# Quantum correlation between a qubit and a relativistic boson in an expanding
spacetime

Shu-Min Wu1, Hao-Sheng Zeng2111Email<EMAIL_ADDRESS>Tonghua Liu3 1
Department of Physics, Liaoning Normal University, Dalian 116029, China
2 Department of Physics, Hunan Normal University, Changsha 410081, China
3 Department of Physics, Yangtze University, Jingzhou 434023, China

###### Abstract

We use the quantum correlation of both logarithmic negativity and mutual
information between a qubit and a relativistic boson to analyze the dynamics
of universe expansion. These dynamical quantum correlations can encode the
information about underlying spacetime structure, which suggests a promising
application in observational cosmology. We find that the dynamics of both
logarithmic negativity and mutual information between the qubit and the boson
are very similar. They decrease monotonically with the growth of the expansion
volume and the expansion rate. Smaller momentum and medium-sized mass of boson
are more favourable for extracting the information about history of universe
expansion. The quantum correlation between the qubit and the antiboson however
has very different behavior: The logarithmic negativity is always zero and the
mutual information can be generated through the expansion of universe. Smaller
momentum and medium-sized mass of antiboson are beneficial for the production
of mutual information. Finally, the trigger phenomenon and conservation for
mutual information are witnessed.

Keywords: logarithmic negativity, mutual information, expanding spacetime

###### pacs:

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

##  I Introduction

Quantum entanglement plays an important role in quantum information theory. It
is a resource for quantum information tasks, such as quantum communication,
teleportation and various computational tasks, and has applications in quantum
control and quantum simulations L1 ; L2 ; L3 ; L4 . Though the study about
quantum entanglement is originated initially from the nonrelativistic realm,
extending it to the relativistic domain is ultimately necessary because the
world can never escape the influence of gravity. In fact, many settings, such
as photons in the realization of quantum information L5 ; L6 ; L7 ; L8 ; L9 ;
L10 ; wsm1 ; L11 ; L12 ; L13 ; L14 ; L15 ; L16 ; L17 ; LL15 ; wsm2 ; wsm3 ,
involve relativistic effects. Moreover, quantum entanglement plays a prominent
role in the information loss problem and in black hole thermodynamics L18 ;
L19 ; L20 ; L21 .

In fact, the notion of relativistic quantum information includes two branches:
the special relativity effect Moradpour2015  and the general relativity
(acceleration and gravity) effect. In the latter case, people mainly concern
two aspects: (i) Study the influence of gravity or acceleration on quantum
resources or quantum settings, such as quantum entanglement, quantum
coherence, quantum key distribution and quantum communication L5 ; L6 ; L7 ;
L8 ; L9 ; L10 ; wsm1 ; L11 ; L12 ; L13 ; L14 ; L15 ; L16 ; L17 ; LL15 ; wsm2 ;
wsm3 ; (ii) Probe the property of spacetime via quantum effect L22 ; L23 ; L24
. Quantum entanglement subject to gravitational effect shows that conceptually
important qualitative differences from the nonrelativistic perspective arise.
For example, quantum entanglement depends on the motional state of the
observer and degrades from the perspective of accelerated observers L5 ; L6 ;
L7 ; L8 ; L9 ; L10 . These results imply that quantum resources in
relativistic settings might not be an invariant concept.

In this paper, we consider an entangled state between a qubit and a
relativistic boson. The latter is transformed, via the expansion of a
Robertson-Walker spacetime, to a superposition of different number states of
boson and antiboson. We then study, in the final state, the pairwise quantum
correlation (i.e.,logarithmic negativity and mutual information) between the
qubit and the boson, or between the qubit and the antiboson. The dynamics,
redistribution and conservation of the underlying quantum correlations in the
expansion of universe are our research focuses. Such researches not only
deepen our understanding about quantum correlations but also provide the
prospect of employing quantum correlations as a tool to study the property of
curved spacetime, because the dynamical quantum correlations contain
information about the history of the expanding spacetime, affording the
possibility of deducing cosmological parameters of the underlying spacetime.
This novel way of obtaining information about cosmological parameters could
both theoretically and experimentally provide new insight into the early
universe LL24 ; LLL24 . Actually, it has been shown that quantum correlations
play an important role in the thermodynamic properties of Robertson-Walker
type spacetime, and can be used to distinguish different spacetimes, and probe
spacetime fluctuations L25 ; L26 ; L27 .

The paper is organized as follows. In Sec. II, we present curtly the
quantization of scalar fields in expanding spacetime. Sec. III is the main
part of the paper, where the dynamics, redistribution and conservation of
quantum correlations in the expanding spacetime are investigated. The last
section is devoted to the brief conclusion.

##  II Quantization of scalar fields in expanding spacetime

Let us consider a two-dimensional Robertson-Walker spacetime with the line
element described by the metric L28 ; L29 ; L30

$ds^{2}=dt^{2}-[a(t)]^{2}dx^{2}\,,$ (1)

where $a(t)$ is the scale factor. Introducing the conformal time $\eta$ that
is related to the cosmological time $t$ by

$\eta=\int_{0}^{t}\frac{\text{d}\tau}{a(\tau)}\,,$ (2)

then the metric has the form

$ds^{2}=[a(\eta)]^{2}(d\eta^{2}-dx^{2})\,.$ (3)

Following the standard notation, the conformal scale factor
$C(\eta)=[a(\eta)]^{2}$ has specific form

$C(\eta)=1+\epsilon(1+\tanh(\rho\eta))\,,$ (4)

where $\epsilon$ is positive real parameter controlling the total expansion
volume, and $\rho$ is positive real parameter controlling the expansion rate.
The conformal scale factor $C(\eta)$ conveniently shows that the spacetime
undergoes a period of smooth expansion for the universe, and becomes flat in
the distant past $\eta\rightarrow-\infty$ [$C(\eta\rightarrow-\infty)=1$] and
in the far future $\eta\rightarrow\infty$
[$C(\eta\rightarrow\infty)=1+2\epsilon$]. For convenience, the distant past
and far future are referred to as the asymptotic “in”-region and “out”-region,
respectively.

