# Does Hawking effect always degrade fidelity of quantum teleportation in
Schwarzschild spacetime?

Shu-Min<EMAIL_ADDRESS>Xiao-Wei<EMAIL_ADDRESS>(corresponding author), Rui-Di Wang1, Hao-Yu Wu1, Xiao-Li Huang1333
<EMAIL_ADDRESS>(corresponding author), Hao-Sheng
<EMAIL_ADDRESS>1 Department of Physics, Liaoning Normal
University, Dalian 116029, China
2 Department of Physics, Hunan Normal University, Changsha 410081, China

###### Abstract

Previous studies have shown that the Hawking effect always destroys quantum
correlations and the fidelity of quantum teleportation in the Schwarzschild
black hole. Here, we investigate the fidelity of quantum teleportation of
Dirac fields between users in Schwarzschild spacetime. We find that, with the
increase of the Hawking temperature, the fidelity of quantum teleportation can
monotonically increase, monotonically decrease, or non-monotonically increase,
depending on the choice of the initial state, which means that the Hawking
effect can create net fidelity of quantum teleportation. This striking result
banishes the extended belief that the Hawking effect of the black hole can
only destroy the fidelity of quantum teleportation. We also find that quantum
steering cannot fully guarantee the fidelity of quantum teleportation in
Schwarzschild spacetime. This new unexpected source may provide a new idea for
the experimental evidence of the Hawking effect.

###### pacs:

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

##  I Introduction

Quantum teleportation, first proposed by Bennett $et$ $al.$ L1 , is one of the
most crucial applications for quantum information, which was experimentally
proved by Bouwmeester $et$ $al.$ L2  by using single photons. Quantum
teleportation is a basic protocol for transmitting quantum information from
one object to another object by shared quantum entanglement, where the
spatially separated sender and receiver can only perform local operations and
communicate between themselves via a classical channel L3 . Quantum
teleportation is the foundation of quantum information and an important part
of practical quantum technology, which has attracted wide attention H1 ; H2 ;
H3 ; H4 ; H5 ; H6 ; H7 ; H8 ; H9 ; H10 . In addition, quantum steering,
formalized from the viewpoint of quantum information theory L4 ; L5 , is a
concept first introduced by Schrödinger in 1935 L6 ; L7 . Quantum steering
refers to the impossibility of describing one party’s conditional state by a
local hidden state model in the modern view. In other words, quantum steering
allows one observer to control a remote subsystem of another observer owned by
measuring his subsystem. Thus, quantum steering represents the quantum
correlation between quantum entanglement and Bell nonlocality. Unlike quantum
entanglement, quantum steering has richer properties in quantum systems, such
as two-way steering, one-way steering, and no-way steering, which have been
experimentally demonstrated L8 ; L9 ; L10 ; L11 . Because quantum steering is
a crucial resource, the problem of describing the optimal two-qubit state of
quantum teleportation under a fixed steering amount is interesting and open
L12 .

From the perspective of Einstein’s theory, the gravitational collapse of
sufficiently massive stars creates black holes that are fascinating objects in
our universe. With the development of astronomy, the existence of black holes
has been indirectly or directly confirmed. For example, (i): the advanced LIGO
detector and Virgo detector have detected gravitational waves for the first
time in a binary black hole merger system L13 ; (ii): the first image of a
supermassive black hole has been taken by the Event Horizon Telescope at the
center of the giant elliptical galaxy M87 L14 ; L15 ; L16 ; L17 ; L18 ; L19 ;
(iii): the Event Horizon Telescope has photographed Sgr A* L20 . Black hole
physics, while making some progress, is still shrouded in mystery, such as the
black hole information paradox. As is well-known, the two pillars of modern
physics are general relativity and quantum mechanics, of which the unification
remains an open question. In order to solve this contradiction, relativistic
quantum information attempts to bridge the gap between general relativity and
quantum mechanics. Recently, Pan and his team have used the “Micius” quantum
communication satellite to complete the quantum optical test of the
gravitational decoherence effect, indicating that the study of relativistic
quantum information has entered the stage of precision experimental
verification L21 . On the simulation side, there is a general interest in
simulating the Hawking radiation of the black hole and the cosmological
particle generation in quantum systems L22 ; L23 ; L24 ; L25 ; L26 ; L27 ; L28
; L29 . In theory, the Hawking effect of the black hole always has a negative
influence on quantum steering, entanglement, discord, coherence, and the
fidelity of quantum teleportation of bosonic fields under the case in curved
spacetime L38 ; L39 ; L40 ; L41 ; L42 ; L43 ; L44 ; L46 ; L47 ; L49 ; L50 ;
L51 ; L52 ; L53 ; L54 ; L55 ; L56 ; L57 ; QLQ57 . Therefore, one of our
motivations is to investigate whether the Hawking effect of the black hole
always reduces the fidelity of quantum teleportation of Dirac fields. Another
motivation is to discuss whether the Hawking effect has the same effect on
quantum steering and the fidelity of quantum teleportation.

In this paper, we investigate quantum teleportation of Dirac fields between
users in Schwarzschild spacetime. We assume that Alice and Bob initially share
an X-type state and they apply a standard teleportation scheme (STS) to send
the unknown state from Alice to Bob. Here, the sender Alice stays stationary
at an asymptotically flat region, while the receiver Bob hovers near the event
horizon of the black hole. Pan and Jing have found that the fidelity of
quantum teleportation of bosonic fields decreases with the increase of the
Hawking temperature L39 . However, we find that the Hawking effect of the
black hole has both positive and negative effects on the fidelity of quantum
teleportation of Dirac fields; this means that the Hawking effect can not only
reduce the fidelity of quantum teleportation but also increase the fidelity in
Schwarzschild spacetime. We also find that the influence of the Hawking effect
on quantum teleportation is not the same as that on quantum steering, showing
that quantum steering cannot guarantee the fidelity of quantum teleportation
in curved spacetime.

The structure of the paper is as follows. In Sec. II, we briefly introduce the
fidelity of quantum teleportation for the X-type state. In Sec. III, we
describe the quantization of Dirac fields in Schwarzschild spacetime. In Sec.
IV, we study the influence of the Hawking effect on the fidelity of quantum
teleportation and quantum steering in Schwarzschild spacetime. The last
section is devoted to the summary.

##  II The fidelity of quantum teleportation for X-type state

In this paper, we consider the universally common X-type state of the
bipartite system, and its density matrix can be written as

$\displaystyle\rho_{X}=\left(\\!\\!\begin{array}[]{cccc}\rho_{11}&0&0&-\rho_{14}\\\
0&\rho_{22}&-\rho_{23}&0\\\ 0&-\rho_{23}&\rho_{33}&0\\\
-\rho_{14}&0&0&\rho_{44}\end{array}\\!\\!\right).$ (5)

Eq.(5) describes the effective quantum states satisfying the unit trace and
positive conditions:$\rho_{11}+\rho_{22}+\rho_{33}+\rho_{44}=1$,
$\rho_{22}\rho_{33}\geqslant|\rho_{23}|^{2}$, and
$\rho_{11}\rho_{44}\geqslant|\rho_{14}|^{2}$. If
$\rho_{22}\rho_{33}<|\rho_{14}|^{2}$ or $\rho_{11}\rho_{44}<|\rho_{23}|^{2}$,
the X-type state is entangled.

In general teleportation scheme, we assume that Alice and Bob initially share
an X-type state $\rho_{X}$ in an asymptotically flat region. The unknown pure
state that can be teleported from Alice to Bob is represented by
$|\phi\rangle$. Alice and Bob can use some trace-preserving and local quantum
operations and classical communication (LOCC) operations for their respective
systems. After these operations, the final state of Bob takes the form

$\rho_{B}={\rm Tr}_{A,C}[M(|\phi\rangle\langle\phi|\otimes\rho_{X})],$

where $M$ denotes the trace-preserving LOCC operation.

Note that the dimension of Hilbert space $H_{A}\otimes H_{B}=C^{d}\otimes
C^{d}$ is $d$. Therefore, the fidelity of quantum teleportation that is
considered as a measure of the quality of quantum teleportation reads L58

$\displaystyle F=\langle\phi|\rho_{B}|\phi\rangle=\frac{fd+1}{d+1},$ (6)

where $f$ is the fully entangled fraction. The fidelity of quantum
teleportation achievable can be entirely decided by the fully entangled
fraction of the bipartite state in the STS, which is written as L59

$\displaystyle f(\rho)=\max_{\varphi}\langle\varphi|\rho|\varphi\rangle,$ (7)

where $|\varphi\rangle$ covers all maximally entangled states. STS requires
the X-type state to meet the condition $f>1/d$ for providing better fidelity
than classical communication, i.e., the quantum region. In this paper, we only
focus on this state in this region.

If the elements of the density matrix in Eq.(5) satisfy the conditions
$\rho_{22}+\rho_{33}\geqslant\frac{1}{2}$ and
$\rho_{23}\geqslant\frac{1}{2}(1-\rho_{22}-\rho_{33})$, the fully entangled
fraction can be expressed as L60

$\displaystyle
f(\rho_{X})=\frac{1}{2}(\rho_{22}+\rho_{33}+2\rho_{23})\geqslant\frac{1}{2}.$
(8)

Because we only pay attention to the quantum region with $f>1/d=1/2$, the
above conditions are supposed for the X-type state in the following.

##  III Quantization of Dirac fields in Schwarzschild spcetime

Firstly, we briefly review the vacuum structure of Dirac particles in
Schwarzschild spacetime. The metric of Schwarzschild spacetime can be given as
L40

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle-(1-\frac{2M}{r})dt^{2}+(1-\frac{2M}{r})^{-1}dr^{2}$ (9)
$\displaystyle+r^{2}(d\theta^{2}+\sin^{2}\theta d\varphi^{2}),$

where $M$ and $r$ are the mass and radius of the black hole, respectively. We
take $c,G,\hbar$ and $k$ as unity for simplicity in this paper. The Dirac
equation L62  $[\gamma^{a}e_{a}{}^{\mu}(\partial_{\mu}+\Gamma_{\mu})]\Phi=0$
in Schwarzschild spacetime can be expressed as follows

$\displaystyle-\frac{\gamma_{0}}{\sqrt{1-\frac{2M}{r}}}\frac{\partial\Phi}{\partial
t}+\gamma_{1}\sqrt{1-\frac{2M}{r}}\bigg{[}\frac{\partial}{\partial
r}+\frac{1}{r}+\frac{M}{2r(r-2M)}\bigg{]}\Phi$
$\displaystyle+\frac{\gamma_{2}}{r}(\frac{\partial}{\partial\theta}+\frac{\cot\theta}{2})\Phi+\frac{\gamma_{3}}{r\sin\theta}\frac{\partial\Phi}{\partial\varphi}=0,$
(10)

where $\gamma_{i}$ ($i=0,1,2,3$) represent the Dirac matrices L63 ; L64 .
Having solved the Dirac equation near the event horizon of the black holes, we
gain positive frequency outgoing solutions outside and inside regions of the
event horizon as

$\displaystyle\Phi^{+}_{{\mathbb{k}},{\rm out}}\sim\phi(r)e^{-i\omega u},$
(11) $\displaystyle\Phi^{+}_{{\mathbb{k}},{\rm in}}\sim\phi(r)e^{i\omega u},$
(12)

where $\phi(r)$ represents four-component Dirac spinor, the retarded
coordinate $u=t-r_{*}$ with the tortoise coordinate
$r_{*}=r+2M\ln\frac{r-2M}{2M}$ L63 ; L64 . Here, $\omega$ and $\mathbb{k}$
represent the frequency and wave vector, respectively, which fulfill
$\omega=|\mathbf{k}|$ for the massless Dirac field. Using Eqs.(11) and (12),
the Dirac field $\Phi$ can be expanded as

$\displaystyle\Phi$ $\displaystyle=$ $\displaystyle\int
d\mathbb{k}[\hat{a}^{\rm out}_{\mathbb{k}}\Phi^{+}_{\rm
out,\mathbb{k}}+\hat{b}^{\rm out{\dagger}}_{\mathbb{-}k}\Phi^{-}_{\rm
out,\mathbb{k}}+\hat{a}^{\rm in}_{\mathbb{k}}\Phi^{+}_{\rm
in,\mathbb{k}}+\hat{b}^{\rm in{\dagger}}_{\mathbb{-}k}\Phi^{-}_{\rm
in,\mathbb{k}}],$ (13)

where $a^{\rm out}_{\mathbb{k}}$ and $b^{\rm out{\dagger}}_{\mathbb{k}}$ are
the fermionic annihilation and antifermionic creation operators which
correspond to the state in the exterior regions of the event horizon,
respectively, and $a^{\rm in}_{\mathbb{k}}$ and $b^{\rm
in{\dagger}}_{\mathbb{k}}$ are the fermionic annihilation and antifermionic
creation operators which correspond to the state in the interior regions of
the event horizon, respectively L42 ; L50 .

