# Scrambling time for analogue black holes embedded in AdS space

Qing-Bing Wang${}^{~{}1}$, Ming-Hui Yu${}^{~{}1}$, Xian-Hui Ge${}^{~{}1}$111
Corresponding author<EMAIL_ADDRESS>
1Department of Physics, College of Sciences, Shanghai University, 200444
Shanghai, China

###### Abstract:

We propose a gedanken experiment on realizing thermofield double state (TFD)
by using analog black holes and provide an approach to test the scrambling
time. Through this approach, we demonstrate clearly how shock wave changes the
TFD state as time evolves. As the whole system evolves forward in time, the
perturbation of space-time geometry will increase exponentially. Finally, it
will destroy the entanglement between the two states of the thermal field, and
the mutual information between them is reduced to zero in the time scale of
scrambling. The results show that for perturbations of analogue black holes
embedded in AdS space, the scale of the scrambling time is closely related to
the logarithm of entropy of the black hole. The results provide further
theoretical argument for the scrambling time, which can be further falsified
in experiments.

##  1 Introduction

Black holes are perhaps the most interesting and mysterious objects in the
universe. In recent years, there is some great progress in black hole physics,
such as gravitational waves observed by LIGO and VIRGO [1, 2], the image of
black holes observed by the Event Horizon Telescope [3]. These progresses not
only inject new impetus into the development of physics, but also undoubtedly
provide strong evidence that black holes are real astrophysical objects in the
sky. However, considering the quantum aspects of black holes, one may need
reply more on the tabletop experiments.

There are a lot of experimental progresses in the study of simulated black
holes or simulated gravity in recent years [4, 5, 6, 7, 8, 9, 10, 11, 12]. In
particular, the study of analogue black holes offers new viewpoints. In fact,
as early as in 1981, Unruh proposed the possibility of simulating black holes
by using fluid [13]. After that, physicists carried out observations of
thermal Hawking radiation in cold atoms [4] and completed the simulation of
the Penrose process of black holes in the fluid experiment [5].

In addition, there are also many ways to mimic black holes or curved spacetime
backgrounds [14, 15, 16, 17], such as lasers with optical metamaterials. For
simulating black holes, Cheng et al. have realized an artificial
“electromagnetic black hole” in microwave band by using an artificial split-
ring resonator (SRR) as a metamaterial [17]. Sheng et al. constructed
waveguides with nonuniform refractive index according to centrosymmetric, and
realized the equivalent potential field similar to the gravitational field of
black holes [16]. The gravitational bending effect of black holes on light
propagation can be well simulated. Koke et al. used waveguide arrays to
simulate quantum field theory. By connecting the Dirac equation of curved
space-time with coupled waveguide arrays, they studied the generation of
positive and negative ion pairs in curved space-time and the Zitterbewegung
effect (trembling motion) of general relativity [18]. Wang et al. studied the
generation of positive and negative ion pairs near the event horizon of black
holes [19]. Tian et al. use the tunneling effect of trapped ions, which is
analogous to Hawking radiation, to test the upper limit of the speed of
scrambling [20].

For simulation of curved spacetime, related work proves that the nonlinear
optics of hyperbolic metamaterials allow holographic dual description [21].
The wave equation describing the propagation of light through metamaterials
has a 2+1 Lorentz symmetry. The role of time in the corresponding 3D Minkowski
spacetime is played by spatial coordinates aligned with the optical axis of
the material. Other similar work [22, 23, 24] proposed the simulation of the
de Sitter (dS) spacetime and used metamaterials to simulate changes in the
structure of space and time.

On the other hand, most of the analogy and simulation studies about black
holes or related involve quantum many body theory, in which entanglement plays
an important role, and its quantification is called entanglement entropy.
Considering a quantum many-body system with $N$ degrees of freedom in the pure
state, for the subsystem with $m$ degrees of freedom less than $N/2$, the
entanglement entropy of these subsystems will reach its maximum value when
they are in thermal equilibrium, and the total system is called scrambled. One
can simply understand that the information in the system is so evenly and
thoroughly dissipated that we need to study at least $N/2$ degrees of freedom
to capture it [25, 26]. If we add a small amount of information or injection
of a small amount of new degrees of freedom to the system, in the beginning,
only the local degrees of freedom in the system are affected. One can restore
the information with only a small number of degrees of freedom. But in the
end, the perturbation will spread to the whole system, the system returns to
the scrambled state. In this process, the period of time from adding
perturbation til the system returns to the equilibrium state again is called
the scrambling time.

The determination of scrambling time not only plays an important role in
quantum information, but also has far-reaching significance in the black hole
spacetime. According to the Hayden-Preskill experiment [27], for a black hole
has evaporated for a time bigger than half of its lifetime, the order of time
required to reconstruct the information of the matter falling into the black
hole is roughly the same as the scrambling time [25]. Therefore, here we are
committed to providing some useful exploration for the experimental
measurement of the scrambling time in the context of curved spacetime.

In order to realize the holographic description of the above process in analog
gravity, a pratical method is to embed acoustic black holes in Anti de-Sitter
(AdS) spacetime. First of all, analogue black holes are easier to observe in
experiments than astrophysical black holes. Secondly, it is convenient to
calculate related physical quantities in AdS spacetime, such as the geodesic
distance and the entanglement entropy. Thirdly, we find that when acoustic
black holes are considered in AdS space-time, the geometry structure
(explained in the next section of this paper) is similar to BTZ black holes.
At the same time, it may also provide a new analogue for the experimental
simulation of rotating black holes.

For acoustic black holes embedded in the AdS space, we can obtain a metric
expressed by the Hadamard product derived from the relativistic Gross-
Pitaevskii theory or simply from the Yang-Mills theory [28]. Then, the
(2+1)-dimensional acoustic black hole geometry of curved spacetime can be
expressed as a product of the background metric and the acoustic metric [28,
29]

$ds^{2}=(g^{GR}_{\mu\nu}\ast
g^{ABH}_{\mu\nu})dx^{\mu}dx^{\nu}=\mathcal{G}_{tt}dt^{2}+\mathcal{G}_{rr}dr^{2}+\mathcal{G}_{\phi\phi}d\phi^{2},$
(1)

where $g^{GR}_{\mu\nu}$ is the background metric line-elements, and
$g^{ABH}_{\mu\nu}$ is the acoustic metric line-elements. Specifically in the
fluid for the general acoustic metric [13, 30], we take

$\displaystyle ds_{ABH}^{2}$ $\displaystyle=$ $\displaystyle-
g_{tt}dt^{2}+g_{rr}dr^{2}+g_{\phi\phi}d\phi^{2}$ (2) $\displaystyle=$
$\displaystyle\\!\frac{\rho_{0}}{c_{s}}\left[-(c_{s}^{2}-v^{2})dt^{2}+\frac{c_{s}^{2}}{c_{s}^{2}-v^{2}}dr^{2}+r^{2}d\phi^{2}\right]\\!.$

The density ratio of the fluid to the velocity of sound in the fluid
$\frac{\rho_{0}}{c_{s}}$ can be absorbed to the left of the equation. Assumed
that the flow velocity of the fluid $v$ goes in radially, in (2+1)-dimensional
geometry, it can be set to $v=-\lambda/r$, where $\lambda$ is a parameter and
the negative sign indicates that the fluid flows radially to the central
region. The metric (2) can be rewritten as

$ds_{ABH}^{2}=-c_{s}^{2}\left(1-\frac{\lambda^{2}}{c_{s}^{2}r^{2}}\right)dt^{2}+\frac{1}{1-\frac{\lambda^{2}}{c_{s}^{2}r^{2}}}dr^{2}+r^{2}d\phi^{2}\\!.$
(3)

On the other hand, the (2+1)-dimensional Schwarschild-AdS metric is given by

$ds_{GR}^{2}=-r^{2}\left(1-\frac{r_{0}^{2}}{r^{2}}\right)dt^{2}+\frac{1}{r^{2}\left(1-\frac{r_{0}^{2}}{r^{2}}\right)}dr^{2}+r^{2}d\phi^{2}.$
(4)

Notice that we take the AdS radius $\ell$ to be 1 and set $M=G=c=1$ for
convenience. Considering the ratio between the sound velocity $c_{s}$ and the
speed of light $c$, we take the value of $c_{s}$ to be $1/\sqrt{3}$. Thus, we
consider a (2+1)-dimensional acoustic black hole embedded in (2+1)-dimensional
Schwarschild-AdS spacetime with the following line elements

$\displaystyle\mathcal{G}_{tt}$ $\displaystyle=$
$\displaystyle-\frac{r^{2}}{3}\left(1-\frac{3\lambda^{2}}{r^{2}}\right)\left(1-\frac{r_{0}^{2}}{r^{2}}\right),$
$\displaystyle\mathcal{G}_{rr}$ $\displaystyle=$
$\displaystyle\frac{1}{r^{2}\left(1-\frac{3\lambda^{2}}{r^{2}}\right)\left(1-\frac{r_{0}^{2}}{r^{2}}\right)},$
(5) $\displaystyle\mathcal{G}_{\phi\phi}$ $\displaystyle=$ $\displaystyle
r^{4}.$

The acoustic event horizon $r_{s}$ locates at $r_{s}=\sqrt{3}\lambda$ and is
larger than the event horizon $r_{0}$. For simplicity, we set
$f(r)=\frac{r^{2}}{\sqrt{3}}\left(1-\frac{r_{s}^{2}}{r^{2}}\right)\left(1-\frac{r_{0}^{2}}{r^{2}}\right)$.
In turn, the metric (1) can be recast as

$ds^{2}=-f(r)dt^{2}+\frac{1}{f(r)}dr^{2}+\sqrt{3}r^{4}d\phi^{2}.$ (6)

Compared (1) with (5), here we multiply a constant $\sqrt{3}$ on the last
term. The acoustic Hawking temperature at the acoustic horizon is
$T=f^{\prime}(r_{s})/(4\pi)=\frac{r_{s}^{2}-r_{0}^{2}}{2\sqrt{3}\pi r_{s}}$.

Now we construct a thermofield double state (TFD) by exploring the Penrose
diagram of acoustic black holes (see Appendix A for the definition of TFD).
The schematic diagram is shown in Figure 1, in which I denotes the region
outside the acoustic horizon, II the region between the acoustic horizon and
the event horizon, and III the region inside the event horizon.

Figure 1: The Penreos diagram of the acoustic AdS black hole [32]. The left
and right blue lines and yellow lines respectively correspond to the AdS
boundary and singularity. The dotted green lines correspond to the acoustic
horizon, and the upper and lower oblique cyan lines stand for the event
horizon.

The degree of entanglement between subregions $A$ and $B$ on the boundary of
the AdS space-time can be quantified by the entropy $S_{A\cup B}$ or the
mutual information $I(A;B)$, where the mutual information between subregions
$A$ and $B$ is defined as $I(A;B)=S_{A}+S_{B}-S_{A\cup B}$. The entropy
$S_{A\cup B}$ represents the entropy of the whole system composed of $A$ and
$B$. The larger the entropy difference between the sum of the entropy of
regions $A$ and $B$ and the entropy of the whole system, the higher the
entanglement degree of the two regions. That is the greater the mutual
information, the greater the entanglement in regions $A$ and $B$. To obtain
the scrambling time in this context, one can do the following. At the initial
time $t=0$, assuming that the subsystems of the region $A$ and region $B$ are
maximally entangled. Then we can turn on a perturbation in the AdS space-time
on the left that evolves over time and eventually affects the whole system.
After the perturbation decays, the mutual information between two subsystems
drops to zero, which means that the entanglement disappears. The time cost in
this process corresponds to the scrambling time.