In the background of the expanding spacetime, a bosonic field obeys the Klein-
Gordon equation

$[\partial_{\eta}^{2}+k^{2}+C(\eta)m^{2}]\Phi=0.$ (5)

Solving this equation, we obtain two sets of complete modes in the asymptotic
past and the asymptotic future

$\displaystyle u^{\text{in}}_{k}(x,\eta)$
$\displaystyle=\sigma^{\text{in}}(\eta)\exp\left[i(kx-\omega_{{}_{+}}\eta)-\frac{i\omega_{-}}{\rho}\ln[2\cosh(\rho\eta)]\right]\,,$
(6) $\displaystyle u^{\text{out}}_{k}(x,\eta)$
$\displaystyle=\sigma^{\text{out}}(\eta)\exp\left[i(kx-\omega_{{}_{+}}\eta)-\frac{i\omega_{-}}{\rho}\ln[2\cosh(\rho\eta)]\right]\,,$
(7)

where

$\displaystyle\sigma^{\text{in}}(\eta)$
$\displaystyle=\frac{1}{2\sqrt{\pi\omega_{\text{in}}}}F_{1}\left[1+\frac{i\omega_{-}}{\rho},\frac{i\omega_{-}}{\rho};1-\frac{i\omega_{\text{in}}}{\rho};\frac{1}{2}[1-\tanh(\rho\eta)]\right]\,,$
(8) $\displaystyle\sigma^{\text{out}}(\eta)$
$\displaystyle=\frac{1}{2\sqrt{\pi\omega_{\text{out}}}}F_{1}\left[1+\frac{i\omega_{-}}{\rho},\frac{i\omega_{-}}{\rho};1-\frac{i\omega_{\text{out}}}{\rho};\frac{1}{2}[1-\tanh(\rho\eta)]\right]\,,$
(9)

with $F_{1}$ the hypergeometric function, and

$\omega_{\text{in}}=\sqrt{k^{2}+m^{2}}\,,\qquad\omega_{\text{out}}=\sqrt{k^{2}+m^{2}(1+2\epsilon)}\,,\qquad\omega_{\pm}=\frac{1}{2}(\omega_{\text{out}}\pm\omega_{\text{in}})\,.$
(10)

The in and out modes are related by the Bogoliubov transformationL29 ;
Birrell1982

$u^{\text{in}}_{k}(x,\eta)=\alpha_{k}u^{\text{out}}_{k}(x,\eta)+\beta_{k}u^{{\text{out}*}}_{-k}(x,\eta)\,,$
(11)

with Bogoliubov coefficients given by

$\displaystyle\alpha_{k}$
$\displaystyle=\sqrt{\frac{\omega_{\text{out}}}{\omega_{\text{in}}}}\frac{\Gamma([1-(i\omega_{\text{in}}/\rho)])\Gamma(-i\omega_{\text{out}}/\rho)}{\Gamma([1-(i\omega_{+}/\rho)])\Gamma(-i\omega_{+}/\rho)}\,,$
(12) $\displaystyle\beta_{k}$
$\displaystyle=\sqrt{\frac{\omega_{\text{out}}}{\omega_{\text{in}}}}\frac{\Gamma([1-(i\omega_{\text{in}}/\rho)])\Gamma(i\omega_{\text{out}}/\rho)}{\Gamma([1+(i\omega_{-}/\rho)])\Gamma(i\omega_{-}/\rho)}\,.$
(13)

In accordance with Eq.(11), the Bogoliubov relationship between the operators
of the in and out modes is

$b_{\text{in},k}=\alpha_{k}^{*}b_{\text{out},k}-\beta_{k}^{*}b_{\text{out},{-k}}^{\dagger},$
(14)

where $b_{\text{in},k}$ is the bosonic annihilation operator acting on the
state in the asymptotic past, and $b_{\text{out},k}$ and
$b_{\text{out},{-k}}^{\dagger}$ are the bosonic annihilation and antibosonic
creation operators acting on the states in the asymptotic future. This allows
us to write the in-vacuum state as L28 ; L29 ; L30 ; Birrell1982

$|0_{k}\rangle_{\text{in}}=\sqrt{1-|\frac{\beta_{k}}{\alpha_{k}}|^{2}}\sum_{n=0}^{\infty}(\frac{\beta^{*}_{k}}{\alpha^{*}_{k}})^{n}|n_{k}\rangle_{\text{out}}|n_{-k}\rangle_{\text{out}},$
(15)

where $n_{k}$ represents the $n$ bosons and $n_{-k}$ represents the $n$
antibosons. This equation clearly shows that the in-vacuum state evolves into
a two-mode entangled state in the asymptotic future. Acting the Hermitian
conjugate of Eq.(14) on the above in-vacuum state, one obtains L31

$|1_{k}\rangle_{\text{in}}=\frac{1}{|\alpha|\alpha^{*}}\sum_{n=0}^{\infty}(\frac{\beta^{*}_{k}}{\alpha^{*}_{k}})^{n}\sqrt{n+1}|(n+1)_{k}\rangle_{\text{out}}|n_{-k}\rangle_{\text{out}}.$
(16)