According to the suggestion of Domour-Ruffini, we can use Kruskal modes to
make analytic continuations for Eqs.(11) and (12) HS1 . However, the Kruskal
observer can be free to create excitations in any accessible mode. Therefore,
the single-frequency Kruskal mode cannot be mapped to a group of single-
frequency Schwarzschild modes HL1 . To avoid this incongruity, we can adopt
the Unruh mode HL1 ; HL2 ; HL3 ; HL4 ; HL5 , which provides an intermediate
bridge between the Kruskal and Schwarzschild modes. The Unruh operators have
the simple Bogoliubov transformations with Schwarzschild modes, which take the
forms as

$\displaystyle\tilde{c}_{\mathbb{k},R}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{e^{-\frac{\omega}{T}}+1}}\hat{a}^{\rm
out}_{\mathbb{k}}-\frac{1}{\sqrt{e^{\frac{\omega}{T}}+1}}\hat{b}^{\rm
in{\dagger}}_{\mathbb{-k}},$ $\displaystyle\tilde{c}_{\mathbb{k},L}$
$\displaystyle=$
$\displaystyle\frac{1}{\sqrt{e^{-\frac{\omega}{T}}+1}}\hat{a}^{\rm
in}_{\mathbb{k}}-\frac{1}{\sqrt{e^{\frac{\omega}{T}}+1}}\hat{b}^{\rm
out{\dagger}}_{\mathbb{-k}},$
$\displaystyle\tilde{c}^{{\dagger}}_{\mathbb{k},R}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{e^{-\frac{\omega}{T}}+1}}\hat{a}^{\rm
out{\dagger}}_{\mathbb{k}}-\frac{1}{\sqrt{e^{\frac{\omega}{T}}+1}}\hat{b}^{\rm
in}_{\mathbb{-k}},$ $\displaystyle\tilde{c}^{{\dagger}}_{\mathbb{k},L}$
$\displaystyle=$
$\displaystyle\frac{1}{\sqrt{e^{-\frac{\omega}{T}}+1}}\hat{a}^{\rm
in{\dagger}}_{\mathbb{k}}-\frac{1}{\sqrt{e^{\frac{\omega}{T}}+1}}\hat{b}^{\rm
out}_{\mathbb{-k}}.$ (14)

Here, the subscripts $R$ and $L$ represent the ”right” and ”left” modes,
respectively. Using the operator ordering $\hat{a}^{\rm
out}_{\mathbb{k}}\hat{b}^{\rm in}_{\mathbb{-k}}\hat{b}^{\rm
out}_{\mathbb{-k}}\hat{a}^{\rm in}_{\mathbb{k}}$, the Unruh vacuum is given by

$\displaystyle|0\rangle_{\rm U}$ $\displaystyle=$
$\displaystyle\frac{1}{e^{-\frac{\omega}{T}}+1}|0000\rangle-\frac{1}{\sqrt{e^{\frac{\omega}{T}}+e^{-\frac{\omega}{T}}+2}}|0101\rangle$
(15)
$\displaystyle+\frac{1}{\sqrt{e^{\frac{\omega}{T}}+e^{-\frac{\omega}{T}}+2}}|1010\rangle-\frac{1}{e^{\frac{\omega}{T}}+1}|1111\rangle,$

where $T=\frac{1}{8\pi M}$ is the Hawking temperature HL6 , and
$|mm^{\prime}n^{\prime}n\rangle=|m_{\mathbb{k}}\rangle^{+}_{\rm
out}|m^{\prime}_{\mathbb{-k}}\rangle^{-}_{\rm
in}|n^{\prime}_{\mathbb{-k}}\rangle^{-}_{\rm
out}|n_{\mathbb{k}}\rangle^{+}_{\rm in}$. Here,
$\\{|n_{\mathbb{k}}\rangle^{+}_{\rm out}\\}$ and
$\\{|n_{\mathbb{-k}}\rangle^{-}_{\rm in}\\}$ are the orthonormal bases for the
outside and inside regions of the Schwarzschild black hole, respectively. The
superscript $\\{+,-\\}$ represents the fermion and antifermion. For the
Schwarzschild observer hovering outside the event horizon, the Hawking
radiation spectrum from the perspective of an outside observer can be written
as $N_{F}==\frac{1}{e^{\frac{\omega}{T}}+1}$ L64 . We can see that the Unruh
vacuum observed by the Schwarzschild observer would be detected as a number of
the generated fermions $N_{F}$ corresponding to a thermal Fermion-Dirac
statistics of fermions. This is known as the Hawking radiation. Each fermionic
mode has only the first excited state due to the Pauli exclusion principle.
The Unruh excited state of the fermionic mode can be expanded as

$\displaystyle|1\rangle_{\rm U}$ $\displaystyle=$
$\displaystyle[q_{R}(\tilde{c}^{{\dagger}}_{\mathbb{k},R}\bigotimes
I_{L})+q_{L}(I_{R}\bigotimes\tilde{c}^{{\dagger}}_{\mathbb{k},L})]|0\rangle_{\rm
U}$ (16) $\displaystyle=$ $\displaystyle
q_{R}[\frac{1}{\sqrt{e^{-\frac{\omega}{T}}+1}}|1000\rangle-\frac{1}{\sqrt{e^{\frac{\omega}{T}}+1}}|1101\rangle]$
$\displaystyle+q_{L}[\frac{1}{\sqrt{e^{-\frac{\omega}{T}}+1}}|0001\rangle+\frac{1}{\sqrt{e^{\frac{\omega}{T}}+1}}|1011\rangle],$

with $|q_{R}|^{2}+|q_{L}|^{2}=1$.

The operator $\tilde{c}^{{\dagger}}_{\mathbb{k},R}$ in Eq.(III) represents the
creation of an antifermion in the interior vacuum and a fermion in the
exterior vacuum of the black hole, respectively. Similarly, the operator
$\tilde{c}^{{\dagger}}_{\mathbb{k},L}$ in Eq.(III) means that a fermion and an
antifermion are created inside and outside the event horizon of the black
hole, respectively. Hawking radiation is generated by quantum fluctuations
near the event horizon that spontaneously produce pairs of fermion and
antifermion. The fermion and antifermion can radiate toward the inside and
outside regions randomly from the event horizon with the total probability
$|q_{R}|^{2}+|q_{L}|^{2}=1$. Therefore, $q_{R}=1$ represents that all the
fermion moves to the outside of the event horizon of the black hole, while all
the antifermion moves toward the inside of the event horizon of the black
hole. This means that only fermion can be detected as Hawking radiation.
Analogously, $q_{L}=1$ means that only antifermion escapes from the event
horizon. Therefore, when only fermion (antifermion) is detected, the single
mode approximation for $q_{R}=1$ ($q_{L}=1$) is a special situation. When we
discuss our fidelity of quantum teleportation beyond the single mode
approximation, we investigate different kinds of Unruh modes with different
values of $q_{R}$.

##  IV Hawking effect on fidelity of quantum teleportation and quantum
steering in Schwarzschild spacetime

We assume that Alice and Bob initially share an X-type state for two Unruh
modes at an asymptotically flat region of the Schwarzschild black hole. Then,
Alice still stays stationary at an asymptotically flat region, while Bob
hovers near the event horizon of the black hole. Bob will detect the thermal
Fermi-Dirac particle distribution with his excited detector. According to
Eqs.(15) and (16), we can rewrite Eq.(5). Since Bob cannot access the modes
inside the event horizon of the black hole, we trace over the inaccessible
modes and obtain the reduced density matrix $\rho^{AB_{out}}$ (for detail
please see Appendix A).

We assume that Bob’s detector is sensitive only to the fermionic modes,
showing that the antifermionic modes cannot be excited in a single detector
when a fermion was detected. Therefore, we should trace out the antifermionic
mode $\\{|n_{\mathbb{-k}}\rangle^{-}_{\rm out}\\}$ outside the event horizon
of the Schwarzschild black hole

$\displaystyle\rho^{S}_{X}=\left(\\!\\!\begin{array}[]{cccccccc}\rho^{S}_{11}&0&0&-\rho^{S}_{14}\\\
0&\rho^{S}_{22}&-\rho^{S}_{23}&0\\\ 0&-\rho^{S}_{23}&\rho^{S}_{33}&0\\\
-\rho^{S}_{14}&0&0&\rho^{S}_{44}\end{array}\\!\\!\right),$ (21)

where

$\displaystyle\rho^{S}_{11}$ $\displaystyle=$
$\displaystyle(e^{-\frac{\omega}{T}}+1)^{-1}\rho_{11}+|q_{L}|^{2}(e^{-\frac{\omega}{T}}+1)^{-1}\rho_{22},$
$\displaystyle\rho^{S}_{22}$ $\displaystyle=$
$\displaystyle(e^{\frac{\omega}{T}}+1)^{-1}\rho_{11}+[1-|q_{L}|^{2}(e^{-\frac{\omega}{T}}+1)^{-1}]\rho_{22},$
$\displaystyle\rho^{S}_{33}$ $\displaystyle=$
$\displaystyle(e^{-\frac{\omega}{T}}+1)^{-1}\rho_{33}+|q_{L}|^{2}(e^{-\frac{\omega}{T}}+1)^{-1}\rho_{44},$
$\displaystyle\rho^{S}_{44}$ $\displaystyle=$
$\displaystyle(e^{\frac{\omega}{T}}+1)^{-1}\rho_{33}+[1-|q_{L}|^{2}(e^{-\frac{\omega}{T}}+1)^{-1}]\rho_{44},$
$\displaystyle\rho^{S}_{14}$ $\displaystyle=$ $\displaystyle
q_{R}(e^{-\frac{\omega}{T}}+1)^{-\frac{1}{2}}\rho_{14},$
$\displaystyle\rho^{S}_{23}$ $\displaystyle=$ $\displaystyle
q_{R}(e^{-\frac{\omega}{T}}+1)^{-\frac{1}{2}}\rho_{23}.$ (22)

We assume that the state $\rho^{S}_{X}$ satisfies the condition

$(e^{\frac{\omega}{T}}+1)^{-1}\rho_{11}+\rho_{22}+(e^{-\frac{\omega}{T}}+1)^{-1}\rho_{33}\geqslant\frac{1}{2}.$

Therefore, we obtain

$\displaystyle f(\rho^{S}_{X})$ $\displaystyle=$
$\displaystyle\frac{1}{2}\big{\\{}(e^{\frac{\omega}{T}}+1)^{-1}\rho_{11}+\big{[}1-|q_{L}|^{2}(e^{-\frac{\omega}{T}}+1)^{-1}\big{]}\rho_{22}+(e^{-\frac{\omega}{T}}+1)^{-1}\rho_{33}$
(23)
$\displaystyle+|q_{L}|^{2}(e^{-\frac{\omega}{T}}+1)^{-1}\rho_{44}+2q_{R}(e^{-\frac{\omega}{T}}+1)^{-\frac{1}{2}}\rho_{23}\big{\\}}.$

The change of $f(\rho^{S}_{X})$ related to the Hawking temperature can be
expressed as

$\displaystyle\Delta_{T}f(\rho^{S}_{X}(T))$ $\displaystyle\equiv$
$\displaystyle f(\rho^{S}_{X}(T=T_{0}))-f(\rho^{S}_{X}(T=0))$ (24)
$\displaystyle=$
$\displaystyle\frac{1}{2}\big{\\{}(e^{\frac{\omega}{T}}+1)^{-1}\rho_{11}+q^{2}_{L}(e^{\frac{\omega}{T}}+1)^{-1}\rho_{22}-(e^{\frac{\omega}{T}}+1)^{-1}\rho_{33}$
$\displaystyle-q^{2}_{L}(e^{\frac{\omega}{T}}+1)^{-1}\rho_{44}-2q_{R}\rho_{23}\big{[}1-(e^{-\frac{\omega}{T}}+1)^{-\frac{1}{2}}\big{]}\big{\\}}.$

The derivative of $f(\rho^{S}_{X})$ with respect to Hawking temperature $T$
can be written as $\frac{\partial f(\rho^{S}_{X})}{\partial T}$. We can easily
obtain that $\frac{\partial f(\rho^{S}_{X})}{\partial T}|_{T_{0}}>0$ means
$\Delta f(\rho^{S}_{X}(T_{0}))>0$, and $\Delta f(\rho^{S}_{X}(T_{0}))<0$ means
$\frac{\partial f(\rho^{S}_{X})}{\partial T}|_{T_{0}}<0$.

In order to investigate whether quantum steering can guarantee the fidelity of
quantum teleportation, we calculate quantum steering from Alice to Bob
$S^{A\rightarrow B}(T)$ and quantum steering from Bob to Alice
$S^{B\rightarrow A}(T)$ (for detail please see Appendix B). There is no
ambiguous map related to the anticommutation properties of field operators in
the quantum teleportation and steering for the same initial X-type state
$\rho_{X}$ in Schwarzschild spacetime (for detail please see Appendix C) HL7 ;
HL8 ; HL9 ; HL10 ; HL11 . In Fig.1-3, we plot the fully entangled fraction
$f(\rho^{S}_{X})$, quantum steering $S^{A\rightarrow B}(T)$ and
$S^{B\rightarrow A}(T)$ between two fermions as a function of the Hawking
temperature $T$ for different $\omega$, $q_{R}$, and initial parameters.