The purpose of this paper is to study the mutual information $I$ between
regions $A$ and $B$ with unconnected boundaries in the maximally extended
eternal acoustic-AdS space-time. In the simplest case of the largest extended
Penrose diagram of the acoustic-AdS black hole, which has two causally
disconnected boundaries, one can be interpreted as an identical copy of the
thermofield double state on the second boundary [31]. In [28, 29], some of us
considered the realization of acoustic black holes in curved spacetime. Since
such relativistic analogue black holes may be realized in high energy process,
it is interesting to examine that how two acoustic black holes are first
entangled from the beginning and how their mutual information decays away
later.

This paper is organized as follows. In section 2, we take the
(2+1)-dimensional acoustic black holes embedded in AdS space as a toy model
and study the maximum extension of the whole space-time by introducing the
Kruskal-Szekeres coordinates. We then consider the effects of shock waves on
the bulk spacetime and regard the geodesic distance as a physical quantity to
reflect the shock wave impact on the space-time geometry. In section 3, we
apply the Ryu-Takayanagi formula to calculate the mutual information between
the left and right subsystems. The results show that, under a proper physical
approximation, when the shock wave duration reaches the order of the
scrambling time, the entanglement between subsystems disappears and the mutual
information is vanishing. The discussion and the conclusion are given in
section 4. We briefly review the TFD in Appendix A.

##  2 The Set up

In this section, we discuss the causal structure of the (2+1)-dimensional
acoustic black hole embedded in AdS spacetime. The metric is given in equation
(6). In order to obtain the maximum extension of the two-side geometry, we
find that it is useful to employ the Kruskal transformation. We first focus on
the geometry in the Kruskal-Szekeres coordinates and then consider shock waves
in this background. The geodesic length will also be calculated.

###  2.1 Kruskal-Szekeres coordinates

To facilitate the calculation of the entanglement degree between the two
regions at the space-time boundary about acoustic black holes in AdS, one
needs to perform some coordinate transformations. In this subsection, we can
consider using Kruskal-Szekeres coordinates to describe acoustic black holes.
The tortoise coordinate is defined as

$r_{*}=\int\frac{dr}{f(r)}=\frac{\sqrt{3}}{2\left(r_{s}^{2}-r_{0}^{2}\right)}\left(r_{0}\ln\frac{r+r_{0}}{|r-r_{0}|}+r_{s}\ln\frac{|r-r_{s}|}{r+r_{s}}\right).$
(7)

Considering that $r_{0}$ and $r_{s}$ are basically of the same order of
magnitude, we make the following approximation

$\displaystyle r_{*}$ $\displaystyle=$
$\displaystyle\frac{1}{2\kappa_{s}}\left(\frac{r_{0}}{r_{s}}\ln\frac{r+r_{0}}{r-r_{0}}+\ln\frac{r-r_{s}}{r+r_{s}}\right)$
(8) $\displaystyle\approx$
$\displaystyle\frac{1}{2\kappa_{s}}\ln\frac{\left(r-r_{s}\right)(r+r_{0})}{\left(r+r_{s}\right)(r-r_{0})},$

where the surface gravity is given by $\kappa_{s}=2\pi
T=2\pi/\beta=f^{\prime}(r_{s})/2=\frac{r_{s}}{\sqrt{3}}\left(1-\frac{r_{0}^{2}}{r_{s}^{2}}\right)$.
We can further introduce the null light Kruskal coordinates $U$ and $V$

$U=-e^{-\kappa_{s}u},~{}~{}~{}V=e^{\kappa_{s}v},$ (9)

where $u=t-r^{*}$, $v=t+r^{*}$.

The Kruskal-Szekeres coordinates are denoted by $(V,U,\phi)$, where $\phi$ is
the coordinate same as equation (6), but $t$ and $r$ are replaced by $V$ and
$U$ based on the following coordinate transformation:

$\left.\begin{aligned}
U_{R}&=-\sqrt{\frac{\left(r-r_{s}\right)(r+r_{0})}{\left(r+r_{s}\right)(r-r_{0})}}\
e^{-\kappa_{s}t}\\\
V_{R}&=\sqrt{\frac{\left(r-r_{s}\right)(r+r_{0})}{\left(r+r_{s}\right)(r-r_{0})}}\
e^{\kappa_{s}t}\end{aligned}\right\\}r>r_{s}\ (-+),$ (10a)
$\left.\begin{aligned}
U_{L}&=\sqrt{\frac{\left(r-r_{s}\right)(r+r_{0})}{\left(r+r_{s}\right)(r-r_{0})}}\
e^{-\kappa_{s}t}\\\
V_{L}&=-\sqrt{\frac{\left(r-r_{s}\right)(r+r_{0})}{\left(r+r_{s}\right)(r-r_{0})}}\
e^{\kappa_{s}t}\end{aligned}\right\\}r>r_{s}\ (+-).$ (10b)

In terms of the Kruskal-Szekers coordinate, the metric given in (6) can be
recast as

$ds^{2}=-\frac{\sqrt{3}r_{s}^{2}(r-r_{0})^{2}(r+r_{s})^{2}}{r^{2}\left(r_{s}^{2}-r_{0}^{2}\right)^{2}}dUdV+r^{4}d\phi^{2}\\!,$
(11)

where $r$ is considered as a function $r=r(V,U)$, satisfying the following
relation

$VU=-\frac{\left(r-r_{s}\right)(r+r_{0})}{\left(r+r_{s}\right)(r-r_{0})},$
(12)

which can be derived from equation (10). The Kruskal-Szekeres metric (11) does
not has singularity at $r=r_{s}$ and $r=r_{0}$. When $r_{s}=r_{0}$, this
metric has singularity, but not physically allowed. This indicates that the
singularity in (6) is just a coordinate singularity rather than a physical
singularity. From (12), we can see that lines of constant $r$ are hyperbolas
of constant $UV$ in the $U\mbox{-}V$ plane. Similarly, isochrones can be
expressed as curves in the Kruskal diagram (Figure 2). From (9) and (10), one
can obtain

$\frac{V}{U}=-e^{2\kappa_{s}t}.$ (13)

Figure 2: The $U$-$V$ plane of the Kruskal extension: The region I, II, III
corresponds to region $r>r_{s}$, $r_{0}<r<r_{s}$ and $0<r<r_{0}$ respectively.
The green solid line represents the acoustic horizon $r=r_{s}$. The blue
dotted line represents the location $r=0$ and $r=\infty$. The location
$r=r_{0}$ is at infinity. We can see that the isochronal line $t=0$ is a
horizontal line through the origin. For $t>0$, an isochron is a straight line
sloping upward from the left region to the right region. When
$t\rightarrow\infty$, the isochronal line is a line approaching the $V$ axis.
For $t<0$, the situation is just the opposite.

Thus lines of constant $V/U$ are lines of constant $t$-straight lines through
the origin. From (11), we can see that sound cones correspond $d\phi=0$
(radial) and $ds^{2}=0$ (null). There are curves of sound cones

$V=\text{const},\ U=\text{const}$ (14)

in Figure 2. The acoustic horizon at $r=r_{s}$ is the line with $V=0$ or
$U=0$. Replacing coordinates $U$ and $V$ in (12) with two new coordinates
$U^{\prime}$ and $V^{\prime}$, we have

$\displaystyle V^{\prime}$ $\displaystyle=\tan^{-1}V,$ (15) $\displaystyle
U^{\prime}$ $\displaystyle=\tan^{-1}U.$ (16)

The Penrose diagram is shown in the $U^{\prime}V^{\prime}$ plane as
demonstrated in Figure 1 (see also [32]).

###  2.2 Acoustic shock waves

Assuming that the total system is in its thermal equilibrium at temperature
$T$, now we have a thermofield double state (TFD) of the two CFTs. For a
review of the TFD, one may refer to Appendix A. We are interested in how the
shock waves affect the TFD and how spacetime structure will be modified. A
simply way to turn on the shock waves can be realized adding a few
quasiparticles say “phonons” at the boundary and leting them fall into the
acoustic black hole.

We release a quasiparticle with energy $E_{0}$ from the boundary, at time
$t_{p}$ in the past, so at time $t=0$ it has energy $E_{P}$

$E_{P}\sim E_{0}e^{\kappa_{s}t_{p}}.$ (17)

If the time $t_{p}$ is long enough, then the energy $E_{P}$ will be large
enough that we need to consider its reaction to space-time geometry. We
consider gluing geometry of mass $M$ $(M\propto r_{s}^{2})$ to geometry of
mass $M+E_{0}$ through the null surface with $U_{p}=e^{-\kappa_{s}t_{p}}$
[33]. Here $E_{0}$ is the asymptotic energy of perturbation, which is small
compared to $M$. Considered the metric after the perturbation, as demonstrated
in Figure 3, the right and the left of the region I are chosen with
coordinates $U,V$ and $\widetilde{U},\widetilde{V}$ respectively. The shell
$U_{p}=e^{-\kappa_{s}t_{p}},\widetilde{U}_{p}=e^{-\widetilde{\kappa}_{s}t_{p}}$
propagates at the infinity. For small $E_{0}$, we approximate
$U_{p}\simeq\widetilde{U}_{p}$. The matching condition for $\widetilde{V}$ to
$V$ via
$\widetilde{U}_{p}\widetilde{V}=-e^{-\widetilde{\kappa}_{s}\widetilde{r}_{*}},U_{p}V=-e^{-\kappa_{s}r_{*}}$.
Because
$UV=-\frac{\left(r-r_{s}\right)(r+r_{0})}{\left(r+r_{s}\right)(r-r_{0})},V/U=-e^{2\kappa_{s}t}$,
we have [33]

$\frac{\left(r-r_{s}\right)(r+r_{0})}{\left(r+r_{s}\right)(r-r_{0})}=\left(r-r_{s}\right)C\left(r\right),$
(18)

where $C$ is smooth at the acoustic horizon $r=r_{s}$ and $C(r_{s})\neq 0$. We
obtain

$C\left(r\right)=\frac{r+r_{0}}{\left(r+r_{s}\right)(r-r_{0})}.$ (19)

According to [33], for linear order in $E_{0}$ and large $t_{p}$, we find the
shift of the shock wave to the geometry $\alpha=\widetilde{V}-V$ and obtain

$\displaystyle\alpha$ $\displaystyle=$
$\displaystyle\frac{E_{0}}{U_{p}}\frac{dr_{s}}{dM}C(r_{s})$ (20)
$\displaystyle=$
$\displaystyle\frac{E_{0}}{U_{p}}\frac{dr_{s}}{dM}\frac{r_{s}+r_{0}}{2r_{s}(r_{s}-r_{0})}.$

Considering the entropy of acoustic black hole $S=\pi r_{s}/2$, $dS=\pi/2\
dr_{s}$ 222The entropy of acoustic black hole is considered as contributed by
the generalized entropy of the Hawking radiation of the quasiparticles
(phonons)., we then have $dM=TdS=\frac{\kappa_{s}}{4}dr_{s}$ and
$dr_{s}/dM=4/\kappa_{s}$. Thus the shift of spacetime metric is

$\displaystyle\alpha$ $\displaystyle=$
$\displaystyle\frac{E_{0}}{U_{p}}\frac{dr_{s}}{dM}\frac{r_{s}+r_{0}}{2r_{s}(r_{s}-r_{0})}$
(21) $\displaystyle=$
$\displaystyle\frac{4E_{0}}{\kappa_{s}}\frac{r_{s}+r_{0}}{2r_{s}(r_{s}-r_{0})}e^{\kappa_{s}t}$
$\displaystyle=$
$\displaystyle\frac{2\sqrt{3}E_{0}}{\left(r_{s}-r_{0}\right)^{2}}e^{\kappa_{s}t}.$

Using discontinuous coordinates $\mathcal{U}=U$,
$\mathcal{V}=V+\alpha\theta(U)$, we can obtain a shock wave geometry with the
form

$ds^{2}=-\frac{\sqrt{3}r_{s}^{2}(r-r_{0})^{2}(r+r_{s})^{2}}{r^{2}\left(r_{s}^{2}-r_{0}^{2}\right)^{2}}[d\mathcal{U}d\mathcal{V}+\alpha\delta(\mathcal{U})d\mathcal{U}^{2}]+r^{4}d\phi^{2}\\!.$
(22)

Figure 3:  The Kruskal-Szekeres and the Penrose schematic diagrams for the
geometry with a shock wave from the left boundary. The green line represents
the acoustic horizon $r=r_{s}$. The blue line in the Kruskal figure represents
the location of $r=0$ and $r=\infty$. We can see the green lines with $UV=0$
and acoustic horizons is staggered by a displacement $\alpha$. The regions $A$
and $B$ are in the AdS boundary, which is black line sketched in Penrose
diagram and the length of the black line corresponds to the distance
$d^{\prime}$ in formula (30). The distance $d$ in formula (29) corresponds to
the yellow dotted line in the figure.

