displaystyle\rho_{A\bar{B}}=\frac{1}{2}\left(\\!\\!\begin{array}[]{cccccccc}\mathcal{C}^{2}&0&0&0\\\
0&\mathcal{S}^{2}&\mathcal{S}&0\\\ 0&\mathcal{S}&1&0\\\
0&0&0&0\end{array}\\!\\!\right).$ (67)

Following the calculation steps in above subsection, we obtain the fermionic
steering between Alice and Anti-Bob, as well as the corresponding steering
asymmetry for the two types of quantifications as

$\displaystyle S^{A\rightarrow\bar{B}}(\rho_{A\bar{B}})$ $\displaystyle=$
$\displaystyle\max\bigg{\\{}0,\frac{1}{4}\big{[}2(1+\mathcal{S})\log(1+\mathcal{S})+2(1-\mathcal{S})\log(1-\mathcal{S})$
(68) $\displaystyle+$
$\displaystyle\mathcal{C}^{2}\log(\mathcal{C}^{2})+\mathcal{S}^{2}\log(\mathcal{S}^{2})\big{]}\bigg{\\}},$
$\displaystyle S^{\bar{B}\rightarrow A}(\rho_{AB})$ $\displaystyle=$
$\displaystyle\max\bigg{\\{}0,\frac{1}{4}\big{[}2(1+\mathcal{S})\log(1+\mathcal{S})+2(1-\mathcal{S})\log(1-\mathcal{S})$
(69) $\displaystyle-$
$\displaystyle(1+\mathcal{C}^{2})\log(1+\mathcal{C}^{2})+\mathcal{C}^{2}\log(\mathcal{C}^{2})\big{]}\bigg{\\}},$
$\displaystyle
S_{A\bar{B}}^{\Delta}=|S^{A\rightarrow\bar{B}}(\rho_{A\bar{B}})-S^{\bar{B}\rightarrow
A}(\rho_{A\bar{B}})|,$ (70)

and

$\displaystyle
T^{A\rightarrow\bar{B}}(\rho_{A\bar{B}})=\max\bigg{\\{}0,\mathcal{S}^{2}(1-\frac{1}{\sqrt{3}}\mathcal{C}^{2})\bigg{\\}},$
(71) $\displaystyle T^{\bar{B}\rightarrow
A}(\rho_{A\bar{B}})=\max\bigg{\\{}0,\mathcal{S}^{2}-\frac{1}{\sqrt{3}}\mathcal{C}^{2}\bigg{\\}}$
(72) $\displaystyle
T_{A\bar{B}}^{\Delta}=|T^{A\rightarrow\bar{B}}(\rho_{A\bar{B}})-T^{\bar{B}\rightarrow
A}(\rho_{A\bar{B}})|.$ (73)

Figure 2: The fermionic steerability and steering asymmetry between Alice and
Anti-Bob as functions of the Hawking temperature $T$. The parameter $\omega$
is fixed as $\omega=1$.

Fig.2 shows how the Hawking temperature $T$ of the black hole influences the
fermionic steerability and steering asymmetry between Alice and Anti-Bob. We
see that results for the two type of quantifications are basically similar.
Hawking radiation can generate fermionic steering between Alice and Anti-Bob,
while the temperature for generating $A\rightarrow\bar{B}$ steering is always
lower than the temperature for generating $\bar{B}\rightarrow A$ steering. As
the temperatures and speeds for generating $A\rightarrow\bar{B}$ steering and
$\bar{B}\rightarrow A$ steering are different, the steering asymmetry appears.
The steering asymmetry firstly increases to the maximum and then decreases to
the nonzero asymptotic value with the increase of the Hawking temperature $T$.
The maximal steering asymmetry takes place at the temperature where the
$\bar{B}\rightarrow A$ steering births, i.e., takes place at the transition
point from one-way steering to two-way steering. For the parameters taken in
the figure, the Hawking temperatures for the maximal steering asymmetry are
$T_{s}\approx 5.8021\omega$ [Fig.2(a)] and $T_{t}=\frac{\omega}{\ln\sqrt{3}}$
[Fig.2(b)], respectively. The figure also shows that the steerability between
Alice and Anti-Bob approaches to the finite asymptotic value when
$T\rightarrow\infty$, which fulfills the intriguing relations,

$\lim_{T\rightarrow\infty}S^{A\rightarrow\bar{B}}=\lim_{T\rightarrow\infty}S^{A\rightarrow
B},\lim_{T\rightarrow\infty}S^{\bar{B}\rightarrow
A}=\lim_{T\rightarrow\infty}S^{B\rightarrow A},$
$\lim_{T\rightarrow\infty}T^{A\rightarrow\bar{B}}=\lim_{T\rightarrow\infty}T^{A\rightarrow
B},\lim_{T\rightarrow\infty}T^{\bar{B}\rightarrow
A}=\lim_{T\rightarrow\infty}T^{B\rightarrow A}.$

This means that the steering status for Bob and Anti-Bob are the same when
$T\rightarrow\infty$.