Figure 1:  The fully entangled fraction $f(\rho^{S}_{X})$, quantum steering
$S^{A\rightarrow B}(T)$ and $S^{B\rightarrow A}(T)$ as a function of the
Hawking temperature $T$ for different $\omega$ and $q_{R}$. The initial
parameters are fixed as $\rho_{11}=\rho_{44}=\rho_{14}=0$, and
$\rho_{22}=\rho_{33}=\rho_{23}=\frac{1}{2}$.

In Fig.1, we can see that $f(\rho^{S}_{X})$, $S^{A\rightarrow B}(T)$, and
$S^{B\rightarrow A}(T)$ decrease monotonically with the increase of the
Hawking temperature $T$. It is worth noting that $f(\rho^{S}_{X})$,
$S^{A\rightarrow B}(T)$, and $S^{B\rightarrow A}(T)$ depend on the choice of
Unruh modes. For example, quantum steering for $q_{R}=1$ and $q_{R}=0.9$
decreases to a fixed value with the Hawking temperature $T$, while quantum
steering for $q_{R}=0.8$ suffers from sudden death with $T$. It means that
quantum steering cannot fully guarantee the fully entangled fraction in
Schwarzschild spacetime. We find that an Unruh mode with $q_{R}=1$ is always
optimal to teleport the unknown pure state to Bob and is optimal for quantum
steering between Alice and Bob. We can also see that the fully entangled
fraction and quantum steering are monotonically increasing functions of the
frequency $\omega$. The results show that we protect the fully entangled
fraction and quantum steering by choosing the high-frequency mode for
maximally entangled states in Schwarzschild spacetime.

Figure 2:  The fully entangled fraction $f(\rho^{S}_{X})$, quantum steering
$S^{A\rightarrow B}(T)$ and $S^{B\rightarrow A}(T)$ as a function of the
Hawking temperature $T$ for different $\omega$ and $q_{R}$. The initial
parameters are fixed as $\rho_{11}=\sqrt{2}-1$, $\rho_{22}=\frac{1}{2}$,
$\rho_{33}=\frac{3-2\sqrt{2}}{2}$, $\rho_{44}=\rho_{14}=0$, and
$\rho_{23}=\frac{\sqrt{2}-1}{2}$.

In Fig.2, we find that, with the growth of the Hawking temperature $T$,
$f(\rho^{S}_{X})$ increases monotonically, while quantum steering from Alice
to Bob $S^{A\rightarrow B}(T)$ is always zero, and quantum steering from Bob
to Alice $S^{B\rightarrow A}(T)$ for $q_{R}=1$ and $q_{R}=0.9$ first decreases
and then suffers from a “sudden death”. This means that the Hawking effect of
the black hole has a positive influence on fully entangled fraction and a
negative influence on quantum steering. Therefore, the Hawking effect can
create net fidelity of quantum teleportation, and quantum steering cannot
fully guarantee the fully entangled fraction. However, previous papers have
shown that the Hawking effect destroys the fidelity of quantum teleportation
and quantum correlation in Schwarzschild spacetime L38 ; L39 ; L40 ; L41 ; L42
; L43 ; L44 . Therefore, the Hawking effect of the black hole cannot be simply
considered as thermal noise that can only destroy the fidelity of quantum
teleportation. We will use this special type of quantum state to
experimentally explore the Hawking effect in the future. We also find that
$f(\rho^{S}_{X})$ and $S^{B\rightarrow A}(T)$ increase as $q_{R}$ increases.
In addition, quantum steering from Bob to Alice for $q_{R}=0.8$ is always zero
in Schwarzschild spacetime. This again demonstrates that the fully entangled
fraction and quantum steering depend on the choice of Unruh modes, and the
Unruh mode with $q_{R}=1$ is always optimal for the fully entangled fraction
and quantum steering between Alice and Bob. Interestingly, increasing the
frequency $\omega$ has a negative effect on the fidelity of quantum
teleportation and a positive effect on quantum steering for this type of
special quantum state. Therefore, we should use low-frequency mode to protect
the fidelity of quantum teleportation, while we use quantum steering of high-
frequency mode to handle relativistic quantum information tasks. For the first
time, we found their different dependence on frequency in Schwarzschild
spacetime. These results contribute to our more comprehensive understanding of
the Hawking effect of the black hole.

Figure 3: The fully entangled fraction $f(\rho^{S}_{X})$ as a function of the
Hawking temperature $T$ for different $\omega$ and $q_{R}$. The initial
parameters are fixed as $\rho_{11}=\frac{\sqrt{2}-1}{2}$,
$\rho_{22}=\frac{\sqrt{2}}{2}$, $\rho_{33}=\frac{3-2\sqrt{2}}{2}$,
$\rho_{44}=\rho_{14}=0$, and $\rho_{23}=\frac{\sqrt{2}-1}{4}$.

In Fig.3, we find that, for $q_{R}=1$, $f(\rho^{S}_{X})$ first increases from
the initial value to the maximum value and then monotonically decreases with
the growth of the Hawking temperature $T$. Through the simple calculation, we
can obtain $S^{A\rightarrow B}(T)=S^{B\rightarrow A}(T)=0$ in this case. This
shows that the Hawking effect has the positive and negative influence on the
fully entangled fraction for the single mode approximation, and quantum
steering fully cannot guarantee the fully entangled fraction. We can see that
the maximum fidelity of quantum teleportation depends on the Hawking
temperature $T$ and frequency $\omega$. However, for $q_{R}=0.9$ and
$q_{R}=0.8$, $f(\rho^{S}_{X})$ increases monotonically with the increase of
the Hawking temperature $T$. For different kinds of Unruh modes, the fully
entangled fraction exhibits completely different properties with the Hawking
temperature $T$.

##  V  Conclutions

In this paper, we have studied the effect of the Hawking effect on the
fidelity of quantum teleportation of Dirac fields between users beyond the
single-mode approximation in Schwarzschild spacetime. Alice and Bob initially
share an X-type state and they apply a standard teleportation scheme (STS) to
send the unknown state from Alice to Bob. Here, Alice stays stationary at an
asymptotically flat region, while Bob hovers near the event horizon of the
black hole. We find that the fidelity of quantum teleportation of Dirac fields
can monotonically increase, monotonically decrease, or non-monotonically
increase, depending on the choice of the initial state with the increase of
the Hawking temperature, meaning that the Hawking effect can enhance and
create net fidelity of quantum teleportation. This makes sharp a contrast with
quantum correlation (quantum steering, entanglement, and discord) and the
fidelity of quantum teleportation of bosonic fields, which decrease
monotonically with the growth of the Hawking temperature in Schwarzschild
spacetime L38 ; L39 ; L40 ; L41 ; L42 ; L43 ; L44 . The reduction of
physically accessible fidelity of quantum teleportation and steering by the
Hawking effect can be attributed to the increase of physically inaccessible
fidelity of quantum teleportation and steering by the Hawking effect.

In addition, the fidelity of quantum teleportation and quantum steering depend
on the choice of Unruh modes. We showed that the Unruh mode with $q_{R}=1$ is
always optimal to teleport the unknown pure state to Bob and $q_{R}=0$ is
optimal for quantum teleportation with anti-Bob inside the event horizon of
the black hole HL2 . For different kinds of Unruh modes, the fidelity of
quantum teleportation exhibits completely different properties with the
Hawking temperature in curved spacetime (please refer to Fig.3 for detail). We
also find that quantum steering cannot guarantee the fidelity of quantum
teleportation in Schwarzschild spacetime. Interestingly, the low-frequency
mode may be beneficial for protecting the fidelity of quantum teleportation
and harmful to quantum steering. These surprising results overturn the
extended belief that the Hawking effect of the black hole can only destroy the
fidelity of quantum teleportation and provides a new and unexpected source for
finding experimental evidence of the Hawking effect in curved spacetime.

##  Appendix A $\rho^{AB_{out}}$

An X-type state is initially shared by Alice and Bob at an asymptotically flat
region. Then, we let Bob hover near the event horizon of the black hole.
According to Eqs.(15) and (16), we can rewrite Eq.(5). Since the exterior
region and the interior region are causally disconnected, we should trace over
the inaccessible modes and obtain

$\displaystyle\rho^{AB_{out}}=\left(\\!\\!\begin{array}[]{cccccccc}\rho^{AB_{out}}_{11}&0&0&-\rho^{AB_{out}}_{14}&0&\rho^{AB_{out}}_{16}&-\rho^{AB_{out}}_{17}&0\\\
0&\rho^{AB_{out}}_{22}&0&0&\rho^{AB_{out}}_{25}&0&0&-\rho^{AB_{out}}_{28}\\\
0&0&\rho^{AB_{out}}_{33}&0&-\rho^{AB_{out}}_{35}&0&0&\rho^{AB_{out}}_{38}\\\
-\rho^{AB_{out}}_{14}&0&0&\rho^{AB_{out}}_{44}&0&-\rho^{AB_{out}}_{46}&\rho^{AB_{out}}_{47}&0\\\
0&\rho^{AB_{out}}_{25}&-\rho^{AB_{out}}_{35}&0&\rho^{AB_{out}}_{55}&0&0&-\rho^{AB_{out}}_{58}\\\
\rho^{AB_{out}}_{16}&0&0&-\rho^{AB_{out}}_{46}&0&\rho^{AB_{out}}_{66}&0&0\\\
-\rho^{AB_{out}}_{17}&0&0&\rho^{AB_{out}}_{47}&0&0&\rho^{AB_{out}}_{77}&0\\\
0&-\rho^{AB_{out}}_{28}&\rho^{AB_{out}}_{38}&0&-\rho^{AB_{out}}_{58}&0&0&\rho^{AB_{out}}_{88}\\\
\end{array}\\!\\!\right),$ (33)

where

$\displaystyle\rho^{AB_{out}}_{11}$ $\displaystyle=$
$\displaystyle(e^{-\frac{\omega}{T}}+1)^{-2}\rho_{11}+|q_{L}|^{2}(e^{-\frac{\omega}{T}}+1)^{-1}\rho_{22},$
$\displaystyle\rho^{AB_{out}}_{22}$ $\displaystyle=$
$\displaystyle(e^{-\frac{\omega}{T}}+1)^{-1}(e^{\frac{\omega}{T}}+1)^{-1}\rho_{11},$
$\displaystyle\rho^{AB_{out}}_{33}$ $\displaystyle=$
$\displaystyle(e^{-\frac{\omega}{T}}+1)^{-1}(e^{\frac{\omega}{T}}+1)^{-1}\rho_{11}+\big{[}|q_{R}|^{2}(e^{-\frac{\omega}{T}}+1)^{-1}+|q_{L}|^{2}(e^{\frac{\omega}{T}}+1)^{-1}\big{]}\rho_{22},$
$\displaystyle\rho^{AB_{out}}_{44}$ $\displaystyle=$
$\displaystyle(e^{\frac{\omega}{T}}+1)^{-2}\rho_{11}+|q_{R}|^{2}(e^{\frac{\omega}{T}}+1)^{-1}\rho_{22},$
$\displaystyle\rho^{AB_{out}}_{55}$ $\displaystyle=$
$\displaystyle(e^{-\frac{\omega}{T}}+1)^{-2}\rho_{33}+|q_{L}|^{2}(e^{-\frac{\omega}{T}}+1)^{-1}\rho_{44},$
$\displaystyle\rho^{AB_{out}}_{66}$ $\displaystyle=$
$\displaystyle(e^{-\frac{\omega}{T}}+1)^{-1}(e^{\frac{\omega}{T}}+1)^{-1}\rho_{33},$
$\displaystyle\rho^{AB_{out}}_{77}$ $\displaystyle=$
$\displaystyle(e^{-\frac{\omega}{T}}+1)^{-1}(e^{\frac{\omega}{T}}+1)^{-1}\rho_{33}+\big{[}|q_{R}|^{2}(e^{-\frac{\omega}{T}}+1)^{-1}+|q_{L}|^{2}(e^{\frac{\omega}{T}}+1)^{-1}\big{]}\rho_{44},$
$\displaystyle\rho^{AB_{out}}_{88}$ $\displaystyle=$
$\displaystyle(e^{\frac{\omega}{T}}+1)^{-2}\rho_{33}+|q_{R}|^{2}(e^{\frac{\omega}{T}}+1)^{-1}\rho_{44},$
$\displaystyle\rho^{AB_{out}}_{14}$ $\displaystyle=$ $\displaystyle
q_{R}q_{L}(e^{-\frac{\omega}{T}}+1)^{-\frac{1}{2}}(e^{\frac{\omega}{T}}+1)^{-\frac{1}{2}}\rho_{22},$
$\displaystyle\rho^{AB_{out}}_{16}$ $\displaystyle=$ $\displaystyle
q_{L}(e^{-\frac{\omega}{T}}+1)^{-1}(e^{\frac{\omega}{T}}+1)^{-\frac{1}{2}}\rho_{23},$
$\displaystyle\rho^{AB_{out}}_{17}$ $\displaystyle=$ $\displaystyle
q_{R}(e^{-\frac{\omega}{T}}+1)^{-\frac{3}{2}}\rho_{14},$
$\displaystyle\rho^{AB_{out}}_{25}$ $\displaystyle=$ $\displaystyle
q_{L}(e^{-\frac{\omega}{T}}+1)^{-1}(e^{\frac{\omega}{T}}+1)^{-\frac{1}{2}}\rho_{14},$
$\displaystyle\rho^{AB_{out}}_{28}$ $\displaystyle=$ $\displaystyle
q_{R}(e^{-\frac{\omega}{T}}+1)^{-\frac{1}{2}}(e^{\frac{\omega}{T}}+1)^{-1}\rho_{14},$
$\displaystyle\rho^{AB_{out}}_{35}$ $\displaystyle=$ $\displaystyle
q_{R}(e^{-\frac{\omega}{T}}+1)^{-\frac{3}{2}}\rho_{23},$
$\displaystyle\rho^{AB_{out}}_{38}$ $\displaystyle=$ $\displaystyle
q_{L}(e^{\frac{\omega}{T}}+1)^{-\frac{3}{2}}\rho_{23},$
$\displaystyle\rho^{AB_{out}}_{46}$ $\displaystyle=$ $\displaystyle
q_{R}(e^{\frac{\omega}{T}}+1)^{-1}(e^{-\frac{\omega}{T}}+1)^{-\frac{1}{2}}\rho_{23},$
$\displaystyle\rho^{AB_{out}}_{47}$ $\displaystyle=$ $\displaystyle
q_{L}(e^{\frac{\omega}{T}}+1)^{-\frac{3}{2}}\rho_{14},$
$\displaystyle\rho^{AB_{out}}_{58}$ $\displaystyle=$ $\displaystyle
q_{R}q_{L}(e^{-\frac{\omega}{T}}+1)^{-\frac{1}{2}}(e^{\frac{\omega}{T}}+1)^{-\frac{1}{2}}\rho_{44}.$