###  2.3 Geodesics

For the purpose of observing the effect of the perturbation on the
entanglement entropy between subsystems, we need to calculate the entanglement
entropy through the RT formula. According to the RT formula, the entanglement
entropy between a given space-like subregion $A$ and its complementary in a
$d$-dimensional CFT is given by the smallest surface area homologous to $A$ in
the dual $(d+1)$-dimensional bulk spacetime [34]. While for the case of
$(2+1)$-dimensional spacetime, the entanglement entropy of two regions $A$ and
$B$ respectively in left and right AdS spacetime is related to its geodesic
distance.

For computing the geodesic distances $d$ between points in the AdS geometry,
we can use the distance formula for the geodesic distance in pure AdS2+1

$\cosh
d=T_{1}T_{1}^{\prime}+T_{2}T_{2}^{\prime}-X_{1}X_{1}^{\prime}-X_{2}X_{2}^{\prime},$
(23)

On the right hand, the embedded coordinates are defined as follows

$\displaystyle T_{1}$
$\displaystyle=\frac{V+U}{1+UV}=\frac{\sqrt{\left(r^{2}-r_{s}^{2}\right)(r^{2}-r_{0}^{2})}\sinh\kappa_{s}t}{r\left(r_{s}-r_{0}\right)},$
$\displaystyle T_{2}$
$\displaystyle=\frac{1-UV}{1+UV}\cosh\kappa_{s}\phi=\frac{\left(r_{s}r_{0}-r^{2}\right)\cosh\kappa_{s}\phi}{r\left(r_{s}-r_{0}\right)},$
$\displaystyle X_{1}$
$\displaystyle=\frac{V-U}{1+UV}=\frac{\sqrt{\left(r^{2}-r_{s}^{2}\right)(r^{2}-r_{0}^{2})}\cosh\kappa_{s}t}{r\left(r_{s}-r_{0}\right)},$
$\displaystyle X_{2}$
$\displaystyle=\frac{1-UV}{1+UV}\sinh\kappa_{s}\phi=\frac{\left(r_{s}r_{0}-r^{2}\right)\sinh\kappa_{s}\phi}{r\left(r_{s}-r_{0}\right)}.$
(24)

The coordinates $T_{1},T_{2},X_{1},X_{2}$ satisfy the following relationship

$T_{1}^{2}+T_{2}^{2}-X_{1}^{2}-X_{2}^{2}\equiv 1.$ (25)

In the left asymptotic region, one need to add the imaginary part $i\beta/2$
(i.e. $i\pi/\kappa_{s}$) to time $t$ in (24), which means that $U$ and $V$ in
the Kruskal coordinates have extra minus sign (i.e. use formula (10b) to
replace formula (10a) in formula (24)).

Taking $T_{1}^{\prime}=X_{1}^{\prime}=V$,
$T_{2}^{\prime}=\cosh\kappa_{s}\phi$, $X_{2}^{\prime}=\sinh\kappa_{s}\phi$ in
(23), we obtain the geodesic distance $d_{2}$ from right boundary to the
acoustic event horizon. That is

$\cosh
d_{2}=\frac{e^{-\kappa_{s}t_{R}}v\sqrt{\left(r^{2}-r_{s}^{2}\right)(r^{2}-r_{0}^{2})}-r^{2}+r_{0}r_{s}}{r(r_{0}-r_{s})}.$
(26)

Similarly, the geodesic distance $d_{1}$ from left boundary to the acoustic
event horizon is

$\cosh
d_{1}=\frac{e^{-\kappa_{s}t_{L}}(\alpha+v)\sqrt{\left(r^{2}-r_{s}^{2}\right)(r^{2}-r_{0}^{2})}+r^{2}-r_{0}r_{s}}{r(r_{s}-r_{0})}.$
(27)

The total geodesic distance $d$ between two points on different boundaries is

$d=2\log\left(\frac{\alpha
e^{-\frac{1}{2}\kappa_{s}(t_{L}+t_{R})}\sqrt{\left(r^{2}-r_{0}^{2}\right)\left(r^{2}-r_{s}^{2}\right)}+\left(r^{2}-r_{0}r_{s}\right)\cosh\frac{\kappa_{s}(t_{L}-t_{R})}{2}}{r(r_{s}-r_{0})}\right).$
(28)

In the large $r$ limit, we approximate

$\displaystyle d$ $\displaystyle\approx$ $\displaystyle
2\log\left(\frac{\alpha
e^{-\frac{1}{2}k(t_{L}+t_{R})}r^{2}+2r^{2}\cosh\left(\frac{1}{2}k(t_{L}-t_{R})\right)}{r(r_{s}-r_{0})}\right)$
(29) $\displaystyle=$ $\displaystyle 2\log\left(\frac{\alpha
e^{-\frac{1}{2}k(t_{L}+t_{R})}r+2r\cosh\left(\frac{1}{2}k(t_{L}-t_{R})\right)}{(r_{s}-r_{0})}\right)$
$\displaystyle=$ $\displaystyle
2\log\left(\frac{2r}{r_{s}-r_{0}}\right)+2\log\left(\frac{\alpha
e^{-\frac{1}{2}k(t_{L}+t_{R})}}{2}+\cosh\frac{\kappa_{s}(t_{L}-t_{R})}{2}\right).$

The above situation corresponds to the distance from $A$ to $B$ in Figure 3
the yellow dotted line. Similarly, on the same boundary, the geodesic distance
$d^{\prime}$ between two equal-time points $(r,t,\phi_{1})$ and
$(r,t,\phi_{2})$ is

$d^{\prime}\approx
2\log\left(\frac{2r\sinh\frac{\kappa_{s}(\phi_{1}-\phi_{2})}{2}}{r_{s}-r_{0}}\right)=2\log\left(\frac{2r}{r_{s}-r_{0}}\right)+2\log\left(\sinh\frac{\kappa_{s}(\phi_{1}-\phi_{2})}{2}\right).$
(30)

Because in 2 + 1-dimensional geometry, the regions on the AdS boundary is one-
dimensional. At the same time, the extremum surface becomes a line. This
distance corresponds to the length of $A$ or $B$ (the black line in Figure 3).

##  3 Mutual information and scrambling time

Up to now, we have constructed the TFD through a toy model and obtained
geodesic distances (29) and (30) in the geometry after applying the
perturbation. Now the primary goal is to obtain the mutual information
$I(A;B)$ between the subsystems $A$ and $B$ (see Figure 3). We first focus on
the entanglement entropy of the subsystem $S_{A}$, or equivalently $S_{B}$.
From the expression (29) and (30), there are two suitable extemal surfaces
(distance). However, the RT formual tells us that the smallest surface is the
most proper choice. Therefore, we chose (30) to obtain the smaller value when
$\phi<\pi$, i.e.

$\displaystyle S_{A}=S_{B}$ $\displaystyle=$
$\displaystyle\text{Min}\bigg{[}\frac{\text{Area}(\gamma)}{4G_{N}}\bigg{]}$
(31) $\displaystyle=$
$\displaystyle\frac{d^{\prime}}{4G_{N}}=\frac{1}{2G_{N}}\left[\log\left(\frac{2r}{r_{s}-r_{0}}\right)+\log\left(\sinh\frac{\kappa_{s}(\phi_{1}-\phi_{2})}{2}\right)\right].$

Note that the position of the area $A$ and $B$ we selected are the same on
their respective boundaries. Secondly, consider the entanglement entropy of
the total system $S_{A\cup B}$. The first choice of the total entropy is to
add the two region’s entanglement entropy $S_{A\cup B}^{(1)}=S_{A}+S_{B}$,
giving the result as $I(A;B)=\big{[}S_{A}+S_{B}\big{]}-S_{A\cup B}^{(1)}\equiv
0$, which is trivial. But for the second choice, using (29) under the equal-
time condition $t_{L}=t_{R}$, we connect a pair of geodesic from $A$ to $B$,
which lead to the minimal value

$\displaystyle S_{A\cup B}^{(2)}$ $\displaystyle=$ $\displaystyle
2\times\frac{d}{4G_{N}}$ $\displaystyle=$
$\displaystyle\frac{1}{G_{N}}\left[\log\left(\frac{2r}{r_{s}-r_{0}}\right)+\log\left(\frac{\alpha}{2}+1\right)\right].$

Thus the mutual information is obtained as

$I(A;B)=S_{A}+S_{B}-S_{A\cup B}^{(2)}\\\
=\frac{1}{G_{N}}\left[\log\left(\sinh\frac{\kappa_{s}(\phi_{1}-\phi_{2})}{2}\right)-\log\left(\frac{\alpha}{2}+1\right)\right].$
(33)

So, when $\sinh\frac{\kappa_{s}(\phi_{1}-\phi_{2})}{2}<1$, i.e. a small
regions with $A$ and $B$, there is $S_{A\cup B}^{(2)}>S_{A\cup B}^{(1)}$,
which means $I(A;B)=0$ for any value of $\alpha$. But for larger regions and a
small $\alpha$, $S_{A\cup B}^{(2)}$ is smaller than $S_{A\cup B}^{(1)}$, and
the expression (33) represents the evolution of the entanglement between the
left and right subsystems with perturbation in the thermofield double state.
The perturbations eventually affect the whole spacetime, resulting in complete
purification of the entanglement between subsystems. When the mutual
information is vanishing, we have

$\sinh\frac{\kappa_{s}(\phi_{1}-\phi_{2})}{2}=\frac{\alpha}{2}+1.$ (34)

Invoking the following approximation

$\sinh\frac{\kappa_{s}(\phi_{1}-\phi_{2})}{2}\approx\frac{1}{2}e^{\frac{\kappa_{s}(\phi_{1}-\phi_{2})}{2}},$
(35)

and assuming that $\alpha\gg 1$, we have

$\frac{1}{2}e^{\frac{\kappa_{s}(\phi_{1}-\phi_{2})}{2}}=\frac{\sqrt{3}E_{0}}{\left(r_{s}-r_{0}\right)^{2}}e^{\kappa_{s}t}.$
(36)

Finally, we obtain

$t=\frac{\phi_{1}-\phi_{2}}{2}+\frac{1}{\kappa_{s}}\ln\frac{2\sqrt{3}E_{0}}{\left(r_{s}-r_{0}\right)^{2}}.$
(37)

In the above expression, we have taken the entropy of acoustic black hole with
following equation

$\frac{2S}{\beta}=\frac{\left(r_{s}+r_{0}\right)\left(r_{s}-r_{0}\right)}{2\sqrt{3}}.$
(38)

In the limit $r_{s}\gg r_{0}$, the scrambling time is obtained as

$t_{w}\approx\frac{1}{\kappa_{s}}\ln\frac{2S}{\beta
E_{0}}\sim\frac{\beta}{2\pi}\ln S,$ (39)

where $E_{0}\sim T=1/\beta$. The results are consistent with the Hayden-
Preskill experiment considered in [27, 33]. In other words, after the
perturbation, the time for the system to return to the equilibrium state is
directly proportional to the logarithm of the entropy of the acoustic black
hole. The scrambling time is inversely proportional to the temperature, that
is, the lower the temperature, the longer the scrambling time. This is
consistent with the existing thermodynamic experience. Similar analyses can
yield a similar result with $\phi>\pi$, the specific details can be found in
[33].