##  III Logarithmic negativity and mutual information in an expanding
spacetime

Consider an entangled state between a qubit $p$ (shared by Alice) and a boson
of momentum $k$ (shared by Bob) in the asymptotic past

$\Psi_{p,k}=\chi|\uparrow\rangle|0_{k}\rangle_{\text{in}}+\sqrt{1-\chi^{2}}|\downarrow\rangle|1_{k}\rangle_{\text{in}},$
(17)

where $\chi$ is a state parameter which runs from 0 to 1, and
$|\uparrow\rangle$ and $|\downarrow\rangle$ denote the basis states of the
qubit. In the simulation experiment, we consider a single qubit (A) and the
boson (B) in their respective cavities. The two cavities are assumed to be the
same otherwise except that they are separated by a spacelike distance. Then,
we close Alice’s cavity through the switching function and turn on Bob’s
switching function. When the boson evolves from the asymptotic past to the
asymptotic future in a cavity, the bipartite state becomes mixed and can be
measured by the decoherence factor. Using Eqs.(15) and (16), we can rewrite
the entangled state as

$\Psi_{p,k}=\frac{1}{|\alpha_{k}|}\sum_{n=0}^{\infty}(\frac{\beta^{*}_{k}}{\alpha^{*}_{k}})^{n}[\chi|\uparrow\rangle|n_{k}\rangle_{\text{out}}|n_{-k}\rangle_{\text{out}}+\sqrt{1-\chi^{2}}\frac{\sqrt{n+1}}{\alpha_{k}^{*}}|\downarrow\rangle|(n+1)_{k}\rangle_{\text{out}}|n_{-k}\rangle_{\text{out}}].$
(18)

In this way, the entangled state encodes historical information concerning the
expanding spacetime. We can understand the properties of the underlying
spacetime structure by analysing the quantum properties of this state, such as
logarithmic negativity and mutual information. For simplicity, we omit the
subscript “out” below.

###  III.1 Logarithmic negativity and mutual information between qubit and
boson

We first study the quantum correlations between the qubit $p$ and the boson
with momentum $k$. By tracing over the antiboson mode $-k$, we obtain the
density operator $\rho_{p,k}$ for the system of qubit and boson

$\displaystyle\rho_{p,k}$ $\displaystyle=$
$\displaystyle\frac{\chi^{2}}{|\alpha_{k}|^{2}}\sum_{n=0}^{\infty}\gamma^{2n}|\uparrow,n_{k}\rangle\langle\uparrow,n_{k}|+\frac{1-\chi^{2}}{|\alpha_{k}|^{4}}\sum_{n=0}^{\infty}\gamma^{2n}(n+1)|\downarrow,(n+1)_{k}\rangle\langle\downarrow,(n+1)_{k}|$
(19) $\displaystyle+$
$\displaystyle\frac{\chi\sqrt{1-\chi^{2}}}{|\alpha_{k}|^{2}\alpha_{k}}\sum_{n=0}^{\infty}\gamma^{2n}\sqrt{n+1}|\uparrow,n_{k}\rangle\langle\downarrow,(n+1)_{k}|$
$\displaystyle+$
$\displaystyle\frac{\chi\sqrt{1-\chi^{2}}}{|\alpha_{k}|^{2}\alpha_{k}^{*}}\sum_{n=0}^{\infty}\gamma^{2n}\sqrt{n+1}|\downarrow,(n+1)_{k}\rangle\langle\uparrow,n_{k}|,$

with
$\gamma^{2}=|\frac{\beta_{k}}{\alpha_{k}}|^{2}=\frac{\sinh^{2}(\pi\frac{\omega_{-}}{\rho})}{\sinh^{2}(\pi\frac{\omega_{+}}{\rho})}$.

There are many special approaches to the detection of quantum entanglement,
among them the Peres-Horodecki positive partial transpose (PPT) criterion
Peres1996 ; Horodecki1996  and the afterwards logarithmic negativityL32  are
widely used, due to the simplicity of calculations. In this paper, we use the
logarithmic negativity as the detection of entanglement, which is defined as
$N(\rho)=\log_{2}\|\rho^{T}\|_{1}$, where the trace norm is given by
$\|A\|_{1}={\rm tr}\sqrt{A^{{\dagger}}A}$ and $\rho^{T}$ denotes the partial
transpose of the bipartite density matrix $\rho$ with respect to its one
party. It is worthwhile to point out that logarithmic negativity is only the
sufficient condition of entanglement and is not necessarily a guarantee of
detecting all the entanglement in any case. However, it is indeed a feasible
indicator for the conjecture of universe expansion.

Performing partial transpose on density matrix $\rho_{p,k}$ with respect to
the qubit, then the transposed density matrix has the eigenvalues

$\frac{\chi^{2}}{|\alpha_{k}|^{2}},~{}~{}~{}~{}\frac{\gamma^{2n}}{2|\alpha_{k}|^{2}}\left[\gamma^{2}\chi^{2}+\frac{n(1-\chi^{2})}{|\beta_{k}|^{2}}\pm\sqrt{\left(\gamma^{2}\chi^{2}+\frac{n(1-\chi^{2})}{|\beta_{k}|^{2}}\right)^{2}+\frac{4\chi^{2}(1-\chi^{2})}{|\alpha_{k}|^{2}}}\right]$

with $n=0,1,2,\cdots$. Due to $\omega_{-}<\omega_{+}$, thus $\gamma^{2}<1$ and
$|\alpha_{k}|^{2}=\frac{1}{1-\gamma^{2}}\neq\infty$, which means that there
must be negative eigenvalues and the state $\rho_{p,k}$ is always entangled.
The logarithmic negativity $N(\rho_{p,k})$ of the state $\rho_{p,k}$ is

