# Schwinger effect of Gaussian correlations in constant electric fields

Shu-Min Wu1 Hao-Sheng Zeng1<EMAIL_ADDRESS>Department of Physics,
Synergetic Innovation Center for Quantum Effects, and Key Laboratory of Low
Dimensional Quantum Structuresand Quantum Control
of Ministry of Education, Hunan Normal University, Changsha, Hunan 410081,
China.

###### Abstract

We study the Schwinger effect of Gaussian correlations (quantum entanglement,
discord and mutual information) of the continuous-variable two-mode squeezed
states shared by Alice and Bob, paying special attention to the difference of
the Schwinger effect of correlations between modes of fermion-fermion and
qubit-bosonic fields studied previously. We also study the redistribution and
conservativeness of the correlations under the Schwinger effect.

Schwinger effect and Gaussian state and quantum correlations

###### pacs:

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

##  I Introduction

In recent years, the topic of relativistic quantum information has received
much attention. The behaviours of quantum correlations (entanglement, discord
and mutual information) in non-inertial frames wd10 ; wd11 ; Qiang2018 ;
Sun2017 ; Richter2015  or in the background of black holes wd12 ; wd13 ;
zeng2019 ; Wang2016  have been investigated extensively. In these settings,
the Unruh effect or the Hawking effect that involves the production of
particle-antiparticle pairs plays the central role for the description of
quantum correlations.

Schwinger predicted that the quantum vacuum is unstable under the influence of
a strong electromagnetic field and decays into particle-antiparticle pairswd17
. The Schwinger effect, in the sense of the production of particle-
antiparticle pairs, is similar to the Unruh effect or Hawking radiation, so
that a unified form of Bogoliubov transformations that relates the $input$ to
the $output$ states exists. However the transformation coefficients have
different structures in different effects, which may lead to different
influences to the quantum correlations. The authors in references wd32 ; wd31
revealed the influence of Schwinger effect to the pairwise quantum
correlations of fermion-fermion system and qubit-boson systems respectively.
For the sake of completeness, we here want to further study the Schwinger
effect of quantum correlations of continuous-variable boson-boson systems. Our
main motivation is to reveal the relationship between the Schwinger effects of
quantum correlations of boson and fermion fields, and the relationship between
Schwinger effect and Unruh effect or Hawking effect of quantum correlations.

Generally speaking, there are various types of continuous variable systems and
also various types of states for the given systems. Here we limit our
attention to the relevant but specific case of position and momentum
continuous spectrum, and choose the two-mode squeezed state as the research
object. This type of bosonic state is a typical example of continuous-variable
entangled state that approximates to an arbitrarily good extent the EPR pair
wd1 , and especially it is within the reach of current laboratories wd19 . As
will be shown in the text, our research method may be straightforwardly apply
to the study of Schwinger effect of other states, such as coherent states and
thermal states, as well as Fock states and their simple superposition.

The paper is organized as follows. In Sec. II, we introduce the measure of
quantum entanglement, discord and mutual information for the two-mode Gaussian
states, where two different sets of measure are involved. In Sec. III, we
present the brief description of Schwinger effect of continuous-variable
bosonic fields. Sec. IV is the main body of our research work, in which the
behaviours of various kinds of pairwise quantum correlations (entanglement,
discord and mutual information), under the influence of Schwinger effect, are
investigated. Finally we give our conclusions and discussions in Sec. V.

##  II Description of correlations for Gaussian states

A Gaussian state is defined as any state whose characteristic function and
quasiprobability distribution are Gaussian functions on the quantum phase
space. The properties of a Gaussian state are thus completely specified by the
first and second statistical moments of the quadrature operators. As the first
moments can be arbitrarily adjusted by marginal displacements, which leave all
informationally relevant properties (such as entropy or any measure of
correlations) invariant, we can simply set them to be zero. Therefore, the
second moments become the unique elements for describing Gaussian states.
Combined with our research, we consider a two-mode Gaussian state $\rho_{AB}$
shared by Alice and Bob. We introduce a vector of quadrature operators as
$\hat{R}=(\hat{X}_{A},\hat{P}_{A},\hat{X}_{B},\hat{P}_{B})^{\sf T}$, where
$\hat{X}_{i}$ and $\hat{P}_{i}$ ($i=A,B)$ are the position and momentum of
each subsystem. The components of $R$ satisfy the canonical commutation
relations $[{{{\hat{R}}_{k}},{{\hat{R}}_{l}}}]=i{\Omega_{kl}}$ with
$\Omega={{\ 0\ \ 1}\choose{-1\ 0}}^{\oplus{2}}$ being the symplectic matrix.
Now we can build a covariance matrix (CM) $\sigma$ with elements
${\sigma_{ij}}={\rm
Tr}\big{[}{{{\\{{{{\hat{R}}_{i}},{{\hat{R}}_{j}}}\\}}_{+}}\
{\rho_{AB}}}\big{]}$. For a physically legitimate Gaussian state, its CM
$\sigma_{AB}$ must fulfill the uncertainty relation ${\sigma_{AB}}+i\Omega\geq
0$. The CM of any two-mode Gaussian state can always be put into a block form
wd23 ; wd24 ; wd25

$\sigma_{AB}=\left(\begin{array}[]{cc}\mathcal{A}&\mathcal{X}\\\
\mathcal{X}^{\rm T}&\mathcal{B}\\\ \end{array}\right),$ (1)

where the sub-matrices have the forms $\mathcal{A}={\rm diag}(a,a)$,
$\mathcal{B}={\rm diag}(b,b)$, and $\mathcal{X}={\rm diag}(c_{1},c_{2})$. The
symplectic invariants are defined by $J_{1}=a^{2}$, $J_{2}=b^{2}$,
$J_{3}=c_{1}c_{2}$ and $J_{4}=\det\sigma_{AB}=(ab-c_{1}^{2})(ab-c_{2}^{2})$.

In this paper, we will study the Schwinger effect of quantum entanglement,
discord and mutual information of the two-mode Gaussian states. There are
several methods for measuring these correlations, and we will adopt two
metrics for our comparative study. The first is the most widely used set of
entanglement logarithmic negativity, discord and mutual information based on
the von Neumann entropy. The intention by this set of measure is to make
comparisons with the results of previous researches for fermion-fermion or
qubit-boson systems. The second set of measure may be called the “Rényi-2
measure of correlations”, because all the measures of correlations are based
on the concept of Rényi-2 entropy. It is introduced for making a comparison
between the two sets of measure of correlations.

###  II.1 The first set of measure of correlation

In this set of measure, the entanglement is measured by the most widely used
logarithmic negativity which quantifies the violation of the positivity of the
partial transposition of a density matrix Peres1996 . This measure has the
advantage of computability for high-dimensional or even continuous-variable
systems. For the two-mode Gaussian state $\sigma_{AB}$ expressed in Eq.(1),
the logarithmic negativity that quantifies the degree of quantum entanglement
between the two modes can be rewritten as wd25

$N({\sigma_{AB}})=\max\bigg{\\{}0,\,-\log{\mu}_{-}\bigg{\\}}\,,$ (2)

where $\mu_{-}$ is the symplectic eigenvalue of the partial transpose of
$\sigma_{AB}$ and is given by
$2{\mu}_{-}^{2}=\delta-\sqrt{\delta^{2}-4\det\sigma_{AB}}$ with
$\delta=J_{1}+J_{2}-2J_{3}$.

The mutual information describes the total correlation between subsystems A
and B, including quantum and classical correlations. The mutual information
based on the von Neumann entropy is defined as
$I(\rho_{AB})=S(\rho_{A})+S(\rho_{B})-S(\rho_{AB})$ with $S(\rho)$ the von
Neumann entropy of quantum state $\rho$. For the two-mode Gaussian state with
symplectic $\sigma_{AB}$ in Eq.(1), the mutual information may be calculated
through expressionwd27 ,

$I({\sigma_{AB}})=f(a)+f(b)-f({\nu}_{-})-f({\nu}_{+}),$ (3)

where the function is defined as
$f(x)=[\frac{x+1}{2}]\log[{\frac{x+1}{2}}]-[\frac{x-1}{2}]\log[{\frac{x-1}{2}}]$,
and $2{\nu}_{\mp}^{2}=\Delta\mp\sqrt{\Delta^{2}-4\det\sigma_{AB}}$ with
$\Delta=J_{1}+J_{2}+2J_{3}$ are the symplectic eigenvalues of CM
$\sigma_{AB}$.

Quantum discord is defined as the difference of mutual information and
classical correlation, i.e.,
$D({\sigma_{AB}})=I({\sigma_{AB}})-C({\sigma_{AB}})$. Unlike entanglement,
quantum discord is asymmetrical with respect to the two subsystems. The
“quantum $A$ discord” based on the Alice’s conditional information obtained by
Bob’s measurements has the form for a two-mode Gaussian state wd25 ; wd26

$D({\sigma_{A|B}})=f(b)-f({\nu}_{-})-f({\nu}_{+})+f(\varepsilon),$ (4)

with parameter $\varepsilon$ defined by
$\varepsilon=\frac{a+ab+c_{1}c_{2}}{b+1}$. With the exchanging of
$a\rightarrow b$, one can pass from the $A$ discord to the $B$ discord.