##  4 Discussion and conclusion

In summary, we study the time evolution of the entanglement entropy and the
mutual information of subsystems in the thermofield double state under the
perturbation and use acoustic black holes embedded in AdS space to simulate
this process. We first construct the TFD by embedding acoustic black holes
into the AdS spacetime (6). Next, we calculate the geodesic distance (29) and
(30) in the whole black hole spacetime after the Kruskal extension. The last
result (39) shows the influence of the shock wave on the scrambling time. The
correlation between the two regions $A$ and $B$ in the AdS boundary of the TFD
decays exponentially with time after the shock wave. It’s shows the
entanglement between dual spacetime is destroyed, and this is related to the
entropy of the central acoustic black hole. After adding shock waves, the more
information the central acoustic black hole contains, the longer the
entanglement of dual space-time can last.

For this result (37), we can also see that the order of the scrambling time is
related to the size of the acoustic horizon and the optical horizon. If we
substitute $\kappa_{s}$ into equation (37), then there is
$t\propto\frac{\sqrt{3}r_{s}}{r_{s}^{2}-r_{0}^{2}}\ln\frac{2\sqrt{3}E_{0}}{\left(r_{s}-r_{0}\right)^{2}}$.
Within the physical allowable range, the larger the acoustic horizon $r_{s}$,
the smaller the scrambling time. The size of the real event horizon $r_{0}$
has the opposite effect on the scrambling time. If the real event horizon
$r_{0}$ increases, the scrambling time will be longer. We think this is so
interesting. As for the relationship between $t$ and $r_{0}$, we may simply
understand that the negative temperature $\beta$ at the acoustic horizon is
positively correlated with $r_{0}$ ($\beta=\frac{2\sqrt{3}\pi
r_{s}}{r_{s}^{2}-r_{0}^{2}}$). That is in formula (39), the larger $r_{0}$,
the larger $\beta$, so the final scrambling time will be longer. Although the
entropy at the acoustic horizon $S$ is related to $r_{s}$ is positively
correlated ($S=\frac{\pi r_{s}}{2}$), but $\beta$ is negatively correlated
with $r_{s}$, so in the final result of formula (39), the larger $r_{s}$ is,
the smaller the scrambling time is. It requires further experimental
verification.

On the other hand, we find that there is a similar scrambling time for
(2+1)-dimensional acoustic black holes, which provides some theoretical
information for simulating black hole entanglement and black hole information
loss problem in subsequent acoustic experiments. Of course, some discrete
models may need to be considered in the experiment, which needs defining the
relevant experimental design in the future. For acoustic black holes in AdS
spacetime, the scrambling time have a negative correlation with the radius of
the acoustic horizon and a positive correlation with the real event horizon.
In this regard, this paper provides a theoretical result for the expectation
of subsequent experiments. Therefore, as an attempt and exploration, we
considered whether there are similar possible effects in table-top acoustic
black holes to the black hole information problem, so as to provide some hints
for subsequent theoretical and experimental studies. At the last, our toy
model is similar to the rotating BTZ in metric factor (in formula (6)), and we
recently find some studies about shock waves in the rotating BTZ in [35]. Our
main result (39) is similar with the result formula (4.17) in this paper
($t_{*}=\frac{\beta(1-\mu)}{\pi}\log\frac{S}{\beta E_{eff}}$). Because form of
the metric factor for acoustic balck holes embedded in AdS space, this may
provide a new way and idea for laboratory simulation of rotating BTZ. This
also inspires us to consider simulating rotating BTZ in the form of acoustic
black holes in the future.

## Acknowledgements

We would like to thank Yu-Qi Lei, Cheng-Yuan Lu and Hai-Ming Yuan for helpful
discussions. This work is partly supported by NSFC, China (Grant No.11875184
$\&$ No.11805117).

##  A Thermofield double state

In this appendix, we review the thermofield double state. The TFD can be
prepared by using two identical copies of a quantum mechanical system. That is
to say, preparing two identical copies of a quantum system described by
Hamiltonians $H_{1}=H_{2}$, with eigenstates $|n\rangle_{1}$ and
$|n\rangle_{2}$ of a common eigenenergy $E_{n}$. A thermofield double state
with standard representation is defined as

$|TFD\rangle=\frac{1}{\sqrt{Z}}\sum_{n}e^{-\frac{\beta
E_{n}}{2}}|n\rangle_{L}\otimes|n\rangle_{R}.$ (40)

Notice that the relation $(H_{L}-H_{R})|TFD\rangle=0$ is satisfied for TFD and
these two systems are maximally entangled.

We consider an operator $V$ at the time $t=0$ and let the system evolves until
time $t$, we impose the operator $W_{L}$ on the $L$-system. The physical state
becomes

$|\psi(t)\rangle=W_{L}(t)|TFD\rangle.$ (41)

After that, we evolve the state inversely. The two-point correlator function
of the $L$ and $R$ systems is then given by

$F^{OTO}_{\beta}(t)=\frac{1}{Z}{\rm tr}\bigg{(}e^{-\frac{\beta
H_{L}}{2}}W^{{\dagger}}_{L}(t)V^{{\dagger}}_{L}(0)W_{L}(t)e^{-\frac{\beta
H_{L}}{2}}V_{L}(0)\bigg{)}.$ (42)

## References

  * [1] B. P. Abbott, T. Adams, R. Bonnand, et al. (LIGO Scientific Collaboration and Virgo Collaboration) GW151226: Observation of gravitational waves from a 22-solar-mass binary black hole coalescence, Physical Review Letters, 116, 241103 (2016).
  * [2] B. P. Abbott, R. Abbott, T. D. Abbott, et al. GW170817: Observation of gravitational waves from a binary neutron star inspiral, Physical Review Letters, 119, 161101 (2017).
  * [3] The Event Horizon Telescope Collaboration, First M87 Event Horizon Telescope Results. I. The Shadow of the Supermassive Black Hole, Astrophys. J. 875, L1 (2019).
  * [4] J. R. Nova, K. Golubkov, V. I. Kolobov, J. Steinhauer, Observation of thermal hawking radiation and its temperature in an analogue black hole, Nature. 569, 688 (2019).
  * [5] M. Cromb, G. M. Gibson, E. Toninelli, M. J. Padgett, E. M. Wright, D. Faccio, Amplification of waves from a rotating body, Nature Physics, 1 (2020).
  * [6] P. D. Nation, M. P. Blencowe, A. J. Rimberg, E. Buks, Analogue hawking radiation in a dc-squid array transmission line, Physical Review Letters, 103(8), 087004 (2009).
  * [7] S. Giovanazzi, Hawking radiation in sonic black holes, Physical Review Letters, 94(6), 061302 (2004).
  * [8] T. A. Jacobson, G. E. Volovik, Event horizons and ergoregions in 3He, Physical Review D, 58(6) (1998).
  * [9] L. J. Garay, J. R. Anglin, J. I. Cirac, P. Zoller, Sonic black holes in dilute bose-einstein condensates, Physical Review A, 63(2), 184 (2000).
  * [10] J. Hu, L. Feng, Z. Zhang, C. Cheng, Quantum simulation of unruh radiation, Nature Physics, 15(8), 1 (2019).
  * [11] B. Horstmann, B. Reznik, S. Fagnocchi, J. I. Cirac, Hawking radiation from an acoustic black hole on an ion ring, Physical Review Letters, 104, 250403 (2010).
  * [12] T. G. Philbin, C. Kuklewicz, S. Robertson, S. Hill, F. Konig, U. Leonhardt, Fiber-optical analog of the event horizon, Science, 319 (2008).
  * [13] W. G. Unruh, Experimental black-hole evaporation, Physical Review Letters, 46(21), 1351 (1981).
  * [14] J. Drori, Y. Rosenberg, D. Bermudez, Y. Silberberg, U. Leonhardt, Observation of Stimulated Hawking Radiation in an Optical Analogue, Physical Review Letters, 122, 010404 (2019).
  * [15] C. Sheng, H. Liu, S. Zhu, D. A. Genov, Omnidirectional optical attractor in structured gap-surface plasmon waveguide, Scientific Reports, 6, 23514 (2016).
  * [16] C. Sheng, H. Liu, Y. Wang, et al. Trapping light by mimicking gravitational lensing, Nature Photon, 7, 902 (2013).
  * [17] Q. Cheng, T. J. Cui, W. X. Jiang, B. G. Cai, An omnidirectional electromagnetic absorber made of metamaterials, New Journal of Physics, 12, 063006 (2010).
  * [18] C. Koke, C. Noh, D. G. Angelakis, Dirac equation in 2-dimensional curved spacetime, particle creation, and coupled waveguide arrays, Annals of Physics, 374, 162 (2016).
  * [19] Y. Wang, C. Sheng, Y. Lu, J. Gao, Y. Chang, X. Pang, T. Yang, S. Zhu, H. Liu, X. Jin, Quantum simulation of particle pair creation near the event horizon, National Science Review, 7(9), 1476 (2020).
  * [20] Z. Tian, Y. Lin, U. R. Fischer, J. Du, Testing the upper bound on the speed of scrambling with an analogue of Hawking radiation using trapped ions, The European Physical Journal C, 82, 212 (2022).
  * [21] I. I. Smolyaninov, Holographic duality in nonlinear hyperbolic metamaterials, Journal of Optics, 16: 075101 (2014).
  * [22] L. Miao, R. X. Miao, P. Yi, More studies on metamaterials mimicking de sitter space, Optics Express, 18(9), 9026 (2009).
  * [23] M. Li, R. X. Miao, Y. Pang, Casimir energy, holographic dark energy and electromagnetic metamaterial mimicking de sitter, Physics Letters B, 689(2-3), 55 (2010).
  * [24] I. I. Smolyaninov, E. E. Narimanov, Metric signature transitions in optical metamaterials, Physical Review Letters, 105(6), 067402 (2010).
  * [25] Y. Sekino, L. Susskind, Fast scramblers, Journal of High Energy Physics, 2008(10) (2008).
  * [26] D. N. Page, Average Entropy of a Subsystem, Physical Review Letters, 71, 1291 (1993).
  * [27] P. Hayden, J. Preskill, Black holes as mirrors: quantum information in random subsystems, Journal of High Energy Physics, 2007(9), 887 (2007).
  * [28] X. H. Ge, M. Nakahara, S. J. Sin, Y. Tian, S. F. Wu, Acoustic black holes in curved spacetime and the emergence of analogue minkowski spacetime, Physical Review D, 99(10), 104047 (2019).
  * [29] Q. B. Wang, X. H. Ge, Geometry outside of acoustic black holes in $(2+1)$-dimensional spacetime, Physical Review D, 102(10), 104009 (2020).
  * [30] M. Visser, Acoustic black holes: horizons, ergospheres, and hawking radiation, Classical and Quantum Gravity, 15(6), 1767 (25) (1997).
  * [31] J. Maldacena, Eternal black holes in anti-de sitter, Journal of High Energy Physics, 2003 (2003).
  * [32] M. Banados, M. Henneaux, C. Teitelboim, J. Zanelli, Geometry of the $(2+1)$-black hole, Physical Review D, 48(4), 1506 (1993).
  * [33] S. H. Shenker, D. Stanford, Black holes and the butterfly effect, Journal of High Energy Physics, 2014, 67 (2014).
  * [34] S. Ryu, T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Physical Review Letters, 96, 181602 (2006).
  * [35] V. Malvimat, R. R. Poojary, Fast scrambling due to rotating shockwaves in BTZ, (2021). [arXiv:2112.14089 [hep-th]]