$N(\rho_{p,k})=\log_{2}\left[\frac{\chi^{2}}{|\alpha_{k}|^{2}}+\sum_{n=0}^{\infty}\frac{\gamma^{2n}}{|\alpha_{k}|^{2}}\sqrt{\left(\gamma^{2}\chi^{2}+\frac{n(1-\chi^{2})}{|\beta_{k}|^{2}}\right)^{2}+\frac{4\chi^{2}(1-\chi^{2})}{|\alpha_{k}|^{2}}}\right].$
(20)

We can also study the total correlation in the state $\rho_{p,k}$ by
calculating the mutual information, which is defined as L33

$I(\rho_{p,k})=S(\rho_{p})+S(\rho_{k})-S(\rho_{p,k}),$ (21)

where $S(\rho)=-\rm Tr[\rho\log_{2}(\rho)]$ is the von Neumann entropy of the
density matrix $\rho$. From the expression $\rho_{p,k}$ of Eq.(19), we obtain
the von Neumann entropies for the joint system and each subsystems as

$S(\rho_{p,k})=-\sum_{n=0}^{\infty}\frac{\gamma^{2n}}{|\alpha_{k}|^{2}}\left(\chi^{2}+\frac{(n+1)(1-\chi^{2})}{|\alpha_{k}|^{2}}\right)\log_{2}\left[\frac{\gamma^{2n}}{|\alpha_{k}|^{2}}\left(\chi^{2}+\frac{(n+1)(1-\chi^{2})}{|\alpha_{k}|^{2}}\right)\right],$
(22)
$S(\rho_{p})=-\chi^{2}\log_{2}\left[\chi^{2}\right]-(1-\chi^{2})\log_{2}\left[(1-\chi^{2})\right],$
(23)

and

$S(\rho_{k})=-\sum_{n=0}^{\infty}\frac{\gamma^{2n}}{|\alpha_{k}|^{2}}\left(\chi^{2}+\frac{n(1-\chi^{2})}{|\beta_{k}|^{2}}\right)\log_{2}\left[\frac{\gamma^{2n}}{|\alpha_{k}|^{2}}\left(\chi^{2}+\frac{n(1-\chi^{2})}{|\beta_{k}|^{2}}\right)\right],$
(24)

respectively. Due to $\gamma^{2}<1$, we thus have $S(\rho_{k})>S(\rho_{p,k})$
and $I(\rho_{p,k})>S(\rho_{p})$.

Eqs.(20) and (21) suggest that logarithmic negativity and mutual information
depend on the cosmological parameters $\epsilon$ and $\rho$, as well as the
bosonic mass $m$ and momentum $k$. Assuming that we live in the universe
corresponding to the asymptotic future of the spacetime. Then we can find the
historical information about the spacetime expansion by inspecting the
evolution of logarithmic negativity and mutual information. The historical
information shows a promising application in observational cosmology and
presents a way to simulate the expansion of universe.

Figure 1:  Logarithmic negativity $N(\rho_{p,k})$ (left) and mutual
information $I(\rho_{p,k})$ (right) as functions of the expansion volume
$\epsilon$ and the expansion rate $\rho$, where $\chi=\frac{1}{\sqrt{2}}$ and
$m=k=1$.

In Fig.1, we show the logarithmic negativity (left panel) and mutual
information (right panel) between qubit and boson as functions of the
expansion volume $\epsilon$ or the expansion rate $\rho$. It is shown that the
change of mutual information $I(\rho_{p,k})$ is very similar to the change of
logarithmic negativity $N(\rho_{p,k})$. They reduce monotonically with respect
to both the expansion volume $\epsilon$ and the expansion rate $\rho$.

Figure 2: Logarithmic negativity $N(\rho_{p,k})$ (left) and mutual information
$I(\rho_{p,k})$ (right) as functions of the momentum $k$ for different
expansion volume $\epsilon$ or different expansion rate $\rho$. Where we have
fixed $m=1$ and $\chi=\frac{1}{\sqrt{2}}$. $\rho=10$ in (a) and (b), and
$\epsilon=10$ in (c) and (d).

Fig.2 shows the dependency of logarithmic negativity and mutual information on
the momentum $k$, where the expansion volume $\epsilon$ and expansion rate
$\rho$ are set to be three typical values according to Fig.1. We find that
both the logarithmic negativity $N(\rho_{p,k})$ and the mutual information
$I(\rho_{p,k})$ increase monotonically with the growth of momentum $k$. For
infinite $k$, we have

$\lim_{k\rightarrow\infty}N(\rho_{p,k})=\log_{2}\left[1+2\chi\sqrt{1-\chi^{2}}\right],~{}~{}~{}~{}\lim_{k\rightarrow\infty}I(\rho_{p,k})=2S(\rho_{p}).$

Note that $\log_{2}\left[1+2\chi\sqrt{1-\chi^{2}}\right]$ and $2S(\rho_{p})$
are respectively the initial values of logarithmic negativity and mutual
information in the state of Eq.(17), we thus conclude that, for boson with
very large momentum, its logarithmic negativity and mutual information with
the qubit are very little affected by the expanding spacetime. If we want to
extract historical information about the expanding spacetime, the boson modes
with smaller momentum are most preferable.

Figure 3: Logarithmic negativity $N(\rho_{p,k})$ (left) and mutual information
$I(\rho_{p,k})$ (right) as functions of the mass $m$ for different expansion
volume $\epsilon$ or different expansion rate $\rho$. Where we have fixed
$k=1$ and $\chi=\frac{1}{\sqrt{2}}$. $\rho=10$ in (a) and (b), and
$\epsilon=10$ in (c) and (d).