###  II.2 The second set of measure of correlations

Rényi-$\alpha$ entropy is a useful quantity in quantum science. We can use it
to construct the measure of correlations for both discrete and continuous
variable systems. Rényi-$\alpha$ entropy is defined as

$S_{\alpha}(\rho)=\frac{1}{1-\alpha}\ln{\rm tr}(\rho^{\alpha}),$ (5)

which reproduces the von Neumann entropy in the limit $\alpha\rightarrow 1$.
For the Gaussian states, the special Rényi-2 entropy can be computed very
easily

$S_{2}(\rho)=-\ln{\rm tr}(\rho^{2})=\frac{1}{2}\ln(\det\sigma),$ (6)

with $\sigma$ the CM of the Gaussian state with density matrix $\rho$.

Following the definition of entanglement of formation Wootters1996 , by
replacing the entanglement monotone with the Rényi-2 entropy, a measure of
entanglement for a Gaussian state with covariance matrix $\sigma_{AB}$ is
constructedZZ1 ,

$N_{2}({\sigma_{AB}})=\inf_{\gamma_{AB}}\frac{1}{2}\ln(\det\gamma_{A}),$ (7)

where $\gamma_{A}$ is the marginal CM of subsystem $A$ by partially tracing
$\sigma_{AB}$ over the modes of subsystem $B$, and the minimization is over
all pure Gaussian states with CM $\gamma_{AB}$ that fulfil
$0<\gamma_{AB}\leq\sigma_{AB}$ and $\det\gamma_{AB}=1$. [For two real
symmetric matrices $M$ and $N$, the statement $M\geq N$ means that the matrix
$M-N$ has all non-negative eigenvalues.] For the two-mode Gaussian states with
CM given by Eq.(1), this Rényi-2 entanglement can be reduced to a more compact
form ZZ3 ; ZZ4

$N_{2}({\sigma_{AB}})=\frac{1}{2}\ln(\inf_{\theta\in[0,2\pi]}m_{\theta}),$ (8)

with

$\begin{array}[]{c}m_{\theta}=1+\bigg{[}c_{1}(ab-
c_{2}^{2})-c_{2}+\cos\theta\sqrt{[a-b(ab-c_{2}^{2})][b-a(ab-
c_{2}^{2})]}\bigg{]}^{2}\\\ \times\bigg{\\{}2(ab-
c_{2}^{2})(a^{2}+b^{2}+2c_{1}c_{2})+\sin\theta(a^{2}-b^{2})\sqrt{1-\frac{[c_{1}(ab-
c_{2})+c_{2}]^{2}}{[a-b(ab-c_{2}^{2})][b-a(ab-c_{2}^{2})]}}\\\
-\frac{\cos\theta[2abc_{2}^{3}+(a^{2}+b^{2})c_{1}c_{2}^{2}+((1-2b^{2})a^{2}+b^{2})c_{2}-ab(a^{2}+b^{2}-2)c_{1}]}{\sqrt{[a-b(ab-
c_{2}^{2})][b-a(ab-c_{2}^{2})]}}\bigg{\\}}^{-1}.\end{array}$

By replacing the von Neumann entropy with Rényi-2 entropy in the definition of
mutual information and discord of Eqs.(3) and (4), one obtains the
corresponding measures of correlations based on the Rényi-2 entropy. For the
two-mode Gaussian states with CM given by Eq.(1), the Rényi-2 “quantum $A$
discord” $D_{2}({\sigma_{A|B}})$ has the expression wd27 ; ZZ1 ; ZZ3

$D_{2}({\sigma_{A|B}})=\ln(b)-\frac{1}{2}\ln(\det\sigma_{AB})+\frac{1}{2}\ln(\gamma),$
(9)

where $\gamma=[2|c_{1}c_{2}|\sqrt{G}+G+c_{1}^{2}c_{2}^{2}](b^{2}-1)^{-2}$
with$G=(ab^{2}-a-bc_{2}^{2})(ab^{2}-a-bc_{1}^{2})$ for
$[ab^{2}c_{2}^{2}-c_{1}^{2}(a+bc_{2}^{2})][ab^{2}c_{1}^{2}-c_{2}^{2}(a+bc_{1}^{2})]\geq
0$; and $\gamma=a(a-\frac{c_{1}^{2}}{b})$ otherwise. With the exchanging
$a\rightarrow b$ and $c_{1}\rightarrow c_{2}$, we can obtain “quantum $B$
discord” $D_{2}({\sigma_{B|A}})$.

Finally the Rényi-2 mutual information for the two-mode Gaussian state Eq.(1)
is given by ZZ1

$I_{2}({\sigma_{AB}})=\frac{1}{2}\ln(\det\mathcal{A})+\frac{1}{2}\ln(\det\mathcal{B})-\frac{1}{2}\ln(\det\sigma_{AB}).$
(10)

##  III Schwinger effect of bosonic fields

Schwinger effect of quantum vacuum has been clearly described by many authors
wd32 ; wd31 ; wd20 . For the sake of self consistency, we present a brief
review of the process. Consider an electric field along the z-direction, whose
gauge potential satisfies $E_{z}(t)=-\partial A_{z}(t)/\partial t$. For a
constant electric field $E_{z}(t)=E_{0}$, the quadri-potential in the four
dimensional Minkowski spacetime with the metric $(+,-,-,-)$ is given by
$A_{\mu}=(0,0,0,-E_{0}t)$. A scalar particle with mass $m$ and charge $q$
coupled to this constant electric field has the equation of motion (Klein-
Gordon equation),

$[(\partial_{\mu}-ieA_{\mu})(\partial^{\mu}-ieA^{\mu})+m^{2}]\Psi(t,x)=0,$
(11)

where $\Psi(t,x)$ is the scalar field.

When $t^{\rm in}=-\infty$ and $t^{\rm out}=+\infty$, the Klein-Gordon equation
has the asymptotic complete sets of mode functions $\\{\mu^{\rm in},\nu^{\rm
in}\\}$ and $\\{\mu^{\rm out},\nu^{\rm out}\\}$ respectivelywd20 . Here the
superscripts ‘in’ and ‘out’ mean the input and output of the bosonic field,
corresponding to respectively the absence and presence of the electric field.
Quantizing scalar field $\Psi(x)$ in terms of the input and output mode
functions respectively, one can obtain the Bogoliubov transformations between
the input and output operators wd20 ; wd28 ; wd29 ,

$a^{\rm in}_{\mathbf{k}}=\alpha_{\mathbf{k}}^{*}a^{\rm
out}_{\mathbf{k}}-\beta^{*}_{\mathbf{k}}b^{{\rm out}{\dagger}}_{\mathbf{k}},$
(12) $b^{\rm in}_{\mathbf{k}}=\alpha_{\mathbf{k}}^{*}b^{\rm
out}_{\mathbf{k}}-\beta^{*}_{\mathbf{k}}a^{{\rm out}{\dagger}}_{\mathbf{k}},$
(13)

where $a_{\mathbf{k}}$ and $b_{\mathbf{k}}$ are the annihilation operators of
particles and antiparticles with momentum $\mathbf{k}$, and

$\begin{array}[]{cc}\alpha_{\mathbf{k}}=\frac{\sqrt{2\pi}}{\Gamma(-\eta)}e^{\frac{-i\pi(\eta+1)}{2}},&\beta_{\mathbf{k}}=e^{-i\pi\eta},\end{array}$
(14)

with $\eta=-\frac{1}{2}-i\frac{\zeta}{2}$ and
$\zeta=\frac{m^{2}+k_{\bot}}{eE_{0}}$ fulfilling
$|\alpha_{\mathbf{k}}|^{2}-|\beta_{\mathbf{k}}|^{2}=1$. The transverse
momentum $k_{\bot}$ is defined by $k_{\bot}^{2}=k_{x}^{2}+k_{y}^{2}$. Through
the Bogoliubov transformations, one can find that an input vacuum state after
Schwinger effect becomes as wd20 ; wd28 ; wd29

$|0_{\mathbf{k}},0_{-\mathbf{k}}\rangle^{\rm
in}=\frac{1}{\alpha_{\mathbf{k}}}\sum_{n=0}^{\infty}\left(\frac{\beta_{\mathbf{k}}^{*}}{\alpha_{\mathbf{k}}^{*}}\right)^{n}|n_{\mathbf{k}},n_{-\mathbf{k}}\rangle^{\rm
out},$ (15)

where $|n_{\mathbf{k}}\rangle$ and $|n_{-\mathbf{k}}\rangle$ represent
respectively the number states of particle and antiparticle with momentum
$\mathbf{k}$ and the minus refers to antiparticle.

