[1]Russelle Guadalupe

[1]Institute of Mathematics, University of the Philippines, Diliman, Quezon
City, 1101, Philippines

# A note on the exact formulas for certain $2$-color partitions

<EMAIL_ADDRESS>*

###### Abstract

Let $p\leq 23$ be a prime and $a_{p}(n)$ count the number of partitions of $n$
using two colors where one of the colors only has parts divisible by $p$.
Using a result of Sussman, we derive the exact formula for $a_{p}(n)$ and
obtain an asymptotic formula for $\log a_{p}(n)$. Our results partially extend
the work of Mauth, who proved the asymptotic formula for $\log a_{2}(n)$
conjectured by Banerjee et al.

###### keywords:

circle method, $\eta$-quotients, partitions, asymptotic formula

###### pacs:

[

MSC Classification]Primary 11P55, 11P82, 05A16

##  1 Introduction

Throughout this paper, we denote $(a;q)_{\infty}=\prod_{n\geq 1}(1-aq^{n-1})$
for complex numbers $a$ and $|q|<1$. Recall that a partition of a positive
integer $n$ is a nonincreasing finite sequence of positive integers, known as
its parts, whose sum is $n$. We define $p(n)$ as the number of partitions of
$n$, which can be seen as the coefficients of its generating function given by

$\displaystyle\dfrac{1}{(q;q)_{\infty}}=\sum_{n=0}^{\infty}p(n)q^{n},\quad|q|<1.$

Hardy and Ramanujan [1] proved the following asymptotic formula for $p(n)$
given by

$\displaystyle
p(n)\sim\dfrac{1}{4n\sqrt{3}}\exp\left(\pi\sqrt{\dfrac{2n}{3}}\right),\quad
n\rightarrow\infty$

using the celebrated circle method. Rademacher [2] refined the Hardy-Ramanujan
circle method and derived the exact formula

$\displaystyle
p(n)=\dfrac{1}{\pi\sqrt{2}}\sum_{k=1}^{\infty}A_{k}(n)\sqrt{k}\dfrac{d}{dn}\left(\dfrac{\sinh(\frac{\pi}{k}\sqrt{\frac{2}{3}(n-\frac{1}{24})})}{\sqrt{n-\frac{1}{24}}}\right),$

where

$\displaystyle A_{k}(n):=\sum_{\begin{subarray}{c}0\leq h<k\\\
\gcd(h,k)=1\end{subarray}}\exp\left[\pi
i\left(s(h,k)-\dfrac{2nh}{k}\right)\right]$

and

$\displaystyle
s(h,k):=\sum_{j=1}^{k-1}\dfrac{j}{k}\left(\left\\{\dfrac{hj}{k}\right\\}-\dfrac{1}{2}\right)$

is the Dedekind sum, with $\\{t\\}$ the fractional part of $t$. Recently,
Banerjee et al. [3] proved the following refined asymptotic formula for $p(n)$
given by

$\displaystyle\log p(n)\sim\pi\sqrt{\dfrac{2n}{3}}-\log n-\log
4\sqrt{3}-\dfrac{0.44\cdots}{\sqrt{n}},\quad n\rightarrow\infty,$

via a general double inequality for $p(n)$, which was used to prove a
refinement of the inequality of DeSalvo and Pak [4], and Chen, Wang and Xie
[5] given by

$\displaystyle\left(1+\dfrac{\pi}{24n^{3/2}}-\dfrac{1}{n^{2}}\right)p(n-1)p(n+1)<p(n)^{2}<\left(1+\dfrac{\pi}{24n^{3/2}}\right)p(n-1)p(n+1)$

for $n\geq 120$.

Let $a_{k}(n)$ be the number of partitions of $n$ using two colors where one
of the colors appears only in parts that are divisible by $k$. For $k=1$, we
have $a_{1}(n)=p(n)$ and for $k=2$, $a_{2}(n)$ also counts the number of cubic
partitions (see [6]). The generating function for $a_{k}(n)$ is given by [7]

$\displaystyle\dfrac{1}{(q;q)_{\infty}(q^{k};q^{k})_{\infty}}=\sum_{n=0}^{\infty}a_{k}(n)q^{n},\quad|q|<1.$

Kotesovec [8] found the following asymptotic formula for $a_{2}(n)$ given by

$\displaystyle a_{2}(n)\sim\dfrac{1}{8n^{5/4}}\exp(\pi\sqrt{n}),\quad
n\rightarrow\infty.$

Banerjee et al. [3] conjectured that $a_{2}(n)$ satisfies the following
asymptotic formula

$\displaystyle\log a_{2}(n)\sim\pi\sqrt{n}-\dfrac{5}{4}\log n-\log
8-\dfrac{0.79\cdots}{\sqrt{n}},\quad n\rightarrow\infty.$ (1)

Recently, Mauth [9] proved (1) by finding the exact formula of Rademacher type
for $a_{2}(n)$ using a result of Zuckerman [10].

In this paper, we use a result of Sussman [11] to derive Rademacher type
formulas for $a_{p}(n)$ when $p\leq 23$ is a prime, and deduce their
asymptotic formulas in the spirit of (1), which can be considered as a partial
extension of Mauth’s work. We give our main results as follows.

######  Theorem 1.1.

Let $p\leq 23$ be a prime. Then for $n\geq 1$ we have

$\displaystyle a_{p}(n)$
$\displaystyle=2\pi\sqrt{p}\left(\dfrac{1+p^{-1}}{24n-p-1}\right)\sum_{j=1}^{p-1}\sum_{\begin{subarray}{c}m=1\\\
m\equiv
j\bmod{p}\end{subarray}}^{\infty}I_{2}\left(\dfrac{\pi}{6m}\sqrt{(1+p^{-1})(24n-p-1)}\right)\dfrac{b_{m}(n)}{m}$
$\displaystyle+2\pi\left(\dfrac{1+p}{24n-p-1}\right)\sum_{\begin{subarray}{c}m=1\\\
p\mid
m\end{subarray}}^{\infty}I_{2}\left(\dfrac{\pi}{6m}\sqrt{(1+p)(24n-p-1)}\right)\dfrac{b_{m}(n)}{m},$

where

$\displaystyle b_{k}(n):=\sum_{\begin{subarray}{c}0\leq h<k\\\
\gcd(h,k)=1\end{subarray}}\exp\left[-\dfrac{2\pi nhi}{k}+\pi s(h,k)i+\pi
s\left(\dfrac{ph}{\gcd(p,k)},\dfrac{k}{\gcd(p,k)}\right)i\right]$

and $I_{2}(s)$ is the second modified Bessel function of the first kind (see
Section 2).

######  Corollary 1.2.

For $p\leq 23$ prime, we have

$\displaystyle\log
a_{p}(n)\sim\pi\sqrt{\dfrac{2n(1+p^{-1})}{3}}-\dfrac{5}{4}\log
n+\log\dfrac{2\sqrt{3p}(1+p^{-1})^{3/4}}{24^{5/4}}-\dfrac{c_{p}}{24\sqrt{6n}},\quad
n\rightarrow\infty$

where

$\displaystyle
c_{p}:=\dfrac{135}{\pi\sqrt{1+p^{-1}}}+\pi\sqrt{\dfrac{(1+p)^{3}}{p}}.$

Setting $p=2$ in Corollary 1.2, we deduce the asymptotic formula due to Mauth
[9] given by

$\displaystyle\log a_{2}(n)\sim\pi\sqrt{n}-\dfrac{5}{4}\log n-\log
8-\left(\dfrac{15}{8\pi}+\dfrac{\pi}{16}\right)\dfrac{1}{\sqrt{n}},\quad
n\rightarrow\infty,$

which immediately yields (1).

The paper is organized as follows. We provide in Section 2 the full statement
of Sussman’s result on the exact formulas for the Fourier coefficients of a
class of $\eta$-quotients. In Section 3, we apply Sussman’s result to prove
Theorem 1.1, and using the asymptotic expansion of $I_{2}(s)$ due to Banerjee
[12], we deduce Corollary 1.2.

##  2 Fourier coefficients of a class of $\eta$-quotients

We consider in this section the exact formulas for the Fourier coefficients of
the following class of holomorphic functions on the open disk given by

$\displaystyle
G(q):=\prod_{r=1}^{R}(q^{m_{r}};q^{m_{r}})_{\infty}^{\delta_{r}}=\sum_{n=0}^{\infty}g(n)q^{n},$

where $\mathbf{m}=(m_{1},\ldots,m_{R})$ is a sequence of $R$ distinct positive
integers and $\mathbf{\delta}=(\delta_{1},\ldots,\delta_{R})$ is a sequence of
$R$ nonzero integers. The functions $G(q)$ can be seen as $\eta$-quotients
since $(q;q)_{\infty}=q^{-1/24}\eta(\tau)$, where $\eta(\tau)$ is the Dedekind
eta function and $q=e^{2\pi i\tau}$ with $\tau$ lying in the complex upper
half plane. Given a particular $G(q)$ with $\sum_{r=1}^{R}\delta_{r}<0$,
Sussman [11] gave a Rademacher-type exact formula for $g(n)$, which is a
special case of the work of Bringmann and Ono [13] on the coefficients of
harmonic Maass forms. Sussman’s proof follows the original approach of
Rademacher [2] on the Hardy-Ramanujan circle method. In the case where
$\sum_{r=1}^{R}\delta_{r}\geq 0$, Chern [14] obtained an analogous formula
with an error term using the method of O-Y. Chan [15].

Before we state Sussman’s result, we need some definitions. Suppose $h$ and
$k$ are coprime positive integers. We set

$\displaystyle\Delta_{1}$
$\displaystyle=-\dfrac{1}{2}\sum_{r=1}^{R}\delta_{r},\quad\Delta_{2}=\sum_{r=1}^{R}m_{r}\delta_{r},$
$\displaystyle\Delta_{3}(k)$
$\displaystyle=-\sum_{r=1}^{R}\dfrac{\gcd(m_{r},k)^{2}}{m_{r}}\delta_{r},\quad\Delta_{4}(k)=\prod_{r=1}^{R}\left(\dfrac{\gcd(m_{r},k)}{m_{r}}\right)^{\delta_{r}/2},\quad\text{
and }$ $\displaystyle\widehat{A}_{k}(n)$
$\displaystyle=\sum_{\begin{subarray}{c}0\leq h<k\\\
\gcd(h,k)=1\end{subarray}}\exp\left(-\dfrac{2\pi nhi}{k}-\pi
i\sum_{r=1}^{R}\delta_{r}s\left(\dfrac{m_{r}h}{\gcd(m_{r},k)},\dfrac{k}{\gcd(m_{r},k)}\right)\right),$

where $s(h,k)$ is the Dedekind sum defined in Section 1. We also set $L$ as
the least common multiple of $m_{1},\ldots,m_{R}$, and partition the set
$\\{1,\ldots,L\\}$ into two disjoint subsets

$\displaystyle\mathcal{L}_{>0}$ $\displaystyle:=\\{1\leq l\leq
L:\Delta_{3}(l)>0\\},$ $\displaystyle\mathcal{L}_{\leq 0}$
$\displaystyle:=\\{1\leq l\leq L:\Delta_{3}(l)\leq 0\\}.$

######  Theorem 2.1 ([11]).

If $\Delta_{1}>0$ and the inequality

$\displaystyle\min_{1\leq r\leq
R}\dfrac{\gcd(m_{r},l)^{2}}{m_{r}}\geq\dfrac{\Delta_{3}(l)}{24}$ (2)

holds for $1\leq l\leq L$, then for positive integers $n>-\Delta_{2}/24$, we
have

$\displaystyle
g(n)=2\pi\sum_{l\in\mathcal{L}_{>0}}\Delta_{4}(l)\left(\dfrac{\Delta_{3}(l)}{24n+\Delta_{2}}\right)^{(\Delta_{1}+1)/2}\sum_{\begin{subarray}{c}k=1\\\
k\equiv
l\bmod{L}\end{subarray}}^{\infty}I_{\Delta_{1}+1}\left(\dfrac{\pi}{6k}\sqrt{\Delta_{3}(l)(24n+\Delta_{2})}\right)\dfrac{\widehat{A}_{k}(n)}{k},$

where

$\displaystyle
I_{\nu}(s):=\sum_{m=0}^{\infty}\dfrac{(\frac{s}{2})^{\nu+2m}}{m!\Gamma(\nu+m+1)}$

is the $\nu$th modified Bessel function of the first kind, and
$\Gamma(s)=\int_{0}^{\infty}e^{-t}t^{s-1}\,dt$ is the gamma function.