Fig.3 shows the change of logarithmic negativity and mutual information versus
the mass $m$ for different expansion volume $\epsilon$ and different expansion
rate $\rho$. It is shown that logarithmic negativity $N(\rho_{p,k})$ and
mutual information $I(\rho_{p,k})$ firstly decrease to the minimum and then
asymptotically recover to their initial values with the increasing of mass
$m$. The fact that logarithmic negativity and mutual information
asymptotically recover to their initial values can also be demonstrate through
the analytical calculations

$\lim_{m\rightarrow
0}N(\rho_{p,k})=\lim_{m\rightarrow\infty}N(\rho_{p,k})=\log_{2}\left[1+2\chi\sqrt{1-\chi^{2}}\right],$
$\lim_{m\rightarrow
0}I(\rho_{p,k})=\lim_{m\rightarrow\infty}I(\rho_{p,k})=2S(\rho_{p}).$

This result suggests that logarithmic negativity and mutual information
between the qubit and the boson with very large or very small mass are less
affected by the expanding spacetime. There is an optimal mass that is most
prone to extract the information about the expanding spacetime. Note that the
optimal mass decreases slightly with the growth of the expansion volume
$\epsilon$, and increases slightly with the growth of the expansion rate
$\rho$.

###  III.2 Logarithmic negativity and mutual information between qubit and
antiboson

In order to study the redistribution of quantum correlations, we need to
calculate logarithmic negativity and mutual information between qubit $p$ and
the antiboson $-k$. Tracing over the mode $k$ in state of Eq.(18), we obtain
the density matrix

$\displaystyle\rho_{p,-k}$ $\displaystyle=$
$\displaystyle\frac{\chi^{2}}{|\alpha_{k}|^{2}}\sum_{n=0}^{\infty}\gamma^{2n}|\uparrow,n_{-k}\rangle\langle\uparrow,n_{-k}|+\frac{1-\chi^{2}}{|\alpha_{k}|^{4}}\sum_{n=0}^{\infty}\gamma^{2n}(n+1)|\downarrow,n_{-k}\rangle\langle\downarrow,n_{-k}|$
(25) $\displaystyle+$
$\displaystyle\frac{\chi\sqrt{1-\chi^{2}}\beta_{k}^{*}}{|\alpha_{k}|^{4}}\sum_{n=0}^{\infty}\gamma^{2n}\sqrt{n+1}|\uparrow,(n+1)_{-k}\rangle\langle\downarrow,n_{-k}|$
$\displaystyle+$
$\displaystyle\frac{\chi\sqrt{1-\chi^{2}}\beta_{k}}{|\alpha_{k}|^{4}}\sum_{n=0}^{\infty}\gamma^{2n}\sqrt{n+1}|\downarrow,n_{-k}\rangle\langle\uparrow,(n+1)_{-k}|.$

The partial transpose of this density matrix with respect to qubit has the
eigenvalues

$\displaystyle\frac{1-\chi^{2}}{|\alpha_{k}|^{4}},~{}~{}~{}~{}\frac{1}{2|\alpha_{k}|^{2}}\gamma^{2n}\left[D\pm\sqrt{D^{2}-\frac{4\chi^{2}(1-\chi^{2})|\beta_{k}|^{2}}{|\alpha_{k}|^{4}}}\right]$

with $D=\chi^{2}+\frac{(n+2)(1-\chi^{2})|\beta_{k}|^{2}}{|\alpha_{k}|^{4}}$
and $n=0,1,2,\cdots.$ Thus the logarithmic negativity between the qubit and
antiboson is

$N(\rho_{p,-k})=\log_{2}\left[\frac{1-\chi^{2}}{|\alpha_{k}|^{4}}+\sum_{n=0}^{\infty}\frac{\gamma^{2n}}{|\alpha_{k}|^{2}}\left(\chi^{2}+\frac{(n+2)(1-\chi^{2})|\beta_{k}|^{2}}{|\alpha_{k}|^{4}}\right)\right]=0.$
(26)

This means that spacetime expanding can not generate logarithmic negativity
between the qubit and antiboson. However, this does not mean that quantum
state between qubit and antiboson is not without quantum correlation. The
correlated quantum coherence is nonlocal coherence and is regarded as a kind
of quantum correlation. Through simple calculation, we find that the
correlated quantum coherence can be generated by the spacetime expanding wsm32
.

The mutual information between the qubit and the antiboson is given by

$I(\rho_{p,-k})=S(\rho_{p})+S(\rho_{-k})-S(\rho_{p,-k}),$ (27)

where $S(\rho_{p})$ is given by Eq.(23), and $S(\rho_{-k})$ and
$S(\rho_{p,-k})$ are given by

$S(\rho_{-k})=-\sum_{n=0}^{\infty}\frac{\gamma^{2n}}{|\alpha_{k}|^{2}}\left(\chi^{2}+\frac{(n+1)(1-\chi^{2})}{|\alpha_{k}|^{2}}\right)\log_{2}\left[\frac{\gamma^{2n}}{|\alpha_{k}|^{2}}\left(\chi^{2}+\frac{(n+1)(1-\chi^{2})}{|\alpha_{k}|^{2}}\right)\right],$

and

$S(\rho_{p,-k})=-\sum_{n=0}^{\infty}\frac{\gamma^{2n}}{|\alpha_{k}|^{2}}\left(\chi^{2}+\frac{n(1-\chi^{2})}{|\beta_{k}|^{2}}\right)\log_{2}\left[\frac{\gamma^{2n}}{|\alpha_{k}|^{2}}\left(\chi^{2}+\frac{n(1-\chi^{2})}{|\beta_{k}|^{2}}\right)\right].$

Eq.(27) is positive in general, meaning that spacetime expanding can generate
mutual information between the qubit and the antiboson.