Here, $\lambda \in \mathbb{C}^*$, $l^{(a)}_{i} \in \mathbb{Z}$ are the toric charges and $S$ is a subset  which is fixed by a subgroup of $(\mathbb{C}^*)^k$. The resolved conifold geometry corresponds to the toric variety associated to the toric charge vector

\begin{equation}
\begin{array}{ccccc}
\end{array}\, .
\end{equation}

To specify the mirror geometry we first consider the variables $y_i \in \mathbb{C}^*,i=0,\dots,3$, subject to the constraint

\begin{equation}
\prod_{i=0}^{3} y_i^{l_i}= \frac{y_0 y_1}{y_2 y_3}=1\,,
\end{equation}

and the polynomial

\begin{equation}
P(a,y)=\sum_{i=0}^3 a_i y_i\,, \quad a_i \in \mathbb{C}\,, i=0,\dots,3\,,
\end{equation}

which enters the definition of the Landau-Ginzburg potential of the mirror. This is given by

\begin{equation}
W= U^2+ V^2 + P(a,y)\,,
\end{equation}

where the additional $U,V \in \mathbb{C}$ variables are an artifact of local mirror symmetry, see e. g. <cit.>. There is a freedom to rescale $W$ by a non-zero complex number, which we can use to set one of the $y$ variables to 1, w.l.o.g we set $y_0=1$. The mirror non-compact CY threefold of the resolved conifold is then given by

\begin{equation}
\check{X}= \left\{ (U,V,y_1,y_2) \in \mathbb{C}^2\times (\mathbb{C}^*)^2 |W=U^2+V^2 + a_0 + a_1 y_1 + a_2 y_2 + a_3  y_1 y_2^{-1}=0 \right\}\,.
\end{equation}

The $a_i, i=0,\dots,3$ are complex parameters which determine the complex structure of $\check{X}$. The rescaling of $W$ and $y_1,y_2$ can be further used to show that the complex structure of $\check{X}$ only depends on the combination

$$z:=\frac{a_0 a_1}{a_2 a_3} \,.$$

Keeping the explicit dependence on the $a_i$ is however more convenient for the derivation of the Picard-Fuchs equations from a GKZ <cit.>
system of differential equations annihilating periods of the unique holomorphic $(3,0)$ form on $\check{X}$. The latter is given by

\begin{equation}
\Omega = \textrm{Res}_{W=0}\, \frac{1}{W} dU\, dV\, \frac{dy_1}{y_1}\, \frac{dy_2}{y_2}\,.
\end{equation}

The periods of $\Omega$ are annihilated by the GKZ operator

\begin{equation}
\frac{\partial}{\partial a_0}  \frac{\partial}{\partial a_1}- \frac{\partial}{\partial a_2} \frac{\partial}{\partial a_3}\,,
\end{equation}

which translates into the Picard-Fuchs operator expressed in $z$, namely

\begin{equation}
L= (1-z)\theta^2\,, \quad \theta=z\frac{d}{dz}\,.
\end{equation}

This operator has the following solutions:

\begin{equation}
\varpi^0=1\,, \quad \varpi^1=\frac{1}{2\pi i}\log z\,.
\end{equation}

These correspond to periods of $\Omega$ over appropriately defined compact three-cycles in $H_3(\check{X},\mathbb{Z})$. The mirror map is identified as

\begin{equation}
t= \frac{1}{2\pi i}\log z\,.
\end{equation}

A familiar phenomenon of mirror symmetry for local CY is that the Picard-Fuchs system of the mirror does not have enough solutions to recover the expected ingredients of an affine special Kähler geometry. Generically it is missing expressions for periods of non-compact three-cycles. One way to recover these is to carefully define non-compact three-cycles on the geometry as was done in <cit.>. Alternatively one may extend the PF operators, guided by the expectation of its general form in compact CY, when it is formulated in terms of the distinguished coordinates corresponding to the mirror map. This was done in <cit.>, which we will outline here. The guiding principle is the expected form of the PF operator in terms of the special (flat) coordinate $t$ in the case when the moduli space is complex one-dimensional. It is given by, see e. g. <cit.>

\begin{equation}
L = \partial_t^2 \, C_{ttt}^{-1} \, \partial_t^2
\end{equation}

where $C_{ttt}:=\frac{\partial^3}{\partial t^3} F_0$. This leads to the extended PF operator in $z$ coordinate <cit.>:

\begin{equation}
L= \theta^2 (1-z) \theta^2\,.
\end{equation}

This operator has the following solutions:

\begin{eqnarray}\label{periods}
\varpi^0 &=& 1 \, ,\nonumber\\
\varpi^1 &=&\frac{1}{2\pi i} \log z \, ,\nonumber\\
\varpi^2 &=&\frac{1}{(2\pi i)^2} \left( \frac{1}{2} (\log z)^2 + \textrm{Li}_2(z) \right) \, ,\nonumber\\
\varpi^3 &=&\frac{1}{(2\pi i)^3}\left(- \frac{1}{6} (\log z)^3 -\log z \textrm{Li}_2(z) +2  \textrm{Li}_3(z)\right)\,.
\end{eqnarray}

We can identify the additional solutions with

\begin{equation}
\varpi^2=: F_t\,,\quad \varpi^3= 2F_0 - t F_t\,,
\end{equation}

where $F_t:=\partial_t F_0$ and where the prepotential $F_0$ reads

\begin{equation}\label{prepot} F_0 =\frac{1}{(2\pi i)^3}\left( \frac{1}{3!} (\log z)^3+ \textrm{Li}_3(z)\right)\end{equation}

matching the expected generating function of the genus zero, degree non-zero GW invariants of the resolved conifold. We note that this basis of solutions does not have integral monodromy around $z=0$, but rather transforms as

\begin{equation}
\varpi \rightarrow \left(
\begin{array}{cccc}
\end{array}\right) \, \cdot \varpi\,.
\end{equation}

A different choice of the basis of solutions however does lead to an integral monodromy.

§.§ Acknowledgements:
The authors would like to give special thanks to Jörg Teschner for many helpful discussions and suggestions related to this work.
We have also benefited from discussions with Vicente Cortés, Timo Weigand, and Alexander Westphal around common research projects within the Cluster of Excellence “Quantum Universe". While work on this project was being finished, we learned from Sergei Alexandrov and Boris Pioline that they were also working on a closely related project <cit.>. We would like to thank them for agreeing on the coordination of the arXiv submission. The work of I.T. is funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy – EXC 2121 Quantum Universe – 390833306.
The work of M.A and A.S. is supported through the DFG Emmy Noether grant AL 1407/2-1.


#  Reverse PAC Codes: Look-ahead List Decoding

Xinyi Gu, Mohammad Rowshan, Member, IEEE, and Jinhong Yuan, Fellow, IEEE
School of Electrical Eng. and Telecom., University of New South Wales (UNSW),
Sydney, Australia
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Convolutional precoding in polarization-adjusted convolutional (PAC) codes is
a recently introduced variant of polar codes. It has demonstrated an effective
reduction in the number of minimum weight codewords (a.k.a error coefficient)
of polar codes. This reduction has the potential to significantly improve the
error correction performance. From a codeword formation perspective, this
reduction has limitations. Capitalizing on the understanding of the
decomposition of minimum-weight codewords, this paper studies reverse
precoding that can effectively reduce minimum-weight codewords more than in
PAC codes. We propose a look-ahead list decoding for the reverse PAC codes,
which has the same order of complexity as list decoding in PAC codes. Through
numerical analysis, we demonstrate a notable reduction in error coefficients
compared to PAC codes and polar codes, resulting in a remarkable improvement
in the block error rate, in particular at high code rates.

###### Index Terms:

Polar codes, PAC codes, convolutional codes, sphere decoding, ordered
statistics decoding, minimum weight codewords, precoding, pre-transformation,
error coefficient, minimum-weight codewords.

##  I INTRODUCTION

Polarization-adjusted convolutional (PAC) codes [1] are a variant of polar
codes [2] resulting from the convolutional pretransformation before polar
coding. The pre-transformation in PAC coding can reduce the number of minimum
weight codewords of underlying polar codes due to the impact on the formation
of minimum weight codewords [3] and involvement of forzen coordinates carrying
non-zero values. This reduction is expected to improve the performance of PAC
codes under (near) ML decoders, such as the list decoder [4, 5, 6], sequential
decoders [5] and sphere decoder [7], and could be considered for adaptation in
6G [8]. The coset-wise study on the reduction of minimum weight codewords in
PAC coding [9] revealed that there are limitations to this reduction,
particularly, when the contribution of frozen rows in codeword formation is
lacking, or they do not possess a special property. One approach to overcome
this drawback is to design a pre-coder that breaks these limitations.
Different pre-transformations (or precoders) have been suggested in the
literature such as the ones based on dynamic frozen bits or parity bits, the
ones based on CRC bits, and finally, the ones based on the idea of pre-
transformation in PAC coding [3, 10, 11, 12, 13, 14]. However, they do not
perform the precoding from the perspective of the minimum-weight codeword
formation. A detailed introduction of polar codes, PAC codes, and their
variations can be found in [8, Section VII]. In our previous work [15], we
introduced a precoding scheme for polar codes that can reduce the number of
minimum weight codewords, considerably more than the PAC coding. The main idea
is to involve more frozen rows in the formation of codewords through row
combinations, particularly in cosets that are characterized as incapable of
reducing the minimum weight codewords through precoding [9, Lemma 2].

In this work, we further elaborate on the proposed precoding scheme in [15]
and name it reverse PAC (RPAC) codes. The shortcoming of [15] was the lack of
a low-complexity decoding algorithm for these codes. Since de-mapping of outer
code (which is the inverse of precoding) is performed in the oppoite direction
of decoding of inner code, the conventional successive cancellation list (SCL)
decoding [4] cannot be used for the RPAC codes. To address this issue, we
propose a look-ahead list (LA-SCL) decoding for RPAC codes which employs two
decoding trees: 1) a tree similar to the list decoding of polar codes
representing pre-transformed information, and 2) a tree representing the
information sequence before the pre-transformation. The latter tree, called a
look-ahead tree, considers a few steps ahead of the current stage, taking into
account the branch expansions corresponding to the reverse convolution.
Numerical results demonstrate that with the LA-SCL decoder, RPAC codes
outperform CRC-Polar codes and PAC codes under conventional list decoding for
high-rate short codes, while maintaining the same order of complexity.

##  II PRELIMINARIES

Notations: We denote the set of indices where vector
$\mathbf{e}\in\mathbb{F}_{2}^{n}$ has a nonzero coordinate by support
$\operatorname{supp}(\mathbf{e})$. The _weight_ of $\mathbf{e}$ is
$\operatorname{w}(\mathbf{e})\triangleq|\operatorname{supp}(\mathbf{e})|$. The
all-one vector $\mathbf{1}$ and the all-zero vector $\mathbf{0}$ are defined
as vectors with all identical elements of 1 or 0, respectively. The (binary)
representation of $i\in[0,2^{n}-1]$ in $\mathbb{F}_{2}$ is defined as
$\operatorname{bin}(i)=i_{n-1}...i_{1}i_{0}$, where $i_{0}$ is the least
significant bit, that is $i=\sum_{a=0}^{n-1}i_{a}2^{a}$. We use the operator
$\backslash$ in $\mathcal{A}\backslash\mathcal{B}$ to subtract elements of the
set $\mathcal{B}$ from $\mathcal{A}$.

###  II-A Polar Codes and PAC Codes

Polar codes of length $N=2^{n}$ are constructed based on the $n$-th Kronecker
power of binary Walsh-Hadamard matrix
$\mathbf{G}_{2}={\footnotesize\begin{bmatrix}1&0\\\ 1&1\end{bmatrix}}$, that
is, $\boldsymbol{G}_{N}=\mathbf{G}_{2}^{\otimes
n}=[\mathbf{g}_{0}\;\;\mathbf{g}_{2}\;;\cdots\;\;\mathbf{g}_{N-1}]^{T}$ which
we call it polar transform throughout this paper. A generator matrix of the
polar code is formed by selecting the rows $\mathbf{g}_{i},i\in\mathcal{I}$ of
$\boldsymbol{G}_{N}$. Then, $\mathcal{C}(\mathcal{I})$ denotes such a linear
code. Note that $\mathcal{I}\subseteq[0,N-1]=[0,2^{n}-1]$. The
characterization of the information set $\mathcal{I}$ for polar codes is based
on the channel polarization theorem [2] and the concept of bit-channel
reliability. The indices in $\mathcal{I}$ are dedicated to information bits,
while the indices in $\mathcal{I}^{c}\triangleq[0,N-1]\setminus\mathcal{I}$
are used to transmit a known value, ’0’ by default, which are called _frozen
bits_ and the corresponding rows are frozen rows.