Comparing Fig.2(a) and (b), we find that the two types of quantifications for
steering also have some tiny difference. For quantification of steering based
on entropy uncertainty relation, both $S^{A\rightarrow\bar{B}}$ and
$S^{\bar{B}\rightarrow A}$ appear “sudden birth” behavior with the growth of
the Hawking temperature. For the quantification of steering on quantum
entanglement, however, only the steerability $T^{\bar{B}\rightarrow A}$ appear
as “sudden birth”. Further, the temperature for generating two-way steering
for entanglement-based quantification is lower than the case for the
quantification based on entropy uncertainty relation. In this sense, we can
say that the quantification of steering based on entanglement is more
sensitive than the quantification of steering based on entropy uncertainty
relation.

It has been shown that Hawking radiation for bosonic fields cannot generate
entanglement and steering between Alice and Anti-BobQ40 ; Q54 ; QWE59 , which
makes sharp contrast with the fermionic steering discussed above. We may
understand the reason as follows: The Hawking radiation for bosonic fields is
equivalent to a local operation on the subsystem of Bob and Anti-Bob, which
cannot generate bosonic entanglement or steering between Alice and Anti-Bob.
For the fermionic fields, however, the Pauli exclusion principle inhibits the
production of more than one fermion in one mode. This limitation forms an
interaction between Alice and Bob (or Ant-Bob). Therefore, the entanglement
(or steering) between Alice and Anti-Bob can be produced Q60 .

(ii) The fermionic steering between Bob and Anti-Bob. Tracing over the mode
$A$, we obtain the density matrix for subsystem of Bob and Anti-Bob

$\displaystyle\rho_{B\bar{B}}=\frac{1}{2}\left(\\!\\!\begin{array}[]{cccccccc}\mathcal{C}^{2}&0&0&\mathcal{C}\mathcal{S}\\\
0&0&0&0\\\ 0&0&1&0\\\
\mathcal{C}\mathcal{S}&0&0&\mathcal{S}^{2}\end{array}\\!\\!\right).$ (78)

The fermionic steering between Bob and Anti-Bob are thus calculated as

$\displaystyle S^{B\rightarrow\bar{B}}(\rho_{A\bar{B}})$ $\displaystyle=$
$\displaystyle\max\bigg{\\{}0,\frac{1}{4}\big{[}2(1+\mathcal{C}\mathcal{S})\log(1+\mathcal{C}\mathcal{S})+2(1-\mathcal{C}\mathcal{S})\log(1-\mathcal{C}\mathcal{S})$
(79) $\displaystyle-$
$\displaystyle(1+\mathcal{S}^{2})\log(1+\mathcal{S}^{2})+\mathcal{S}^{2}\log(\mathcal{S}^{2})\big{]}\bigg{\\}}=0,$
$\displaystyle S^{\bar{B}\rightarrow B}(\rho_{A\bar{B}})$ $\displaystyle=$
$\displaystyle\max\bigg{\\{}0,\frac{1}{4}\big{[}2(1+\mathcal{C}\mathcal{S})\log(1+\mathcal{C}\mathcal{S})+2(1-\mathcal{C}\mathcal{S})\log(1-\mathcal{C}\mathcal{S})$
(80) $\displaystyle-$
$\displaystyle(1+\mathcal{C}^{2})\log(1+\mathcal{C}^{2})+\mathcal{C}^{2}\log(\mathcal{C}^{2})\big{]}\bigg{\\}}=0,$

and

$\displaystyle
T^{B\rightarrow\bar{B}}(\rho_{B\bar{B}})=\max\bigg{\\{}0,\mathcal{S}^{2}\big{(}\mathcal{C}^{2}-\frac{1}{\sqrt{3}}\big{)}\bigg{\\}},$
(81) $\displaystyle T^{\bar{B}\rightarrow
B}(\rho_{B\bar{B}})=\max\bigg{\\{}0,\mathcal{C}^{2}\big{(}\mathcal{S}^{2}-\frac{1}{\sqrt{3}}\big{)}\bigg{\\}}=0.$
(82)