Figure 4: Mutual information $I(\rho_{p,-k})$ as a function of the expansion
volume $\epsilon$ and the expansion rate $\rho$, where $m=k=1$ and
$\chi=\frac{1}{\sqrt{2}}$.

In Fig.4, we plot the mutual information between the qubit and the antiboson
as a function of the expansion volume $\epsilon$ and the expansion rate
$\rho$. It is shown that the mutual information $I(\rho_{p,-k})$ increases
monotonically with both $\epsilon$ and $\rho$, but the change is more
sensitive to the beginning of $\epsilon$ and $\rho$. This means that smaller
$\epsilon$ and $\rho$ are more favorable from the consideration of extracting
information.

Figure 5: Mutual information $I(\rho_{p,-k})$ as functions of the momentum $k$
or the mass $m$ for different expansion volume $\epsilon$ or different
expansion rate $\rho$, where $\chi=\frac{1}{\sqrt{2}}$. Other parameters are
chosen as: (a) $m=1$ and $\rho=10$; (b) $m=1$ and $\epsilon=10$; (c) $k=1$ and
$\rho=10$; (d) $k=1$ and $\epsilon=10$.

Fig.5 shows the change of mutual information $I(\rho_{p,-k})$ versus the
momentum $k$ or mass $m$, where the expansion volume $\epsilon$ and expansion
rate $\rho$ are set as the typical values according to Fig.4. We can see that
$I(\rho_{p,-k})$ reduces monotonically with the growth of the momentum $k$
[see Fig.5(a) and 5(b)]. In the limit of infinite momentum, we have
$\lim_{k\rightarrow\infty}I(\rho_{p,-k})=0$ (through analytic calculation).
This means that the smaller the momentum we pick out, the more the generated
mutual information $I(\rho_{p,-k})$ is, which is more favourable for
extracting information of spacetime expansion.

From Fig.5(c) and 5(d), we see that the generated mutual information
$I(\rho_{p,-k})$ increases from zero to a maximum and then reduces again to
zero with the growth of the mass $m$, i.e., satisfying

$\lim_{m\rightarrow
0}I(\rho_{p,-k})=\lim_{m\rightarrow\infty}I(\rho_{p,-k})=0.$

This means that there are some medium-sized masses which can produce more
mutual information between the qubit and the antiboson and thus are more
conducive to extract information about the expanding of universe. As displayed
by Fig.3, the optimal mass decreases with the growth of the expansion volume
$\epsilon$, and increases with the growth of the expansion rate $\rho$. In
fact, we can demonstrate (see III.3 below) that the optimal mass displayed
here is actually consistent with that in Fig.3.

Figure 6: Logarithmic negativity and mutual information as functions of the
initial parameter $\chi$, where $\epsilon=\rho=10$, $k=0.5$ and $m=1$.

###  III.3 Trigger and monogamy of mutual information

From the above discussion, we find that the expansion of universe can not
produce logarithmic negativity $N(\rho_{p,-k})$, but can produce mutual
information $I(\rho_{p,-k})$ between the qubit and the antiboson. However, the
production of $I(\rho_{p,-k})$ requires some initial conditions. In Fig.6, we
plot the logarithmic negativity $N(\rho_{p,k})$, the mutual information
$I(\rho_{p,k})$ and $I(\rho_{p,-k})$ as functions of the initial parameter
$\chi$. It is shown that the production of the mutual information
$I(\rho_{p,-k})$ needs the trigger of the initial logarithmic negativity
$N(\rho_{p,k})$ or initial mutual information $I(\rho_{p,k})$. With no these
triggers, the mutual information $I(\rho_{p,-k})$ can not be generated. The
more the trigger logarithmic negativity $N(\rho_{p,k})$ or the mutual
information $I(\rho_{p,k})$ is, the more the generated mutual information
$I(\rho_{p,-k})$ is.

In fact, the trigger phenomenon of mutual information is obviously. We can
view the boson and antiboson as a bipartite system. If there is no initial
mutual information between the qubit and the boson, then also no initial
mutual information between the qubit and the bipartite system. As no
interaction exists between the qubit and the bipartite system in the expanding
of spacetime, the mutual information between them always remains zero. Thus no
mutual information exists between the qubit and the antiboson. What’s
interesting is that the more the trigger mutual information, the more the
generated mutual information $I(\rho_{p,-k})$, and this trigger phenomenon
does not take place in the logarithmic negativity.

From Figs.1-5, we find that the changes of $I(\rho_{p,k})$ and
$I(\rho_{p,-k})$ are always opposite: Whenever the mutual information between
the qubit and the boson $I(\rho_{p,k})$ degrades, then the mutual information
between the qubit and the antiboson $I(\rho_{p,-k})$ increases. After careful
inspection of Eqs.(21) and (27), we find the monogamy relationship

$I(\rho_{p,k})+I(\rho_{p,-k})=2S(\rho_{p}),$ (28)

where $2S(\rho_{p})$ is the initial mutual information for the state of
Eq.(17). This implies that the mutual information is re-distributable: The
initial mutual information is redistributed into $\rho_{p,k}$ and
$\rho_{p,-k}$, but the total mutual information is conserved in the expansion
of universe. This is evidenced by the numerical simulations, i.e., the value
of $I(\rho_{p,k})$ in Fig.3(b)(d) plus the value of $I(\rho_{p,-k})$ in
Fig.5(c)(d), for the same $\epsilon$ and $\rho$, is equal to 2. This
conservation also demonstrates that the optimal mass appeared in Fig.3(b)(d)
and Fig.5(c)(d) coincide.