##  Appendix B Quantification of quantum steering for X-type state in
Schwarzschild spacetime

As is well-known, the bipartite entanglement can be effectually measured by
the concurrence. For the X-state $\rho_{X}$ of Eq.(5), the concurrence is
given by L66

$\displaystyle
C(\rho_{X})=2\max\\{|\rho_{14}|-\sqrt{\rho_{22}\rho_{33}},|\rho_{23}|-\sqrt{\rho_{11}\rho_{44}}\\}.$
(34)

For any bipartite state $\rho_{AB}$ between Alice and Bob, quantum steering
from Bob to Alice can be recognized if the density matrix $\tau_{AB}^{1}$ that
becomes

$\displaystyle\tau_{AB}^{1}=\frac{\rho_{AB}}{\sqrt{3}}+\frac{3-\sqrt{3}}{3}(\rho_{A}\otimes\frac{I}{2}),$
(35)

is entangled L67 ; L68 , where $\rho_{A}$ is Alice’s reduced density matrix,
$\rho_{A}={\rm Tr}_{B}(\rho_{AB})$, and $I$ is the two-dimension identity
matrix of Bob’s subsystem. Analogously, quantum steering from Alice to Bob can
be proved if the state $\tau_{AB}^{2}$ that reads

$\displaystyle\tau_{AB}^{2}=\frac{\rho_{AB}}{\sqrt{3}}+\frac{3-\sqrt{3}}{3}(\frac{I}{2}\otimes\rho_{B}),$
(36)

is entangled, where $\rho_{B}={\rm Tr}_{A}(\rho_{AB})$. Therefore, the matrix
$\tau_{AB}^{1}$ for the X-state $\rho_{X}$ of Eq.(5) can be specifically
expressed as

$\displaystyle\tau_{AB}^{1,x}=\left(\\!\\!\begin{array}[]{cccc}\frac{\sqrt{3}}{3}\rho_{11}+r&0&0&-\frac{\sqrt{3}}{3}\rho_{14}\\\
0&\frac{\sqrt{3}}{3}\rho_{22}+r&-\frac{\sqrt{3}}{3}\rho_{23}&0\\\
0&-\frac{\sqrt{3}}{3}\rho_{23}&\frac{\sqrt{3}}{3}\rho_{33}+s&0\\\
-\frac{\sqrt{3}}{3}\rho_{14}&0&0&\frac{\sqrt{3}}{3}\rho_{44}+s\end{array}\\!\\!\right),$
(41)

with $r=\frac{(3-\sqrt{3})}{6}(\rho_{11}+\rho_{22})$ and
$s=\frac{(3-\sqrt{3})}{6}(\rho_{33}+\rho_{44})$. Using Eq.(9), the state
$\tau_{AB}^{1,x}$ is entangled as long as one of the conditions
$|\rho_{14}|^{2}>\mathcal{F}_{a}-\mathcal{F}_{b}$ and
$|\rho_{23}|^{2}>\mathcal{F}_{c}-\mathcal{F}_{b}$ is satisfied, where

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

Thus the steering from Bob to Alice is proved. Similarly, the steering from
Alice to Bob can also be proved by one of the inequalities,

$\displaystyle|\rho_{14}|^{2}>\mathcal{F}_{a}+\mathcal{F}_{b},$ (42)

or

$\displaystyle|\rho_{23}|^{2}>\mathcal{F}_{c}+\mathcal{F}_{b}.$ (43)

Next, we introduced quantities

$\displaystyle S^{A\rightarrow
B}={\rm{max}}\bigg{\\{}0,\frac{8}{\sqrt{3}}(|\rho_{14}|^{2}-\mathcal{F}_{a}-\mathcal{F}_{b}),\frac{8}{\sqrt{3}}(|\rho_{23}|^{2}-\mathcal{F}_{c}-\mathcal{F}_{b})\bigg{\\}},$
(44)

and

$\displaystyle S^{B\rightarrow
A}={\rm{max}}\bigg{\\{}0,\frac{8}{\sqrt{3}}(|\rho_{14}|^{2}-\mathcal{F}_{a}+\mathcal{F}_{b}),\frac{8}{\sqrt{3}}(|\rho_{23}|^{2}-\mathcal{F}_{c}+\mathcal{F}_{b})\bigg{\\}},$
(45)

to quantify the steerability from Alice to Bob and from Bob to Alice,
respectively. Here, the factor $\frac{8}{\sqrt{3}}$ is to ensure that the
maximum steering is $1$.

For the X-type state in Eq.(21), using Eqs.(44) and (45), we obtain the
quantum steering from Alice to Bob

$\displaystyle S^{A\rightarrow B}(T)$ $\displaystyle=$
$\displaystyle{\rm{max}}\bigg{\\{}0,\frac{8}{\sqrt{3}}\big{[}|-q_{R}(e^{-\frac{\omega}{T}}+1)^{-\frac{1}{2}}\rho_{14}|^{2}-\mathcal{F}_{a}(T)-\mathcal{F}_{b}(T)\big{]},$
(46)
$\displaystyle\frac{8}{\sqrt{3}}\big{[}|-q_{R}(e^{-\frac{\omega}{T}}+1)^{-\frac{1}{2}}\rho_{23}|^{2}-\mathcal{F}_{c}(T)-\mathcal{F}_{b}(T)\big{]}\bigg{\\}},$

and the quantum steering from Bob to Alice

$\displaystyle S^{B\rightarrow A}(T)$ $\displaystyle=$
$\displaystyle{\rm{max}}\bigg{\\{}0,\frac{8}{\sqrt{3}}\big{[}|-q_{R}(e^{-\frac{\omega}{T}}+1)^{-\frac{1}{2}}\rho_{14}|^{2}-\mathcal{F}_{a}(T)+\mathcal{F}_{b}(T)\big{]},$
(47)
$\displaystyle\frac{8}{\sqrt{3}}\big{[}|-q_{R}(e^{-\frac{\omega}{T}}+1)^{-\frac{1}{2}}\rho_{23}|^{2}-\mathcal{F}_{c}(T)+\mathcal{F}_{b}(T)\big{]}\bigg{\\}},$

where

$\displaystyle\mathcal{F}_{a}(T)$
$\displaystyle=\frac{2-\sqrt{3}}{2}[(e^{-\frac{\omega}{T}}+1)^{-1}\rho_{11}+(1-|q_{R}|^{2})(e^{-\frac{\omega}{T}}+1)^{-1}\rho_{22}][(e^{\frac{\omega}{T}}+1)^{-1}\rho_{33}$
(48)
$\displaystyle+\rho_{44}-(1-|q_{R}|^{2})(e^{-\frac{\omega}{T}}+1)^{-1}\rho_{44}]+\frac{2+\sqrt{3}}{2}[(e^{\frac{\omega}{T}}+1)^{-1}\rho_{11}+\rho_{22}$
$\displaystyle-(1-|q_{R}|^{2})(e^{-\frac{\omega}{T}}+1)^{-1}\rho_{22}][(e^{-\frac{\omega}{T}}+1)^{-1}\rho_{33}+(1-|q_{R}|^{2})(e^{-\frac{\omega}{T}}+1)^{-1}\rho_{44}]$
$\displaystyle+\frac{1}{4}[(e^{-\frac{\omega}{T}}+1)^{-1}\rho_{11}+(1-|q_{R}|^{2})(e^{-\frac{\omega}{T}}+1)^{-1}\rho_{22}+(e^{\frac{\omega}{T}}+1)^{-1}\rho_{33}+\rho_{44}$
$\displaystyle-(1-|q_{R}|^{2})(e^{-\frac{\omega}{T}}+1)^{-1}\rho_{44}][(e^{\frac{\omega}{T}}+1)^{-1}\rho_{11}+\rho_{22}-(1-|q_{R}|^{2})(e^{-\frac{\omega}{T}}+1)^{-1}\rho_{22}$
$\displaystyle+(e^{-\frac{\omega}{T}}+1)^{-1}\rho_{33}+(1-|q_{R}|^{2})(e^{-\frac{\omega}{T}}+1)^{-1}\rho_{44}],$
$\displaystyle\mathcal{F}_{b}(T)$ $\displaystyle=$
$\displaystyle\frac{1}{4}[(e^{-\frac{\omega}{T}}+1)^{-1}\rho_{11}+(1-|q_{R}|^{2})(e^{-\frac{\omega}{T}}+1)^{-1}\rho_{22}-(e^{\frac{\omega}{T}}+1)^{-1}\rho_{33}-\rho_{44}$
(49)
$\displaystyle+(1-|q_{R}|^{2})(e^{-\frac{\omega}{T}}+1)^{-1}\rho_{44}][(e^{\frac{\omega}{T}}+1)^{-1}\rho_{11}+\rho_{22}-(1-|q_{R}|^{2})(e^{-\frac{\omega}{T}}+1)^{-1}\rho_{22}$
$\displaystyle-(e^{-\frac{\omega}{T}}+1)^{-1}\rho_{33}-(1-|q_{R}|^{2})(e^{-\frac{\omega}{T}}+1)^{-1}\rho_{44}],$
$\displaystyle\mathcal{F}_{c}(T)$ $\displaystyle=$
$\displaystyle\frac{2+\sqrt{3}}{2}[(e^{-\frac{\omega}{T}}+1)^{-1}\rho_{11}+(1-|q_{R}|^{2})(e^{-\frac{\omega}{T}}+1)^{-1}\rho_{22}][(e^{\frac{\omega}{T}}+1)^{-1}\rho_{33}$
(50)
$\displaystyle+\rho_{44}-(1-|q_{R}|^{2})(e^{-\frac{\omega}{T}}+1)^{-1}\rho_{44}]+\frac{2-\sqrt{3}}{2}[(e^{\frac{\omega}{T}}+1)^{-1}\rho_{11}+\rho_{22}$
$\displaystyle-(1-|q_{R}|^{2})(e^{-\frac{\omega}{T}}+1)^{-1}\rho_{22}][(e^{-\frac{\omega}{T}}+1)^{-1}\rho_{33}+(1-|q_{R}|^{2})(e^{-\frac{\omega}{T}}+1)^{-1}\rho_{44}]$
$\displaystyle+\frac{1}{4}[(e^{-\frac{\omega}{T}}+1)^{-1}\rho_{11}+(1-|q_{R}|^{2})(e^{-\frac{\omega}{T}}+1)^{-1}\rho_{22}+(e^{\frac{\omega}{T}}+1)^{-1}\rho_{33}+\rho_{44}$
$\displaystyle-(1-|q_{R}|^{2})(e^{-\frac{\omega}{T}}+1)^{-1}\rho_{44}][(e^{\frac{\omega}{T}}+1)^{-1}\rho_{11}+\rho_{22}-(1-|q_{R}|^{2})(e^{-\frac{\omega}{T}}+1)^{-1}\rho_{22}$
$\displaystyle+(e^{-\frac{\omega}{T}}+1)^{-1}\rho_{33}+(1-|q_{R}|^{2})(e^{-\frac{\omega}{T}}+1)^{-1}\rho_{44}].$