Eq.(15) can be rewritten as $|0_{\mathbf{k}},0_{-\mathbf{k}}\rangle^{\rm
in}=U_{\mathbf{k}}|0_{\mathbf{k}},0_{-\mathbf{k}}\rangle^{\rm out}$, where
$U_{\mathbf{k}}=S_{\mathbf{k}}P_{\mathbf{k}}$ with the phase operator
$P_{\mathbf{k}}$ and the two-mode squeezing operator $S_{\mathbf{k}}$ given by

$P_{\mathbf{k}}=\exp[i\theta_{\mathbf{k}}(a^{{\rm
out}{\dagger}}_{\mathbf{k}}a^{\rm out}_{\mathbf{k}}+b^{{\rm
out}{\dagger}}_{\mathbf{k}}b^{\rm out}_{\mathbf{k}}+1)],$
$S_{\mathbf{k}}=\exp[r_{\mathbf{k}}(a^{{\rm out}{\dagger}}_{\mathbf{k}}b^{{\rm
out}{\dagger}}_{\mathbf{k}}e^{2i\vartheta_{\mathbf{k}}}-a^{\rm
out}_{\mathbf{k}}b^{\rm out}_{\mathbf{k}}e^{-2i\vartheta_{\mathbf{k}}})],$

here the squeeze parameter $r_{\mathbf{k}}$, the squeeze angle
$\vartheta_{\mathbf{k}}$, and the phase angle $\theta_{\mathbf{k}}$ are
determined by $\alpha_{\mathbf{k}}=e^{-i\theta_{\mathbf{k}}}\cosh
r_{\mathbf{k}}$ and
$\beta^{*}_{\mathbf{k}}=-e^{-i\theta_{\mathbf{k}}}(e^{2i\vartheta_{\mathbf{k}}}\sinh
r_{\mathbf{k}}).$ Observing that
$P_{\mathbf{k}}|0_{\mathbf{k}},0_{-\mathbf{k}}\rangle^{\rm
out}=\exp(i\theta_{\mathbf{k}})|0_{\mathbf{k}},0_{-\mathbf{k}}\rangle^{\rm
out}$, thus up to an overall phase factor, $U_{\mathbf{k}}$ is equivalent to
$S_{\mathbf{k}}$. Further, by absorbing the phases into the Bosonic operators,
i.e., performing transformations $a^{\rm out}_{\mathbf{k}}\rightarrow a^{\rm
out}_{\mathbf{k}}e^{-i\vartheta_{\mathbf{k}}}$ and $b^{\rm
out}_{\mathbf{k}}\rightarrow b^{\rm
out}_{\mathbf{k}}e^{-i\vartheta_{\mathbf{k}}}$, which is equivalent to
rotating the squeezing direction of the input fields by an angle
$\vartheta_{\mathbf{k}}$, we have

$S_{\mathbf{k}}=\exp[r_{\mathbf{k}}(a^{{\rm out}{\dagger}}_{\mathbf{k}}b^{{\rm
out}{\dagger}}_{\mathbf{k}}-a^{\rm out}_{\mathbf{k}}b^{\rm
out}_{\mathbf{k}})].$ (16)

In the phase space defined in the above section, $S_{\mathbf{k}}$ may be
written as

$\displaystyle
S_{\mathbf{k}}=\left(\begin{array}[]{cc}|\alpha_{\mathbf{k}}|I_{2}&|\beta_{\mathbf{k}}|Z_{2}\\\
|\beta_{\mathbf{k}}|Z_{2}&|\alpha_{\mathbf{k}}|I_{2}\\\ \end{array}\right),$
(19)

where $I_{2}$ is the unity matrix in $2\times 2$ space and $Z_{2}$ the third
Pauli matrix. The deduction suggests that the influence of Schwinger effect on
the bosonic fields is equivalent to a bosonic amplication channel described by
the squeezing operator $S_{\mathbf{k}}$.

Having the transformation of Eq.(16) or Eq.(19) in hand, one can study in
principle the Schwinger effect of any quantum states, including Gaussian
states (coherent state, squeezed state and thermal state), as well as Fock
states and their superposition. As an exemplary example, we will study the
Schwinger effect of a two-mode squeezed state shared by Alice and Bob, with
the momentum of each mode given by $\mathbf{p}$ and $\mathbf{q}$ respectively.
We will consider two cases: (A) unilateral Schwinger effect, i.e., only Bob’s
mode encounters the influence of a constant electric field; (B) bilateral
Schwinger effect, i.e., both Alice’s and Bob’s modes encounter the influence
of electric fields. For simplicity, we assume that the modes $\mathbf{p}$ and
$\mathbf{q}$ for the bilateral Schwinger effect have the same magnitude and
direction, and encounter the same constant electric fields, so that
$p_{\bot}=q_{\bot}$ and $\alpha_{\mathbf{p}}=\alpha_{\mathbf{q}}\equiv\alpha$,
$\beta_{\mathbf{p}}=\beta_{\mathbf{q}}\equiv\beta$.

##  IV Schwinger effect of Gaussian correlations

With the above preparations, we can now study the influence of Schwinger
effect on the Gaussian correlations. The input state we considered is a two-
mode squeezed vacuum state with CM L23 ; CV

$\displaystyle\sigma_{AB}(\mathbf{p},\mathbf{q})^{\rm
in}=\left(\\!\\!\begin{array}[]{cc}\cosh(2s)I_{2}&\sinh(2s)Z_{2}\\\
\sinh(2s)Z_{2}&\cosh(2s)I_{2}\end{array}\\!\\!\right),$ (22)

where $s$ is the squeezing parameter, and $\mathbf{p}$ and $\mathbf{q}$ denote
respectively the momenta of the two modes shared by Alice and Bob. Observing
that the input modes of the antiparticles $-\mathbf{p}$ and $-\mathbf{q}$ are
in vacuum states, Eq.(22) also can be rewritten as

$\sigma_{AB}(\mathbf{p},\mathbf{q},-\mathbf{p},-\mathbf{q})^{\rm
in}=\sigma_{AB}(\mathbf{p},\mathbf{q})^{\rm in}\oplus
I(-\mathbf{p},-\mathbf{q}),$ (23)

where $I(-\mathbf{p},-\mathbf{q})$ denotes the CM of vacuum states. Now we go
on our process in two cases.

###  IV.1 Unilateral Schwinger effect

When only Bob’s mode $\mathbf{q}$ suffers from a constant electric field, then
the output state of the whole system becomes L23 ; CV

$\displaystyle\sigma_{AB}(\mathbf{p},\mathbf{q},-\mathbf{p},-\mathbf{q})^{\rm
out}$ $\displaystyle=$ $\displaystyle\big{[}I(\mathbf{p},-\mathbf{p})\oplus
S(\mathbf{q},-\mathbf{q})]\big{[}\sigma_{AB}(\mathbf{p},\mathbf{q},-\mathbf{p},-\mathbf{\mathbf{q}})^{\rm
in}\big{]}$ (24) $\displaystyle\big{[}I(\mathbf{p},-\mathbf{p})\oplus
S(\mathbf{q},-\mathbf{q})\big{]}\,^{\sf T},$

where operation $S(\mathbf{q},-\mathbf{q})$ is given by Eq.(19).

For unilateral Schwinger effect, we are interested in the correlation between
modes of $(\mathbf{p},\mathbf{q})$ and modes of $(\mathbf{p},\mathbf{-q})$ for
the output state. By tracing over the field modes of
$(\mathbf{-p},\mathbf{-q})$ on Eq.(24), we obtain the CM for modes of
$(\mathbf{p},\mathbf{q})$ of the output fields as

$\sigma_{AB}(\mathbf{p},\mathbf{q})=\left({\begin{array}[]{*{20}{c}}\mathcal{A}_{\mathbf{p},\mathbf{q}}&\mathcal{X}_{\mathbf{p},\mathbf{q}}\\\
{{\mathcal{X}_{\mathbf{p},\mathbf{q}}^{\sf
T}}}&\mathcal{B}_{\mathbf{p},\mathbf{q}}\\\ \end{array}}\right),$ (25)

where $\mathcal{A}_{\mathbf{p},\mathbf{q}}=[\cosh(2s)]I_{2}$,
$\mathcal{B}_{\mathbf{p},\mathbf{q}}=[\cosh(2s)|\alpha|^{2}+|\beta|^{2}]I_{2}$
and $\mathcal{X}_{\mathbf{p},\mathbf{q}}=[|\alpha|\sinh(2s)]Z_{2}$, with
$\alpha$ and $\beta$ given by Eq.(14). For brevity, we have omitted the
superscript ‘out’.

If tracing over the field modes $(\mathbf{-p},\mathbf{q})$, we obtain CM for
field modes $(\mathbf{p},\mathbf{-q})$ as

$\sigma_{AB}(\mathbf{p},\mathbf{-q})=\left({\begin{array}[]{*{20}{c}}\mathcal{A}_{\mathbf{p},\mathbf{-q}}&\mathcal{X}_{\mathbf{p},\mathbf{-q}}\\\
{{\mathcal{X}_{\mathbf{p},\mathbf{-q}}^{\sf
T}}}&\mathcal{B}_{\mathbf{p},\mathbf{-q}}\\\ \end{array}}\right),$ (26)

where $\mathcal{A}_{\mathbf{p},\mathbf{-q}}=[\cosh(2s)]I_{2}$,
$\mathcal{B}_{\mathbf{p},\mathbf{-q}}=[|\alpha|^{2}+\cosh(2s)|\beta|^{2}]I_{2}$
and $\mathcal{X}_{\mathbf{p},\mathbf{-q}}=[|\beta|\sinh(2s)]I_{2}$.