###  III.4 Entanglement entropy of boson

Ball $et$ $al$ firstly studied the entanglement entropy of bosonic field in an
expanding spacetime L28 . Then, the comparison between fermionic and bosonic
entanglement entropy for free field modes was studied L8 ; L29 . It is shown
that the more information about cosmological parameters can be extracted by
using fermionic entanglement entropy. Next, Li $et$ $al$ studied decoherence
and disentanglement of scalar fields for Unruh-Wald qubit detector model in an
expanded spacetime L31 . The results show that the expansion of the spacetime
in its history destroys the entanglement between the qubits. Based on these
work, Moradi $et$ $al$ investigated spin-particles entanglement between two
modes of Dirac field in Robertson-Walker spacetime wsm33 . The analysis
highlight the role of polarization of particles and compare the polarization
results with those obtainable without accounting for it. Mohammadzadeh $et$
$al$ studied entropy production due to Lorentz invariance violation (LIV) in
an expanding spacetime wsm34 , and Liu $et$ $al$ used the optimal estimation
of parameters for scalar fields to extract information in expanding universe
exhibiting LIV L30 . The extracted information about the past existence of LIV
might be useful in recovering the quantum structure of gravity. Really, the
occurrence of a peak in the behavior of entanglement entropy for a specific
momentum can provide information about the expansion parameters. Therefore,
the information about the LIV parameter is codified in this peak.

In III.1, we use the logarithmic negativity and mutual information between the
qubit and the boson to detect the law of universe expansion, because they
contain the information about universe expansion. In fact, we find from
Eq.(24) that the entanglement entropy (i.e. von Neumann entropy) $S(\rho_{k})$
of the boson also contains the information about universe expansion, thus it
also can be used to detect the law of universe expansion. As the whole system
(qubit, boson and antiboson) is always a pure state in the universe expansion,
$S(\rho_{k})$ describes actually the quantum correlation of the boson with the
qubit and the antiboson. In Fig.7, we plot the entanglement entropy
$S(\rho_{k})$ as a function of the expansion volume $\epsilon$ and the
expansion rate $\rho$, where the parameters are chosen as the same as in
Fig.4. It is shown that the evolution of $S(\rho_{k})$ is also similar to that
of $I(\rho_{p,-k})$ in Fig.4. $S(\rho_{k})$ increases monotonically with both
$\epsilon$ and $\rho$, and the change is more sensitive to the beginning of
$\epsilon$ and $\rho$, meaning that smaller $\epsilon$ and $\rho$ are more
favorable for the extraction of information.

Figure 7: Entanglement entropy $S(\rho_{k})$ as a function of the expansion
volume $\epsilon$ and the expansion rate $\rho$, where $m=k=1$ and
$\chi=\frac{1}{\sqrt{2}}$.

Fig.8 shows the change of entanglement entropy $S(\rho_{k})$ versus the
momentum $k$ or mass $m$, where the parameters are chosen as the same as in
Fig.5. We see that Fig.8 is also very similar to Fig.5. $S(\rho_{k})$ reduces
monotonically with the growth of the momentum $k$ [Fig.8(a) and 8(b)]. In the
limit of infinite momentum, we have $\lim_{k\rightarrow\infty}S(\rho_{k})=0$
(through analytic calculation). This means that the smaller the momentum we
pick out, the more the generated entanglement entropy $S(\rho_{k})$ is, which
is more favourable for extracting information of spacetime expansion.

From Fig.8(c) and 8(d), we see that the generated entanglement entropy
$S(\rho_{k})$ increases from zero to a maximum and then reduces again to zero
with the growth of the mass $m$, i.e., satisfying

$\lim_{m\rightarrow 0}S(\rho_{k})=\lim_{m\rightarrow\infty}S(\rho_{k})=0.$

This means that there are some medium-sized masses which can produce more
entanglement entropy for the boson and thus are more conducive to extract
information about the universe expansion. Also, the optimal mass decreases
slightly with the growth of the expansion volume $\epsilon$, and increases
slightly with the growth of the expansion rate $\rho$. However, this optimal
mass is conceptually not the same thing as that in Fig.5(c)(d) for
$I(\rho_{p,-k})$.

Figure 8: Entanglement entropy $S(\rho_{k})$ as functions of the momentum $k$
or the mass $m$ for different expansion volume $\epsilon$ or different
expansion rate $\rho$, where $\chi=\frac{1}{\sqrt{2}}$. Other parameters are
chosen as: (a) $m=1$ and $\rho=10$; (b) $m=1$ and $\epsilon=10$; (c) $k=1$ and
$\rho=10$; (d) $k=1$ and $\epsilon=10$.