##  Appendix C Ambiguity of quantum teleportation and steering in
Schwarzschild spacetime

Ambiguity is related to the ordering criterion of the creation and
annihilation operators and has gone unnoticed in fermionic quantum
teleportation and steering in Schwarzschild spacetime. Below, we briefly
introduce ambiguity. We consider a two-mode fermionic system associated with
fermionic creation operators $a^{{\dagger}}$ and $b^{{\dagger}}$ acting on a
vacuum state $|0\rangle$. Therefore, the relevant Hilbert space is four
dimensional. The Hilbert space basis of our toy model can be represented as

$\displaystyle|00\rangle=|0\rangle,\quad|10\rangle=a^{{\dagger}}|0\rangle,\quad|01\rangle=b^{{\dagger}}|0\rangle,\quad|11\rangle=a^{{\dagger}}b^{{\dagger}}|0\rangle.$
(51)

Based on this basis, we can endow the Hilbert space with a tensor product
structure, which allows us to consider two qubits, where the first label
corresponds to one qubit and the second label corresponds to the other qubit.
We may change the quantum entanglement of state by making nonlocal changes to
the basis: we swap the positions of $a^{{\dagger}}$ and $b^{{\dagger}}$ in
Eq.(51) HL7 ; HL8 ; HL9 ; HL10 ; HL11 . The new basis can be obtained as

$\displaystyle|00\rangle^{\prime}=|00\rangle,\quad|01\rangle^{\prime}=|01\rangle,\quad|10\rangle^{\prime}=|10\rangle,\quad|11\rangle^{\prime}=b^{{\dagger}}a^{{\dagger}}|0\rangle=-|11\rangle.$
(52)

Therefore, we can obtain a new basis in this specific case. Interestingly,
choosing these two different types of bases may result in a separable state
being the Bell state.

In this paper, we use the operator ordering $\hat{a}^{\rm
out}_{\mathbb{k}}\hat{b}^{\rm out}_{\mathbb{-k}}\hat{b}^{\rm
in}_{\mathbb{-k}}\hat{a}^{\rm in}_{\mathbb{k}}$ to rewrite Eq.(15) as

$\displaystyle|0\rangle_{\rm U}^{\prime}$ $\displaystyle=$
$\displaystyle\frac{1}{e^{-\frac{\omega}{T}}+1}|0000\rangle-\frac{1}{\sqrt{e^{\frac{\omega}{T}}+e^{-\frac{\omega}{T}}+2}}|0101\rangle$
(53)
$\displaystyle+\frac{1}{\sqrt{e^{\frac{\omega}{T}}+e^{-\frac{\omega}{T}}+2}}|1010\rangle+\frac{1}{e^{\frac{\omega}{T}}+1}|1111\rangle.$

However, Eq.(16) remains unchanged. Similarly, we can rewrite Eq.(5). Then, we
trace over its inaccessible modes and obtain

$\displaystyle\tilde{\rho}^{AB_{out}}=\left(\\!\\!\begin{array}[]{cccccccc}\rho^{AB_{out}}_{11}&0&0&-\rho^{AB_{out}}_{14}&0&\rho^{AB_{out}}_{16}&-\rho^{AB_{out}}_{17}&0\\\
0&\rho^{AB_{out}}_{22}&0&0&\rho^{AB_{out}}_{25}&0&0&-\rho^{AB_{out}}_{28}\\\
0&0&\rho^{AB_{out}}_{33}&0&-\rho^{AB_{out}}_{35}&0&0&-\rho^{AB_{out}}_{38}\\\
-\rho^{AB_{out}}_{14}&0&0&\rho^{AB_{out}}_{44}&0&-\rho^{AB_{out}}_{46}&-\rho^{AB_{out}}_{47}&0\\\
0&\rho^{AB_{out}}_{25}&-\rho^{AB_{out}}_{35}&0&\rho^{AB_{out}}_{55}&0&0&-\rho^{AB_{out}}_{58}\\\
\rho^{AB_{out}}_{16}&0&0&-\rho^{AB_{out}}_{46}&0&\rho^{AB_{out}}_{66}&0&0\\\
-\rho^{AB_{out}}_{17}&0&0&-\rho^{AB_{out}}_{47}&0&0&\rho^{AB_{out}}_{77}&0\\\
0&-\rho^{AB_{out}}_{28}&-\rho^{AB_{out}}_{38}&0&-\rho^{AB_{out}}_{58}&0&0&\rho^{AB_{out}}_{88}\\\
\end{array}\\!\\!\right).$ (62)

From Eqs.(33) and (62), we can see that the density matrix
$\tilde{\rho}^{AB_{out}}$ is different from the density matrix
$\rho^{AB_{out}}$. Then, we trace out the antifermionic mode of
$\tilde{\rho}^{AB_{out}}$ outside the event horizon of the black hole and
obtain

$\displaystyle\tilde{\rho}^{S}_{X}=\left(\\!\\!\begin{array}[]{cccccccc}\rho^{S}_{11}&0&0&-\rho^{S}_{14}\\\
0&\rho^{S}_{22}&-\rho^{S}_{23}&0\\\ 0&-\rho^{S}_{23}&\rho^{S}_{33}&0\\\
-\rho^{S}_{14}&0&0&\rho^{S}_{44}\end{array}\\!\\!\right).$ (67)

From Eqs.(21) and (67), we find that quantum state $\tilde{\rho}^{S}_{X}$ is
the same as quantum state ${\rho}^{S}_{X}$. Therefore, there is no ambiguous
map in the quantum teleportation and steering for the same initial X-type
state $\rho_{X}$ in the Schwarzschild black hole.

###### Acknowledgements.

This work is supported by the National Natural Science Foundation of China
(Grant No. 12205133), LJKQZ20222315 and 2021BSL013.

## References

  * (1) C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, and W. K. Wootters, Phys. Rev. Lett. 70, 1895 (1993).
  * (2) D. Bouwmeester, J. W. Pan, K. Mattle, M. Eibl, H. Weinfurter, and A. Zeilinger, Nature 390, 575 (1997).
  * (3) C. H. Bennett, G. Brassard, S. Popescu, B. Schumacher, J. A. Smolin, and W. K. Wootters, Phys. Rev. Lett. 76, 722 (1996).
  * (4) N. Friis, A. R. Lee, K. Truong, C. Sabín, E. Solano, G. Johansson, and I. Fuentes, Phys. Rev. Lett. 110, 113602 (2013).
  * (5) P. Lipka-Bartosik and P. Skrzypczyk, Phys. Rev. Lett. 127, 080502 (2021).
  * (6) U. Marzolino and A. Buchleitner, Phys. Rev. A 91, 032316 (2015).
  * (7) C. Noh, A. Chia, H. Nha, M. J. Collett, and H. J. Carmichael, Phys. Rev. Lett. 102, 230501 (2009).
  * (8) X. Chen and K. W. C. Chan, Phys. Rev. A 99, 022334 (2019).
  * (9) R. Fortes and G. Rigolin, Phys. Rev. A 93, 062330 (2016).
  * (10) C. Q. Xu and D. L. Zhou, Phys. Rev. A 106, 012413 (2022).
  * (11) X. M. Hu, C. Zhang, B. H. Liu, Y. Cai, X. J. Ye, Y. Guo, W. B. Xing, C. X. Huang, Y. F. Huang, C. F. Li, and G. C. Guo, Phys. Rev. Lett. 125, 230501 (2020).
  * (12) S. Gangopadhyay, T. Wang, A. Mashatan, and S. Ghose, Phys. Rev. A 106, 052433 (2022).
  * (13) B. Yoshida and N. Y. Yao, Phys. Rev. X 9, 011006 (2019).
  * (14) H. M. Wiseman, S. J. Jones, and A. C. Doherty, Phys. Rev. Lett. 98, 140402 (2007).
  * (15) S. J. Jones, H. M. Wiseman, and A. C. Doherty, Phys. Rev. A 76, 052116 (2007).
  * (16) E. Schrödinger, Proc. Cambridge Philos. Soc. 31, 555 (1935).
  * (17) E. Schrödinger, Proc. Cambridge Philos. Soc. 32, 446 (1936).
  * (18) D. J. Saunders, S. J. Jones, H. M. Wiseman, and G. J. Pryde, Nat. Phys. 6, 845 (2010).
  * (19) V. Händchen, T. Eberle, S. Steinlechner, A. Samblowski, T. Franz, R. F. Werner, and R. Schnabel, Nat. Photonics 6, 598 (2012).
  * (20) S. Wollmann, N. Walk, A. J. Bennet, H. M. Wiseman, and G. J. Pryde, Phys. Rev. Lett. 116, 160403 (2016).
  * (21) Y. Xiao, X. J. Ye, K. Sun, J. S. Xu, C. F. Li, and G. C. Guo, Phys. Rev. Lett. 118, 140404 (2017).
  * (22) A. Ghosal, D. Das, S. Roy, and S. Bandyopadhyay, Phys. Rev. A 101, 012304 (2020).
  * (23) B. P. Abbott $et$ $al.$, Phys. Rev. Lett. 116, 061102 (2016).
  * (24) The Event Horizon Telescope Collaboration, Astrophys. J. Lett. 875, L1 (2019).
  * (25) The Event Horizon Telescope Collaboration, Astrophys. J. Lett. 875, L2 (2019).
  * (26) The Event Horizon Telescope Collaboration, Astrophys. J. Lett. 875, L3 (2019).
  * (27) The Event Horizon Telescope Collaboration, Astrophys. J. Lett. 875, L4 (2019).
  * (28) The Event Horizon Telescope Collaboration, Astrophys. J. Lett. 875, L5 (2019).
  * (29) The Event Horizon Telescope Collaboration, Astrophys. J. Lett. 875, L6 (2019).
  * (30) The Event Horizon Telescope Collaboration, Astrophys. J. Lett. 930, L12 (2022).
  * (31) P. Xu, $et$ $al.$, Science 366, 132 (2019).
  * (32) X. Busch, R. Parentani, Phys. Rev. D 89, 105024 (2014).
  * (33) Z. Tian, J. Jing, A. Dragan, Phys. Rev. D 95, 125003 (2017).
  * (34) Z. Tian, J. Du, Eur. Phys. J. C 79, 994 (2019).
  * (35) J. Drori, Y. Rosenberg, D. Bermudez, Y. Silberberg, and U. Leonhardt, Phys. Rev. Lett. 122, 010404 (2019).
  * (36) M. Isoard, N. Pavloff, Phys. Rev. Lett. 124, 060401 (2020).
  * (37) Z. Tian, L. Wu, L. Zhang, J. Jing, J. Du, Phys. Rev. D 106, L061701 (2022).
  * (38) J. Steinhauer, $et$ $al.$, Nature Communications 13, 2890 (2022).
  * (39) C. Viermann, $et$ $al.$, Nature 611, 260 (2022).
  * (40) S. Xu, X. K. Song, J. D. Shi, and L. Ye, Phys. Rev. D 89, 065022 (2014).
  * (41) Q. Pan, J. Jing, Phys. Rev. D 78, 065015 (2008).
  * (42) E. Martín-Martínez, L. J. Garay, and J. León, Phys. Rev. D 82, 064006 (2010).
  * (43) J. Wang, Q. Pan, J. Jing, Phys. Lett. B 692, 202 (2010).
  * (44) S. M. Wu, H. S. Zeng, Eur. Phys. J. C 82, 4 (2022).
  * (45) J. Wang, H. Cao, J. Jing, and H. Fan, Phys. Rev. D 93, 125011 (2016).
  * (46) S. Bhattacharya, N. Joshi, Phys. Rev. D 105, 065007 (2022).
  * (47) I. Fuentes-Schuller, and R. B. Mann, Phys. Rev. Lett. 95,120404 (2005).
  * (48) B. N. Esfahani, M. Shamirzaie, and M. Soltani, Phys. Rev. D 84, 025024 (2011).
  * (49) S. M. Wu, Y. T. Cai, W. J. Peng, H. S. Zeng, Eur. Phys. J. C 82, 412 (2022).
  * (50) P. M. Alsing, I. Fuentes-Schuller, R. B. Mann, and T. E. Tessier, Phys. Rev. A 74, 032326 (2006).
  * (51) J. L. Huang, W. C. Gan, Y. Xiao, F. W. Shu, M. H. Yung, Eur. Phys. J. C 78, 545 (2018).
  * (52) J. Wang, J. Jing, and H. Fan, Phys. Rev. D 90, 025032 (2014).
  * (53) D. E. Bruschi, A. Datta, R. Ursin, T. C. Ralph, and I. Fuentes, Phys. Rev. D 90, 124001 (2014).
  * (54) D. E. Bruschi, T. C. Ralph, I. Fuentes, T. Jennewein, and M. Razavi, Phys. Rev. D 90, 045041 (2014).
  * (55) J. He, S. Xu, L. Ye, Phys. Lett. B 756, 278 (2016).
  * (56) N. Hosseinidehaj, and R. Malaney, Phys. Rev. A 91, 022304 (2015).
  * (57) F. Shahbazi, S. Haseli, H. Dolatkhaha and S. Salimi, JCAP 10, 47 (2020).
  * (58) S. M. Wu, H. S. Zeng, Eur. Phys. J. C 82 716 (2022).
  * (59) S. Popescu, Phys. Rev. Lett 72, 797 (1994).
  * (60) C. H. Bennett, D. P. DiVincenzo, J. A. Smolin, and W. K. Wootters, Phys. Rev. A 54, 3824 (1996).
  * (61) P. Badziag, M. Horodecki, P. Horodecki, and R. Horodecki, Phys. Rev. A 62, 012311 (2000).
  * (62) D. R. Brill and J. A. Wheeler, Rev. Mod. Phys. 29, 465 (1957).
  * (63) J. Jing, Phys. Rev. D 70, 065004 (2004).
  * (64) J. Wang, Q. Pan, J. Jing, Annals Phys. 325, 1190 (2010).
  * (65) T. Damoar, R. Ruffini, Phys. Rev. D 14, 332 (1976).
  * (66) D. E. Bruschi, J. Louko, E. Martín-Martínez, A. Dragan, and I. Fuentes, Phys. Rev. A 82, 042332 (2010).
  * (67) E. Martín-Martínez, D. Hosler, and M. Montero, Phys. Rev. A 86, 062307 (2012).
  * (68) J. Chang and Y. Kwon, Phys. Rev. A 85, 032302 (2012).
  * (69) E. Martín-Martínez and I. Fuentes, Phys. Rev. A 83, 052306 (2011).
  * (70) D. E. Bruschi, A. Dragan, I. Fuentes, and J. Louko, Phys. Rev. D 86, 025026 (2012).
  * (71) S. A. Hayward, R. Di Criscienzo, M. Nadalini, L. Vanzo, and S. Zerbini, Classical Quantum Gravity 26, 062001 (2009).
  * (72) M. Montero and E. Martín-Martínez, Phys. Rev. A 83, 062323 (2011).
  * (73) K. Br$\acute{a}$dler, and Rocío J$\acute{a}$uregui, Phys. Rev. A 85, 016301 (2012).
  * (74) M. Montero and E. Martín-Martínez, Phys. Rev. A 85, 016302 (2012).
  * (75) M. Montero and E. Martín-Martínez, Phys. Rev. A 85, 024301 (2012).
  * (76) N.Friis, A. R. Lee, D. E. Bruschi, Phys. Rev. A 87, 022338 (2013).
  * (77) S. M. Hashemi Rafsanjani, M. Huber, C. J. Broadbent, and J. H. Eberly, Phys. Rev. A 86, 062303 (2012).
  * (78) D. Das, S. Sasmal, and S. Roy, Phys. Rev. A 99, 052109 (2019).
  * (79) K. Zhang, J. Wang, Phys. Rev. A 104, 042404 (2021).