If we use the first set of measures, according to Eq.(2)-(4), we have the
logarithmic negativity, von-Neumann discord and von-Neumann mutual
information,

$\displaystyle N_{1}(\sigma_{AB}(\mathbf{p},\mathbf{q}))$ $\displaystyle=$
$\displaystyle\frac{1}{2}-\frac{1}{2}\log\\{\cosh^{2}(2s)+2|\alpha|^{2}\sinh^{2}(2s)+\zeta^{2}$
(27) $\displaystyle-$ $\displaystyle
2[\cosh(2s)+\zeta]\sqrt{\cosh^{4}(s)|\beta|^{4}+|\alpha|^{2}\sinh^{2}(2s)}\\},$
$D_{1}(\sigma_{A|B}(\mathbf{p},\mathbf{q}))=f(\zeta)-f(\eta),$ (28)
$D_{1}(\sigma_{B|A}(\mathbf{p},\mathbf{q}))=f[\cosh(2s)]+f(1+2|\beta|^{2})-f(\eta),$
(29)
$I_{1}(\sigma_{AB}(\mathbf{p},\mathbf{q}))=f[\cosh(2s)]+f(\zeta)-f(\eta),$
(30)

for the correlations between modes $(\mathbf{p},\mathbf{q})$, and

$\displaystyle N_{1}(\sigma_{AB}(\mathbf{p},\mathbf{-q}))=0,$ (31)
$D_{1}(\sigma_{A|B}(\mathbf{p},\mathbf{-q}))=f(\xi)-f({\nu}_{-})-f(\nu_{+})+f(\varepsilon_{1}),$
(32)
$D_{1}(\sigma_{B|A}(\mathbf{p},\mathbf{-q}))=f[\cosh(2s)]-f({\nu}_{-})-f({\nu}_{+})+f(\varepsilon_{2}),$
(33)
$I_{1}(\sigma_{AB}(\mathbf{p},\mathbf{-q}))=f[\cosh(2s)]-f({\nu}_{-})-f({\nu}_{+})+f(\xi),$
(34)

for the correlations between modes $(\mathbf{p},\mathbf{-q})$. Where
$\zeta=\cosh(2s)|\alpha|^{2}+|\beta|^{2}$, ${\eta}=1+2|\beta|^{2}\cosh(s)$,
$\xi=|\alpha|^{2}+|\beta|^{2}\cosh(2s)$,
$\varepsilon_{1}=\frac{|\beta|^{2}+(|\beta|^{2}+2)\cosh(2s)}{2+[1+\cosh(2s)]|\beta|^{2}}$,
$\varepsilon_{2}=2|\beta|^{2}+1$, and

${\nu}_{\mp}=\frac{1}{\sqrt{2}}\left\\{\cosh^{2}(2s)+\xi^{2}-2|\beta|^{2}\sinh^{2}(2s)\mp\sqrt{[\cosh^{2}(2s)+\xi^{2}-2|\beta|^{2}\sinh^{2}(2s)]^{2}-4\zeta}\right\\}^{\frac{1}{2}}.$

These correlations in Eqs.(27)-(34) depend on both the strength of the
electric field $E_{0}$ and the squeezing parameter $s$. When $E_{0}=0$, they
reduce to their input values.

Figure 1: Entanglement, discord, and mutual information as functions of the
dimensionless parameters $\frac{eE_{0}}{p_{\bot}^{2}+m^{2}}$ for the
unilateral Schwinger effect, where the squeezing parameter $s=1$. (a) and (b)
for the first set of measure of correlations, and (c) and (d) for the second
one.

In Fig.1(a) and (b), we plot the correlations as functions of the
dimensionless electric field $\frac{eE_{0}}{p_{\bot}^{2}+m^{2}}$, for fixed
squeezing parameter $s=1$. We see that all the correlations between modes
$(\mathbf{p},\mathbf{q})$ decrease monotonically with the increase of $E_{0}$,
but have nonzero asymptotic values when $E_{0}\rightarrow\infty$. At the same
time, the correlations between modes $(\mathbf{p},\mathbf{-q})$, except for
the entanglement $N_{1}(\sigma_{AB}(\mathbf{p},\mathbf{-q}))$, increase from
zero, and reach bounded asymptotic values when $E_{0}\rightarrow\infty$. A
point needed to emphasize is that the unilateral Schwinger effect can produce
asymmetry of quantum discord, i.e.,
$D(\sigma_{A|B}(\mathbf{p},\mathbf{q}))\neq
D(\sigma_{B|A}(\mathbf{p},\mathbf{q}))$ and
$D(\sigma_{A|B}(\mathbf{p},\mathbf{-q}))\neq
D(\sigma_{B|A}(\mathbf{p},\mathbf{-q}))$ in general.

For the second set of measures, correlations may be calculated in principle
through Eq.(8)-(10). However they are too tedious and here we only give the
analytical expressions of the mutual information, and show the entanglement
and discord numerically. The Rényi-2 entropy mutual information between modes
$(\mathbf{p},\mathbf{q})$ or modes $(\mathbf{p},\mathbf{-q})$ reads
respectively,

$I_{2}(\sigma_{AB}(\mathbf{p},\mathbf{q}))=\ln[\frac{\zeta\cosh(2s)}{\xi}],$
(35)
$I_{2}(\sigma_{AB}(\mathbf{p},\mathbf{-q}))=\ln[\frac{\xi\cosh(2s)}{\zeta}],$
(36)

with $\zeta$ and $\xi$ as given previously.

In Fig.1(c) and (d), we plot the Rényi-2 entropy correlations between modes
$(\mathbf{p},\mathbf{q})$ and between modes $(\mathbf{p},\mathbf{-q})$ as
functions of the dimensionless electric field
$\frac{eE_{0}}{p_{\bot}^{2}+m^{2}}$, for fixed squeezing parameter $s=1$. We
see that the correlation behaviors for this case are very similar to that for
the first set of measures.

It is worthwhile to point out that the authors in reference wd31  studied a
model in which a free qubit couples to a bosonic mode with Schwinger effect.
They studied, by the first set of measures, the Schwinger effect to the
influence of entanglement and mutual information. We find that the
entanglement and mutual information’s behaviors presented by Fig.1 are very
similar to the one of reference wd31 , though a free qubit is now replaced by
the continuous-variable bosonic mode.

Figure 2: Monogamy scores of discord and mutual information as functions of
the dimensionless parameters $\frac{eE_{0}}{p_{\bot}^{2}+m^{2}}$ for the
unilateral Schwinger effect, where the squeezing parameter $s=1$. (a) and (b)
are respectively corresponding to the first and the second sets of measure.

The monogamy or redistribution of correlations is an interesting problem in
the dynamics of correlations. To study the problem, we define the monogamy
score of the correlation $R$ for the case of unilateral Schwinger effect as

$\delta R_{j}(AB)=R_{j}(\sigma_{AB}(\mathbf{p},\mathbf{q})^{\rm
in})-R_{j}(\sigma_{AB}(\mathbf{p},\mathbf{q})^{\rm
out})-R_{j}(\sigma_{AB}(\mathbf{p},\mathbf{-q})^{\rm out}).$ (37)

This quantity describes the change or/and redistribution of the correlation
$R$. If $\delta R_{j}(AB)=0$, we say that the correlation $R$ is conservative
and part of which is redistributed form $(\mathbf{p},\mathbf{q})$ to
$(\mathbf{p},\mathbf{-q})$ through the Schwinger effect. Otherwise the
correlation is said to be nonconservative. In this definition, the quantity
$R$ may be the $A$-discord $D(A|B)$, $B$-discord $D(B|A)$, or the mutual
information $I(AB)$, and the subscript $j=1,2$ denotes the first or the second
set of measure of correlations. For clarity, we recover the superscripts to
clearly mark the input and output states.

From Fig.1, we can see already that the discord and mutual information for the
both sets of measures can be partially redistributed from modes
$(\mathbf{p},\mathbf{q})$ to modes $(\mathbf{p},\mathbf{-q})$. However, the
entanglement obviously can not be redistributed, because the entanglement
between modes $(\mathbf{p},\mathbf{-q})$ remains always zero.

Fig.2 shows the evolution of the correlation monogamy versus the dimensionless
electric field. We see that the discord for both sets of measure is
nonconservative and is lossy, but the mutual information has different
behaviour: The von-Neumann mutual information is lossy, while the Rényi-2
entropy mutual information is conservative. This result contrasts with that of
unilateral Schwinger effect of qubit-boson systemwd31 , where the von-Neumann
mutual information is conservative. The conservativeness of the Rényi-2
entropy mutual information also can be verified analytically. According to
Eqs.(35)-(36), one easily obtains

$I_{2}(\sigma_{AB}(\mathbf{p},\mathbf{q}))+I_{2}(\sigma_{AB}(\mathbf{p},\mathbf{-q}))=2\ln[\cosh(2s)],$
(38)

with $2\ln[\cosh(2s)]$ is the initially mutual information for Eq.(22).