From Eqs.(79)-(82) and combined with Fig.3, we find that Hawking radiation
cannot produce fermionic steering based on entropy uncertainty relation
between Bob and Anti-Bob, but can produce the entanglement-based steering
$T^{B\rightarrow\bar{B}}$. It again suggests that the quantification of
steering based on entanglement is a more sensitive than that based on entropy
uncertainty relation.

Figure 3: The fermionic steerability and steering asymmetry between Bob and
Anti-Bob as functions of the Hawking temperature $T$. The parameter $\omega$
is fixed as $\omega=1$.

Fig.3(b) shows that the fermionic steering $T^{B\rightarrow\bar{B}}$ increases
from zero to the maximum and then decreases to zero again (sudden death) with
the increasing of Hawking temperature. The Hawking temperature for the maximal
steering is $T_{max}=\frac{\omega}{\ln(\sqrt{3}+1)-\ln(\sqrt{3}-1)}$, and for
sudden death is $T_{dea}=\frac{-\omega}{\ln(\sqrt{3}-1)}$. This result is
quite different from the behavior of fermionic entanglement between Bob to
Anti-Bob in a relativistic setting, where fermionic entanglement increases
monotonically Q60 .

##  V  Monogamous relation between fermionic steering and entanglement in
Schwarzschild spacetime

It is well known that quantum steering is an intermediate form of quantum
inseparabilities in between Bell nonlocality and quantum entanglement Q3 .
Quantum states that show Bell nonlocality form a strict subset of quantum
states showing quantum steering, the latter also form a strict subset of
entangled states. Quantum steering is a good potential object for connecting
Bell nonlocality and quantum entanglement. In this section, we try to
establish the relations between fermionic steering and entanglement in
Schwarzschild spacetime.

From above study, we find that with the increase of the Hawking temperature
$T$, the physically accessible fermionic steering between Alice and Bob
decreases monotonically, and at the same time the physically inaccessible
fermionic steering between Alice (Bob) and Anti-Bob increases monotonically or
nonmonotonically. Also note that the entanglement has similar properties: The
physically accessible fermionic entanglement decreases monotonically and the
physically inaccessible fermionic entanglement increases monotonically with
the growth of the Hawking temperature Q60 . A question arises naturally: Are
there relations between the physically accessible steering (entanglement) and
the physically inaccessible steering (entanglement)? The answer is yes.
Through analytical calculation, we find some relations of this types, which
can help us to understand more deeply the fermionic entanglement and steering
in a relativistic setting.

From Eqs.(56), (67) and (78), we can calculate the concurrence of entanglement
$C(\rho_{AB})$ between Alice and Bob, the concurrence $C(\rho_{A\bar{B}})$
between Alice and Anti-Bob, as well as the concurrence $C(\rho_{B\bar{B}})$
between Bob and Anti-Bob as,

$C(\rho_{AB})=\frac{1}{\sqrt{e^{-\frac{\omega}{T}}+1}},$
$C(\rho_{A\bar{B}})=\frac{1}{\sqrt{e^{\frac{\omega}{T}}+1}},$

and

$C(\rho_{B\bar{B}})=\frac{1}{\sqrt{e^{\frac{\omega}{T}}+e^{-\frac{\omega}{T}}+2}}.$

After carefully inspection, we find the following monogamy relations between
fermionic steering and entanglement,

$\displaystyle T^{A\rightarrow
B}(\rho_{AB})-T^{A\rightarrow\bar{B}}(\rho_{A\bar{B}})=C^{2}(\rho_{AB})-C^{2}(\rho_{A\bar{B}}),$
(83) $\displaystyle T^{A\rightarrow
B}(\rho_{AB})+T^{A\rightarrow\bar{B}}(\rho_{A\bar{B}})=C^{2}(\rho_{AB})+C^{2}(\rho_{A\bar{B}})-\frac{2}{\sqrt{3}}C^{2}(\rho_{B\bar{B}}),$
(84) $\displaystyle\frac{3-\sqrt{3}}{2}[T^{B\rightarrow
A}(\rho_{AB})-T^{\bar{B}\rightarrow
A}(\rho_{A\bar{B}})]=C^{2}(\rho_{AB})-C^{2}(\rho_{A\bar{B}}),~{}{\rm for}$
$\displaystyle\frac{\omega}{\ln\sqrt{3}}<T,$ (85)
$\displaystyle\frac{3+\sqrt{3}}{2}[T^{B\rightarrow
A}(\rho_{AB})+T^{\bar{B}\rightarrow
A}(\rho_{A\bar{B}})]=C^{2}(\rho_{AB})+C^{2}(\rho_{A\bar{B}}),~{}{\rm for}$
$\displaystyle\frac{\omega}{\ln\sqrt{3}}<T.$ (86)