# On Exponential Convergence of Random Variables

Dawid Tarłowski<EMAIL_ADDRESS><EMAIL_ADDRESS>ORCID 0000-0002-6824-4568 Faculty of Mathematics and Computer Science,
Jagiellonian University, Łojasiewicza 6, Kraków, 30-348, Poland

###### Abstract

Given the discrete-time sequence of nonnegative random variables, general
dependencies between the exponential convergence of the expectations,
exponential convergence of the trajectories and the logarithmic growth of the
corresponding expected hitting times are analysed. The applications are
presented: the general results are applied to the areas of optimization, risk
sensitive stochastic control and estimation.

###### keywords:

exponential convergence, stopping times, optimization, risk sensitive
stochastic control

###### pacs:

[

2010 MSC Classification]60G07

##  1 Introduction

Various forms of exponential convergence of random variables appear naturally
in many applications of Markov chains [5], supermartingales [10],[11], and
more general stochastic processes. The applications often involve various
problems from optimization and control theory. This paper compares several
definitions of exponential convergence under general assumptions, and next
applies the results to some problems of optimization, control, and estimation.
Section 2 presents the general theory. It is assumed that $X_{t}\in[0,\infty)$
is some discrete-time sequence of nonnegative random variables, and most
attention is paid to the exponential convergence of $X_{t}$ to zero, although
some of the results may be applied to the exponential growth of $X_{t}$.
Section2 shows, among other results, that the convergence rate of the
expectations defined by

$A:=\limsup\limits_{t\to\infty}\sqrt[t]{E[X_{t}]}$ (1)

bounds from above the convergence rate of the trajectories

$\mathbb{P}[\limsup\limits_{t\to\infty}\sqrt[t]{X_{t}}\leq A]=1$ (2)

and that, if $A\in(0,1)$, the stopping moment
$\tau_{\varepsilon}=inf\\{t\in\mathbb{N}\colon X_{t}<\varepsilon\\}$ is
integrable and satisfies

$\limsup\limits_{\varepsilon\to
0^{+}}\frac{E[\tau_{\varepsilon}]}{|\log(\varepsilon)|}\leq\frac{-1}{\log(A)}.$
(3)

The above statements are very general and at the same time the author of this
paper did not manage to find the proof of (2) and (3) in the literature (the
special case of $\eqref{E1}$ is proved in [16] in the context of continuous
optimization). Statement (2) does not assume that $A\leq 1$ and thus may be
used to bound the exponential growth of the trajectories. It is worth to
mention that in the literature the exponential decrease of
$\mathbb{P}[X_{n}\geq\varepsilon]$ is often studied, for instance in the
analysis of large deviations where one usually considers the sequences of the
form $X_{n}=|\frac{S_{n}-E[S_{n}]}{n}|$, [10], [15]. Generally, if for any
$\varepsilon>0$ the sequence $\mathbb{P}[X_{t}>\varepsilon],{t\in\mathbb{N}},$
decreases exponentially fast then $X_{t}\to 0$ almost sure however the
exponential decrease of the trajectories of $X_{t}$ is not forced. Note also
that by Chebyshev inequality for any $\varepsilon>0$ we have
$E[X_{t}]\geq\varepsilon\cdot\mathbb{P}[X_{t}\geq\varepsilon]$ which implies
that for any $\varepsilon>0$,
$\limsup\limits_{t\to\infty}\sqrt[t]{E[X_{t}]}\geq\limsup\limits_{t\to\infty}\sqrt[t]{\mathbb{P}[X_{t}\geq\varepsilon]}$
and hence:

$A=\limsup\limits_{t\to\infty}\sqrt[t]{E[X_{t}]}\geq\lim\limits_{\varepsilon\to
0^{+}}\limsup\limits_{t\to\infty}\sqrt[t]{\mathbb{P}[X_{t}\geq\varepsilon]}.$
(4)

Thus, the exponential convergence of the expectations is a very strong
convergence mode and the value of $A$ given by (1) bounds the convergence rate
in case of other exponential convergence types (2), (3), (4). Section 2
introduces general definitions precisely and proves (2),(3) and related
results. Section 3 applies those results to the areas of optimization and
control. In the context of optimization we usually deal with the sequences of
the form $X_{t}=f(Z_{t})$, where $Z_{t}$ is some sequence of random variables
which represents an optimization process and $f$ represents some real-valued
problem function. In such case it is an important question how fast the
sequence $f(Z_{t})$ approaches the global extremum of $f$ and all the above
mentioned types of exponential convergence (linear convergence) are being
considered, see [1], [7],[8],[11],[13],[14],[16],[17],[18], [19],[20]. On the
other hand, if $X_{t}\in\mathbb{R}$ describes the wealth process of an
investor in the context of portfolio optimization, [2],[3],[4],[9],[12], then
we are rather interested in how fast the process $X_{t}$ grows. Let $W_{t}$,
$t\in\mathbb{N}$, denote the wealth process and let $\gamma<0$ denote the
risk-averse parameter. Section 3 shows that under general assumptions (any
ergodicity-type assumptions on the log-wealth process are not involved) the
following limit of the long-run risk sensitive criterion

$C=\liminf\limits_{t\to\infty}\frac{1}{t}\frac{1}{\gamma}\log(E[(W_{t})^{\gamma}])$

determines the log-growth rate of the trajectories of $W_{t}$ according to

$\liminf\limits_{t\to\infty}\frac{1}{t}\log W_{t}\geq C\ a.s.$

and that, if $C>0$, the barrier hitting times
$T_{b}=\inf\\{t\in\mathbb{N}\colon W_{t}>b\\},\ b\in\mathbb{R}^{+},$ are
integrable and satisfy

$\limsup\limits_{b\to+\infty}\frac{E[T_{b}]}{\log(b)}\leq\frac{1}{C}.$

At the end it is illustrated how the results of Section 2 can be applied to
the mean squared error in the context of estimation.

##  2  Exponential convergence of random variables

In the whole paper we assume that $(\Omega,\Sigma,\mathbb{P})$ is a
probability space and that $X_{t}\colon\Omega\to\mathbb{R}^{+}$,
$t\in\mathbb{N}$, is a sequence of random variables, where
$\mathbb{R}^{+}=[0,+\infty)$. We start with the following definition.

######  Definition 1.

Given the sequence $x_{t}\in\mathbb{R}^{+}$, $t\in\mathbb{N}$, define:

$\overline{R}(x_{t})=\limsup\limits_{t\to\infty}\sqrt[t]{x_{t}}.$

We are mostly interested in the convergence to $0$ and thus we will usually
refer to $\overline{R}(x_{t})$ as convergence rate. If $x_{t}\to 0$ then
$\overline{R}(x_{t})\in[0,1]$. Condition $\overline{R}(x_{t})\in(0,1)$
determines the exponential convergence of $x_{t}$ to zero. We will sometimes
slightly abuse the nomenclature and we will refer to $\overline{R}(x_{t})$ as
convergence rate regardless of whether the $x_{t}$ converges or not. Condition
$\overline{R}(x_{t})=1$ excludes the exponential convergence and does not
determine the convergence of $x_{t}$. Condition $\overline{R}(x_{t})>1$
implies that some subsequence of $x_{t}$ diverges exponentially fast. The
following observation presents general characterisation of exponential
convergence and is rather easy to prove.

######  Observation 1.

Given $C\in\mathbb{R}^{+}$ and a sequence $x_{t}\in\mathbb{R}^{+}$, the
following conditions are equivalent:

  1. 1.

$C\geq\limsup\limits_{t\to\infty}\sqrt[t]{x_{t}}$,

  2. 2.

$\log C\geq\limsup\limits_{t\to\infty}\frac{1}{t}\log(x_{t})$, where
$\log(\cdot)=\exp^{-1}(\cdot)$ and $\log(0):=-\infty$,

  3. 3.

for any $R>C$, $\sup\limits_{t\in\mathbb{N}}\frac{x_{t}}{R^{t}}<+\infty$.

  4. 4.

for any $R>C$, $\lim\limits_{t\to\infty}\frac{x_{t}}{R^{t}}=0$,

  5. 5.

for any $R>C$, $\sum\limits_{t\in\mathbb{N}}\frac{x_{t}}{R^{t}}<\infty$,

  6. 6.

for any $R>C$,
$\sum\limits_{t\in\mathbb{N}}\sum\limits_{i=t}^{\infty}\frac{x_{i}}{R^{i}}<\infty$.

If additionally $C\in(0,1)$ and
$\tau_{\varepsilon}:=\inf\\{t\in\mathbb{N}\colon x_{t}<\varepsilon\\},\
\varepsilon>0,$ then any of the above conditions implies that

$\limsup\limits_{\varepsilon\to
0^{+}}\frac{\tau_{\varepsilon}}{|\log(\varepsilon)|}\leq\frac{1}{|\log(C)|}.$
(5)

Observation 1 implies that the value of the limit
$\overline{R}(x_{t})=\limsup\limits_{t\to\infty}\sqrt[t]{x_{t}}$ is the
infimum of constants $C\geq 0$ such that for any $R>C$:

$x_{t}\leq M_{R}\cdot R^{t},\ t\in\mathbb{N},\mbox{ where
}M_{R}:=\sup\limits_{t\in\mathbb{N}}\frac{x_{t}}{R^{t}}<\infty.$ (6)

Equation (6) is satisfied for any $R>\overline{R}(x_{t})$.

Given any measurable sequence $X_{t}\geq 0$, $t\in\mathbb{N}$, we may consider
the random variable $C\colon\Omega\to[0,+\infty]$ given by

$C:=\overline{R}(X_{t})=\limsup\limits_{t\to\infty}\sqrt[t]{X_{t}}.$ (7)

Below, if $E[X_{t}]=\infty$ then we put $\sqrt[t]{\infty}:=\infty$. Theorem 2
does not assume that $X_{t}\to 0$ and may describe the divergence of $X_{t}$
to infinity.