##  3 Proofs of Theorem 1.1 and Corollary 1.2

In this section, we use Theorem 2.1 to prove Theorem 1.1. As a consequence, we
obtain Corollary 1.2 using the asymptotic expansion of the modified Bessel
function of the first kind due to Banerjee [12].

###### Proof of Theorem 1.1.

Writing

$\displaystyle\dfrac{1}{(q;q)_{\infty}(q^{p};q^{p})_{\infty}}=\sum_{n=0}^{\infty}a_{p}(n)q^{n},$

we have $\mathbf{m}=(1,p),\mathbf{\delta}=(-1,-1)$ and $L=p$. We compute
$\Delta_{1}=1$ and $\Delta_{2}=-p-1$. Recalling that $p$ is a prime, we have
that for $l\in\\{1,\ldots,p\\}$,

$\displaystyle\Delta_{3}(l)=\begin{cases}1+\dfrac{1}{p},&l\neq p\\\
1+p,&l=p\end{cases},\qquad\Delta_{4}(l)=\begin{cases}\sqrt{p},&l\neq p\\\
1,&l=p\end{cases},$

so that $L_{>0}=\\{1,\ldots,p\\}$. Since $p\leq 23$, we see that condition (2)
holds for $l\in\\{1,\ldots,p\\}$. Applying Theorem 2.1 with
$\widehat{A}_{k}(n)=b_{k}(n)$, we obtain

$\displaystyle
a_{p}(n)=2\pi\sum_{l\in\mathcal{L}_{>0}}\Delta_{4}(l)\left(\dfrac{\Delta_{3}(l)}{24n-p-1}\right)\sum_{\begin{subarray}{c}m=1\\\
m\equiv
l\bmod{p}\end{subarray}}^{\infty}I_{2}\left(\dfrac{\pi}{6m}\sqrt{\Delta_{3}(l)(24n-p-1)}\right)\dfrac{b_{m}(n)}{m}$

for $n>(p+1)/24$. Plugging in the values of $\Delta_{3}(l)$ and
$\Delta_{4}(l)$ yields the desired result. ∎

We see that the series for $a_{p}(n)$ in Theorem 1.1 converges rapidly and
that the term $m=1$ contributes signficantly to the series. We thus obtain the
asymptotic formula

$\displaystyle a_{p}(n)\sim
2\pi\sqrt{p}\left(\dfrac{1+p^{-1}}{24n-p-1}\right)I_{2}\left(\dfrac{\pi}{6}\sqrt{(1+p^{-1})(24n-p-1)}\right),\quad
n\rightarrow\infty.$ (3)

To prove Corollary 1.2, we use the following asymptotic formula [12, eq.
(2.11)]

$\displaystyle I_{\nu}(s)\sim\dfrac{e^{s}}{\sqrt{2\pi
s}}\sum_{m=0}^{\infty}\dfrac{(-1)^{m}d_{m}(\nu)}{x^{m}},\quad
s\rightarrow\infty,$ (4)

where

$\displaystyle
d_{m}(\nu)=\dfrac{\binom{\nu-1/2}{m}(\nu+\frac{1}{2})_{m}}{2^{m}}$

with

$\displaystyle\dbinom{a}{m}$
$\displaystyle=\begin{cases}\dfrac{a(a-1)\cdots(a-m+1)}{m!},&m\in\mathbb{Z}_{>0}\\\
1,&m=0\end{cases},$ $\displaystyle(a)_{m}$
$\displaystyle=\begin{cases}a(a+1)\cdots(a+m-1),&m\in\mathbb{Z}_{>0}\\\
1,&m=0\end{cases}$

for $a\in\mathbb{R}$.

###### Proof of Corollary 1.2.

From (4) we get

$\displaystyle I_{2}(s)=\dfrac{e^{s}}{\sqrt{2\pi
s}}\left(1-\dfrac{15}{8s}+O\left(\dfrac{1}{s^{2}}\right)\right).$

In view of (3), we have

$\displaystyle a_{p}(n)\sim 2\sqrt{3p}$
$\displaystyle\cdot\dfrac{(1+p^{-1})^{3/4}}{(24n-p-1)^{5/4}}$
$\displaystyle\cdot\exp\left(\dfrac{\pi}{6}\sqrt{(1+p^{-1})(24n-p-1)}\right)\left(1-\dfrac{45}{8\pi\sqrt{6n(1+p^{-1})}}+O\left(\dfrac{1}{n}\right)\right).$
(5)

Since

$\displaystyle\dfrac{1}{(24n-p-1)^{5/4}}$
$\displaystyle=\dfrac{1}{(24n)^{5/4}}\left(1+O\left(\dfrac{1}{n}\right)\right),$
$\displaystyle\exp\left(\dfrac{\pi}{6}\sqrt{(1+p^{-1})(24n-p-1)}\right)$
$\displaystyle=\exp\left(\pi\sqrt{\dfrac{2n(1+p^{-1})}{3}}\right)\left(1-\dfrac{p^{-1/2}(1+p)^{3/2}\pi}{24\sqrt{6n}}+O\left(\dfrac{1}{n}\right)\right),$

we infer from (5) that

$\displaystyle a_{p}(n)\sim 2\sqrt{3p}$
$\displaystyle\cdot\dfrac{(1+p^{-1})^{3/4}}{(24n)^{5/4}}\left(1+O\left(\dfrac{1}{n}\right)\right)\cdot\exp\left(\pi\sqrt{\dfrac{2n(1+p^{-1})}{3}}\right)$
$\displaystyle\cdot\left(1-\dfrac{p^{-1/2}(1+p)^{3/2}\pi}{24\sqrt{6n}}+O\left(\dfrac{1}{n}\right)\right)\left(1-\dfrac{45}{8\pi\sqrt{6n(1+p^{-1})}}+O\left(\dfrac{1}{n}\right)\right),$

and taking logarithms yields the desired conclusion. ∎

We remark that when $p>23$ is a prime, condition (2) of Theorem 2.1 fails.
Nonetheless, by numerical experiments via Mathematica, we conjecture that
Corollary 1.2 also holds for such primes.

## References

  *   * Hardy and Ramanujan [1918] Hardy, G.H., Ramanujan, S.: Asymptotic formulae in combinatory analysis. Proc. London Math. Soc. s2-17, 75–115 (1918) https://doi.org/10.1112/plms/s2-17.1.75
  * Rademacher [1938] Rademacher, H.: On the partition function $p(n)$. Proc. London Math. Soc. s2-43, 241–254 (1938) https://doi.org/10.1112/plms/s2-43.4.241
  * Banerjee et al. [2023] Banerjee, K., Paule, P., Radu, C.-S., Zeng, W.: New inequalities for $p(n)$ and $\log p(n)$. Ramanujan J. 61, 1295–1338 (2023) https://doi.org/10.1007/s11139-022-00653-6
  * DeSalvo and Pak [2015] DeSalvo, S., Pak, I.: Log-concavity of the partition function. Ramanujan J. 38, 61–73 (2015) https://doi.org/10.1007/s11139-014-9599-y
  * Chen et al. [2016] Chen, W.Y.C., Wang, L.X.W., Xie, G.Y.B.: Finite differences of the logarithm of the partition function. Math. Comp. 85, 825–847 (2016) https://doi.org/10.1090/mcom/2999
  * Kim [2008] Kim, B.: A crank analog on a certain kind of partition function arising from the cubic continued fraction. preprint, 13 pp (2008)
  * Ahmed et al. [2015] Ahmed, Z., Baruah, N.D., Dastidar, M.G.: New congruences modulo 5 for the number of 2-color partitions. J. Number Theory 157, 184–198 (2015) https://doi.org/10.1016/j.jnt.2015.05.002
  * Kotesovec [2015] Kotesovec, V.: A method of finding the asymptotics of q-series based on the convolution of generating functions. preprint at https://arxiv.org/abs/1509.08708 (2015)
  * Mauth [2023] Mauth, L.: Exact formula for cubic partitions. preprint at https://arxiv.org/abs/2305.03396 (2023)
  * Zuckerman [1939] Zuckerman, H.: On the coefficients of certain modular forms belonging to subgroups of the modular group. Trans. Amer. Math. Soc. 45, 298–321 (1939) https://doi.org/%****␣twocolor.bbl␣Line␣175␣****10.2307/1990119
  * Sussman [2017] Sussman, E.: Rademacher Series for $\eta$-quotients. preprint at https://arxiv.org/abs/1710.03415 (2017)
  * Banerjee [2023] Banerjee, K.: Inequalities for the modified Bessel function of first kind of non-negative order. J. Math. Anal. Appl. 524, 127082 (2023) https://doi.org/10.1016/j.jmaa.2023.127082
  * Bringmann and Ono [2012] Bringmann, K., Ono, K.: Coefficients of harmonic Maass forms. In: Alladi, K., Garvan, F. (eds.) Partitions, $q$-Series and Modular Forms, pp. 23–38. Springer, New York (2012)
  * Chern [2019] Chern, S.: Asymptotics for the Fourier coefficients of eta-quotients. J. Number Theory 199, 168–191 (2019) https://doi.org/10.1016/j.jnt.2018.11.008
  * Chan [2005] Chan, O.-Y.: Some asymptotics for cranks. Acta Arith. 120, 107–143 (2005) https://doi.org/10.4064/aa120-2-1


Further author information: Send correspondence to D.S.D
D.S.D.: E-mail<EMAIL_ADDRESS>
# Laboratory demonstration of the triple-grating vector vortex coronagraph

David S. Doelman  Leiden Observatory, Leiden University, P.O. Box 9513, 2300
RA Leiden, The Netherlands  SRON Netherlands Institute for Space Research,
Niels Bohrweg 4, 2333 CA, Leiden, The Netherlands  Mireille Ouellet  Leiden
Observatory, Leiden University, P.O. Box 9513, 2300 RA Leiden, The Netherlands
Axel Potier  Jet Propulsion Laboratory, California Institute of Technology,
Pasadena, California 91109, USA  Garreth Ruane  Jet Propulsion Laboratory,
California Institute of Technology, Pasadena, California 91109, USA  Kyle van
Gorkom  University of Arizona, Steward Observatory, Tucson, Arizona, United
States  Sebastiaan Y. Haffert  University of Arizona, Steward Observatory,
Tucson, Arizona, United States  Ewan S. Douglas  University of Arizona,
Steward Observatory, Tucson, Arizona, United States  Frans Snik  Leiden
Observatory, Leiden University, P.O. Box 9513, 2300 RA Leiden, The Netherlands

###### Abstract

The future Habitable Worlds Observatory aims to characterize the atmospheres
of rocky exoplanets around solar-type stars. The vector vortex coronagraph
(VVC) is a main candidate to reach the required contrast of $10^{-10}$.
However, the VVC requires polarization filtering and every observing band
requires a different VVC. The triple-grating vector vortex coronagraph (tgVVC)
aims to mitigate these limitations by combining multiple gratings that
minimize the polarization leakage over a large spectral bandwidth. In this
paper, we present laboratory results of a tgVVC prototype using the In-Air
Coronagraphic Testbed (IACT) facility at NASA’s Jet Propulsion Laboratory and
the Space Coronagraph Optical Bench (SCoOB) at the University of Arizona Space
Astrophysics Lab (UASAL). We study the coronagraphic performance with
polarization filtering at 633 nm and reach a similar average contrast of
$2\times 10^{-8}$ between 3-18 $\lambda/D$ at the IACT, and $6\times 10^{-8}$
between 3-14 $\lambda/D$ at SCoOB. We explore the limitations of the tgVVC by
comparing the testbed results. We report on other manufacturing errors and
ways to mitigate their impact.