###  IV.2 Bilateral Schwinger effect

When both Alice’s mode $\mathbf{p}$ and Bob’ mode $\mathbf{q}$ are influenced
by the same electric fields, the output state of the whole system is L23 ; CV

$\displaystyle\sigma_{AB}(\mathbf{p},\mathbf{q},-\mathbf{p},-\mathbf{q})^{\rm
out}$ $\displaystyle=$ $\displaystyle\big{[}S(\mathbf{p},-\mathbf{p})\oplus
S(\mathbf{q},-\mathbf{q})]\big{[}\sigma_{AB}(\mathbf{p},\mathbf{q},-\mathbf{p},-\mathbf{\mathbf{q}})^{\rm
in}\big{]}$ (39) $\displaystyle\big{[}S(\mathbf{p},-\mathbf{p})\oplus
S(\mathbf{q},-\mathbf{q})\big{]}\,^{\sf T}.$

Now we are interested in four sets of correlations, $(\mathbf{p},\mathbf{q})$,
$(\mathbf{p},\mathbf{-q})$, $(\mathbf{-p},\mathbf{q})$ and
$(\mathbf{-p},\mathbf{-q})$. Considering the symmetry of the problem, we need
only to calculate the three sets of correlations, $(\mathbf{p},\mathbf{q})$,
$(\mathbf{p},\mathbf{-q})$, and $(\mathbf{-p},\mathbf{-q})$. The corresponding
reduced CM are respectively

$\sigma_{AB}(\mathbf{p},\mathbf{q})=\left({\begin{array}[]{*{20}{c}}\mathcal{A}_{\mathbf{p},\mathbf{q}}&\mathcal{X}_{\mathbf{p},\mathbf{q}}\\\
{{\mathcal{X}_{\mathbf{p},\mathbf{q}}^{\sf
T}}}&\mathcal{B}_{\mathbf{p},\mathbf{q}}\\\ \end{array}}\right),$ (40)
$\sigma_{AB}(\mathbf{p},\mathbf{-q})=\left({\begin{array}[]{*{20}{c}}\mathcal{A}_{\mathbf{p},\mathbf{-q}}&\mathcal{X}_{\mathbf{p},\mathbf{-q}}\\\
{{\mathcal{X}_{\mathbf{p},\mathbf{-q}}^{\sf
T}}}&\mathcal{B}_{\mathbf{p},\mathbf{-q}}\\\ \end{array}}\right),$ (41)
$\sigma_{AB}(\mathbf{-p},\mathbf{-q})=\left({\begin{array}[]{*{20}{c}}\mathcal{A}_{\mathbf{-p},\mathbf{-q}}&\mathcal{X}_{\mathbf{-p},\mathbf{-q}}\\\
{{\mathcal{X}_{\mathbf{-p},\mathbf{-q}}^{\sf
T}}}&\mathcal{B}_{\mathbf{-p},\mathbf{-q}}\\\ \end{array}}\right),$ (42)

where
$\mathcal{A}_{\mathbf{p},\mathbf{q}}=\mathcal{B}_{\mathbf{p},\mathbf{q}}=\mathcal{A}_{\mathbf{p},\mathbf{-q}}=[\cosh(2s)|\alpha|^{2}+|\beta|^{2}]I_{2}$,
$\mathcal{B}_{\mathbf{p},\mathbf{-q}}=\mathcal{A}_{\mathbf{-p},\mathbf{-q}}=\mathcal{B}_{\mathbf{-p},\mathbf{-q}}=[|\alpha|^{2}+\cosh(2s)|\beta|^{2}]I_{2}$,
$\mathcal{X}_{\mathbf{p},\mathbf{q}}=[|\alpha|^{2}\sinh(2s)]Z_{2}$,
$\mathcal{X}_{\mathbf{p},\mathbf{-q}}=[|\alpha||\beta|\sinh(2s)]I_{2}$, and
$\mathcal{X}_{\mathbf{-p},\mathbf{-q}}=[|\beta|^{2}\sinh(2s)]Z_{2}$.

For the first set of measure, we calculate the logarithmic negativity, discord
and mutual information as,

$\displaystyle
N_{1}(\sigma_{AB}(\mathbf{p},\mathbf{q}))=-\log[\zeta-\sinh(2s)|\alpha|^{2}],$
(43) $\displaystyle
D_{1}(\sigma_{A|B}(\mathbf{p},\mathbf{q}))=D_{1}(\sigma_{B|A}(\mathbf{p},\mathbf{q}))=f(\zeta)+f(|\alpha|^{2}+|\beta|^{2})-2f(\tau),$
(44) $I_{1}(\sigma_{AB}(\mathbf{p},\mathbf{q}))=2f(\zeta)-2f(\tau),$ (45)

for the correlation between modes $(\mathbf{p},\mathbf{q})$, and

$\displaystyle N_{1}(\sigma_{AB}(\mathbf{p},\mathbf{-q}))$ $\displaystyle=$
$\displaystyle\frac{1}{2}-\frac{1}{2}\log\\{[1+\cosh^{2}(2s)](|\alpha|^{4}+|\beta|^{4})$
$\displaystyle+$
$\displaystyle[\cosh(4s)+4\cosh(2s)-1]|\alpha|^{2}|\beta|^{2}$
$\displaystyle-$ $\displaystyle
2\cosh^{2}(s)\sinh(s)(|\alpha|^{2}+|\beta|^{2})$ $\displaystyle\times$
$\displaystyle\sqrt{\chi-3+\cosh(2s)(1+\chi)}\\},$
$D_{1}(\sigma_{A|B}(\mathbf{p},\mathbf{-q}))=f(\xi)-f({\nu}^{\prime}_{-})-f({\nu}^{\prime}_{+})+f(\varepsilon_{3}),$
(47)
$D_{1}(\sigma_{B|A}(\mathbf{p},\mathbf{-q}))=f(\zeta)-f({\nu}^{\prime}_{-})-f({\nu}^{\prime}_{+})+f(\varepsilon_{2}),$
(48)
$I_{1}(\sigma_{AB}(\mathbf{p},\mathbf{-q}))=f(\zeta)+f(\xi)-f({\nu}^{\prime}_{-})-f({\nu}^{\prime}_{+}),$
(49)

for the correlation between modes $(\mathbf{p},\mathbf{-q})$, and

$\displaystyle
N_{1}(\sigma_{AB}(\mathbf{-p},\mathbf{-q}))=-\log[\xi-\sinh(2s)|\beta|^{2}],$
(50) $\displaystyle
D_{1}(\sigma_{A|B}(\mathbf{-p},\mathbf{-q}))=D_{1}(\sigma_{B|A}(\mathbf{-p},\mathbf{-q}))=f(\xi)+f(\varepsilon_{4})-2f(\tau),$
(51) $\displaystyle
I_{1}(\sigma_{AB}(\mathbf{-p},\mathbf{-q}))=2f(\xi)-2f(\tau),$ (52)

for the correlation between modes $(\mathbf{-p},\mathbf{-q})$. Where $\zeta$,
$\xi$, $\varepsilon_{2}$ are defined as before, and
$\chi=(|\alpha|^{2}+|\beta|^{2})^{2}+4|\alpha|^{4}|\beta|^{4}$,
$\tau=\frac{1}{2}\sqrt{3+8\cosh(2s)|\alpha|^{2}|\beta|^{2}+\chi}$,

${\nu}^{\prime}_{\mp}=\frac{1}{2}\left\\{2+\cosh(4s)+\chi+8\cosh(2s)|\alpha|^{2}|\beta|^{2}\mp
4\sqrt{2}\cosh(s)\sinh^{2}(s)\sqrt{\cosh(2s)+\chi}\right\\}^{\frac{1}{2}},$
$\varepsilon_{3}=\frac{\cosh^{2}(2s)|\alpha|^{2}|\beta|^{2}+\\{1-\frac{1}{2}[\cosh(4s)-3]|\alpha|^{2}\\}|\beta|^{2}+\cosh(2s)(|\alpha|^{2}+|\alpha|^{4}+|\beta|^{4})}{1+\xi},$
$\varepsilon_{4}=\frac{|\alpha|^{4}+|\beta|^{4}+2\cosh(2s)|\alpha|^{2}|\beta|^{2}+\xi}{1+\xi}.$

Figure 3: Entanglement, discord, and mutual information as functions of the
dimensionless parameters $\frac{eE_{0}}{p_{\bot}^{2}+m^{2}}$ for the bilateral
Schwinger effect, where the squeezing parameter $s=1$. (a)-(c) for the first
set of measure of correlations, and (d)-(f) for the second one.

For the second set of measure, the Rényi-2 entropy mutual information between
the interesting modes read respectively,

$I_{2}(\sigma_{AB}(\mathbf{p},\mathbf{q})=\ln\frac{\zeta^{2}}{\xi^{2}+|\beta|^{4}\sinh^{2}(2s)},$
(53)
$I_{2}(\sigma_{AB}(\mathbf{p},\mathbf{-q}))=\ln\frac{\zeta\xi}{|\alpha|^{2}|\zeta+|\beta|^{2}\xi}.$
(54)
$I_{2}(\sigma_{AB}(\mathbf{-p},\mathbf{-q}))=\ln\frac{\xi^{2}}{\xi^{2}+|\beta|^{4}\sinh^{2}(2s)}.$
(55)

In Fig.3, we plot the evolution of the various correlations versus the
dimensionless electric field for the case of bilateral Schwinger effect. It is
shown that in general the behaviours are similar to that of the unilateral
Schwinger effect: The correlation between modes $(\mathbf{p},\mathbf{q})$
degrades monotonically, and at the same time the correlation (discord and
mutual information) between modes $(\mathbf{p},\mathbf{-q})$,
$(\mathbf{-p},\mathbf{q})$ and $(\mathbf{-p},\mathbf{-q})$ produces. In other
words, except for entanglement, discord and mutual information can be
redistributed partially. The difference is that the entanglement of both
$N_{1}(\sigma_{AB}(\mathbf{p},\mathbf{q}))$ and
$N_{2}(\sigma_{AB}(\mathbf{p},\mathbf{q}))$ in the case of bilateral Schwinger
effect takes place sudden death. The position of sudden death of
$N_{1}(\sigma_{AB}(\mathbf{p},\mathbf{q}))$ fulfils

$\frac{eE_{0}}{p_{\bot}^{2}+m^{2}}=\frac{\pi}{\ln[\frac{1+\cosh(2s)-\sinh(2s)}{1-\cosh(2s)+\sinh(2s)}]}.$

Except for entanglement, the discord and mutual information in Fig.LABEL:fig3
decay to nonzero asymptotical values or increase to bounded asymptotical
values when $E_{0}\rightarrow\infty$.