######  Theorem 2.

Let $X_{t}\geq 0$. The convergence rate of the trajectories is bounded from
above by the convergence rate of the expectations in sense

$\mathbb{P}[\limsup\limits_{t\to\infty}\sqrt[t]{X_{t}}\leq\limsup\limits_{t\to\infty}\sqrt[t]{E[X_{t}]}]=1.$

###### Proof.

For the proof it is enough to assume that
$A:=\limsup\limits_{t\to\infty}\sqrt[t]{E[X_{t}]}<\infty$ and it is enough to
show that for any $\varepsilon>0$

$\mathbb{P}[\limsup\limits_{t\to\infty}\sqrt[t]{X_{t}}>A+\varepsilon]=0.$

Fix $\varepsilon>0$ and note that
$\limsup\limits_{t\to\infty}\sqrt[t]{X_{t}}=\lim\limits_{t\to\infty}\sup\limits_{i\geq
t}\sqrt[i]{X_{i}}$ . We thus have

$\mathbb{P}[\limsup\limits_{t\to\infty}\sqrt[t]{X_{t}}>A+\varepsilon]=\mathbb{P}[\lim\limits_{t\to\infty}\sup\limits_{i\geq
t}\sqrt[i]{X_{i}}>A+\varepsilon].$

By the monotonicity $\sup\limits_{i\geq
t+1}\sqrt[i]{X_{i}}\leq\sup\limits_{i\geq t}\sqrt[i]{X_{i}}$, we have

$\\{\lim\limits_{t\to\infty}\sup\limits_{i\geq
t}\sqrt[i]{X_{i}}>A+\varepsilon\\}\subset\bigcap\limits_{t\in\mathbb{N}}\\{\sup\limits_{i\geq
t}\sqrt[i]{X_{i}}>A+\varepsilon\\}$

and

$\mathbb{P}[\lim\limits_{t\to\infty}\sup\limits_{i\geq
t}\sqrt[i]{X_{i}}>A+\varepsilon]\leq\lim\limits_{t\to\infty}\mathbb{P}[\sup\limits_{i\geq
t}\sqrt[i]{X_{i}}>A+\varepsilon]=$
$=\lim\limits_{t\to\infty}\mathbb{P}[\bigcup\limits_{i=t}^{+\infty}\\{X_{i}>(A+\varepsilon)^{i}\\}]\leq\lim\limits_{t\to\infty}\sum\limits_{i=t}^{\infty}\mathbb{P}[X_{i}>(A+\varepsilon)^{i}].$

By Chebyshev inequality

$\lim\limits_{t\to\infty}\sum\limits_{i=t}^{\infty}\mathbb{P}[X_{i}>(A+\varepsilon)^{i}]\leq\lim\limits_{t\to\infty}\sum\limits_{i=t}^{\infty}\frac{E[X_{i}]}{(A+\varepsilon)^{i}},$

and by point 5. of Observation 1 (with $x_{t}:=E[X_{t}]$ and
$R=A+\varepsilon$),

$\lim\limits_{t\to\infty}\sum\limits_{i=t}^{\infty}\frac{E[X_{i}]}{(A+\varepsilon)^{i}}=0.$

∎

######  Conclusion 3.

The convergence rate of the expectations $E[X_{t}]$ is bounded from below by
the essential suppremum of $\overline{R}(X_{t})$, i.e.

$\limsup\limits_{t\to\infty}\sqrt[t]{E[X_{t}]}\geq\inf\\{\hat{C}\in\mathbb{R}\colon\mathbb{P}[\overline{R}(X_{t})\leq\hat{C}]=1\\}.$

Now we will focus on the convergence of the expectations of the following
hitting times

$\tau_{\epsilon}=\inf\\{t\in\mathbb{N}\colon X_{t}<\varepsilon\\},\
\varepsilon>0.$

Assume from now on that $X_{t}\to 0$ a.s. so
$\overline{R}(X_{t})=\limsup\limits_{t\to\infty}\sqrt[t]{X_{t}}\leq 1$ a.s.
Let

$C:=ess\sup\overline{R}(X_{t})$

denote the essential supremum of $\overline{R}(X_{t})$. By point 3. of
Observation 1 for any $R>C$ there is some random variable
$M_{R}\colon\Omega\to\mathbb{R}^{+}$ with

$X_{t}\leq M_{R}\cdot R^{t}\ a.s.$

If $C\in(0,1)$ then again by Observation 1,

$\limsup\limits_{\varepsilon\to
0^{+}}\frac{\tau_{\varepsilon}}{|\log(\varepsilon)|}\leq\frac{-1}{\log(C)}\
a.s.$

At the same time the expectation of $\tau_{\epsilon}$ may be infinite, see a
simple example below.

######  Example 1.

Let $\Omega=\\{0,1,2,\dots\\}$, $\Sigma=\mathcal{P}(\Omega)$ be the family of
all subsets of $\Omega$ and let $p_{n}:=\mathbb{P}[n],n\in\Omega$. Assume that
$X_{t}$ is the characteristic function of set
$\mathbb{N}_{t}:=\\{0,1,\dots,t\\}$, i.e.

$X_{t}(n)=1_{\mathbb{N}_{t}}(n),\ n\in\mathbb{N}.$

Clearly we have $\limsup\limits_{t\to\infty}\sqrt[t]{X_{t}}=0$. At the same
time, for $\varepsilon\in(0,1)$ we have

$\tau_{\varepsilon}(n)={n+1},\ n\in\Omega,$

and

$E[\tau_{\varepsilon}]=\sum\limits_{t\in\mathbb{N}}\mathbb{P}[\tau_{\varepsilon}>t]=\sum\limits_{t\in\mathbb{N}}\mathbb{P}[X_{t}=1]=\sum\limits_{t\in\mathbb{N}}\mathbb{P}[\mathbb{N}_{t}]=\sum\limits_{t\in\mathbb{N}}\sum\limits_{n\leq
t}p_{n}.$

The above sum may be infinite depending on how the probabilities $p_{n}$ are
chosen. At the same time $\overline{R}(X_{t})=0$ and
$\lim\limits_{\varepsilon\to
0^{+}}\frac{\tau_{\varepsilon}}{|\log(\varepsilon)|}=0$ a.s.

If the expectations $E[X_{t}]$ decrease exponentially fast then
$E[\tau_{\varepsilon}]$ is finite and Theorem 4 shows that the convergence
rate of the expectations $\overline{R}(E[X_{t}])$ controls the convergence of
$E[\tau_{\varepsilon}]$ with $\varepsilon\to 0^{+}$.

######  Theorem 4.

If $A:=\overline{R}(E[X_{t}])=\limsup\limits_{t\to\infty}\sqrt[t]{E[X_{t}]}<1$
then

$\limsup\limits_{\varepsilon\to
0^{+}}\frac{E[\tau_{\varepsilon}]}{|\log(\varepsilon)|}\leq\frac{-1}{\log(A)}.$

Above, if $A=0$ then $\frac{-1}{\log(A)}:=0$. The proof of the above theorem
is based on the following lemma which follows from Theorem 2 and Observation
1.

######  Lemma 5.

Assume that $A=\limsup\limits_{t\to\infty}\sqrt[t]{E[X_{t}]}<1$. For any $R>A$
the following random variable

$H_{R}:=\inf\\{t\in\mathbb{N}\colon\sup\limits_{i\geq t}\frac{X_{i}}{R^{i}}\
\leq\ 1\\}$

has finite expectation.

###### Proof.

Fix $R>A$. By Theorem 2 and by point 4. of Observation 1 we have
$\mathbb{P}[H_{R}<\infty]=1$. We need to show $E[H_{R}]<\infty$. We have:

$E[H_{R}]=\sum\limits_{t\in\mathbb{N}}\mathbb{P}[H_{R}>t].$

By definition of $H_{R}$ and by Chebyszev inequality,

$\mathbb{P}[H_{R}>t]=\mathbb{P}[\bigcup\limits_{i=t}^{\infty}\\{X_{i}>R^{i}\\}]\leq\sum\limits_{i=t}^{\infty}\mathbb{P}[X_{i}>R^{i}]\leq\sum\limits_{i=t}^{\infty}\frac{E[X_{i}]}{R^{i}}.$

We have thus shown that

$E[H_{R}]\leq\sum\limits_{t\in\mathbb{N}}\sum\limits_{i=t}^{\infty}\frac{E[X_{i}]}{R^{i}}$

and the above series is finite by point 6. of Observarion 1 applied to the
sequence $x_{t}=E[X_{t}]$. ∎

_Proof of Theorem 4_. Fix $C\in(A,1)$. Under notation of Lemma 5, for any
$t\in\mathbb{N}$ on set $\\{t\geq H_{C}\\}$ we have $X_{t+i}\leq C^{t+i}$ for
all $i\in\mathbb{N}$. In other words, for any $t\geq H_{C}$ we have $X_{t}\leq
C^{t}$. At the same time, as $C<1$, note that for

$\hat{C}:=\frac{-1}{log(C)}$

the ceiling $t=\lceil\hat{C}\cdot|\log(\varepsilon)|\rceil$ is the smallest
natural number with $C^{t}\leq\varepsilon$. This implies that if
$t\geq\max\\{H_{C},\lceil\hat{C}\cdot|\log(\varepsilon)|\rceil\\}$ then
$X_{t}\leq\varepsilon$ and $X_{t+1}<\varepsilon$. Hence,

$\tau_{\varepsilon}\leq\max\\{H_{C},\lceil\hat{C}\cdot|\log(\varepsilon)|\rceil\\}+1$

and

$\frac{\tau_{\varepsilon}}{|\log(\varepsilon)|}\leq\frac{1}{|\log(\varepsilon)|}(\max\\{H_{C},\lceil\hat{C}\cdot|\log(\varepsilon)|\rceil\\}+1).$
(8)

Thus

$E[\frac{\tau_{\varepsilon}}{|\log(\varepsilon)|}]\leq\frac{1}{|\log(\varepsilon)|}E[\max\\{H_{C},\lceil\hat{C}\cdot|\log(\varepsilon)|\rceil\\}]+\frac{1}{|\log(\varepsilon)|}.$
(9)

Lemma 5 allows us to use the Lebesgue’s dominated convergence theorem:

$\frac{1}{|\log(\varepsilon)|}E[\max\\{H_{C},\lceil\hat{C}\cdot|\log(\varepsilon)|\rceil\\}]=E[\max\\{\frac{H_{C}}{|\log(\varepsilon)|},\hat{C}\\}]\to\hat{C}\mbox{
with }\varepsilon\to 0^{+}.$

Equation (9) and the above imply that $\limsup\limits_{\varepsilon\to
0^{+}}\frac{E[\tau]}{|\log(\varepsilon)|}\leq\hat{C}\mbox{ for any }C\in(A,1)$
and thus

$\limsup\limits_{\varepsilon\to
0^{+}}\frac{E[\tau_{\varepsilon}]}{|\log(\varepsilon)|}\leq\hat{A}=\frac{-1}{\log(A)}.$

$\Box$

##  3 Applications.

The results of the previous sections will be applied to some problems of
optimization, control and estimation.