###### keywords:

High-contrast imaging, exoplanets, coronagraphy, vortex coronagraph, direct
imaging

##  1 INTRODUCTION

Detecting and characterizing the atmospheres of rocky exoplanets in the
habitable zone of solar-type stars is recognized as the main scientific goal
of NASA’s future flagship mission, the Habitable Worlds Observatory. To
achieve this immense goal, it is necessary to achieve $10^{-10}$ contrast over
20% bandwidth at separations close to the inner working angle of a 6.5 m
telescope [1]. It has been recognized that most components of high-contrast
imaging systems are close but do not yet have the technological maturity to
meet these requirements
[2]111https://science.nasa.gov/astrophysics/programs/gomap. At the heart of
these systems is the coronagraph, a combination of optical masks that reject
on-axis starlight in a region called the dark zone by many orders of magnitude
while light from off-axis sources is only slightly affected. So far, only Lyot
coronagraphs have demonstrated $4\times 10^{-10}$ in an annular dark zone at
10% bandwidth [3], and the same contrast for 20% bandwidth in a D-shaped dark
zone with larger inner working angle [4]. While this is close, a significant
push is still necessary to reach the required performance to directly image
Earth-twins in the habitable zone.
The vortex coronagraph [5] is another promising coronagraphic architecture
under study for the Habitable Worlds Observatory. This coronagraphic
architecture theoretically has a much improved performance for unobstructed
apertures. The vortex coronagraph consists of a mask with an azimuthal phase
ramp in the focal plane that diffracts the on-axis star light outside of the
pupil in the next pupil plane, where it is blocked by a Lyot stop. Two ways to
manufacture these masks in the focal plane are generally considered. First,
there is the mask based on classical phase, where the phase is accrued by
changing the optical path length, making it chromatic. Manufacturing of such a
mask can be done by etching variations in height [6] or by optimizing pillar
shapes and densities in meta-materials [7]. Because the phase is chromatic,
multiple solutions are considered to improve the chromatic behaviour. Examples
are the changing of the azimuthal phase of the mask [8, 9, 10, 11], or by
selecting pilar heights in meta-materials such that the phase ramp is the same
for a wider wavelength range, except with a overall phase offset [12, 7].
Overall, these methods that use classical phase have decent performance but
chromaticity continues to be an issue.
A second type of mask is based on the achromatic geometric phase. Geometric
phase is accrued when circularly polarized light goes through a half-wave
retarder with a rotated fast-axis [13, 14]. The geometric phase only depends
on the fast-axis orientation and the handedness of the circular polarization
coming in and is independent of wavelength. The vector vortex coronagraph
(VVC) based on the geometric phase is therefore a flat patterned half-wave
retarder where the fast-axis rotates in the azimuthal direction [15, 16]. When
the retardance is not half-wave, a fraction of the light does not acquire any
phase and is unaffected by the mask, i.e. the polarization leakage. The
retardance therefore determines the diffraction efficiency of the mask.
Polarization leakage of the VVC leads to an attenuated copy of the star on the
focal plane, resulting in a performance loss that can only be solved by
filtering one circular polarization state.
In this paper, we describe how combining multiple grating patterns with a VVC
can reduce the impact of polarization leakage by many orders of magnitude in
Section 2. Next, we describe the design and manufacturing of two prototype
triple-grating VVCs in Section 3 and characterize their performance on the In-
Air Coronagraphic Testbed at NASA/JPL and the Space Coronagraph Optical Bench
at the University of Arizona in Section 4. We discuss some manufacturing
limitations in Section 5 and present our conclusions in Section 6

##  2 Minimizing polarization leakage by with multi-grating stacks

To overcome limitation of polarization leakage and the need for leakage
filtering when using the VVC, we introduced the multi-grating concept in
Doelman et al. 2020 [17], which has already successfully been tested on-sky
with the vector Apodizing Phase plate coronagraph [17, 18].

Figure 1:  Concept of the triple-grating vector vortex coronagraph. Left: The
incoming unpolarized light is diffracted in two main directions with opposite
circular polarization, which occur equally in unpolarized light. The beams are
diffracted in the opposite direction by the next grating and back on-axis by
the third. This is only enabled by the polarization gratings diffracting
circularly polarized light with extreme efficiency ($>98\%$) into the $\pm 1$
orders. Middle: We show the main beams in red, while all leakage beams are
shown in gray. The gray scale indicates the relative intensity of the leakage
terms. Right: The simulated on-axis polarized leakage for the tgVVC created
with liquid-crystal layers with three sublayers. This idealized simulation
shows an on-axis leakage lower than $10^{-8}$ over the full visible
wavelength-range, resulting in a $10^{-10}$ intensity at the first Airy ring.
The dotted lines indicate the polarization leakage for each individual
grating. Adopted from Doelman et al. 2020 [17] Figure 2:  Conceptual
comparison of a non-perfect VVC with polarization leakage (L) with and without
filtering to two multi-element VVCs. The focal-plane images are on logarithmic
scale. We show three planets, where P1 has a contrast of $1\times 10^{-5}$,
P2: $1\times 10^{-7}$, and P3: $1\times 10^{-9}$. The two closest planets
(P2,3) are mostly hidden by the polarization leakage for the VVC. The
polarization leakage is diffracted by the double- or triple- grating patterns,
resulting in reduced polarization leakage on-axis and the appearance of P2 and
P3. Adapted from Doelman et al. 2020. [17]

The core idea is that by combining phase ramps with the VVC pattern, the
polarization leakage can be diffracted off-axis and no polarization filtering
is required. We make use of phase ramps from polarization gratings (PGs),
hence the name multi-grating. PGs are another type of geometric phase mask and
consists of a half-wave retarder with continuously rotating fast-axis in one
direction [19]. The rotating fast-axis creates the phase ramp in geometric
phase, and thus optimally separates left- and right-handed circularly
polarized light in the $\pm 1$ diffraction order. Unpolarized stellar light is
diffracted equally in both orders, while deviations from half-wave lead to a
zero-order term from the polarization leakage. The concept of combining three
gratings to make a triple-grating VVC is shown in Figure 1. By combining three
polarization gratings with different grating frequencies, f, in a 1:3:2 ratio,
we minimize the on-axis polarization leakage by diffracting most leakage off-
axis. The main beams are recombined after they are separated by the first
grating, and maintain a high throughput. Triple-grating VVCs can be
manufactured using liquid-crystal technology. Direct-write[20] is the only
method to accurately pattern the VVC and PG patterns, while using self-
aligning multi-layer stacks of liquid-crystals to achromatize the
retardance[21] results in polarization leakages $<1\%$ over a spectral range
of a few hundered nanometer. In the right panel of Figure 1 we demonstrate
that combining three gratings, each manufactured from three-layered liquid-
crystal films, can suppress the on-axis leakage by 8 orders of magnitude.
The implementation of the multi-grating VVC (mgVVC) on system level and the
comparison with a normal vortex is shown in Figure 2. A normal VVC requires
polarization filtering to reach space-based contrasts levels better than
$10^{-6}$ as the on-axis leakage PSF is dominant. For a double-grating VVC
(dgVVC) with three-layered liquid-crystal films, the polarization leakage can
be suppressed more than six orders of magnitude over a bandwidth of 20%. The
diffracted polarization leakage is blocked by the Lyot stop. Half of the
diffracted and blocked polarization leakage has been diffracted by the first
grating and leaked through the second grating, and will form a ring of fire
outside the Lyot stop. To minimize the ring of fire tail leaking through the
Lyot stop opening, the grating periods must be minimized. For the triple-
grating VVC (tgVVC), more diffraction orders are generated and these have
increased angles. Some of these will not be imaged by the lens for slow
optical systems, and should be blocked. However, the on-axis leakage is also
reduced by at least two additional orders of magnitude, as was demonstrated in
Figure 1. Ultimately, these idealized simulated performances need to be tested
and we will describe the manufacturing of the prototype next.

##  3 The triple-grating vortex prototype

The first two triple-grating vortex prototypes have been manufactured by
ImagineOptix in the second half of 2021. Their design is based on requirements
and the F-number of the NASA/JPL In-Air Coronagraphic Testbed (IACT) and
manufacturing limitations.

Figure 3:  Schematic and images of the tgVVC prototypes manufactured by
ImagineOptix in 2021. Left: The tgVVC prototype has five thin substrates in
sequence and has anti-reflection coating on the outside. The amplitude mask is
glued to the forked-grating pattern to cover the central defect. The thee
substrates with a liquid-crystal film (substrate 2,3, and 4) are ordered
keeping the 2:1 distance ratio between the gratings in mind. Right: Images of
the two prototypes labeled 1 and 2, in addition to images between crossed
polarizers.

We minimize the grating pitch within the direct-write constraints to maximize
the diffraction angle of polariziation leakage, reducing the impact of these
diffracted polarization leakage (apodized) pupils. With the selected grating
pitch of 12.65 $\mu$m for the first geometric phase hologram (GPH1) and the
IACT F-number, the pupils will be separated by at least 1.5 pupil distance at
the Lyot stop. The other grating periods follow the 1:3:2 ratio of the triple-
grating concept and their periods are presented in Table 1. All GPH patterns
are generated with 1 micron pixels and we use a charge 6 vortex to decrease
the sensitivity to low-order aberrations [22].

Table 1:  Properties of the five substrates that make the compound triple-grating vector-vortex coronagraph Substrate | Material | Diameter | Thickness | AR-coating | Description | Specification
---|---|---|---|---|---|---
1 | UVFS | 20 mm | 1 mm | BBAR-A | Amplitude mask | 10 $\mu$m dot
2 | UVFS | 25.4 mm | 1 mm | - | GPH1 | VVC6$+$PG, 12.65 $\mu$m pitch
3 | UVFS | 25.4 mm | 1 mm | - | GPH2 | PG, 4.217 $\mu$m pitch
4 | UVFS | 25.4 mm | 1 mm | - | GPH3 | PG, 6.325 $\mu$m pitch
5 | UVFS | 25.4 mm | 1 mm | BBAR-A | End cap | -

We select a single-layer liquid-crystal recipe with a half-wave retardance at
633 nm that are deposited on Thorlabs 1 inch WG41010 substrates. The triple-
gratings consist of five of these substrates, one substrate with an amplitude
dot to cover the central defect, three substrates with the patterned liquid-
crystal films, and an end-cap to cover the last liquid-crystal film. Both
outer surfaces are coated with anti-reflection coating. The full sandwich is
shown in Figure 3 and is 5.5 mm thick. Undersizing the first substrate with
the amplitude dot was necessary to accurately align the 10 $\mu$m amplitude
dot with the central defect of the first liquid-crystal film. Around four mm
was removed from the edges of the liquid-crystal films to enable glass-to-
glass bonding of the Norland NOA-61 adhesive. All three polarization gratings
are carefully aligned in clocking to overlap the main diffraction beams before
curing the adhesive. The NOA-61 epoxy was aged for 12 hours at 50 degrees
Celsius.

Figure 4:  Microscope images of the central part of the first substrate (GPH1)
with the forked-grating pattern between crossed polarizers. Left: We see
exquisite patterning without any defects and a central defect smaller than 5
$\mu$m. Right: The same area after being covered by a chromium dot with a 10
$\mu$m diameter.

Images of the tgVVC prototypes are shown in Figure 3, in addition to images of
the optics between crossed polarizers. The crossed polarizer images should
generate a vortex charge 6 pattern without any residual polarization grating
pattern in the vertical direction. We conclude that there is a slight
rotational alignement error between the gratings, resulting in this pattern.
In Section 5 we will go into manufacturing errors in more detail. Additional
quality checks of the two tgVVC prototypes have been performed by
ImagineOptix. Zero-order leakage measurements as function of wavelength show
that the retardance should be close to half-wave at 633 nm, and far-field
diffraction errors indicate the presence of additional non-standard
diffraction orders. All main diffraction orders agree with the grating pitch,
however, the non-standard diffraction orders include half-orders ($\pm 0.5,\pm
1.5$), with a separation that is between the main orders. Overall, we find
multiple clues of the presence of manufacturing errors that could impact the
performance of the optic.
Therefore, we explore the coronagraphic performance with these possible errors
in mind, while also separately look more into the manufacturing errors.
ImagineOptix has produced a third optic called the ’bonus part’ consisting of
only GPH1 and an amplitude mask, that is ideal for studying the manufacturing
errors. Crossed polarizer microscope images of the amplitude dot alignment of
this bonus part are shown in Figure 4. The images show no patterning defects
except for the central defect, which is extremely small with a diameter of
roughly $5\mu$m. The patterning quality looks good, although this is hard to
verify with these images, and the alignment of the central dot is close to
perfect.