Note that the authors in wd32  studied the bilateral Schwinger effect of
fermion-fermion fields by use of the first set of measures. We find three
differences between the two settings: Firstly, in the degradation of
entanglement between fermion modes $(\mathbf{p},\mathbf{q})$, it would not
take place entanglement sudden death, or even has nonzero asymptotical value
when $E_{0}\rightarrow\infty$. Secondly, the entanglement between fermion
modes $(\mathbf{p},\mathbf{-q})$ and $(\mathbf{-p},\mathbf{q})$ can be
produced. Lastly, the entanglement and mutual information between fermion
modes $(\mathbf{p},\mathbf{-q})$ and $(\mathbf{-p},\mathbf{q})$ change non-
monotonically with the increase of electric field. These behaviours are
contrasted with the corresponding bosonic setting.

Figure 4: Monogamy scores of discord and mutual information as functions of
the dimensionless parameters $\frac{eE_{0}}{p_{\bot}^{2}+m^{2}}$ for the
bilateral Schwinger effect, where the squeezing parameter $s=1$. (a) and (b)
are respectively corresponding to the first and the second sets of measure.

For bilateral Schwinger effect, we now define the monogamy scores of
correlations

$\displaystyle\delta R_{j}(AB)$ $\displaystyle=$ $\displaystyle
R_{j}(\sigma_{AB}(\mathbf{p},\mathbf{q})^{\rm
in})-R_{j}(\sigma_{AB}(\mathbf{p},\mathbf{q})^{\rm
out})-R_{j}(\sigma_{AB}(\mathbf{p},\mathbf{-q})^{\rm out})$ (56)
$\displaystyle-$ $\displaystyle R_{j}(\sigma_{AB}(\mathbf{-p},\mathbf{q})^{\rm
out})-R_{j}(\sigma_{AB}(\mathbf{-p},\mathbf{-q})^{\rm out}),$

here the quantity $R$ may be the $A$-discord $D(A|B)$, $B$-discord $D(B|A)$,
or the mutual information $I(AB)$, and the subscript $j=1,2$ denotes the first
or the second set of measure of correlations. In Fig.4, we plot these monogamy
scores as functions of the dimensionless electric field
$\frac{eE_{0}}{p_{\bot}^{2}+m^{2}}$. It is shown that there is no conservative
quantity of correlations for the case of bilateral Schwinger effect. Like in
case of unilateral Schwinger effect, the entanglement and discord are always
lossy [$\delta N>0$, $\delta D>0$]. Different from the case of unilateral
Schwinger effect, now we find that $\delta I_{2}(A|B)<0$ and $\delta
I_{1}(A|B)$ is also negative when $E_{0}$ is larger than certain critical
value. This is to say the bilateral Schwinger effect can produce new (besides
redistribution) mutual information between Alice and Bob, in sharp contrast
with the case of unilateral Schwinger effect. As the discord is lossy, the
production of mutual information means actually the production of classical
correlation between Alice and Bob.

##  V Conclusions and discussions

We have studied the Schwinger effect of quantum entanglement, discord and
mutual information of continuous-variable two-mode squeezed states in constant
electric fields. In the process of analyses, we have considered the two cases
of unilateral and bilateral Schwinger effect, and have employed two sets of
measures. We have found that macroscopically the behaviours are the same: All
the correlations between the original modes of particle-particle degrade
monotonically, at the same time the correlations (except for entanglement)
between the modes of particle-antiparticle or antiparticle-antiparticle
increase from zero to bounded asymptotic values, with the increase of
Schwinger electric fields. However microscopically there are differences: The
entanglement between the original modes of particle-particle decays to a
nonzero asymptotic value for the case of unilateral Schwinger effect, but
takes place sudden death for the case of bilateral Schwinger effect.

We have also studied the problem of monogamy of correlations. We found that
entanglement, whether measured by Logarithmic negativity or based on Rényi-2
entropy, for both unilateral and bilateral Schwinger effects, can not be
redistributed, while discord and mutual information always can be
redistributed in the considered situations. In the process of redistribution,
the total quantity of discord is always degraded, but the mutual information
has different behavior: The total quantity of mutual information is non-
increasing (not always decreasing) for the unilateral Schwinger effect and can
be produced for the bilateral Schwinger effect. An interesting result is that
the Rényi-2 entropy mutual information for the case of unilateral Schwinger
effect is conservative.

The Schwinger effect of correlations of boson-boson fields has some different
features compared with that of fermion-fermion fields. For example, in the
case of bilateral Schwinger effect of fermion fields wd32 , there is no sudden
death in the degradation of entanglement between the original fermion modes of
particle-particle, and the entanglement between fermion modes of particle-
antiparticle and antiparticle-antiparticle can be produced. Further, the
entanglement and mutual information between the fermion modes of particle-
antiparticle change non-monotonically with the increase of electric field. All
these behaviours are in contrast with the results of the boson-boson Schwinger
effect presented in this paper. The reason that leads to these differences
originates perhaps from the difference in statistics, which leads to the
Bogoliubov coefficients in Eqs.(12)-(13) fulfilling
$|\alpha_{\mathbf{k}}|^{2}-|\beta_{\mathbf{k}}|^{2}=1$ for bosonic fields and
$|\alpha_{\mathbf{k}}|^{2}+|\beta_{\mathbf{k}}|^{2}=1$ for fermion fields wd20
.

There are also some differences for the Schwinger effects of correlation
between boson-boson system and the qubit-boson system wd31 . For example, in
the case of unilateral Schwinger effect of qubit-boson field, the mutual
information based on von-Neumann entropy is conservative. However, for the
corresponding boson-boson system, we have found instead the conservativeness
of the Rényi-2 entropy mutual information.

Finally, we would like to compare the Schwinger effect of correlation induced
by electric fields with the Unruh effect of correlation induced by
accelerations L23 . Both the two effects involve the production of particle-
antiparticle pairs, and thus have very similar behaviors on the quantum
correlations. However, there are still some differences. For example, in the
degradation of correlations, the Rényi-2 entropy discord between modes of
particle-particle asymptotically vanishes in the infinite acceleration limit,
but approaches to a nonzero value in the limit of infinite electric field. The
same result is also valid for the degradation of the Rényi-2 entropy
entanglement in the cases of unilateral Unruh (unilateral Schwinger) effect.
Mathematically, these differences is related to the relation
$|\alpha_{\mathbf{k}}|^{2}-|\beta_{\mathbf{k}}|^{2}=1$, in which
$|\beta_{\mathbf{k}}|$ is less than one for Schwinger effect and arbitrary for
Unruh effect.

###### Acknowledgements.

This work is supported by the National Natural Science Foundation of China
(Grant No. 11275064), and the Construct Program of the National Key
Discipline.

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# Comparative Study of ZF, LMS and RLS Adaptive Equalization for alpha-mu
Fading Channels

Rayyan Abdalla
Department of Electrical, Computer and Energy Engineering
Arizona State University
<EMAIL_ADDRESS>

###### Abstract

Wireless communication systems generally endure severe fading and interference
caused by the time-dispersive channel. Major transmission distortion is
produced by channel multipath propagation and overlap of subsequent symbols.
To counteract channel response, equalization techniques are employed to
operate on channel output and recover transmitted signal at the receiver. This
work investigates equalization of channels undergoing $\alpha-\mu$ fading
distribution, which models multipath propagation in millimeter Wave (mmWave)
and sub-Terahertz (sub-THz) frequencies bands. Three equalization algorithms,
which are non-adaptive Zero-Forcing (ZF), adaptive Least-Mean-Square (LMS) and
adaptive Recursive-Least-Square (RLS) are implemented and addressed in terms
of Bit Error Rate (BER), Convergence speed and implementation complexity.
Monte Carlo simulations are then carried out to compare algorithms and assess
performance by varying multiple parameters such as training lengths, channel
order, equalization taps and diverse fading conditions.