In PAC coding [1], however, we have a pre-transformation stage before polar
coding where the input vector $\mathbf{u}=[u_{0},\ldots,u_{N-1}]$ for polar
coding is obtained by a convolutional transformation using the binary
generator polynomial of degree $s$, with coefficients
$\mathbf{p}=[p_{0},\ldots,p_{s}]$ as follows:

$u_{i}=\sum_{\ell=0}^{s}p_{\ell}v_{i-\ell},$ (1)

where $\mathbf{v}=[v_{0},\ldots,v_{N-1}]$ is the vector constructed based on
$\mathcal{I}$. This coding scheme is called polarization-adjusted
convolutional (PAC) coding. The convolution operation can be represented in
the form of an upper triangular matrix [3] where the rows of the pre-
transformation matrix $\mathbf{P}$ are formed by shifting the vector
$\mathbf{p}=(p_{0},p_{1},\ldots p_{s})$ one element at a row. Note that
$p_{0}=p_{s}=1$ by convention. Then, we can obtain $\mathbf{u}$ by matrix
multiplication as $\mathbf{u}=\mathbf{v}\mathbf{P}$.

Due to this precoding, we would have $u_{i}\in\\{0,1\\}$ for
$i\in\mathcal{I}^{c}$, indicating that $u_{i}$ correpsoding to a frozen
$v_{i}=0,i\in\mathcal{I}^{c}$ may no longer be fixed. Consequently, the vector
$\mathbf{u}$ is mapped to codeword $\mathbf{x}=\mathbf{u}\mathbf{G}_{N}$.
Overall, we obtain $\mathbf{x}=\mathbf{v}\mathbf{P}\mathbf{G}_{N}$. It was
analytically shown in [3, 16] that by convolutional pre-transformation, the
number of minimum weight codewords, a.k.a error coefficient which is denoted
by $A_{\mathrm{w}_{\min}}$ where $\mathrm{w}_{\min}$ is the minimum weight,
may significantly decrease relative to polar codes (without pre-
transformation). Hence, from the union bound [17, Sect. 10.1], we expect that
this reduction potentially improves the block error rate (BLER) of a (near)
maximum likelihood decoding for a binary input additive white Gaussian noise
(BI-AWGN) channel, particularly at high signal-to-noise ratios (SNRs).

###  II-B Minimum Weight Codewords in Cosets

In the conventional PAC coding, forward convolution as per (1) is performed.
Although forward convolution can reduce the number of codewords of minimum
weight relative to polar codes [3], it has its own limitations. To show the
limitations, we first partition all codewords, excluding the all-zero
codeword, of a polar code $\mathcal{C}(\mathcal{I})$ into _cosets_ defined as:

######  Definition 1.

Cosets: Given information set $\mathcal{I}\subseteq[0,N-1]$ for a polar code,
we define the set of codewords
$\mathcal{C}_{i}(\mathcal{I})\subseteq\mathcal{C}(\mathcal{I})$ for each
$i\in\mathcal{I}$ in a coset of the subcode
$\mathcal{C}(\mathcal{I}\setminus[0,i])$ of $\mathcal{C}(\mathcal{I})$ as

$\mathcal{C}_{i}(\mathcal{I})\triangleq\left\\{\mathbf{g}_{i}+\sum_{h\in\mathcal{H}}\mathbf{g}_{h}\colon\mathcal{H}\subseteq\mathcal{I}\setminus[0,i]\right\\}\subseteq\mathcal{C}(\mathcal{I}),$
(2)

where $\mathbf{g}_{i}$ is the _coset leader_. We denote the number of minimum
weight codewords of the coset $\mathcal{C}_{i}$ by
$A_{i,\mathrm{w}_{\min}}(\mathcal{I})$. The total number of minimum weight
codewords for a polar code $\mathcal{C}(\mathcal{I})$ is
$A_{\mathrm{w}_{\min}}=\sum_{i\in\mathcal{I}}A_{i,\mathrm{w}_{\min}}(\mathcal{I})$.

Observe that the coordinate of the first non-zero element in vector
$\mathbf{u}$, $i=\min\\{\operatorname{supp}(\mathbf{u})\\}$, while encoding by
$\mathbf{x}=\mathbf{u}\mathbf{G}_{N}$ plays a key role in classifying the
codewords into cosets. According to [9, Lemma 1], this coordinate remains the
same after precoding by $\mathbf{u}=\mathbf{v}\mathbf{P}$. That is,

$\min\\{\operatorname{supp}(\mathbf{u})\\}=\min\\{\operatorname{supp}(\mathbf{v})\\}.$
(3)

Nevertheless, the resulting $u_{j}$ for $j>i,j\in\mathcal{I}^{c}$ might be
$u_{j}\not=0$, unlike in polar coding. This difference may impact the number
of minimum weight codewords in the cosets due to the inclusion of rows
$\mathbf{g}_{j}$ for $j\in\mathcal{I}^{c}\cap[i,N-1]$ in row combinations.
Note that the weight of the codewords in a coset $\mathcal{C}_{i}$ depends on
the set of row indices $\mathcal{H}=\\{h:h>i\\}$ in (2). Furthermore,
according to [16, Corollary 5]

$\operatorname{w}(\mathbf{g}_{i}+\sum_{h\in\mathcal{H}}\mathbf{g}_{h})\geq\operatorname{w}(\mathbf{g}_{i}),$
(4)

there exists minimum weight codeword $\mathbf{c}\in\mathcal{C}_{i}$, where
$\operatorname{w}(\mathbf{c})=\mathrm{w}_{\min}$, only if $i\in\mathcal{B}$
where set $\mathcal{B}$ is defined as

$\mathcal{B}=\\{j:j\in\mathcal{I},w(\mathbf{g}_{j})=\mathrm{w}_{\min}\\}.$ (5)

Observe that we have $\mathcal{H}\subseteq[i+1,N-1]$ in PAC coding whereas in
polar coding, we have
$\mathcal{H}\subseteq[i+1,N-1]\backslash\mathcal{I}^{c}$. The minimum distance
of the PAC codes is [9, Lemma 1]

$d_{\min}=\mathrm{w}_{\min}=\min(\\{\operatorname{w}(\mathbf{g}_{i}):i\in\mathcal{I}\\}).$
(6)

According to [16, Theorem 1], the minimum weight codewords are uniquely formed
by the following row combinations:

$\operatorname{w}\big{(}\mathbf{g}_{i}+\sum_{j\in\mathcal{J}}\mathbf{g}_{j}+\sum_{m\in\mathcal{M}(\mathcal{J})}\mathbf{g}_{m}\big{)}=\mathrm{w}_{\min},$
(7)

where $\operatorname{w}(\mathbf{g}_{i})=\mathrm{w}_{\min}$,
$\mathcal{J}\subseteq\mathcal{K}_{i}$ and $\mathcal{K}_{i}$ is [16, Lemma 2.a]

$\mathcal{K}_{i}\triangleq\\{j\in\mathcal{I}\backslash[0,i]\colon|\operatorname{supp}(j)\backslash\operatorname{supp}(i)|=1\\}.$
(8)

As a result, every subset of $\mathcal{K}_{i}$ along with other rows in (7)
form a minimum weight codeword. The number of subsets of $\mathcal{K}_{i}$ is
given by $2^{|\mathcal{K}_{i}|}$. Given
$\mathcal{B}\triangleq\\{i\in\mathcal{I}:\operatorname{w}(\mathbf{g}_{i})=\mathrm{w}_{\min}\\}$,
the total number of minimum-weight codewords of the polar code will be
$\sum_{i\in\mathcal{B}}2^{|\mathcal{K}_{i}|}$. The set
$\mathcal{M}(\mathcal{J})$ is a function of the set $\mathcal{J}$ and every
$m\in\mathcal{M}(\mathcal{J})$ has the property (see [16, (9),(10)] for a
detailed definition of $\mathcal{M}(\mathcal{J})$):

$\mathcal{M}(\mathcal{J})\\!\subseteq\\!\\{m\\!>\\!i:|\operatorname{supp}(\operatorname{bin}(m))\backslash\operatorname{supp}(\operatorname{bin}(i))|\\!>\\!1\\}.$
(9)

The relation (7) can be extened such that the sets $\mathcal{J}$ and
$\mathcal{M}$ also intersect with $\mathcal{I}^{c}$ (see [9, (18)]). This is
useful when considering the impact of precoding. Now, let us see the main
limitation of the forward convolution in PAC coding, that forward convolution
cannot reduce their minimum weight codewords.

######  Lemma 1.

([9, Lemma 2]) For any coset $\mathcal{C}_{i}(\mathcal{I})$ where

  1. 1.

$\mathcal{I}^{c}\cap(i,N-1]=\emptyset$, or

  2. 2.

$|\operatorname{supp}(\operatorname{bin}(f))\\!\backslash\\!\operatorname{supp}(\operatorname{bin}(i))|\\!=\\!1,\forall
f\\!\in\\!\left(\mathcal{I}^{c}\\!\cap\\!(i,N\\!-\\!1]\right)$,

we have

$A_{i,\mathrm{w}_{\min}}(\mathbf{G},\mathcal{I})=A_{i,\mathrm{w}_{\min}}(\mathbf{PG},\mathcal{I}).$

In other words, any cosets $\mathcal{C}_{i}$ where there is no frozen row
$\mathbf{g}_{f}$ for $f\in\mathcal{I}^{c}\cap(i,N-1]$ such that
$|\operatorname{supp}(\operatorname{bin}(f))\backslash\operatorname{supp}(\operatorname{bin}(i))|>1$,
we get $A_{i,\mathrm{w}_{\min}}=$ $2^{\left|\mathcal{K}_{i}\right|}$ in the
PAC coding, independently of the choice of $\mathbf{p}$.

##  III Reverse PAC (RPAC) Codes

To tackle the limitations described for the characterised cosets in Lemma 1
and further reduce the total number of minimum weight codewords
$A_{\mathrm{w}_{\min}}$, the general idea is to change the coset leader
$\mathbf{g}_{i}$ in (2) through a different precoding scheme such that the
coordinate of the first non-frozen element in $\mathbf{u}$ is not preserved,
unlike (3). That is, we need a pre-transformation matrix $\mathbf{P}^{\prime}$
that gives $\mathbf{u}=\mathbf{v}\mathbf{P}^{\prime}$ where

$\min\\{\operatorname{supp}(\mathbf{u})\\}\not=\min\\{\operatorname{supp}(\mathbf{v})\\}.$

To further refine the problem, for any pair of $(i,i^{\prime})$ where
$i^{\prime}=\min\\{\operatorname{supp}(\mathbf{v})\\}$ and
$i=\min\\{\operatorname{supp}(\mathbf{u})\\}$, we need to have

  1. 1.

$w(\mathbf{g}_{i})\\!\geq\\!w(\mathbf{g}_{i^{\prime}})$. Observe that if we
have $w(\mathbf{g}_{i})\\!<\\!w(\mathbf{g}_{i^{\prime}})$, according to (4),
the minimum distance $d_{min}$ after precoding might decrease. Note that the
resulting coset may be led by a frozen row
$\mathbf{g}_{i},i\in\mathcal{I}^{c}$.

  2. 2.

$i<i^{\prime}$. Observe that if we have $i>i^{\prime}$, we may lose some of
the information bits.