##  4 Performance of the tgVVC prototypes

We test both tgVVC prototypes from ImagineOptix at two different in-air
coronagraphic testbeds. The testbeds are the In-Air Coronagraphic Testbed[23]
(IACT) at NASA’s JPL and the Space Coronagraphic Optical Bench (SCoOB) at the
University of Arizona Space Astrophysics Lab[24, 25] (UASAL). Both testbeds
have clear apertures and are optimized for characterizing the performance with
VVC coronagraphs. Other similarities are the use of a single deformable
mirror, the ability to filter polarization leakage, and a limited number of
optical elements to improve the performance and keep simplicity. As a result,
the wavefront control only allows for the digging of one-sided dark zones,
although this is not an issue for the prototype testing. The differences are
in the wavefront control, the use of stops in the optical system and the
F-numbers. The IACT is F/82.5 at the focal plane mask, has model-based
electric field conjugation[26, 27] (EFC) and some field stops reduce the
impact of high-order aberrations. The SCoOB testbed is is F/48 at the focal
plane mask, uses the data-driven implicit EFC[28] (iEFC) and did not have
stops when the measurements were taken. The many similarities allow us to
compare results, while the differences between the two testbeds enable us to
differentiate between testbed effects or limitations of the tgVVC.

Figure 5:  Dark hole measurements of the tgVVC1 prototype measured with
monochromatic light at 633 nm and polarization filtering in two high-contrast
imaging testbeds. Left:. The dark hole image of the tgVVC1 prototype as
measured in the In-Air Coronagraphic Testbed at NASA’s Jet Propulsion
Laboratory. The D-shaped dark hole has a size of 3-18 $\lambda/D$, light
outside this dark hole is blocked by a field stop. Model-based EFC was used to
reconstruct the wavefront correction. The average dark zone contrast from 3-14
$\lambda/D$ is $3\times 10^{-8}$. Right: The dark hole image of the tgVVC1
prototype as measured in the Space Coronagraphic Optical Bench at the
University of Arizona Space Astrophysics Lab. The D-shaped dark hole has a
size of 3-14 $\lambda/D$ and no field stop is present. Implicit EFC was used
to reconstruct the wavefront correction. The bright speckle is a ghost from
the quarter-wave plate and the average dark zone contrast without this ghost
is 6$\times 10^{-8}$.

Measurements of the second prototype, tgVVC2, on the IACT testbed revealed
that the performance is worse by a factor of a few. Therefore, no tests of the
tgVVC2 prototype were conducted on the SCoOB testbed. A likely cause for the
reduced performance is an unexpected diffracted beam at 45 degrees from the
diffraction direction with a separation less than a quarter of the first
grating order. We focus on the tgVVC1 prototype and the results at the IACT
and the SCoOB testbed is shown in Figure 5. On the IACT the dark hole is 3-18
$\lambda/D$ and has a mean contrast of $2.1\times 10^{-8}$ in this full dark
zone. The EFC was combined with the beta-bumping technique [29] to reach this
performance, with strong regularization as described in Ruane et al. 2022 [30]
There is a lot of structure in this dark hole, with a ringing structure in
three main stripes extending in the vertical direction, while the corners of
the D-shaped dark hole contain almost no speckles. The speckle structure in
the dark zone on the SCoOB testbed is completely different. The D-shaped dark
hole extends from 3-14 $\lambda/D$, contains a ghost from the quarter-wave
plate, and has a mostly constant speckle field over the full dark zone. The
SCoOB mean contrast for the tgVVC1 prototype is worse, with an average of
6$\times 10^{-8}$ after masking the ghost.
Comparing the contrast for similarly shaped dark zones of 3-14 $\lambda/D$,
gives a contrast of $3\times 10^{-8}$ for IACT compared to the 6$\times
10^{-8}$ of the SCoOB testbed, a difference of a factor of two. With the minor
contrast differences and extreme differences in the speckle field, a direct
comparison has interesting implications. The vertical structures in the IACT
dark zone are brighter than the speckles of the SCoOB dark zone at the same
location. Therefore, it would seem that these structures are not incoherent
structures generated by the tgVVC1 optic, but rather not controlled as
consequence of model inaccuracies of the EFC algorithm. As the EFC model is
based on ideal vortex coronagraphic patterns, some differences from
manufacturing or patterning inaccuracies might make it difficult to control
these speckles with EFC. Because iEFC is empirically calibrated, it is robust
against these situations where optical models might be difficult to realize.
To summarize, the structured IACT speckle field suggests that the tgVVC1 has
some manufacturing errors, although these do not have to impact the contrast
performance.

##  5 Discussion

We present the manufactured a triple-grating vector vortex coronagraph
prototype and its performance in two high-contrast imaging testbeds. While the
contrast does reach a contrast level that is expected for a first prototype on
in-air testbeds, we find that there are clear signs of manufacturing errors.
First, the crossed polarizer images show residual grating patterns from
imperfect grating clocking alignment. Second, polarization leakage filtering
was necessary to reach the reported contrast levels. This could point to
either pattern writing errors resulting in an increased zero-order leakage or
deviations from half-wave retardance. Third, the dark hole structures in the
IACT are a likely result of a model mismatch with a standard vortex pattern,
also a sign of pattern writing errors. Lastly, the far-field diffraction
reported by ImagineOptix shows the presence of half-order diffraction. These
half-orders are also generated when the grating pattern has writing errors.
Overall, these errors warrant a closer inspection if we are to make improved
tgVVC prototypes. Further study of the diffraction efficiency, retardance, and
patterning accuracy with far-field diffraction measurements and imaging
Mueller matrix ellipsometry of the Bonus part is required.

##  6 Conclusion

  * •

The multi-grating concept can suppress polarization leakage terms by many
orders of magnitude by combining phase patterns with polarization gratings.

  * •

We apply this concept to the vector vortex coronagraph and successfully
manufacture two triple-grating prototypes.

  * •

We test these triple-grating vector vortex coronagraphs in two high-contrast
imaging testbeds and demonstrate good performance.

  * •

On the In-Air Coronagraphic Testbed at NASA’s Jet Propulsion Laboratory we
report a contrast of $2.1\times 10^{-8}$ in a D-shaped dark hole of 3-18
$\lambda/D$.

  * •

On the Space Coronagraphic Optical Bench at the University of Arizona Space
Astrophysics Lab we report a contrast of $6\times 10^{-8}$ in a D-shaped dark
hole of 3-14 $\lambda/D$.

  * •

A comparison between these two dark holes suggests that the IACT performance
was limited by EFC model errors.

  * •

We find that there are clear signs of manufacturing errors of the prototype
that can easily be mitigated by more controlled manufacturing. We will do this
in collaboration with ColorLink Japan, ltd.

###### Acknowledgements.

We are very grateful for support from the Netherlands Space Office.

## References

  * [1] National Academies of Sciences, Engineering, and Medicine, [Pathways to Discovery in Astronomy and Astrophysics for the 2020s ], The National Academies Press, Washington, DC (2021).
  * [2] Clery, D., “Future nasa scope would find life on alien worlds,” Science 379(6628), 123–124 (2023).
  * [3] Seo, B.-J., Patterson, K., Balasubramanian, K., Crill, B., Chui, T., Echeverri, D., Kern, B., Marx, D., Moody, D., Prada, C. M., et al., “Testbed demonstration of high-contrast coronagraph imaging in search for earth-like exoplanets,” in [Techniques and Instrumentation for Detection of Exoplanets IX ], 11117, 599–609, SPIE (2019).
  * [4] Allan, G. W., Riggs, A. E., Ruane, G., Mejia Prada, C., Noyes, M., Poon, P. K., Jovanovic, N., Llop-Sayson, J. D., Coker, C., Stark, C. C., Mennesson, B., and Mawet, D., “Demonstration of coronagraph technology for high-contrast point spectroscopy of exoearths,” in [Techniques and Instrumentation for Detection of Exoplanets XI ], 12680, SPIE (2023).
  * [5] Foo, G., Palacios, D. M., Swartzlander Jr, G. A., and Swartzlander, G. A., “Optical vortex coronagraph,” Optics letters 30(24), 3308–3310 (2005).
  * [6] Lee, J. H., Foo, G., Johnson, E. G., and Swartzlander Jr, G. A., “Experimental verification of an optical vortex coronagraph,” Physical review letters 97(5), 053901 (2006).
  * [7] Konig, L., Palatnick, S., Desai, N., Absil, O., Millar-Blanchaer, M., and Mawet, D., “Metasurface-based scalar vortex phase mask in pursuit of 1e-10 contrast,” in [Techniques and Instrumentation for Detection of Exoplanets XI ], 12680, SPIE (2023).
  * [8] Ruane, G., Mawet, D., Riggs, A. E., and Serabyn, E., “Scalar vortex coronagraph mask design and predicted performance,” in [Techniques and Instrumentation for Detection of Exoplanets IX ], 11117, 454–469, SPIE (2019).
  * [9] Galicher, R., Huby, E., Baudoz, P., and Dupuis, O., “A family of phase masks for broadband coronagraphy example of the wrapped vortex phase mask theory and laboratory demonstration,” Astronomy & Astrophysics 635, A11 (2020).
  * [10] Desai, N., Bertrou-Cantou, A., Llop-Sayson, J. D., Ruane, G., Riggs, A. E., Serabyn, E., and Mawet, D., “Achromatizing scalar vortex coronagraphs with radial phase mask dimples,” in [Techniques and Instrumentation for Detection of Exoplanets XI ], 12680, SPIE (2023).
  * [11] Desai, N., Ruane, G., Llop-Sayson, J., Betrou-Cantou, A., Potier, A., Riggs, A. E., Serabyn, E., and Mawet, D., “Laboratory demonstration of the wrapped staircase scalar vortex coronagraph,” Journal of Astronomical Telescopes, Instruments, and Systems 9(2), 025001–025001 (2023).
  * [12] Palatnick, S., Konig, L., Millar-Blanchaer, M., Wallace, J. K., Echeverri, D., Desai, N., Absil, O., and Mawet, D., “Imaging exoplanets with metasurfaces: current prospects and the road to terrestrial planet contrasts,” in [Techniques and Instrumentation for Detection of Exoplanets XI ], 12680, SPIE (2023).
  * [13] Pancharatnam, S., “Generalized theory of interference, and its applications,” in [Proceedings of the Indian Academy of Sciences-Section A ], 44(5), 247–262, Springer (1956).
  * [14] Berry, M., “The Adiabatic Phase and Pancharatnam’s Phase for Polarized Light,” Journal of Modern Optics 34(11), 1401–1407 (1987).
  * [15] Mawet, D., Riaud, P., Absil, O., and Surdej, J., “Annular groove phase mask coronagraph,” The Astrophysical Journal 633(2), 1191 (2005).
  * [16] Mawet, D., Serabyn, E., Liewer, K., Hanot, C., McEldowney, S., Shemo, D., and O?Brien, N., “Optical vectorial vortex coronagraphs using liquid crystal polymers: theory, manufacturing and laboratory demonstration,” Optics Express 17(3), 1902–1918 (2009).
  * [17] Doelman, D. S., Por, E. H., Ruane, G., Escuti, M. J., and Snik, F., “Minimizing the polarization leakage of geometric-phase coronagraphs with multiple grating pattern combinations,” Publications of the Astronomical Society of the Pacific 132(1010), 045002 (2020).
  * [18] Doelman, D., Por, E., Otten, G., Kenworthy, M., et al., “Vector-apodizing phase plate coronagraph: design, current performance, and future development,” Applied optics 60(19), D52–D72 (2021).
  * [19] Oh, C. and Escuti, M. J., “Achromatic diffraction from polarization gratings with high efficiency: erratum,” Optics Letters 34(23), 3637 (2009).
  * [20] Miskiewicz, M. N. and Escuti, M. J., “Direct-writing of complex liquid crystal patterns,” Optics Express 22(10), 12691 (2014).
  * [21] Komanduri, R. K., Lawler, K. F., and Escuti, M. J., “Multi-twist retarders: broadband retardation control using self-aligning reactive liquid crystal layers,” Optics Express 21(1), 404 (2013).
  * [22] Ruane, G., Mawet, D., Mennesson, B., Jewell, J., and Shaklan, S., “Vortex coronagraphs for the habitable exoplanet imaging mission concept: theoretical performance and telescope requirements,” Journal of Astronomical Telescopes, Instruments, and Systems 4(1), 015004–015004 (2018).
  * [23] Baxter, W., Potier, A., Ruane, G., and Prada, C. M., “Design and commissioning of an in-air coronagraph testbed in the hcit facility at nasa’s jet propulsion laboratory,” in [Techniques and Instrumentation for Detection of Exoplanets X ], 11823, 574–585, SPIE (2021).
  * [24] Ashcraft, J. N., Choi, H., Douglas, E. S., Derby, K., Van Gorkom, K., Kim, D., Anche, R., Carter, A., Durney, O., Haffert, S., et al., “The space coronagraph optical bench (scoob): 1. design and assembly of a vacuum-compatible coronagraph testbed for spaceborne high-contrast imaging technology,” in [Space Telescopes and Instrumentation 2022: Optical, Infrared, and Millimeter Wave ], 12180, 2040–2053, SPIE (2022).
  * [25] Van Gorkom, K., Douglas, E. S., Ashcraft, J. N., Haffert, S., Kim, D., Choi, H., Anche, R. M., Males, J. R., Milani, K., Derby, K., et al., “The space coronagraph optical bench (scoob): 2. wavefront sensing and control in a vacuum-compatible coronagraph testbed for spaceborne high-contrast imaging technology,” in [Space Telescopes and Instrumentation 2022: Optical, Infrared, and Millimeter Wave ], 12180, 2054–2061, SPIE (2022).
  * [26] Give’on, A., Kern, B., Shaklan, S., Moody, D. C., and Pueyo, L., “Electric field conjugation-a broadband wavefront correction algorithm for high-contrast imaging systems,” in [American Astronomical Society Meeting Abstracts ], 211, 135–20 (2007).
  * [27] Groff, T. D., Eldorado Riggs, A., Kern, B., and Jeremy Kasdin, N., “Methods and limitations of focal plane sensing, estimation, and control in high-contrast imaging,” Journal of Astronomical Telescopes, Instruments, and Systems 2(1), 011009–011009 (2016).
  * [28] Haffert, S., Males, J., Ahn, K., Van Gorkom, K., Guyon, O., Close, L., Long, J., Hedglen, A., Schatz, L., Kautz, M., et al., “Implicit electric field conjugation: Data-driven focal plane control,” arXiv preprint arXiv:2303.13719 (2023).
  * [29] Seo, B.-J., Cady, E., Gordon, B., Kern, B., Lam, R., Marx, D., Moody, D., Muller, R., Patterson, K., Poberezhskiy, I., et al., “Hybrid lyot coronagraph for wfirst: high-contrast broadband testbed demonstration,” in [Techniques and Instrumentation for Detection of Exoplanets VIII ], 10400, 106–126, SPIE (2017).
  * [30] Ruane, G., Riggs, A. E., Serabyn, E., Baxter, W., Prada, C. M., Mawet, D., Noyes, M., Poon, P. K., and Tabiryan, N., “Broadband vector vortex coronagraph testing at nasa’s high contrast imaging testbed facility,” in [Space Telescopes and Instrumentation 2022: Optical, Infrared, and Millimeter Wave ], 12180, 769–782, SPIE (2022).