###### Index Terms:

Adaptive equalization, alpha-mu fading, LMS, RLS, ZF

##  I Introduction

Modern wireless communication technologies are required to achieve data
transmission at high rates within constrained power resources and complexity.
These requirements impose interference mitigation and reliable operation in
noise environments. As wireless broadband applications increasingly demand
large bandwidth, several challenges are encountered at higher frequency bands
in terms of multipath (small-scale) fading and severely attenuated
communication channel. The resulting dispersion of channel frequency response
causes smeared transmission and inseparable transmitted symbols, which is
referred to as inter-symbol interference (ISI). One possible technique to
reduce the distortion produced by the multipath propagation is channel
equalization at the receiver. A channel equalizer is a digital filter that
inverts the response of the multipath channel in such a way that the BER is
kept low at reasonable Signal to Noise Ratio (SNR). The function of a channel
equalizer quantifies the communication channel at the receiver to recover the
transmitted signal. While a simple solution, ZF equalization for instance [1,
2], suggests implementing an inverse filter with a static (non-adaptive)
transfer function, such equalizer performance is not effective since it
neglects channel variations and noise effect. Therefore, adaptive equalizers
are suited to track the time-varying channel response and compensate for
additive noise in transmission medium. Two common linear adaptive equalizers
are based on LMS and RLS algorithms. The LMS algorithm is simpler and more
robust than the RLS algorithm, however, it has slower convergence compared to
that of the RLS algorithm [3]. The choice between LMS and RLS approaches is
dependent on many factors such as channel coherence time, hardware complexity
and transmitted signal characteristics, which in turn determines algorithm
stability. Since equalizer design depends on the multipath channel statistics,
it is essential to comprehensively model the fading experienced by the
communication channel to address the degree of attenuation and ISI. Main
channel dispersion sources include path propagation loss, shadowing effect and
small-scale fading. While shadowing and path loss can be modeled
deterministically, small-scale fading is stochastic and dominant. Several
fading distributions were proposed to model multipath channel envelop, such as
Rayleigh, Rice, and Nakagami-m distributions. Although widely adopted, these
statistical models capture channel behavior at certain conditions,
particularly at lower frequencies up to the sub-6 GHz band. As new mobile
communication systems are envisioned to utilize available spectrum at
millimeter Wave (mmWave) and sub-THz bands, corresponding fading at these
bands is modeled by means of a generalized Gamma-based distribution known as
$\alpha-\mu$ distribution [4, 5, 6]. That is, this distribution provides
theoretical characterization of coefficients magnitude of the channel impulse
response to be equalized. The problem of channel equalization is a frequently
addressed topic in wireless communication research. It is observed that most
efforts are in the direction of enabling adaptive equalizers. The work in [2]
implemented an LMS-modified ZF algorithm for Rayleigh fading channels. A
similar analysis was introduced using ZF technique aided with interference
cancellation by the authors in [1]. A joint channel estimation approach
combining LMS and RLS algorithms was proposed by [3]. The authors in [7]
analyzed general ISI channel using ZF, LMS and RLS algorithms and provided
interesting results in terms of recommended training symbols and BER
performance. The work in [8] investigated modulation effect on equalizing
Rayleigh distributed channel using LMS and RLS algorithms. Moreover, some
recent efforts related to high-frequency channel equalization with time-
reversal signaling and hardware impairment mitigation were conducted by [9]
and [10] respectively. While most tutorials focus on addressing single or two
equalization algorithms for a specific fading distribution, this work presents
a comprehensive analysis and performance assessment of ZF, LMS and RLS pilot-
aided equalization algorithms designed for a generalized multipath
communication channel following $\alpha-\mu$ distribution in the presence of
additive white Gaussian noise (AWGN). Note that blind equalization is not
covered in this article. The paper organization is follows: section II
illustrates the theory of channel equalization and implementation of the three
algorithms. Section III presents the proposed baseband system model structured
for simulations. Simulation results are discussed in section IV and concluding
remarks are drawn in section V.

##  II Channel Equalization

In a communication system, the multipath channel acts a time-dispersive band-
limited filter that distorts the transmitted signal, which in turns causes bit
errors at the receiver. A channel with ISI can be modeled as a multi-tap
filter with each tap corresponding to a multipath component. The idea of
channel equalization is to introduce a digital filter at the receiver that
takes distorted transmitted signal as an input and produces a delayed version
of the originally transmitted signal. Channel equalization can be implemented
statically, meaning equalizer impulse response is made constant as the inverse
of estimated communication channel impulse response, or adaptively, where
equalizer impulse response continuously adapts with the varying channel
impulse response. Static equalizers, such as ZF equalizers, are normally pre-
set linear filters that use peak distortion criterion to estimate equalizer
coefficients [1, 2, 7]. Adaptive equalization, on the other hand, is based on
transmitting a signal within channel coherence time and known to both
transmitter and receiver, which in turn gets distorted by the multipath
channel and AWGN. The corresponding distorted signal is then utilized by the
receiver to adjust equalizer tabs by attempting to produce a delayed version
of the originally transmitted signal. Once equalizer taps are set, the
predicted filter coefficients can then be used to mitigate both channel
dispersion and noise affecting subsequent transmissions in the same coherence
period. Both LMS and RLS algorithms are popular adaptive system identification
algorithms that are used for equalizing communication channels [3, 8]. Figure
1 shows a block diagram of adaptive channel equalization.

Figure 1: Block diagram of adaptive channel equalization

###  II-A Zero-Forcing Equalization

The method of ZF utilizes estimated ISI channel to design suitable linear
filter with transfer function inverse to channel impulse response. The ZF
equalizer forces ISI components to zero while ignoring the additive noise in
the channel. While ZF equalizer operates ideally in a noise-free case, it is
less efficient in practical conditions since it may significantly amplify the
noise if the channel response has spectral nulls. Moreover, finite-length
channel impulse responses require infinitely long ZF impulse response [7].
Therefore, ZF equalization is restricted to certain applications where ISI
components are dominant, and the additive noise is negligible. Let
$H_{chn}(f)$ be the channel transfer function with phase $\theta_{chn}(f)$,
and $H_{equ}(f)$ be the equalizer transfer function. Both functions are
related according to

$H_{chn}(f)=|H_{chn}(f)|e^{j\theta_{chn}(f)},\hskip 7.22743pt|f|\leq W$ (1)
$H_{equ}(f)=\frac{1}{H_{chn}(f)}=\frac{1}{|H_{chn}(f)|}e^{-j\theta_{chn}(f)},\hskip
3.61371pt|f|\leq W$ (2)

###  II-B Least-Mean-Square Equalization

The adaptive LMS algorithm is a system identification gradient decent method
that minimizes the Mean Square Error (MSE) between desired filter response and
the adaptive filter output. The algorithm iteratively estimates error gradient
vector and uses it to adjust filter weights, attempting to converge on optimal
Weiner solution [7, 8]. The LMS algorithm is widely adopted in several
applications due to low implementation complexity and robustness. For a
desired filter response $d(n)$, adaptive filter output $y(n)$, input vector
$x(n)$ and error signal $e(n)$, and weights vector $b(n)$, the LMS algorithm
is executed in two steps:

  * •

Estimation process: this involves computing the error within iteration n as a
function of desired signal and generated filter output:

$e(n)=d(n)-y(n)$ (3)

This error is used to calculate the gradient vector:

$\textbf{$\nabla$}(n)=-e(n)\textit{$\textbf{x}$}^{*}(n)$ (4)

  * •

Adaptation process: this involves automatic adjustment of the adaptive filter
weights based on gradient vector, i.e., the estimated error:

$\textit{ {b}}(n+1)=\textit{{b}}(n)-\eta\textit{{$\nabla$}}(n)$ (5)
$\textit{{b}}(n+1)=\textit{{b}}(n)+\eta e(n)\textit{{x}}^{*}(n)$ (6)

Where

$\eta$

is the step size that controls convergence speed and thus directly determines
algorithm stability. Note that all vectors are of size

$(L+1)\texttimes 1$

where

$L$

is the adaptive filter order. In a channel equalization case and when adapting
equalizer taps, the pilot signal represents the desired filter response while
the distorted pilot by the channel represents filter input. The equalizer
filter is then desired to cancel channel effect and produce the originally
transmitted signal at the receiver.

###  II-C Recursive Least Square Equalization

The RLS method is a recursive algorithm that is similar to LMS in terms of
minimizing MSE and implementation processes. However, a weighted RLS algorithm
minimizes a least square cost function corresponding to input signal. An RLS
filter whitens the input data by evaluating and utilizing inverse correlation
matrix within every iteration to update filter weights. The rate of RLS
convergence is several times faster than that of LMS, which makes the
algorithm a great fit for fast fading communication channel. However, RLS
involves high computation complexity and may encounter stability challenges
during the calculation of inverse autocorrelation function [3, 8, 7].

For a desired filter response $d(n)$, input vector $\textbf{x}(n)$ and error
signal $e(n)$, and weights vector $\textbf{b}(n)$, time-varying gain vector
k(n), forgetting factor $\gamma$, and input autocorrelation matrix
$\hat{\textbf{R}}$(n), the RLS algorithm, like MSE, is executed in two steps:

  * •

Estimation process: which involves estimating error and gain vector as
follows:

$\textit{$\textbf{k}$(n)}=\frac{\gamma^{-1}\hat{\textbf{R}}^{-1}(n-1)\textbf{x}(n)}{1+\gamma^{-1}\textbf{x}^{H}(n)\hat{\textbf{R}}^{-1}(n-1)\textbf{x}(n)}$
(7)

$e(n)=d(n)-\textbf{b}^{H}(n-1)\textbf{x}(n)$ (8)

With initial conditions $\hat{\textbf{R}}^{-1}(0)=I$ , $\textbf{b}(0)=0$,
$0<\gamma<1$

  * •

Adaptation process: which involves filter taps updates as follows:

$\textit{{b}}(n)=\textit{{b}}(n-1)+\textit{{k}}(n)e^{*}(n)$ (9)

$\hat{\textbf{R}}^{-1}(n)=\gamma^{-1}\hat{\textbf{R}}^{-1}(n-1)-\gamma^{-1}\textit{$\textbf{k}$(n)}\textbf{x}^{H}(n)\hat{\textbf{R}}^{-1}(n)$
(10)

In adaptive channel equalization case and when adapting equalizer taps, the
pilot signal represents the desired filter response while the distorted pilot
by the channel represents filter input. The equalizer filter is then desired
to cancel channel effect and produce the originally transmitted signal at the
receiver.