######  Remark 1.

Letting
$\min\\{\operatorname{supp}(\mathbf{u})\\}<\min\\{\operatorname{supp}(\mathbf{v})\\}$
moves the coset leader from $\mathbf{g}_{i^{\prime}}$ to $\mathbf{g}_{i}$
where $i<i^{\prime}$. This will affect the minimum weight codewords in the
coset $\mathcal{C}_{i}$ as follows:

  1. a)

If
$\operatorname{w}(\mathbf{g}_{i})>\operatorname{w}(\mathbf{g}_{i^{\prime}})$,
according to (4), we have $A_{i,\mathrm{w}_{\min}}=0$.

  2. b)

If
$\operatorname{w}(\mathbf{g}_{i})=\operatorname{w}(\mathbf{g}_{i^{\prime}})$
and the conditions of Lemma 1 are violated, then
$A_{i,\mathrm{w}_{\min}}(\mathbf{PG},\mathcal{I})<A_{i,\mathrm{w}_{\min}}(\mathbf{G},\mathcal{I})$.

###  III-A Practical Approach for Designing the Pre-transform

There might be different ways to implement the general approach proposed
above. In this section, we propose a practical scheme to realize the proposed
approach.

Our scheme is based on designing a mapping function
$\mathbf{v}\rightarrow\mathbf{u}$ such that we get
$\min\\{\operatorname{supp}(\mathbf{u})\\}<\min\\{\operatorname{supp}(\mathbf{v})\\}$,
and then form the corresponding pre-transformation matrix
$\mathbf{P}^{\prime}$. The requirements for the mapping are as follows:

  * •

$u_{i}=1$ for $i<i^{\prime}$ where
$i^{\prime}=\min\\{\operatorname{supp}(\mathbf{v})\\}$,

  * •

$u_{j}\in\\{0,1\\}$ for $j>i$,

  * •

$u_{j}\leftarrow v_{j}+\sum_{k\in\mathcal{K}}p_{k}v_{k}$ where
$\mathcal{K}\subset[i^{\prime},N-1]\backslash\\{j\\}$ and $p_{k}\in\\{0,1\\}$.

To make the scheme tractable, we constrain the scope of
$\mathcal{K}\subset[i^{\prime},N-1]\backslash\\{j\\}$ by selecting
$\mathcal{K}=[j+1,j+s]$ for some $s$. Hence, we can write this mapping
function in $\mathbb{F}_{2}$ as

$u_{i}=\sum_{\ell=0}^{s}p_{\ell}v_{i+\ell}.$ (10)

The vector $\mathbf{p}=[p_{0},p_{1},\cdots,p_{s}]$ is similar to the one used
in the conventional precoding of PAC codes. However, the matrix form of the
proposed precoder has a lower triangular shape; that is,
$\mathbf{P}^{\prime}=\mathbf{P}^{T}$. From linear algebra, we know that the
product of two lower (or upper) triangular matrices is a lower (or upper)
triangular matrix. Hence, knowing $\mathbf{G}_{N}$ is a lower triangular
matrix, we can conclude that $\mathbf{P}^{\prime}\mathbf{G}_{N}$ is also a
lower triangular matrix.

Furthermore, to fulfill the first requirement mentioned in Section III as
$\operatorname{w}(\mathbf{g}_{i})\geq\operatorname{w}(\mathbf{g}_{i^{\prime}})$
and to avoid the reduction in $\mathrm{w}_{\min}$, we need to constrain the
precoder as follows:

$\text{$u_{i}$}=\begin{dcases*}\sum_{\ell=0}^{s}p_{\ell}v_{i+\ell}&if
$\operatorname{w}(g_{i})\geq\mathrm{w}_{\min}$\\\ v_{i}&otherwise\\\
\end{dcases*}.$ (11)

As can be seen in (11), the mapping to $u_{i}$ is based on
$[v_{i}\;v_{i+1}\;\cdots\;v_{i+s}]$ which are placed ahead of coordinate $i$.
Thus, the convolution is performed in _reverse_ direction, the opposite of the
forward direction in the PAC coding. Hence, this scheme is called reverse PAC
or in short RPAC. We use the notion RPAC($s+1$) to denote the constraint
length, $s+1$, of the reverse precoder. Now, let us see a simple example that
illustrates the difference between precoding in PAC codes and the proposed
RPAC codes.

######  Example 1.

Consider the polar code (64,14) with $d_{min}=16$ and
$\mathcal{I}=\\{31,46,47,51,53\\!-\\!55,57\\!-\\!63\\}$ where ’$-$’ indicates
a range of integers. Given a vector $\mathbf{v}$ with $v_{i}=0$ for every
$i\in\mathcal{I}$ except for $v_{54}=1$, then the minimum weight of codewords
obtained by row combinations for polar coding where $\mathbf{p}=[1]$, PAC
coding with $\mathbf{p}=[1\;1\;0\;1\;1\;0\;1\;1\;0\;1]$, and RPAC coding with
the same $\mathbf{p}$ are:

$\text{Polar:}\operatorname{w}(\mathbf{v}\mathbf{P}\mathbf{G}_{N}=\mathbf{g}_{54})=16,\vspace{-5pt}$
$\text{PAC:}\operatorname{w}(\mathbf{v}\mathbf{P}\mathbf{G}_{N}=\mathbf{g}_{54}+\mathbf{g}_{56})=16,\vspace{-5pt}$
$\text{RPAC:}\operatorname{w}(\mathbf{v}\mathbf{P}^{\prime}\mathbf{G}_{N}=\mathbf{g}_{43}+\mathbf{g}_{54})=24.$

Note that in the RPAC coding, the frozen row $\mathbf{g}_{43}$ with the index
$43<54$ is involved in the row combination. This effectively increases the
weight of the RPAC codeword to 24.

##  IV Look-ahead List Decoding for RPAC

Reverse PAC codes have an advantage over PAC codes in terms of the number of
minimum weight codewords. However, this advantage brings about challenges in
decoding due to the opposite directions of pre-transformation and decoding. A
trivial solution is to employ generic decoders to guess the coordinate(s) of
errors. Alternatively, sphere decoding can be considered, which is performed
from the last bit towards the first bit. However, these candidate decoders
exhibit high and variable computational complexity, and consequently high and
non-constant latency and energy consumption. Here, we propose a look-ahead
list decoding algorithm that expands the $\mathbf{v}$ sequence on each path by
$s$ bits ahead of the current stage $i$. We use $L$ to denote the list size of
the LA-SCL decoder.

The look-ahead list decoding can be considered as the expansion of search
trees. We use two trees to represent the expansions: 1) a main tree for
$\mathbf{u}$, corresponding to the tree in conventional list decoding, where
the leaves represent stage $i$. 2) a parallel binary tree for $\mathbf{v}$,
affiliated with the main tree, looks ahead for stage $i+s$. We refer to this
tree as the _look-ahead tree_. This tree maintains the same number of paths as
the main tree but extends $\mathbf{v}$-values for $s$ stages ahead,
corresponding to each path. It’s important to note that the look-ahead tree
only represents binary values of $\mathbf{v}$, while the $L$ independent SC
processes with different intermediate LLRs are retained on the main tree.
During decoding, at stage $i$ of the main tree, the binary tree for
$\mathbf{v}$ expands first by examining stage $i+s$. If $i+s\in\mathcal{I}$,
each path expands with two possibilities: $\hat{v}_{i+s}=0$ and
$\hat{v}_{i+s}=1$. Considering the sequences of:

$[\hat{v}_{i}\;\;\hat{v}_{i+1}\;\;\cdots\;\;\hat{v}_{i+s}],$ (12)

the estimated $\mathbf{v}$ becomes the input to the convolutional transform in
(11). The output $\hat{u}_{i}$ corresponds to the paths on the main tree. The
path metrics are computed based on the values of $\hat{u}_{i}$, which are then
fed back to the SC process to conduct the partial sums for intermediate LLRs.
As we move from stage $i$ to stage $i+1$ on the main tree, we extend the
leaves of the look-ahead part of the tree by one branch from $i+s$ to $i+s+1$
and append the stage $i+1$ to the main tree. The convolution in (11) for
$\hat{u}_{i+1}$ and partial sums for intermediate LLRs are computed
accordingly.

For the LA-SCL decoder, at stage $i=0$, the affiliated tree needs to look
ahead for $s$ stages, requiring initialization. The initialization of the
trees is determined by the number of information bit coordinates within
$v_{0}^{s}$, denoted by $\nu$. This initialization has two conditions: a) If
the first information bit coordinate, denoted by $\mathcal{I}_{0}$, satisfies
$\mathcal{I}_{0}\geq s$, the look-ahead tree undergoes an expansion process
similar to the conventional expansion of the list decoder for PAC codes. When
the looking-ahead sequence $\hat{v}_{i+s}$ encounters $\mathcal{I}_{0}$, the
look-ahead tree initiates expansion into two paths of $\hat{v}_{i+s}=0$ and
$\hat{v}_{i+s}=1$. b) However, if $\mathcal{I}_{0}<s$ in very high code rate
conditions, the look-ahead tree needs to consider all the possible
combinations of $[\hat{v}_{\mathcal{I}_{0}}\;\cdots\;\hat{v}_{s}]$. It will be
initialized into a tree with a depth of $s-\mathcal{I}_{0}+1$, and the minimum
required list size becomes $L_{min}=2^{\nu}$.

Note that the look-ahead list decoding does not increase the list size
compared to the conventional list decoding. Every path on the main tree for
$\hat{u}_{i}$ corresponds to the paths on the look-ahead tree for
$\hat{v}_{i+s}$ in a one-to-one manner. The SC process is performed based on
the main tree and the intermediate and decision LLRs are calculated for the
$L$ paths up to stage $i$. In stage $i+1$ (corresponding to the main tree), we
extend every path of the look-ahead tree by appending
$\hat{v}_{i+s+1}\in\\{0,1\\}$. Then, the corresponding $\hat{u}_{i+1}$ on the
main tree are computed based on the expanded $\hat{v}_{i+s+1}$. The number of
the paths on both trees becomes $2L$. When $L\geq L_{max}$, paths with the
largest path metrics on the main tree, along with the corresponding paths on
the look-ahead tree, are pruned. Therefore, since the tree expansion of the
main tree is similar to conventional list decoding and the look-ahead tree is
expanded in parallel to keep track of $\mathbf{v}$-values ahead of stage $i$,
the order of complexity remains the same, i.e. $O(L\cdot N\log_{2}N)$.

To clarify this matter, let’s consider an example.

######  Example 2.

Given $s=3$ and $\mathbf{p}=[1\;0\;\;1\;\;1]$, suppose that we are in stage
$i=0$ and performing look-ahead list decoding with list size $L=4$. We
consider two initialization conditions with different information sets:
$\mathcal{I}=\\{3,\;4\\}$ and $\mathcal{I}=\\{2,\;3,\;4\\}$. The corresponding
decoding trees are illustrated in Fig. 1.

For $\mathcal{I}=\\{3,\;4\\}$, look-ahead tree initiates expansion until it
reaches $\hat{v}_{3}$. An example of normal initializations of the trees are
depicted in Fig. 1 (a). At stage $i=0$, the initialization of the look-ahead
tree is marked in green. While for $\mathcal{I}=\\{2,\;3,\;4\\}$, two
information bit coordinates are included within the range of $\mathcal{I}_{0}$
to $s$, i.e. $\nu=2$. Consequently, all possible paths of
$[\hat{v}_{2},\;\hat{v}_{3}]$ needs to be listed. This special initialization,
where $\mathcal{I}_{0}<s$, is shown in Fig. 1 (b). The main tree for
$\hat{u}_{i}$ is expanded based on the look-ahead trees, resulting in the same
number of paths. Moving forward to stage $i=1$, $\hat{v}_{4}$ is expanded into
two possible values of $\hat{v}_{4}\in\\{0,\;1\\}$ for all existing paths,
which is the same as the conventional SCL decoder, but with $s$ bits looking
ahead. The expansion for stage $i=1$ is shown in yellow in the figure. For the
example in Fig. 1 (b), at level $i=1$, the number of expanded paths reaches
$2L$ for the main tree and look-ahead tree. Path pruning is then conducted
based on the path metrics. The $L$ paths with the largest path metrics are
pruned for the main tree, as well as for the corresponding look-ahead paths
repreenting $\mathbf{v}$.