utf8

# Dialect Identification in Nuanced Arabic Tweets Using Farasa Segmentation
and AraBERT

Anshul Wadhawan
Flipkart Private Limited
<EMAIL_ADDRESS>
###### Abstract

This paper presents our approach to address the EACL WANLP-2021 Shared Task 1:
Nuanced Arabic Dialect Identification (NADI). The task is aimed at developing
a system that identifies the geographical location(country/province) from
where an Arabic tweet in the form of modern standard Arabic or dialect comes
from. We solve the task in two parts. The first part involves pre-processing
the provided dataset by cleaning, adding and segmenting various parts of the
text. This is followed by carrying out experiments with different versions of
two Transformer based models, AraBERT and AraELECTRA. Our final approach
achieved macro F1-scores of 0.216, 0.235, 0.054, and 0.043 in the four
subtasks, and we were ranked second in MSA identification subtasks and fourth
in DA identification subtasks.

##  1 Introduction

Spoken by about 500 million people around the world, Arabic is the biggest
part of the Semitic language family. Being the official language of almost 22
countries belonging to the Middle-East North Africa (MENA) region, it is not
only an integral member of the six official UN languages, but also fourth most
used language on the Internet Guellil et al. (2019). Middle East contributes
to 164 million internet users and North Africa contributes to 121 million
internet users. Comparing with other languages, Arabic language has received
little attention in modern computational linguistics, despite its religious,
political and cultural significance. However, with rapid development of tools
and techniques delivering state-of-the-art performance in many language
processing tasks, this negligence is being taken care of.

The presence of various dialects and complex morphology are some of the
distinguishing factors prominent in the Arabic language. Also, the informal
nature of conversations on social media and the differences in Modern Standard
Arabic (MSA) and Dialectical Arabic (DA), both significantly increase this
complexity. While DA is used for informal daily communication, MSA is used for
formal writing. Social media is the home for both of these forms, with the
former being the most common form. Lack of data is the primary reason why many
of the Arabic dialects remain understudied. With the availability of diverse
data belonging to 21 Arab countries, this bottleneck can be diminished. The
Nuanced Arabic Dialect Identification (NADI), with this goal, is the task of
automatic detection of the source variety of a given text or speech segment.

Previously, on the lines of Arabic dialect identification, there have been
approaches focusing on coarse-grained regional varieties such as Levantine or
Gulf Elaraby and Abdul-Mageed (2018); Zaidan and Callison-Burch (2014);
Elfardy and Diab (2013) or country level varieties Bouamor et al. (2019);
Zhang and Abdul-Mageed (2019). There have been tasks that involved city level
classification on human translated data Salameh et al. (2018). Some tasks have
focused on country and province level classification simultaneously Abdul-
Mageed et al. (2020).

In this paper, we present our process to tackle the WANLP-2021 Shared Task 1.
The paper is organised in the following way: Section 2 presents the problem
statement and details of the provided dataset. Section 3 describes a
modularised process that we inculcate as part of methodology. Section 4
describes the experiments that were conducted, with detailed statistics about
the dataset, system settings and results of these experiments. A brief
conclusion of the paper with the potential prospects of our study are
presented in Section 5.

##  2 Task Definition

The WANLP-2021 Shared Task 1 Abdul-Mageed et al. (2021) is based on a multi-
class classification problem where the aim is to recognize which country or
province an Arabic tweet in the form of modern standard Arabic or dialect
belongs to. The task targets dialects at the province-level, and also focuses
on naturally-occurring fine-grained dialects at the sub-country level. The
NADI 2021 task promotes efforts made towards distinguishing both modern
standard Arabic (MSA) and dialects (DA) according to their geographical
origin, focusing on fine-grained dialects with new datasets. The provided data
comes from the domain of Twitter and covers 100 provinces from 21 Arab
countries. The task is divided into 4 subtasks as described below:

Subtask 1.1: Country-level MSA identification

Subtask 1.2: Country-level DA identification

Subtask 2.1: Province-level MSA identification

Subtask 2.2: Province-level DA identification

The training dataset has a total of 21,000 tweet, validation and test datasets
have 5,000 tweets each. Every example belongs to one of 100 provinces of 21
Arab countries. Additional 10M unlabeled tweets are provided that can be used
in developing the systems for either or both of the tasks. F-score, Accuracy,
Precision and Recall are the evaluation metrics. However, the official metric
of evaluation is the Macro Averaged F-score.

##  3 Methodology

We present our methodology in two parts. The first part in the methodology is
data preprocessing. This is followed by experimenting with different
transformer based models for the task at hand. Both these parts have been
described in detail in the following sub sections.

###  3.1 Data Pre-Processing

Transformer based models, that we plan to fine tune on our dataset, are pre-
trained on processed rather than raw data. Owing to the variations in
expression of opinions among users belonging to different parts of the world,
the tweets fetched from the website are a clear representation of these
variations. We find these variations on randomly checking the given examples
in different forms. It is common for users to use slang words on the Twitter
platform, and post non-ascii characters like emojis. Also, spelling errors,
user mentions and URLs are prominent in tweets of most users. These parts
within the tweets do not contribute to being informative towards deciding the
geographical location of the tweet as they correspond to noise. Thus, the
given dataset is cleaned in the following ways, so that the data used for fine
tuning has a similar distribution to that used for the pre-training process:

  1. 1.

Perform Farasa segmentation (for select models only) Abdelali et al. (2016).

  2. 2.

Replace all URLs with [ رابط ¿ ], emails with [ بريد ¿ ], mentions with [
مستخدم ¿ ].

  3. 3.

Remove HTML line breaks and markup, unwanted characters like emoticons,
repeated characters ($>$ 2) and extra spaces.

  4. 4.

Insert whitespace before and after all non Arabic digits or English Digits and
Alphabet and the 2 brackets, and between words and numbers or numbers and
words.

###  3.2 Transformer Based Models

The domains of speech recognition Graves et al. (2013) and computer vision
Krizhevsky et al. (2012) have largely utilised different deep learning
techniques and produced significant improvements over the traditional machine
learning techniques. In the domain of natural language processing, most deep
learning based techniques until now utilised word vector representations
Bengio et al. (2003); Yih et al. (2011); Mikolov et al. (2013) for different
classification tasks. Lately, transformer based approaches have shown
significant progress towards many NLP benchmarks Vaswani et al. (2017),
including text classification Chang et al. (2020), owing to their ability to
build proficient language models. As an output of the pre-training process,
embeddings are produced which are utilised for finer tasks.

Model | Size | Pre-Segmentation | Dataset
---|---|---|---
MB | Params | #Sentences | Size | #Words
AraBERTv0.2-base | 543MB | 136M | No | 200M | 77GB | 8.6B
AraBERTv0.2-large | 1.38G | 371M | No | 200M | 77GB | 8.6B
AraBERTv2-base | 543MB | 136M | Yes | 200M | 77GB | 8.6B
AraBERTv2-large | 1.38G | 371M | Yes | 200M | 77GB | 8.6B
AraBERTv0.1-base | 543MB | 136M | No | 77M | 23GB | 2.7B
AraBERTv1-base | 543MB | 136M | Yes | 77M | 23GB | 2.7B
Table 1: Model Pre-training Parameters

####  3.2.1 AraBERT

AraBERT is an Arabic pretrained language model based on Google’s BERT
architecture Antoun et al. . There are six versions of the model:
AraBERTv0.1-base, AraBERTv0.2-base, AraBERTv0.2-large, AraBERTv1-base,
AraBERTv2-base and AraBERTv2-large. For these variations, the model parameters
with respect to the pre-training process have been depicted in Table 1.

####  3.2.2 AraELECTRA

Being a method for self-supervised language representation learning, ELECTRA
has the ability of making use of lesser computations for the task of pre-
training transformers Antoun et al. (2020). Similar to the objective of
discriminator of a Generative Adversarial Network, ELECTRA models are trained
with the goal of distinguishing fake input tokens from the real ones. On the
Arabic QA dataset, AraELECTRA achieves state-of-the-art results.