##  III General System Model

The communication system model employed in this study include Binary Phase
Shift Keying (BPSK) modulated complex baseband system with prominent sections
of transmitter, multipath channel, and receiver. Assuming normalized energy,
the transmitter transmits a signal $u_{t}$, which propagates through ISI
channel $h$ with $K$ multipath components, i.e. channel taps. Note that with
$K=1$, the multipath channel is assumed flat with a constant gain effect. The
receiver signal $u_{r}$ consists of multipath components combined with AWGN
represented by $w$. The system equation at a given time $n$ is given by:

$u_{r}(n)=\sum_{k=1}^{K-1}u_{t}(n)h(n-k)+w(n)\ $ (11)

The channel taps follow $\alpha-\mu$ fading distribution, which comprises a
Gamma-based general distribution. This fading distribution has been proven to
successfully describe small-scale fading in Radio Frequency (RF), mmWave and
sub-THz channels [4, 5, 9]. Moreover, with a certain parameter setting,
$\alpha-\mu$ Probability Distribution Function (PDF) yields many well-known
fading distributions, such as Rayleigh, Weibull, Nakagami-m and one-sided
Gaussian, which makes it a great fit for fading modeling at various channel
conditions. The PDF and Cumulative Density Function (CDF) of $\alpha-\mu$
distribution are shown below:

$f(x)=\frac{\alpha\mu^{\alpha}\left(\frac{x}{\beta}\right)^{\alpha\mu-1}\exp{\left(-\mu\left(\frac{x}{\beta}\right)^{\alpha}\right)}}{\beta\Gamma(\mu)}$
(12)

$F(x)=1-\frac{\Gamma\left(\mu,\mu-\mu\left(\frac{x}{\beta}\right)^{\alpha}\right)}{\Gamma(\mu)}$
(13) $\beta=\sqrt[\alpha]{E\left(X^{\alpha}\right)}$ (14)

Where $\Gamma(.)$ is the complete Gamma function. In order to determine
practical $\alpha-\mu$ parameter values that resemble realistic communication
channel, one can follow the estimations of [5]. These findings are based on
empirical channel measurements for both Line-of-Sight (LOS) and non-LOS sub-
THz transmission links. From figure (1) and Table (1) of [5], three RX-TX
pairs are considered for study, which are RX-TX1 with
$\alpha$$=3.21$,$\mu$$=7.81$ (LOS), RX-TX2 with $\alpha$$=3.13$,$\mu$$=3.76$
and RX-TX5 with $\alpha$$=2.64$,$\mu$$=0.71$ (non-LOS). The PDFs of the
corresponding three RX-TX pairs are shown in figure 2.

Figure 2: Channel PDF for different $\alpha-\mu$ values

Now that system parameters are specified, simulations are carried out using
MATLAB to investigate the performance of ZF, LMS and RLS equalization
algorithms. The first stage of simulation is based on transmitting the pilot
signal to adjust equalizers coefficients. Convergence speed is analyzed in
this stage. The second simulation stage is concerned with running Monte Carlo
simulations by generating several streams to pass through the multipath
channel. These streams are processed at the receiver to mitigate channel and
noise distortion and therefore, performance is assessed in term of BER curves.

##  IV Experiments and Results

This section presents computer simulation results for ZF, LMS and RLS
equalization algorithms. Each algorithm is analyzed in terms of BER
performance, adaptive algorithms are moreover addressed in terms of
convergence speed, behavior according to varying channel parameters and
optimal lengths of training and equalization taps. That is, a major analysis
objective is to illustrate how $\alpha-\mu$ distributed multi-tap channel is
mitigated with channel equalization.

###  IV-A Zero-Forcing Performance

The algorithm is examined in terms of ability to equalize flat fading (single
channel taps) and ISI channel (2-3 channel taps). The LOS RX-TX1 pair of
figure (2) and Table (1) in [5] is assumed for setting channel parameters.
Generated streams for Monte Carlo simulations are of length 1000 symbols. The
equalizer impulse response is the inverse of channel transfer function. The
BER performance for flat versus ISI channel is shown in figure 3.

Figure 3: ZF equalizer performance on flat vs. ISI channel

For a reasonable range of SNR, the ZF equalizer error probability is much
lower for flat fading channel compared to multi-tap channel. Recall that ZF
algorithm ignores the additive noise, meaning that it is only effective when
ISI distortion is dominant under high SNR values.

###  IV-B Least-Mean-Square Performance

The algorithm is evaluated in terms of equalizing flat and ISI channel and
equalizing channel under several $\alpha-\mu$ parameter settings. Different
training lengths and equalization orders and are tested to optimize BER
performance. A minimum number of 10000 streams of length 1000 symbols are
generated for Monte Carlo simulations. The algorithm is run 100 times to
estimate MSE and average BER. For suitable convergence and minimal
misalignment, the learning rate, $\eta$ is selected 0.04. Unless otherwise
specified, all BER and convergence curves are obtained assuming RX-TX1 pair of
figure 2 with a training length of 1000 symbols and 16-tap equalizer. The
analysis is conducted according to the following criterions:

####  IV-B1 Flat vs. ISI Channel Equalization

Figure 4 shows BER performance of single-tap versus multiple-tap LMS channel
equalization with 16 taps. The equalizer performance in a flat $\alpha-\mu$ is
significantly better than in ISI channel. This is clearly exhibited by
algorithm convergence curves for both flat and ISI channels shown in figure 5.

Figure 4: LMS equalizer performance on flat vs. ISI channel Figure 5: LMS
equalizer Convergence for flat vs. ISI channel

While LMS algorithm is anticipated to yield better BER performance for multi-
tap channels, the BPSK training symbols employed are not whitened. This
indicates that the equalizer is selectively canceling channel frequency
response under certain frequencies. Since training length is constrained by
channel coherence time, whitened training symbols may be used to optimize
performance.

####  IV-B2 Impact of Training Length

Training length determination is dependent on channel coherence time. Since
long training length guarantees stable convergence, fast fading channels
constraints applicable lengths and thus affect equalizer accuracy. Figure 6
illustrates how training length impacts error probabilities.

It can be shown that employing a training length of about 500-1000 symbols is
sufficient for algorithm convergence. A shorter training length of 100 symbols
resulted in less effective channel cancellation, proven by high BER values for
the specified SNR range.

Figure 6: LMS equalizer Performance for different training lengths

####  IV-B3 Impact of Equalization Taps

The equalizer filter accounts for both ISI channel taps and additive noise
effect. While it may seem feasible to adapt equalizer filter with multiple
equalization taps more than those of channel, it is yet proven that, for the
same convergence rate and training length, the less the number of equalization
taps, the better the BER performance. This is exemplified in figure 7.

Figure 7: LMS equalizer Performance for different numbers of equalizer taps

####  IV-B4 Different parameter setting of the $\alpha-\mu$ channel

Assuming a flat fading channel and similar equalization parameters, the state
of the channel considerably affects the performance. Different $\alpha-\mu$
values correspond to channel multipath status and fading gain. For a LOS link,
lower error probabilities are obtained than non-LOS case. Figure 8 addresses
this notion through BER curves of the three TX-RX pairs of figure 2.

Figure 8: LMS equalizer Performance for different $\alpha-\mu$ values

###  IV-C Recursive-Least-Square Performance

The RLS algorithm is similar to LMS algorithm in terms of equalizing both
channel and noise distortion. Assuming adapted RLS equalizer filter, same
remarks are drawn when comparing RLS BER performance for different parameter
settings and channel conditions. Figure 9 shows RLS BER performance under flat
and ISI channels. It is noted that the converged RLS yields similar results as
converged LMS.

Figure 9: RLS equalizer Performance for flat vs. ISI channel

The major advantage of RLS algorithm is significantly fast convergence speed
compared to LMS algorithm. This comes at the expense of higher complexity.
That is, for short training lengths suited for channel with short coherence
time, RLS algorithm results in much reduced error probability. This can be
shown in figure 10 where a training length of 100 is used for both RLS and
LMS.

Figure 10: RLS vs LMS equalizer Performance for same training length

Figure 10 result can further be elaborated by addressing RLS versus LMS
convergence for the same training length as illustrated in figure 11. The RLS
algorithm required 10 times fewer samples to converge into optimal filter
parameters as opposed to LMS convergence. Therefore, under less tight latency
constraints, RLS based equalization is much more efficient than that of LMS.

Figure 11: RLS vs LMS Convergence for a flat fading channel

##  V Conclusion

This work considered evaluating channel equalization of multipath wireless
channels following $\alpha-\mu$ fading distribution. The ZF, LMS and RLS
equalization algorithms were analyzed based on error probability, convergence
and implementation complexity. The three algorithms exhibited better
performance for flat fading channels compared to ISI channels. Non-adaptive ZF
is only applicable when channel effect is dominant compared to noise. Adaptive
LMS is studied in terms of suitable training lengths, equalization taps and
response to $\alpha-\mu$ parameters variation. It was proven that LMS and RLS
algorithms demonstrate similar post-convergence performance. Though LMS
algorithm is simpler, it converges much slower than RLS. Fast converging RLS
makes it a good fit for rapidly fading channels, with the consideration of
high complexity and restricted stability.

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