These monogamies reveal the relation between the physically accessible
correlation and the physically inaccessible correlation, as well as the
relation between entanglement and steering. They suggest that the Hawking
radiation can give rise to the transformation between these different types of
quantum correlations.

##  VI Conclusions

In conclusion, we have investigated the influence of Hawking radiation on the
fermionic quantum steering for the setup where Alice resides in the
asymptotically flat region and Bob hovers near the event horizon of a
Schwarzschild black hole. Two different types of quantification for quantum
steering have been employed. The redistribution and transformation between
physically accessible and inaccessible steering induced by Hawking radiation
have been studied. Some monogamy relations between fermionic steering and
entanglement have been found.

Firstly, we have found that Hawking radiation reduces the fermionic
steerability between Alice and Bob, and make it approaching to the nonzero
asymptotic values in the limit of infinite Hawking temperature. The
steerability from Alice to Bob is always bigger than the steerability from Bob
to Alice in the degradation process. The steering asymmetry increases with
Hawking temperature and approaches to the nonzero asymptotic values for
infinite Hawking temperature. For the degradation of steering between Alice
and Bob, the two types of quantification for steerability behave completely
consistent.

Secondly, we have found that Hawking radiation can produce fermionic steering
between Alice and Anti-Bob, which approaches to the finite asymptotic values
in the limit of infinite Hawking temperature. The temperature for generating
$A\rightarrow\bar{B}$ steering is lower than for generating
$\bar{B}\rightarrow A$ steering. Also, the $A\rightarrow\bar{B}$ steerability
is always greater than $\bar{B}\rightarrow A$ steerability. The steering
asymmetry firstly increases, then decreases and finally approaches to a
nonzero asymptotic value when the Hawking temperature changes from zero to
infinite. The maximal steering asymmetry occurs at the transition point from
one-way steering to two-way steering. Note that here the two types of
quantification for steering behave slightly different: The temperature for
generating steering based on entanglement is lower than for generating
steering based on entropy uncertainty relation, meaning that the
quantification of steering based on entanglement is more sensitive than that
based on entropy uncertainty relation.

Thirdly, we have also studied the effect of Hawking radiation on the steering
between Bob and Anti-Bob. We have found that the two types of quantification
for steering in this case behave clearly different. The steering between Bob
and Anti-Bob based on entropy uncertainty relation is always zero, but the
steering from Bob to Anti-Bob based on entanglement can be generated for some
domain of Hawking temperature. This again suggests that the quantification of
steering based on entanglement is more sensitive than that based on entropy
uncertainty relation. On the other hand, quantum steering between Bob hovering
near the event horizon and anti-Bob segregated by the event horizon is
different from the other two forms of quantum steering, possibly because Alice
stays stationary at an asymptotically flat region.

Finally, we have established some monogamy relations between fermionic
steering and fermionic entanglement. These monogamies reveal the regularity
for the redistribution of steering and entanglement between different
subsystems under the Hawking effect, and may be useful for understanding the
information paradox of black holes.

We have also made a comparison between fermionic steering studied in this
context and bosonic steering studied previously under the influence of Hawking
radiation. The previous study suggested that the bosonic steering between
Alice and Bob suffers from sudden death in the process of degradation, and the
$A\rightarrow B$ bosonic steerability is always smaller than the $B\rightarrow
A$ bosonic steerability under the Hawking effect Q53 ; Q54 . Hawking effect
cannot generate the bosonic steering between Alice and Anti-Bob Q40 ; Q54 ;
QWE59 . All these results are opposite to the fermionic steering displayed in
the text.

###### Acknowledgements.

This work is supported by the National Natural Science Foundation of China
(Grant Nos.1217050862, 11275064), and 2021BSL013.

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