For all new AraBERT and AraELECTRA models, the same pretraining data is used.
The dataset that is used for pre-training, before the application of Farasa
Segmentation, has a total of 82,232,988,358 characters or 8,655,948,860 words
or 200,095,961 lines, and has a size of 77GB. Initially, several websites like
OSCAR unshuffled and filtered, Assafir news articles, Arabic Wikipedia dump
from 2020/09/01, The OSIAN Corpus and The 1.5B words Arabic Corpus, were
crawled to create the pre-training dataset. Later, unshuffled OSCAR corpus,
after thorough filtering, was added to the previous dataset used in AraBERTv1
without including the data from the above mentioned crawled websites to create
the new dataset.

Country | DA | MSA
---|---|---
Train | Dev | Train | Dev
Algeria | 1809 | 430 | 1899 | 427
Bahrain | 215 | 52 | 211 | 51
Djibouti | 215 | 27 | 211 | 52
Egypt | 4283 | 1041 | 4220 | 1032
Iraq | 2729 | 664 | 2719 | 671
Jordan | 429 | 104 | 422 | 103
Kuwait | 429 | 105 | 422 | 103
Lebanon | 644 | 157 | 633 | 155
Libya | 1286 | 314 | 1266 | 310
Mauritania | 215 | 53 | 211 | 52
Morocco | 858 | 207 | 844 | 207
Oman | 1501 | 355 | 1477 | 341
Palestine | 428 | 104 | 422 | 102
Qatar | 215 | 52 | 211 | 52
Saudi_Arabia | 2140 | 520 | 2110 | 510
Somalia | 172 | 49 | 346 | 63
Sudan | 215 | 53 | 211 | 48
Syria | 1287 | 278 | 1266 | 309
Tunisia | 859 | 173 | 844 | 170
UAE | 642 | 157 | 633 | 154
Yemen | 429 | 105 | 422 | 88
Table 2: Country Level Data Distribution Parameter | Value
---|---
Learning Rate | 1e-5
Epsilon (Adam optimizer) | 1e-8
Maximum Sequence Length | 256
Batch Size (for base models) | 40
Batch Size (for large models) | 4
#Epochs | 5
Table 3: Parameter Values Model | Subtask 1.1 | Subtask 1.2 | Subtask 2.1 | Subtask 2.2
---|---|---|---|---
F1 | A | F1 | A | F1 | A | F1 | A
AraBERTv0.1-base | 0.283 | 0.324 | 0.338 | 0.390 | 0.024 | 0.028 | 0.025 | 0.037
AraBERTv0.2-base | 0.300 | 0.344 | 0.382 | 0.427 | 0.038 | 0.042 | 0.035 | 0.051
AraBERTv0.2-large | 0.304 | 0.343 | 0.362 | 0.413 | 0.022 | 0.030 | 0.029 | 0.041
AraBERTv1-base | 0.281 | 0.318 | 0.306 | 0.377 | 0.032 | 0.040 | 0.019 | 0.033
AraBERTv2-base | 0.309 | 0.347 | 0.389 | 0.432 | 0.029 | 0.038 | 0.034 | 0.048
AraBERTv2-large | 0.315 | 0.346 | 0.416 | 0.450 | 0.001 | 0.010 | 0.001 | 0.010
AraELECTRA-base-generator | 0.106 | 0.231 | 0.165 | 0.285 | 0.005 | 0.018 | 0.006 | 0.022
AraELECTRA-base-discriminator | 0.192 | 0.281 | 0.280 | 0.375 | 0.007 | 0.020 | 0.006 | 0.026
Table 4: Validation Set Results | M-F1 | A | P | R
---|---|---|---|---
Subtask 1.1 | 0.216 | 0.317 | 0.321 | 0.189
Subtask 1.2 | 0.235 | 0.433 | 0.280 | 0.233
Subtask 2.1 | 0.054 | 0.060 | 0.061 | 0.060
Subtask 2.2 | 0.043 | 0.053 | 0.044 | 0.051
Table 5: Test Set Results

##  4 Experiments

We experiment with eight transformer based models using the given training and
validation sets. We calculate the final test predictions by fine tuning the
most efficient model, which is decided by the scores produced above, with the
concatenated labeled training and validation splits. This is followed by
evaluating the test set on this fine tuned model. This section presents the
Country-level dataset distribution, system settings, results of our research
followed by a descriptive analysis of our system.

###  4.1 Dataset

The country-wise distribution of the provided training and validation splits,
for both the tasks of MSA and DA, are shown in Table 2.

###  4.2 System Settings

We make use of pre-trained AraBERT and AraELECTRA models, with the names of
bert-base-arabert, bert-base-arabertv01, bert-large-arabertv2, bert-base-
arabertv2, bert-large-arabertv02, bert-base-arabertv02, araelectra-base-
generator and araelectra-base-discriminator for fine-tuning the transformer
based models. We use hugging-face111https://huggingface.co/transformers/ API
to fetch the pre-trained transformer based models, and then fine tuned the
same on our dataset. The hyper parameters used for fine tuning these models
have been specified in Table 3.

###  4.3 Results and Analysis

For all subtasks, the performance results of proposed models on the provided
validation set with reference to accuracy(A) and weighted F1 scores(F1) are
shown in Table 4.

From Table 4, we conclude that:

  1. 1.

For most of the subtasks, one of the base models performs almost as good as
the best performing large model.

  2. 2.

AraELECTRA models seem to perform worse than all AraBERT models, possibly due
to their specialization in handling GAN related tasks, which are different
from classification based tasks.

  3. 3.

AraBERTv2-large out performs all other models for subtasks 1.1 and 1.2. For
subtasks 2.1 and 2.2, AraBERTv0.2-base produces the best results on the
validation set.

From the above results, we choose AraBERTv2-large for subtasks 1.1, 1.2 and
AraBERTv0.2-base for subtasks 2.1, 2.2 to be the primary models to fine tune
on the concatenated training and validation set as well as carry out
inferences on the unseen dataset. The final test set results in terms of Macro
F1 Score(M-F1), Recall(R), Accuracy(A) and Precision(P) are specified in Table
5.

##  5 Conclusion and Future Work

In this paper, we present a comprehensive overview of the approach that we
employed to solve the EACL WANLP-2021 Shared Task 1. We tackle the given
problem in two parts. The first part involves pre processing the given data by
modifying various parts of the text. The second part involves experimenting
with different versions of two Transformer based networks, AraBERT and
AraELECTRA, all pre-trained on Arabic text. Our final submissions for the four
subtasks are based on the best performing version of AraBERT model. With Macro
Averaged F1-Score as the final evaluation criteria, our approach fetches a
private leaderboard rank of 2 for MSA identification and 4 for DA
identification. In the future, we aim to utilise other features relevant for
classification tasks like URLs, emoticons, and experiment with ensembles of
transformer based and word vector based input representations.

## References

  * Abdelali et al. (2016) Ahmed Abdelali, Kareem Darwish, Nadir Durrani, and Hamdy Mubarak. 2016.  Farasa: A fast and furious segmenter for Arabic.  In _Proceedings of the 2016 Conference of the North American Chapter of the Association for Computational Linguistics: Demonstrations_ , pages 11–16, San Diego, California. Association for Computational Linguistics.
  * Abdul-Mageed et al. (2020) Muhammad Abdul-Mageed, Chiyu Zhang, Houda Bouamor, and Nizar Habash. 2020.  NADI 2020: The First Nuanced Arabic Dialect Identification Shared Task.  In _Proceedings of the Fifth Arabic Natural Language Processing Workshop (WANLP 2020)_ , Barcelona, Spain.
  * Abdul-Mageed et al. (2021) Muhammad Abdul-Mageed, Chiyu Zhang, AbdelRahim Elmadany, Houda Bouamor, and Nizar Habash. 2021.  NADI 2021: The Second Nuanced Arabic Dialect Identification Shared Task.  In _Proceedings of the Sixth Arabic Natural Language Processing Workshop (WANLP 2021)_.
  * (4) Wissam Antoun, Fady Baly, and Hazem Hajj.  Arabert: Transformer-based model for arabic language understanding.  In _LREC 2020 Workshop Language Resources and Evaluation Conference 11–16 May 2020_ , page 9.
  * Antoun et al. (2020) Wissam Antoun, Fady Baly, and Hazem Hajj. 2020.  Araelectra: Pre-training text discriminators for arabic language understanding.
  * Bengio et al. (2003) Yoshua Bengio, Réjean Ducharme, Pascal Vincent, and Christian Janvin. 2003.  A neural probabilistic language model.  _J. Mach. Learn. Res._ , 3(null):1137–1155.
  * Bouamor et al. (2019) Houda Bouamor, Sabit Hassan, and Nizar Habash. 2019.  The madar shared task on arabic fine-grained dialect identification.  In _Proceedings of the Fourth Arabic Natural Language Processing Workshop_ , pages 199–207.
  * Chang et al. (2020) Wei-Cheng Chang, Hsiang-Fu Yu, Kai Zhong, Yiming Yang, and Inderjit Dhillon. 2020\.  Taming pretrained transformers for extreme multi-label text classification.
  * Elaraby and Abdul-Mageed (2018) Mohamed Elaraby and Muhammad Abdul-Mageed. 2018.  Deep models for Arabic dialect identification on benchmarked data.  In _Proceedings of the Fifth Workshop on NLP for Similar Languages, Varieties and Dialects (VarDial 2018)_ , pages 263–274, Santa Fe, New Mexico, USA. Association for Computational Linguistics.
  * Elfardy and Diab (2013) Heba Elfardy and Mona Diab. 2013.  Sentence level dialect identification in Arabic.  In _Proceedings of the 51st Annual Meeting of the Association for Computational Linguistics (Volume 2: Short Papers)_ , pages 456–461, Sofia, Bulgaria. Association for Computational Linguistics.
  * Graves et al. (2013) Alex Graves, Abdel rahman Mohamed, and Geoffrey Hinton. 2013.  Speech recognition with deep recurrent neural networks.
  * Guellil et al. (2019) Imane Guellil, Houda Saâdane, Faical Azouaou, Billel Gueni, and Damien Nouvel. 2019\.  Arabic natural language processing: An overview.  _Journal of King Saud University - Computer and Information Sciences_.
  * Krizhevsky et al. (2012) Alex Krizhevsky, Ilya Sutskever, and Geoffrey E. Hinton. 2012.  Imagenet classification with deep convolutional neural networks.  In _Proceedings of the 25th International Conference on Neural Information Processing Systems - Volume 1_ , NIPS’12, page 1097–1105, Red Hook, NY, USA. Curran Associates Inc.
  * Mikolov et al. (2013) Tomas Mikolov, Ilya Sutskever, Kai Chen, Greg Corrado, and Jeffrey Dean. 2013.  Distributed representations of words and phrases and their compositionality.  In _Proceedings of the 26th International Conference on Neural Information Processing Systems - Volume 2_ , NIPS’13, page 3111–3119, Red Hook, NY, USA. Curran Associates Inc.
  * Salameh et al. (2018) Mohammad Salameh, Houda Bouamor, and Nizar Habash. 2018.  Fine-grained arabic dialect identification.  In _Proceedings of the 27th International Conference on Computational Linguistics_ , pages 1332–1344.
  * Vaswani et al. (2017) Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N. Gomez, Lukasz Kaiser, and Illia Polosukhin. 2017.  Attention is all you need.
  * Yih et al. (2011) Wen-tau Yih, Kristina Toutanova, John C. Platt, and Christopher Meek. 2011.  Learning discriminative projections for text similarity measures.  In _Proceedings of the Fifteenth Conference on Computational Natural Language Learning_ , pages 247–256, Portland, Oregon, USA. Association for Computational Linguistics.
  * Zaidan and Callison-Burch (2014) Omar F. Zaidan and Chris Callison-Burch. 2014.  Arabic dialect identification.  _Computational Linguistics_ , 40(1):171–202.
  * Zhang and Abdul-Mageed (2019) Chiyu Zhang and Muhammad Abdul-Mageed. 2019.  No army, no navy: BERT semi-supervised learning of Arabic dialects.  In _Proceedings of the Fourth Arabic Natural Language Processing Workshop_ , pages 279–284, Florence, Italy. Association for Computational Linguistics.


date author

# Entanglement Entropy in Quantum Mechanics: An Algebraic Approach

A.F. Reyes-Lega
Departamento de Física  Universidad de los Andes
A.A. 4976-12340  Bogotá  Colombia
<EMAIL_ADDRESS>
###### Abstract

An algebraic approach to the study of entanglement entropy of quantum systems
is reviewed. Starting with a state on a $C^{*}$-algebra, one can construct a
density operator describing the state in the GNS representation state.
Applications of this approach to the study of entanglement measures for
systems of identical particles are outlined. The ambiguities in the definition
of entropy within this approach are then related to the action of unitaries in
the commutant of the representation and their relation to modular theory
explained.