###  3.1 Optimization

Let $(K,d)$ be some metric space and $f\colon K\to\mathbb{R}^{+}$ be a Borel-
measurable function which attains its global minimum $f_{\min}$. Assume that
$X_{t}\in K$, $t\in\mathbb{N}$, represents an optimization process. The
hitting time of the $\varepsilon$-optimal sublevel set $\\{x\in A\colon
f(x)<f_{\min}+\varepsilon\\}$ is defined by

$\tau_{\varepsilon}=\inf\\{t\in\mathbb{N}\colon
f(X_{t})<f_{\min}+\epsilon\\}.$

Theorems 2 and 4 give us the relations between convergence rate of the
trajectories $f(X_{t})$, the convergence rate of the expectations
$E[|f(X_{t})-f_{\min}|]$ and the convergence behaviour of
$E[\tau_{\varepsilon}]$ under general assumptions on the sequence $f(X_{t})$.
In particular, if the following constant

$A=\overline{R}(E[f(X_{t})-f_{\min}],t\in\mathbb{N})=\limsup\limits_{t\to\infty}\sqrt[t]{E[f(X_{t})-f_{\min}]}$

satisfies $A<1$ then by Theorem 4 we have the following upper bound
$\limsup\limits_{t\to\infty}\frac{E[\tau_{\varepsilon}]}{|\log(\varepsilon)|}\leq\frac{-1}{\log(A)}$
and the following control of the asymptotic behaviour of
$E[\tau_{\varepsilon}]$ with $\varepsilon\to 0^{+}$ :

$E[\tau_{\varepsilon}]\leq C(\varepsilon)\cdot|\log(\varepsilon)|,$

where $C\colon(0,1)\to\mathbb{R}^{+}$ satisfies
$\limsup\limits_{\varepsilon\to
0^{+}}C(\varepsilon)\leq\hat{A}=\frac{-1}{\log(A)}$. This strengthens Theorem
7 from [16]. We also have, by (2) and (4),

$\mathbb{P}[\limsup\limits_{t\to\infty}\sqrt[t]{f(X_{t})-f_{min}}\leq
A]=1\mbox{ and }\lim\limits_{\varepsilon\to
0^{+}}\limsup\limits_{t\to\infty}\sqrt[t]{\mathbb{P}[f(X_{t})\geq
f_{\min}+\varepsilon]}\leq A.$

###  3.2 Risk sensitive control

Let $W_{t}\colon\Omega\to\mathbb{R}^{+}$, $t\in\mathbb{N}$, represent the
wealth process of an investor, see [2],[3],[4],[12] for details. Let
$\gamma\neq 0$ represent the risk-averse parameter of an investor so we are
interested in the limit of the long-run risk-sensitive criterion of the
following log wealth growth:

$C=\liminf\limits_{t\to\infty}\frac{1}{t}\frac{1}{\gamma}\log(E[(W_{t})^{\gamma}]).$
(10)

We will assume that $\gamma<0$ which is a common investment criterion in the
context of long run optimization. This section shows how the value of the
risk-sensitive criterion determines the exponential growth of the trajectories
of the wealth process $W_{t}$ and the logarithmic growth of the expected
barrier-hitting time $T_{b}$ with $b\nearrow\infty$. This type of convergence
behaviour is natural under suitable ergodicity-type assumptions on the log-
wealth process, see for instance the asymptotic optimality principle, [2].
This section brings to attention that the log growth rate of the trajectories
of $W_{t}$ and the logarithmic growth of the corresponding expected hitting
times are determined by the limit $\eqref{CCCC}$ under general assumptions on
$W_{t}$ (no ergodicity involved) which follows from the results of Section 2.

######  Theorem 6.

Let $\gamma<0$ and let $W_{t}\colon\Omega\to(0,\infty)$ satisfy
$E[(W_{t})^{\gamma}]<\infty$, $t\in\mathbb{N}$. The constant

$C=\liminf\limits_{t\to\infty}\frac{1}{t}\frac{1}{\gamma}\log(E[(W_{t})^{\gamma}])$

determines the log-growth of $W_{t}$ according to

$\liminf\limits_{t\to\infty}\frac{1}{t}\log W_{t}\geq C\ a.s.$

Additionally, if $C>0$ then the following hitting times
$T_{b}=\inf\\{t\in\mathbb{N}\colon W_{t}>b\\},\ b\in\mathbb{R}^{+},$ are
integrable and satisfy

$\limsup\limits_{b\to+\infty}\frac{E[T_{b}]}{\log(b)}\leq\frac{1}{C}.$

Let $X_{t}:=\frac{1}{W_{t}}$. Recall that

$E[(X_{t})^{p_{1}}]^{\frac{1}{p_{1}}}\leq
E[(X_{t})^{p_{2}}]^{\frac{1}{p_{2}}}\mbox{ for }0<p_{1}<p_{2}<\infty,$ (11)

assuming that the above expectations exist. For $\gamma<0$ let

$C_{\gamma}=\liminf\limits_{t\to\infty}\frac{1}{t}\frac{1}{\gamma}\log(E[(W_{t})^{\gamma}]).$

By elementary calculations and by (11) one may show that

$C_{\gamma_{1}}\leq C_{\gamma_{2}}\mbox{ for }\gamma_{1}<\gamma_{2}<0.$ (12)

See also Lemma 2.1 in [9] for the above. Theorem 6 and Equation (12)
immediately lead to the following.

######  Conclusion 7.

Assume that $W_{t}\colon\Omega\to(0,\infty)$ satisfy
$E[(W_{t})^{\gamma}]<\infty$, $t\in\mathbb{N}$, for any $\gamma<0$ close to
zero. The following limit

$C=\lim\limits_{\gamma\to
0^{-}}\liminf\limits_{t\to\infty}\frac{1}{t}\frac{1}{\gamma}\log(E[(W_{t})^{\gamma}])$

bounds the log-growth of $W_{t}$ according to
$\liminf\limits_{t\to\infty}\frac{1}{t}\log W_{t}\geq C\ a.s.$ Additionally,
if $C>0$ then the hitting times $T_{b}=\inf\\{t\in\mathbb{N}\colon W_{t}>b\\}$
are integrable and satisfy

$\limsup\limits_{b\to+\infty}\frac{E[T_{b}]}{\log(b)}\leq\frac{1}{C}.$

Proof of Theorem 6. As $\gamma<0$, equation (10) implies that

$C\cdot\gamma=\limsup\limits_{t\to\infty}\frac{1}{t}\log(E[(W_{t})^{\gamma}]).$

Hence,

$\exp(C\gamma)=\limsup\limits\limits_{t\to\infty}(E[(W_{t})^{\gamma}])^{\frac{1}{t}}.$
(13)

By Theorem 2,

$\limsup\limits_{t\to\infty}((W_{t})^{\gamma})^{\frac{1}{t}}\leq\exp(C\cdot\gamma)\
a.s.$

As $\gamma<0$, the above determines the growth of $W_{t}$ according to

$\liminf\limits_{t\to\infty}((W_{t})^{|\gamma|})^{\frac{1}{t}}\geq\exp(C\cdot|\gamma|)\
a.s.$

Hence, we have

$\liminf\limits_{t\to\infty}(W_{t})^{\frac{1}{t}}\geq\exp(C)\ a.s.$ (14)

and

$\liminf\limits_{t\to\infty}\frac{1}{t}\log W_{t}\geq C\ a.s.$

which proves the first part of the theorem.

Now we assume that $C>0$ and we will discuss the hitting times
$T_{b}=\inf\\{t\in\mathbb{N}\colon W_{t}>b\\}.$ Let

$\tau_{\varepsilon}=\inf\\{t\in\mathbb{N}\colon(W_{t})^{\gamma}<\varepsilon\\}=\inf\\{t\in\mathbb{N}\colon
W_{t}>(\tfrac{1}{\varepsilon})^{|\gamma|^{-1}}\\}.$

Condition $C>0$ implies $\exp(\gamma\cdot C)<1$. By (13) and Theorem 4, we
have

$\limsup\limits_{\varepsilon\to
0^{+}}\frac{E[\tau_{\varepsilon}]}{|\log(\varepsilon)|}\leq\frac{1}{C\cdot|\gamma|}.$
(15)

Let

$T(\varepsilon)=\inf\\{t\in\mathbb{N}\colon W_{t}>\frac{1}{\varepsilon}\\}$

so we have $\tau_{\varepsilon}=\inf\\{t\in\mathbb{N}\colon
W_{t}>\frac{1}{\varepsilon^{|\gamma|^{-1}}}\\}=T(\varepsilon^{|\gamma|^{-1}})$
and

$\limsup\limits_{\varepsilon\to
0^{+}}\frac{E[\tau_{\varepsilon}]}{|\log(\varepsilon)|}=\limsup\limits_{\varepsilon\to
0^{+}}\frac{E[T(\varepsilon^{\frac{1}{|\gamma|}})]}{|\log(\varepsilon)|}=\limsup\limits_{\varepsilon\to
0^{+}}\frac{E[T(\varepsilon)]}{|\log(\varepsilon^{|\gamma|})|}=\limsup\limits_{\varepsilon\to
0^{+}}\frac{E[T(\varepsilon)]}{|\gamma|\cdot|\log(\varepsilon)|}.$

Hence, by (15), $\limsup\limits_{\varepsilon\to
0^{+}}\frac{E[T(\varepsilon)]}{|\gamma|\cdot|\log(\varepsilon)|}\leq\frac{1}{|\gamma|\cdot
C}$, and thus

$\limsup\limits_{\varepsilon\to
0^{+}}\frac{E[T(\varepsilon)]}{|\log(\varepsilon)|}\leq\frac{1}{C}.$

For $b=\frac{1}{\varepsilon}$ we have $T_{b}=T(\varepsilon)$ and
$|\log(\varepsilon)|=\log(\frac{1}{\varepsilon})=\log(b)$. We thus have

$\limsup\limits_{b\to+\infty}\frac{E[T_{b}]}{\log(b)}\leq\frac{1}{C}.$

$\Box$

###  3.3 Estimation

Assume that a sequence $Z_{t}\in\mathbb{R}$ approaches the unknown parameter
$\theta\in\Theta\subset\mathbb{R}$ and that the mean squared error
$E|Z_{t}-\theta|^{2}$ satisfies

$C=\limsup\limits_{t\to\infty}(E|Z_{t}-\theta|^{2})^{\frac{1}{t}}.$

The results of Section 2 can be applied to the sequence
$X_{t}=|Z_{t}-\theta|^{2}$. In particular, by Theorem 2, we bound the
convergence rate of the trajectories

$\mathbb{P}[\limsup\limits_{t\to\infty}(|Z_{t}-\theta|^{2})^{\frac{1}{t}}\leq
C]=1\mbox{ and hence
}\mathbb{P}[\limsup\limits_{t\to\infty}(|Z_{t}-\theta|)^{\frac{1}{t}}\leq\sqrt{C}]=1.$

More generally, if $(K,d)$ is a metric space, $\theta\in K$ and $Z_{t}\in K$
converges in mean to $\theta$, i.e. $E[d(Z_{t},\theta)]\to 0$, then the
results of Section 2 may be applied to the sequence $X_{t}:=d(Z_{t},\theta)$.

## References

  * [1] Agapie, A. (2022). Evolution strategies under the 1/5 success rule. Mathematics, 11(1), 201.
  * [2] Algoet, P. H., and Cover, T. M. (1988). Asymptotic optimality and asymptotic equipartition properties of log-optimum investment. The Annals of Probability, 876-898.
  * [3] Bielecki, T. R., and Pliska, S. R. (2003). Economic properties of the risk sensitive criterion for portfolio management. Review of Accounting and Finance, 2(2), 3-17.
  * [4] Biswas, A., and Borkar, V. S. (2023). Ergodic risk-sensitive control—a survey. Annual Reviews in Control, 55, 118-141.
  * [5] Douc, R., Moulines, E., Priouret, P., and Soulier, P. (2018). Markov chains, Cham, Switzerland: Springer International Publishing.
  * [6] Dudley, R. M. (2018). Real analysis and probability. CRC Press.
  * [7] Jögersküpper, J., (2007). Algorithmic analysis of a basic evolutionary algorithm for continuous optimization. Theoretical Computer Science, 379(3), 329-347.
  * [8] Jögersküpper, J., (2006). "How the (1+ 1) ES using isotropic mutations minimizes positive definite quadratic forms." Theoretical Computer Science 361.1 , 38-56.
  * [9] Kupper, M., and Schachermayer, W. (2009). Representation results for law invariant time consistent functions. Mathematics and Financial Economics, 2(3), 189-210.
  * [10] Liu, Q., and Watbled, F. (2009). Exponential inequalities for martingales and asymptotic properties of the free energy of directed polymers in a random environment. Stochastic processes and their applications, 119(10), 3101-3132.
  * [11] Morinaga, D., Fukuchi, K., Sakuma, J., and Akimoto, Y. (2023). Convergence Rate of the (1+ 1)-ES on Locally Strongly Convex and Lipschitz Smooth Functions. IEEE Transactions on Evolutionary Computation.
  * [12] Pitera, M., and Stettner, Ł. (2023). Discrete-time risk sensitive portfolio optimization with proportional transaction costs. Mathematical Finance, 33(4), 1287-1313.
  * [13] Rudolph, G. (1994, June). Convergence of non-elitist strategies. In Proceedings of the First IEEE Conference on Evolutionary Computation. IEEE World Congress on Computational Intelligence (pp. 63-66). IEEE.
  * [14] G. Rudolph, (1997). Convergence rates of evolutionary algorithms for a class of convex objective functions. Control and Cybernetics Volume 26 Issue 3 , pp. 375-390.
  * [15] Sun, J. (1997). Exponential convergence for sequences of random variables. Statistics and probability letters, 34(2), 159-164.
  * [16] Tarłowski, D. (2024). On asymptotic convergence rate of random search. Journal of Global Optimization, 89(1), 1-31.
  * [17] Tarłowski, D. (2018). On geometric convergence rate of Markov search towards the fat target. Operations Research Letters, 46(1), 33-36.
  * [18] Tikhomirov, A. S. (2019, October). On the convergence rate of the quasi–Monte Carlo method of search for extremum. In Journal of Physics: Conference Series (Vol. 1352, No. 1, p. 012051). IOP Publishing.
  * [19] Toure, C., Auger, A. Hansen, N. (2023). Global linear convergence of evolution strategies with recombination on scaling-invariant functions. Journal of Global Optimization, 86(1), 163-203.
  * [20] Zhigljavsky, A., and Zilinskas, A. (2021). Bayesian and high-dimensional global optimization. Cham: Springer International Publishing.