Fig. 1: Tree traversal in look-ahead list decoding.

Given the fundamental idea behind the look-ahead list decoding, we review the
computation of path metric, which is identical to the conventional list
decoding [5]. To find a path with the highest probability through the tree
traversal restricted to $L$ paths, the following probability should be
maximized:

$\begin{multlined}P(\hat{\mathbf{u}}|\mathbf{y})=\prod_{i=0}^{N-1}P(\hat{u}_{i}|\hat{u}_{0}^{i-1},y_{0}^{N-1}).\end{multlined}P(\hat{\mathbf{u}}|\mathbf{y})=\prod_{i=0}^{N-1}P(\hat{u}_{i}|\hat{u}_{0}^{i-1},y_{0}^{N-1}).$
(13)

As the addition operation is preferable in practice, we define the path metric
for the $l$-th sequence/path $\hat{u}_{0}^{i-1}$ based on the logarithm of
(13) as $M_{i-1}(l)=-\sum_{j=0}^{i-1}\log
P(\hat{u_{j}}|\hat{u}_{0}^{j-1},y_{0}^{N-1})$, which converts maximization in
(13) to minimization problem. By extending the sequence $\hat{u}_{0}^{i}$, or
equivalently $\hat{v}_{0}^{i}$, by one bit, the path metric for this longer
sequence becomes

$\displaystyle M_{i}(l)$ $\displaystyle=$ $\displaystyle-\sum_{j=0}^{i}\log
P(\hat{u}_{j}|\hat{u}_{0}^{j-1},y_{0}^{N-1})$ (14) $\displaystyle=$
$\displaystyle M_{i-1}(l)+\mu_{i},$ (15)

where the branch metric $\mu_{i}$ of the extended $l$-th path is:

$\begin{multlined}\mu_{i}=-\log
P(\hat{u}_{i}|\hat{u}_{0}^{i-1},y_{0}^{N-1})=\log\left(1+e^{-(1-2\hat{u}_{i})\lambda_{0}^{i}}\right).\end{multlined}\mu_{i}=-\log
P(\hat{u}_{i}|\hat{u}_{0}^{i-1},y_{0}^{N-1})=\log\left(1+e^{-(1-2\hat{u}_{i})\lambda_{0}^{i}}\right).$
(16)

The branch metric can be approximated using [18, Equations (5) and (6)].
Obtaining $M_{i}(l),l\in[1,2L]$, we retain the $L$ paths with the smallest
$M_{i}(l)$. As observed, the main decoding process relies on $\mathbf{u}$.
Expanding the look-ahead sequence
$[\hat{v}_{i+1}\;\;\hat{v}_{i+2}\;\;\cdots\;\;\hat{v}_{i+s}]$ to obtain
$\hat{u}_{i}$ does not increase the decoding complexity in the main decoding
procedure.

##  V Numerical Results and Discussions

The block error rates (BLER) of the codes (64, 50) and (128, 110) are shown in
Figs. 2 and 3, constructed using approximate density evolution method [19]. We
adopt the proposed LA-SCL decoder and the 3rd-order ordered statistic decoder
(OSD [20], denoted by OSD(3), as near-ML decoder to decode the RPAC codes. The
error correction performance of RPAC codes is compared with that of polar and
PAC codes with SCL decoders and CRC-polar codes with SCL decoders. Let
SCL($L$) and LA-SCL($L$) denote the SCL and LA-SCL decoders with list size
$L$. We use PAC($s+1$) to denote conventional PAC codes precoded by
polynomials with lengths $s+1$. The same polynomial mentioned in Example 1 is
used for RPAC(10) and $\mathbf{p}=[1\;1\;0\;1\;1\;0\;1]$ is used for RPAC(7).
The adopted CRC-polar codes use 11 CRC bits with the generator polynomial
$g(x)=x^{11}+x^{10}+x^{9}+x^{5}+1$. Table I gives the minimum distance with
the corresponding error coefficient for the underlying codes.

TABLE I: Minimum weight $\mathrm{w}_{\min}$ and the corresponding error coefficient $A_{\mathrm{w}_{\min}}$ of polar codes. code | (64, 50) | (128, 110)
---|---|---
$\mathrm{w}_{\min}$ | $A_{\mathrm{w}_{\min}}$ | $\mathrm{w}_{\min}$ | $A_{\mathrm{w}_{\min}}$
Polar | 4 | 944 | 4 | 4448
CRC-Polar | 2 | 2 | 4 | -
PAC(10) | 4 | 944 | 4 | 4320
RPAC(10) | 4 | 70 | 4 | 99
Fig. 2: Performance comparison of (64,50) codes.

In Fig. 2, SCL decoding of PAC code (64,50) with list size $L=32,128$ and SCL
decoding of polar code (64,50) with $L=32$ show identical error correction
performance. When adopting the CRC-polar code, the minimum distance of the
code reduces due to the employment of CRC bits and occupation of more bit-
channels, resulting in the degradation of error performance under SCL
decoding, as shown in Fig. 2. With the proposed look-ahead SCL (LA-SCL)
decoding, RPAC code (64,50) outperforms the counterpart PAC code and CRC-polar
code. From Table I, it can be observed that more than 92.5$\%$ of the minimum
weight codewords of RPAC(10) relative to PAC code are eliminated, leading to
the improvement of up to 0.6 dB for LA-SCL decoder (for RPAC code) and SCL
decoder (for CRC-polar code) when $L=128$. In high SNR regimes, the
performance of RPAC code under SCL(128) decoder approaches that of under OSD
decoder and the approximation for BLER under AWGN channel given by normal
approximation (NA) [21].

Fig. 3: Performance comparison of (128,110) codes.

Similar trends are observed for longer codes at high rates, e.g., (128,110).
Table I and Fig. 3 demonstrate a significant decrease in
$A_{\mathrm{w}_{\min}}$ for RPAC(10), enabling the code to outperform the PAC
code counterpart. The LA-SCL decoder for the RPAC code shows improvements over
the CRC-polar code, with gains of up to 0.1 dB achievable with $L=32$. As the
list size of the LA-SCL decoder increases to 64 and 128, the performance
approaches that of OSD for the RPAC code, with gains of up to 0.2 dB.

##  VI CONCLUSION

In this paper, we introduce the reverse PAC coding and its associated look-
ahead list decoding. The RPAC codes have a remarkably smaller number of
minimum-weight codewords compared to polar codes and conventional PAC codes
for high-rate codes. The proposed LA-SCL decoder overcomes the challenge posed
by the opposite directions of demapping and decoding in RPAC codes. With a
similar complexity as conventional list decoding, the LA-SCL decoder for RPAC
codes outperforms CRC-Polar codes and PAC codes under conventional list
decoding for high-rate short codes.

## References

  * [1] E. Arıkan, “From sequential decoding to channel polarization and back again,” arXiv preprint arXiv:1908.09594 (2019).
  * [2] E. Arıkan, “Channel polarization: A method for constructing capacity-achieving codes for symmetric binary-input memoryless channels,” IEEE Trans. Inf. Theory, vol. 55, no. 7, pp. 3051-3073, Jul. 2009.
  * [3] M. Rowshan and E. Viterbo, “On Convolutional Precoding in PAC Codes,” 2021 IEEE Globecom Workshops, Madrid, Spain, 2021, pp. 1-6.
  * [4] I. Tal and A. Vardy, “List Decoding of Polar Codes,” in IEEE Transactions on Information Theory, vol. 61, no. 5, pp. 2213-2226, May 2015.
  * [5] M. Rowshan, A. Burg and E. Viterbo, “Polarization-adjusted Convolutional (PAC) Codes: Fano Decoding vs List Decoding,” in _IEEE Trans. on Vehicular Tech._ , vol. 70, no. 2, pp. 1434-1447, Feb. 2021.
  * [6] H.Yao, A.Fazeli, and A.Vardy, “List decoding of arıkan’s PAC codes,” arXiv preprint arXiv: 2005.13711 (2020).
  * [7] S. Kahraman and M. E. Celebi, “Code based efficient maximum-likelihood decoding of short polar codes,” Proc. IEEE Int. Symp. Inf. Theory, July 2012.
  * [8] M. Rowshan, M. Qiu, Y. Xie, X. Gu and J. Yuan, “Channel Coding Toward 6G: Technical Overview and Outlook,” in _IEEE Open Journal of the Communications Society_ , vol. 5, pp. 2585-2685, 2024.
  * [9] M. Rowshan and J. Yuan, “On the Minimum Weight Codewords of PAC Codes: The Impact of Pre-Transformation,” in _IEEE Journal on Selected Areas in Information Theory_ , vol. 4, pp. 487-498, 2023.
  * [10] W. Liu, L. Chen and X. Liu, “A Weighted Sum Based Construction of PAC Codes, in _IEEE Communications Letters_ , vol. 27, no. 1, pp. 28-31, Jan. 2023.
  * [11] A. Liu, B. Feng, C. Liang, J. Xu and Q. Zhang, “A Novel Hamming Check Concatenated Polarization-Adjusted Convolutional (PAC) Codes,” _2023 IEEE/CIC International Conference on Communications in China (ICCC)_ , Dalian, China, 2023, pp. 1-5.
  * [12] H. Wan, J. Cho and C. J. Zhang, “Polar Codes with Enhanced Weight Distribution,” _2023 IEEE Globecom Workshops (GC Wkshps),_ Kuala Lumpur, Malaysia, 2023, pp. 1237-1242.
  * [13] X. Gu, M. Rowshan, and J. Yuan, “Selective Reverse PAC Coding for Sphere Decoding,” arXiv preprint arXiv:2212.00254 (2022).
  * [14] Zunker, A., Geiselhart, M., Johannsen, L., Kestel, C., Brink, S.T., Vogt, T. and Wehn, N., “Row-Merged Polar Codes: Analysis, Design and Decoder Implementation,” arXiv preprint arXiv:2312.14749 (2023).
  * [15] X. Gu, M. Rowshan and J. Yuan, “Improved Convolutional Precoder for PAC Codes,” _GLOBECOM 2023 - 2023 IEEE Global Communications Conference_ , Kuala Lumpur, Malaysia, 2023, pp. 1836-1841.
  * [16] M. Rowshan, S. H. Dau and E. Viterbo, “On the Formation of Min-Weight Codewords of Polar/PAC Codes and Its Applications,” in _IEEE Transactions on Information Theory_ , vol. 69, no. 12, pp. 7627-7649, Dec. 2023.
  * [17] S. Lin and D. J. Costello, “Error Control Coding,” 2nd Edition, Pearson Prentice Hall, Upper Saddle River, 2004, pp. 395-400.
  * [18] M. Rowshan and E. Viterbo, “List Viterbi Decoding of PAC Codes,” in _IEEE Transactions on Vehicular Technology_ , vol. 70, no. 3, pp. 2428-2435, March 2021.
  * [19] S. B. Korada, A. Montanari, E. Telatar, and R. Urbanke, “An empirical scaling law for polar codes,” in 2010 IEEE Intl Symp. on Inf. Theory (ISIT), 2010, pp. 884-888.
  * [20] M. Fossorier and S. Lin, “Soft-decision decoding of linear block codes based on ordered statistics,” IEEE Transactions on Information Theory, vol. 41, no. 5, pp. 1379–1396, Sep 1995.
  * [21] Y. Polyanskiy, H. V. Poor and S. Verdu, “Channel Coding Rate in the Finite Blocklength Regime,” IEEE Trans. Inf. Theory, vol. 56, no. 5, pp. 2307-2359, May 2010.