##  1 Introduction

Entanglement entropy has increasingly become an important concept in physics.
From black hole physics [1, 2] to conformal field theory and statistical
mechanics [3], to quantum information theory [4], its prominent role cannot be
overemphasized. In the particular case of bipartite quantum systems described
by a tensor product structure of the underlying Hilbert space, the
information-theoretic content of entanglement entropy is well known [5]. In
the case of entanglement measures for systems of identical particles there has
been an intense debate during the last years [6]. In fact, many proposals have
been put forward which try to provide sensible definitions of entanglement
measures which take into account the fact that a separation of such a system
into subsystems does not necessarily entail a factorization of the Hilbert
space into a tensor product. It is the aim of this chapter to review the basic
aspects of an approach to this problem that I have pursued for some time,
together with A.P. Balachandran and collaborators [7, 8, 9, 10].

##  2 Observables and States

Quite generally, it is possible to say that any sensible physical theory
should at least provide a definition of what is to be understood as
_observable_ and what is supposed to be regarded as a _state_ of a given
system. Results of measurements, then can be thought of as the result of a
“pairing” between observables and states.

As an example, let us consider the Hamiltonian description of a classical
system of $n<\infty$ degrees of freedom. Assuming a phase space of the form
$T^{*}Q$, with $Q$ the configuration space manifold, we can use local
coordinates for position ($q^{i}$) and momentum ($p_{i}$). Time evolution is
given by the solution $(q(t),p(t))$ of Hamilton’s equations for a given
Hamiltonian $H(q,p)$, which is a smooth function on phase space, $H\in
C^{\infty}(T^{*}Q)$. In this context we say that the state of the system at a
given time $t_{0}$ is given by the point $(q(t_{0}),p(t_{0}))\in T^{*}Q$. We
are then led to think of points in $T^{*}Q$ as possible states of the system.
On the other hand, observable quantities like energy, momentum, _etc._ , are
given by smooth functions on phase space. The “pairing” we alluded to above is
given, in this case, by the evaluation of an observable $f\in
C^{\infty}(T^{*}Q)$ at the point $(q_{0},p_{0})$ specified by the state. If
the number of degrees of freedom becomes significantly large (as is the case
in classical statistical mechanics) a statistical approach becomes necessary.
In that case, the state of the system is given by a probability distribution
$\rho(q,p)$ and the pairing is now given by the evaluation of the expectation
value of $f(q,p)$,

$\langle f\rangle=\int f(q,p)\rho(q,p)d\mu,$ (1)

where $\mu$ is the integration (Liouville) measure. Notice that this is a more
general notion of state. We can use it to describe, say, the canonical
ensemble, for which $\rho$ is given (up to a normalization factor) by
$\exp(-H(q,p)/k_{\mathrm{B}}T)$. But we can also recover our initial
definition of state, by assigning a Dirac delta distribution to the point
$(q_{0},p_{0})$:

$\rho(q,p)=\delta(q-q_{0},p-p_{0}).$ (2)

In summary, we have the following

  * •

_Observables_ are elements of the algebra $C^{\infty}(T^{*}Q)$.

  * •

_States_ are positive, normalized functionals of the form

$\displaystyle\omega_{\rho}\mathrel{\mathop{\mathchar
58\relax}}C^{\infty}(T^{*}Q)$ $\displaystyle\longrightarrow\;\mathbb{R}$
$\displaystyle f\qquad$ $\displaystyle\longmapsto\omega_{\rho}(f)=\int
f(q,p)\rho(q,p)d\mu.$ (3)

One of the reasons why the algebraic approach to quantum physics is so useful
in order to elucidate the probabilistic structure of quantum theory and its
relation to its classical counterpart is that the pairing between observables
and states runs, essentially, in parallel to what we have just described in
the above example. The difference lies in the types of algebras considered
which, as a rule, are non-commutative ones in quantum theory. Positivity and
normalization of states (defined as functionals with those properties) allow
one to construct probability distributions. Following the analogy naturally
provided by Gelfand duality, the similarities and (most importantly) the
differences between classical and quantum physics are made very clear (the
reader may consult [11] for details).

In quantum physics, $C^{*}$-algebras and von Neumann algebras are the two
kinds of algebras which are most relevant. Since every von Neumann algebra is,
in particular, a $C^{*}$-algebra and since in the finite-dimensional case (the
one we are interested in this chapter) the distinction disappears, we will
only consider $C^{*}$-algebras. Let us remark, though, that in (local) quantum
field theory, it is essential to take into account those properties that are
particular to von Neumann algebras [12].

######  Definition 1.

A $C^{*}$-algebra $(\mathcal{A},\|\cdot\|,*)$ is an involutive Banach algebra
satisfying the following compatibility condition between the involution “$*$”
and the norm “$\|\cdot\|$”:

$\|aa^{*}\|=\|a\|^{2},\qquad\forall a\in\mathcal{A}.$ (4)

######  Definition 2.

A state $\omega$ on a (unital) $C^{*}$-algebra $(\mathcal{A},\|\cdot\|,*)$ is
a positive, normalized linear functional $\omega\mathrel{\mathop{\mathchar
58\relax}}\mathcal{A}\rightarrow\mathbb{C}$.

A fundamental result due to Gelfand, Naimark and Segal (the GNS construction
[12, 13]) allows one to associate a representation of any $C^{*}$-algebra
$\mathcal{A}$ (by bounded operators on a Hilbert space) to each state
$\omega\mathrel{\mathop{\mathchar 58\relax}}\mathcal{A}\rightarrow\mathbb{C}$.
The main idea behind the construction is to take advantage of the vector space
structure of $\mathcal{A}$ and to use the product in order to define an action
of $\mathcal{A}$ (as an algebra) on itself (but this time regarded as a vector
space). In order to endow this vector space with an inner product, one makes
use of the fact that $\omega(a^{*}b)=\overline{\omega(b^{*}a)}$ in order to
define a sesquilinear form $\langle a,b\rangle\mathrel{\mathop{\mathchar
58\relax}}=\omega(a^{*}b)$. Since there might be elements
$a\in\mathcal{A}\setminus\\{0\\}$ for which $\omega(a^{*}a)=0$, it is
necessary to take the quotient by the ideal $\mathcal{N}_{\omega}$ generated
by all such elements. The Hilbert space for the representation is then defined
as the completion of the quotient space:

$\mathcal{H}_{\omega}\mathrel{\mathop{\mathchar
58\relax}}=\overline{\mathcal{A}/\mathcal{N}_{\omega}}.$ (5)

The (bounded) operator representing $a\in\mathcal{A}$ is the operator
$\pi_{\omega}(a)$ defined through

$\pi_{\omega}(a)|[b]\rangle\mathrel{\mathop{\mathchar
58\relax}}=|[ab]\rangle.$ (6)

It is through the GNS construction that we can make contact with the standard
formulation of quantum mechanics in terms of Hilbert spaces and linear
operators acting on them.

##  3 Entanglement Entropy

Let us suppose we are given a composite (bipartite) quantum system described
by a Hilbert space of the form
$\mathcal{H}=\mathcal{H}_{A}\otimes\mathcal{H}_{B}$. If
$|\psi\rangle\in\mathcal{H}$ is a pure state, then it is well known that
$|\psi\rangle$ is an entangled state if and only if the von Neumann entropy of
either of its reduced density matrices is greater than zero. The reduced
density matrices are obtained by partial trace:

$\rho_{A}\mathrel{\mathop{\mathchar
58\relax}}=\operatorname{Tr}_{\mathcal{H}_{B}}(|\psi\rangle\langle\psi|),\qquad\rho_{B}\mathrel{\mathop{\mathchar
58\relax}}=\operatorname{Tr}_{\mathcal{H}_{A}}(|\psi\rangle\langle\psi|).$ (7)

Using the Schmidt decomposition one then shows that $S(\rho_{A})=S(\rho_{B})$
and that, furthermore, this (von Neumann) entropy is zero if and only if the
state $|\psi\rangle$ is separable. The problem with systems of identical
particles stems from the fact that the underlying Hilbert space is not a
tensor product: many-particle states must be either totally symmetric or
totally anti-symmetric tensors. This leads to problems when trying to use
partial trace or the usual Schmidt decomposition in this context. Some time
ago [7, 8] we proposed to use the more general notion of _restriction_ of a
state to a subalgebra instead of partial trace. If
$\omega\mathrel{\mathop{\mathchar 58\relax}}\mathcal{A}\rightarrow\mathbb{C}$
is a state on $\mathcal{A}$ and $\mathcal{A}_{0}\subset\mathcal{A}$ is a
subalgebra of $\mathcal{A}$, then the restriction of $\omega$ to
$\mathcal{A}_{0}$ is a state on $\mathcal{A}_{0}$, that can be used to study
entanglement properties relative to the subsystem decomposition corresponding
to this assignment of a subalgebra. The following example (taken from Refs.
[7, 11]) is very useful in order to clarify the connection between
restrictions and partial traces.

######  Example 3.

Let
$\mathcal{H}=\mathcal{H}_{A}\otimes\mathcal{H}_{B}\equiv\mathbb{C}^{2}\otimes\mathbb{C}^{2}$
and take $\mathcal{A}=M_{2}(\mathbb{C})\otimes M_{2}(\mathbb{C})$ as the
observable algebra. Consider now following state vector,

$|\psi_{\lambda}\rangle=\sqrt{\lambda}|+,-\rangle+\sqrt{(1-\lambda)}|-,+\rangle,$
(8)

where $0\leq\lambda\leq 1$. We can also regard it as an algebraic state
$\omega_{\lambda}$, in the sense of Definition 2, if we put
$\omega_{\lambda}(O)\mathrel{\mathop{\mathchar
58\relax}}=\langle\psi_{\lambda}|O|\psi_{\lambda}\rangle$, for all
$O\in\mathcal{A}$. Let now $\mathcal{A}_{0}\subset\mathcal{A}$ denote the
subalgebra generated by elements of the form $\alpha\otimes{\vvmathbb 1}_{2}$,
with $\alpha\in M_{2}(\mathbb{C})$. By the definition, the restriction of
$\omega_{\lambda}$ to $\mathcal{A}_{0}$ is given by

$\omega_{\lambda,0}(\alpha\otimes{\vvmathbb
1}_{2})=\langle\psi_{\lambda}|\alpha\otimes{\vvmathbb
1}_{2}|\psi_{\lambda}\rangle=\lambda\langle+|\alpha|+\rangle+(1-\lambda)\langle-|\alpha|-\rangle.$
(9)

On the other hand, the use of partial trace (with
$\rho_{A}=\operatorname{Tr}_{B}|\psi_{\lambda}\rangle\langle\psi_{\lambda}|$)
leads to

$\rho_{A}=\left(\begin{array}[]{cc}\lambda&0\\\
0&1-\lambda\end{array}\right).$ (10)

It is readily checked that $\omega_{\lambda,0}(\alpha\otimes{\vvmathbb
1}_{2})=\operatorname{Tr}_{\mathcal{H}_{A}}(\rho_{A}\alpha)$.