The ability to recover information about the cosmological parameters which are
given access the generated mutual information between the qubit and the
antiboson as well as entanglement entropy, were analysed in the limit of small
mass $m\ll 2\rho\sqrt{\epsilon}$. In this limit, we find an approximate
relation between the generated mutual information and the mass $m$

$\gamma^{2}=|\frac{\beta_{k}}{\alpha_{k}}|^{2}\approx\frac{\epsilon
m^{2}}{2(m^{2}+k^{2})}.$

We can see that the generated mutual information is a monotonically increasing
function of the expansion volume $\epsilon$, and is a monotonically decreasing
function of the momentum $k$. We can also see that the generated mutual
information is a increasing function of the mass $m$ for smaller $m$. However,
for bigger mass $m$, the generated mutual information is a decreasing function
of the mass $m$. In this same limit, the expansion rate $\rho$ can be related
to the variation of the generated mutual information

$\rho\approx\pi E\sqrt{\frac{1+\gamma^{2}}{-E\partial_{E}\ln[\gamma^{2}]-4}},$

where $E=\sqrt{m^{2}+k^{2}}$ is the energy of the $k$-momentum mode.
Unfortunately, for ${m\rightarrow 0}$, the generated mutual information is
almost zero due to the conformal symmetry of the theory. Therefore, the
generated mutual information of massive states in our simple toy universe
encodes the complete information about the cosmological parameters.

In the hopes of stimulating further research into the relation between the
generated mutual information and universe expansion, we conclude with some
discussion of further simulation avenues in this area. We discuss the
possibility of the cosmic neutrino background that possesses quantum
entanglement and mutual information from the early universe and surviving all
the way to the present because of its weak coupling to matter. We can also
discuss the importance of analogue experimental models providing a laboratory-
accessible testbed for cosmological models. Finally, we can use quantum
entanglement and mutual information as a tool for theoretical and experimental
cosmology.

Perhaps, it may be applied in the modeling of universe expansion with
superconducting circuit Zehua2017 . We theoretically propose a setup to
realize Jackiw-Teitelboim expansion universe as solitons of sine-Gordon. Note
that our configuration can be similar to the superconducting coplanar
waveguide. However, each capacitor in our configuration is parallel with an
identical superconducting quantum interference devices. It is the added
superconducting quantum interference device providing a nonlinear potential to
the Lagrangian of fluxes in the superconducting coplanar waveguide, which
plays a very crucial role in the simulation of expansion universe. Each
superconducting quantum interference device consists of two identical parallel
Josephson junctions. Besides, the geometric size of superconducting quantum
interference devices loop is assumed to be small enough such that its
selfinductance is negligible compared to its kinetic inductance. In this case,
each superconducting quantum interference device can be referred to as an
effective Josephson junction with a junction capacitance and a tunable
Josephson energy. We can realize the Jackiw-Teitelboim expansion universe as
solitons of sine-Gordon equation in our setup. Thus, it allows us to
understand the mechanics of solitons and especially the duality between
Jackiw-Teitelboim gravity and sine-Gordon soliton from the perspective of
experiment. The analogous dynamic behavior of quantum fields under the
expanding universe in principle could be implemented in the setup. By
appropriately adjusting the external magnetic flux through the superconducting
quantum interference device loop, we can simulate the behavior of the
conformal scale factor. Therefore, the analogous dynamic behavior of quantum
fields under an expanding universe in principle could be implemented in setup.
The in-mode and out-mode are two particular solutions in the past and future,
respectively. Physically, these two particular solutions are a description of
two different vacuum and excited states in the two asymptotic regions. We
consider the boson (B) that is affected by the simulated spacetime expanding
and the qubit (A) entangled with boson in shielding environment.

##  IV  Conclusions

The effects of the expansion universe on quantum correlations (including
logarithmic negativity and mutual information) between a qubit and a
relativistic boson have been investigated. We have found that the dynamics for
both the logarithmic negativity and mutual information between the qubit and
the boson are very similar. Both of them reduce with the growth of the
expansion rate and the expansion volumes. The degradation of quantum
correlations also depends on the nature of the boson. The less the boson
momentum is, the more serious the quantum correlations reduce. The dependency
of quantum correlations on the mass of boson is however non-monotonic. Very
small and very large mass of bosons have little influence, and some medium
mass of bosons induce large reduction of quantum correlations.

We have found that, in the expanding of universe, logarithmic negativity
between the qubit and the antiboson always remains zero, but the mutual
information between them can be generated. With the growth of the expansion
rate and the expansion volume, the generated mutual information becomes large
and large. Smaller momentum and medium mass of bosons are more beneficial to
generate mutual information between the qubit and the antiboson.
Interestingly, the production of the mutual information between the qubit and
the antiboson requires a trigger of initial quantum correlations. The more the
trigger quantum correlations, the more the generated mutual information.

Further study has suggested that the mutual information is re-distributable.
In the expansion of universe, the mutual information between the qubit and the
boson reduce, at the same time the mutual information between the qubit and
the antiboson is generated, where the total mutual information is conserved.
This result is different from logarithmic negativity, which is not re-
distributable and has no conservation property. We have also studied the
effect of universe expansion on the entanglement entropy of the boson. With
the growth of the expansion rate and the expansion volume, the entanglement
entropy of boson is generated and becomes large and large. Smaller momentum
and medium mass of bosons are more beneficial for the generation of the
entanglement entropy.

In the expansion of universe, quantum correlation dynamics encode the
information about the underlying spacetime structure, which suggests a
promising application in observational cosmology. The results in this paper
suggest that small momentum and medium mass of bosons can more sensitively
highlight the changes of quantum correlations and thus are more beneficial for
extracting the historical information about the underlying spacetime
expansion. Of course, the scheme may be restricted by the current technology,
because of the strict assumption that the qubit is shielded from any external
influence in the expanding of universe, except for the initial entanglement
with the boson. We expect our research can present helps for the relativistic
quantum information processing and the observational cosmology.

###### Acknowledgements.

This work is supported by the National Natural Science Foundation of China
(Grant Nos. 1217050862, 11275064), the Construct Program of the National Key
Discipline, and BNUXKJC2017.

$\mathbf{Conflict}$ $\mathbf{of}$ $\mathbf{Interest}$

The authors declare no conflict of interest.

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