As already remarked, the use of partial trace in the case of identical
particles will be problematic, whereas the restriction of a state to a
subalgebra is always a well-defined operation, and physically sensible. Being
a purely algebraic operation, it remains to see how to compute the von Neumann
entropy of the reduced (restricted) state. This can be done using the GNS
construction. The idea is quite simple. If we restrict a state $\omega$ on
$\mathcal{A}$ to a subalgebra $\mathcal{A}_{0}$, then the entropy of
$\omega_{0}\mathrel{\mathop{\mathchar 58\relax}}=\omega|_{\mathcal{A}_{0}}$
can be computed as follows. Construct the GNS space $\mathcal{H}_{\omega_{0}}$
and find a density matrix $\rho_{\omega_{0}}$ such that

$\omega_{0}(\alpha)=\operatorname{Tr}_{\mathcal{H}_{\omega_{0}}}(\rho_{\omega_{0}}\pi_{\omega_{0}}(\alpha)).$
(11)

This can be done by a careful manipulation of the projectors associated with
the decomposition of $\mathcal{H}_{\omega_{0}}$ into irreducible
representations. Then we can define

$S(\omega_{0})\mathrel{\mathop{\mathchar
58\relax}}=-\operatorname{Tr}\left(\rho_{\omega_{0}}\ln\rho_{\omega_{0}}\right).$
(12)

Explicit examples of this procedure, illustrating its usefulness in the
definition of an entanglement measure for systems of identical particles, can
be found in Refs. [7, 9, 11].

##  4 An Unexpected Symmetry

The assignment of an entropy to a state $\omega$ through the GNS construction
is based on the possibility of finding a density operator $\rho_{\omega}$
acting on the representation space $\mathcal{H}_{\omega}$ in such a way that

$\omega(a)=\operatorname{Tr}_{\mathcal{H}_{\omega}}(\rho_{\omega}\pi_{\omega}(a))\quad\mbox{for
all}\quad a\in\mathcal{A}.$ (13)

The relationship between $\omega$ and $\rho_{\omega}$ is, however, not unique.
The choice that was made in [7] was based on the idea of using the projectors
$P^{(k)}$ arising from the decomposition of $\mathcal{H}_{\omega}$ into
irreducible subspaces,
$\mathcal{H}_{\omega}=\bigoplus_{k}\mathcal{H}_{\omega}^{(k)}$, in order to
define

$\rho_{\omega}\mathrel{\mathop{\mathchar
58\relax}}=\sum_{k}P^{(k)}|[{\vvmathbb
1}_{\mathcal{A}}]\rangle\langle[{\vvmathbb 1}_{\mathcal{A}}]|P^{(k)}.$ (14)

This choice might be regarded as a “natural” one, but it is not the only one.
In fact, since the decomposition of $\mathcal{H}_{\omega}$ into irreducible
subspaces is in general not unique (this can only happen if each irreducible
component appears with multiplicity at most one), there are many choices of
projectors, which in turn lead to different choices of density operators
compatible with (14). The ensuing entropy ambiguities were studied in Refs.
[14, 15]. In particular, in Ref. [15] it was suggested that the ambiguity
could be naturally explained in the context of Tomita–Takesaki modular theory.
Although the natural context of modular theory is that of von Neumann
algebras, an analysis of the entropy ambiguity for finite dimensional algebras
using modular theory really sheds light on the problem. Such an analysis,
valid for any finite dimensional $C^{*}$-algebra, has been carried out in
[10]. In the following, I will present a simple example [10, 16] that
illustrates the general idea.

######  Example 4.

Let $\mathcal{A}=M_{n}(\mathbb{C})$ and consider the state $\omega$ defined
through

$\omega(a)\mathrel{\mathop{\mathchar
58\relax}}=\sum_{i}^{n}\lambda_{i}a_{ii},$ (15)

where the coefficients $\lambda_{i}$ are assumed to be strictly positive and
$\sum_{i=1}^{n}\lambda_{i}=1$. The fact that $\lambda_{i}>0$ for all $i$
implies that there is no null space ($\mathcal{N}_{\omega}=\\{0\\}$), so that
vectors in $\mathcal{H}_{\omega}$ have only one representative. We may,
therefore, write simply $|a\rangle$ to denote elements in
$\mathcal{H}_{\omega}$. The inner product is given by $\langle
a|b\rangle=\omega(a^{*}b)$. As $\omega(a)=\langle{\vvmathbb
1}_{n}|\pi_{\omega}(a)|{\vvmathbb 1}_{n}\rangle$, we see that the state vector
$|{\vvmathbb 1}_{n}\rangle$ gives us a _purification_ of the state $\omega$.
The algebra $\mathcal{A}$ (system $\mathbf{A}$) is now faithfully represented
as a subalgebra $\pi_{\omega}(\mathcal{A})$ of
$\mathcal{L}(\mathcal{H}_{\omega})$ –an emergent system we will denote as
system $\mathbf{C}$– and which in turn contains the commutant
$\pi_{\omega}(\mathcal{A})^{\prime}$ of $\pi_{\omega}(\mathcal{A})$, denoted
here as system $\mathbf{B}$. According to modular theory, observables in
$\mathbf{B}$ are obtained from observables in $\mathbf{A}$ by the action of
the modular conjugation $J$ [16]. Indeed, if
$\alpha\in\pi_{\omega}(\mathcal{A})$, then $J\alpha
J\in\pi_{\omega}(\mathcal{A})^{\prime}$. In our simple example, $J$ can be
given described explicitly in terms of its action on a basis of matrix units
$e_{ij}$ for $\mathcal{A}$:

$J|e_{ij}\rangle=\sqrt{\lambda_{j}/\lambda_{i}}|e_{ji}\rangle.$ (16)

We refer the reader to [10, 16] for details. Let now $|\hat{e}_{ij}\rangle$
denote the vector $|e_{ij}\rangle$, properly normalized with respect to the
GNS inner product. It follows that the projectors entering Eq. (14) can in
this case be written as follows:

$P^{(k)}=\sum_{i=1}^{n}|\hat{e}_{ik}\rangle\langle\hat{e}_{ik}|.$ (17)

As remarked above, any other orthogonal decomposition of
$\mathcal{H}_{\omega}$ providing an equivalent representation of $\mathcal{A}$
will lead to a density operator satisfying (13). In particular, given any
unitary element of $\mathcal{A}$ ($g\in\mathcal{A},\;g^{*}g={\vvmathbb
1}_{n}$), we can induce an equivalent decomposition by defining
$|\hat{e}_{ik}(g)\rangle\mathrel{\mathop{\mathchar
58\relax}}=U(g)|\hat{e}_{ik}\rangle$, where

$U(g)\mathrel{\mathop{\mathchar
58\relax}}=J\pi_{\omega}(g)J\in\pi_{\omega}(\mathcal{A})^{\prime}$ (18)

is a unitary operator that belongs to the commutant and which, therefore, can
be regarded as a _gauge transformation_.

Let us denote with $\rho_{\omega}(g)$ the density operator obtained by
replacing the projectors $\\{P^{(k)}\\}_{k=1,\ldots,n}$ in (14) by the
following ones,

$P_{g}^{(k)}\mathrel{\mathop{\mathchar
58\relax}}=J\pi_{\omega}(gp^{(k)}g)J,\quad k=1,\ldots,n,$ (19)

where $g$ is a unitary element of $\mathcal{A}$ and
$p^{(k)}\mathrel{\mathop{\mathchar 58\relax}}=e_{kk}$. It can be readily shown
that $\rho_{\omega}(g)$ still satisfies (13) and can be used to describe the
original state $\omega$ in system $\mathbf{A}$. But its entropy will in
general differ from that of $\rho_{\omega}\equiv\rho_{\omega}({\vvmathbb
1}_{n})$. In fact, we have (_cf._ Theorem 2 in [10]):

$S(\rho_{\omega}(g))\geq S(\rho_{\omega}).$ (20)

The above example can be generalized to the case of arbitrary finite
dimensional $C^{*}$-algebras [10]. The emergent gauge symmetry appearing
through the action of the unitaries $U(g)$ has been related to the action of
the gauge group of the fiber bundle describing “molecular shapes” in [15]. A
generalization of this approach to the case of homogeneous spaces of compact
Lie groups has been studied by Tabban in [16].

######  Remark 5.

Recently, an alternative approach to the study of the entropy ambiguities
discussed in this section has been proposed by Facchi and collaborators [17].
It would be interesting to relate the two approaches.

##  5 Working with Bal

I met Bal for the first time during a conference on the connection between
spin and statistics that took place in Trieste back in 2008. During the last
decade I have had the pleasure and the honor of collaborating with him. The
lessons learnt from his way of doing research are countless, but there is one
which has been particularly important for me and my group in Bogotá, namely
the group discussions. Although I did not take part in the legendary “Room-316
meetings”, I was lucky enough to enjoy many discussions on different occasions
that I adopted as part of my way of doing research with my students, some of
whom are now my younger collaborators. Seeing how fast they can learn –and
become accomplished researchers– has shown me the value that a generous and
lively exchange of ideas (something that Bal has always fostered) can have.

## References

  * [1] L. Bombelli, R. K. Koul, J. Lee, and R. Sorkin.  Quantum source of entropy for black holes.  Phys. Rev. D, 34:373–383, 1986.
  * [2] S. N. Solodukhin.  Entanglement entropy of black holes.  Living Rev. Relativ., 14(1):1–96, 2011.
  * [3] G. Vidal, J. I. Latorre, E. Rico, and A. Kitaev.  Entanglement in Quantum Critical Phenomena.  Phys. Rev. Lett., 90:227902, Jun 2003.
  * [4] M. A. Nielsen and I. L. Chuang.  Quantum computation and quantum information.  Cambridge University Press, 2010.
  * [5] D. Janzing.  Entropy of Entanglement, pp. 205–209.  In D. Greenberger, K. Hentschel, and F. Weinert (eds.), Compendium of Quantum Physics, pp. 205–209. Springer Berlin, Heidelberg, 2009.
  * [6] F. Benatti, R. Floreanini, F. Franchini, and U. Marzolino.  Entanglement in indistinguishable particle systems.  Phys. Rep., 878:1–27, 2020.
  * [7] A. P. Balachandran, T. R. Govindarajan, A. R. de Queiroz, and A. F. Reyes-Lega.  Entanglement and particle identity: A unifying approach.  Phys. Rev. Lett., 110:080503, 2013.
  * [8] A. P. Balachandran, T. R. Govindarajan, A. R. de Queiroz, and A. F. Reyes-Lega.  Algebraic approach to entanglement and entropy.  Phys. Rev. A, 88(2):022301, 2013.
  * [9] A. Balachandran, T. Govindarajan, A. R. De Queiroz, and A. Reyes-Lega.  Algebraic theory of entanglement.  Il Nuovo Cimento C, 36(3):27–33, 2013.
  * [10] A. P. Balachandran, I. M. Burbano, A. F. Reyes-Lega, and S. Tabban.  Emergent gauge symmetries and quantum operations.  J. Phys. A, 53(6):06LT01, 2020.
  * [11] A. F. Reyes-Lega.  Some Aspects of Operator Algebras in Quantum Physics.  In L. Cano, A. Cardona, H. Ocampo, and A. Reyes-Lega, editors, Geometric, Algebraic and Topological Methods for Quantum Field Theory, pp. 1–74. World Scientific, 2016.
  * [12] R. Haag.  Local Quantum Physics.  Texts and Monographs in Physics. Springer-Verlag, 2nd edition, 1996.
  * [13] D. Werner.  Funktionalanalysis.  Springer, 6 edition, 2007.
  * [14] A. P. Balachandran, A. R. de Queiroz, and S. Vaidya.  Entropy of Quantum States: Ambiguities.  Eur. Phys. J. Plus, 128:112, 2013.
  * [15] A. P. Balachandran, A. Queiroz, and S. Vaidya.  Quantum entropic ambiguities: Ethylene.  Phys. Rev. D, 88:025001, 2013.
  * [16] S. Tabban.  Modular theory and algebraic quantum physics.  PhD thesis, Universidad de los Andes, 2022.
  * [17] P. Facchi, G. Gramegna, and A. Konderak.  Entropy of Quantum States.  Entropy, 23(6), 2021.